Factors that determine web performance

Broadly, the factors that determine web performance to consider during optimization fall into two categories: loading performance and rendering performance.

HTML loading performance

• The time from the initial page request over the network to when the browser receives the HTML document and starts rendering
• Determines how quickly the page starts to display
• Optimize by minimizing redirects, caching HTML responses, compressing resources, and using an appropriate CDN

Rendering performance

• The time it takes the browser to paint what users see and make it interactive
• Determines how smoothly and quickly the screen is drawn
• Optimize by removing unnecessary CSS and JS, avoiding delayed loading of fonts and thumbnails, offloading heavy computations to a separate Web Worker to minimize main-thread occupancy, and optimizing animations

Web Performance Metrics (Web Vitals)

This post follows Google’s web.dev and the Chrome Developers docs. Unless there’s a special reason, aim for overall improvement rather than focusing on a single metric, and identify which part of the target page is the performance bottleneck. If you have real-user data, it’s better to focus on lower-quartile (Q1) values rather than the top or average, and verify that your targets are still met in those cases and improve accordingly.

Core Web Vitals

As we’ll cover shortly, there are many Web Vitals. Among them, Google highlights three metrics that are tightly tied to user experience and can be measured in the field rather than only in lab conditions; these are called the Core Web Vitals. Because Google incorporates Core Web Vitals into its search ranking, site owners should pay close attention to these for SEO.

• Largest Contentful Paint (LCP): reflects loading performance; should be within 2.5 s
• Interaction to Next Paint (INP): reflects responsiveness; should be ≤ 200 ms
• Cumulative Layout Shift (CLS): reflects visual stability; should be ≤ 0.1

Core Web Vitals are primarily field metrics, but the other two besides INP can also be measured in lab tools like Chrome DevTools or Lighthouse. INP requires actual user input, so it can’t be measured in a lab; in such cases, TBT is highly correlated with INP and serves as a close proxy, and improving TBT usually improves INP as well.

Performance score weights in Lighthouse 10

The Lighthouse performance score is a weighted average of metric scores, using the following weights.

FCP (First Contentful Paint)

• Measures the time from page request to the first render of DOM content
• Counts images, non-white <canvas> elements, and SVG as DOM content; excludes content inside iframes

One factor that significantly affects FCP is font loading. For optimization tips, the Chrome Developers docs recommend this related post.

Lighthouse scoring thresholds

According to the Chrome Developers docs, Lighthouse uses the following thresholds:

LCP (Largest Contentful Paint)

• Measures the time it takes to render the largest element (image, text block, video, etc.) within the initial viewport when the page first opens
• The larger the on-screen area it occupies, the more likely users will perceive it as primary content
• If the LCP is an image, you can break the time down into four sub-intervals; identify where the bottleneck occurs:
  1. Time to First Byte (TTFB): time from the start of page load to receipt of the first byte of the HTML response
  2. Load delay: the difference between when the browser starts loading the LCP resource and the TTFB
  3. Load time: the time to load the LCP resource itself
  4. Render delay: the time from finishing the LCP resource load until the LCP element is fully rendered

Lighthouse scoring thresholds

According to the Chrome Developers docs, Lighthouse uses the following thresholds:

TBT (Total Blocking Time)

• Measures the total time the page is unable to respond to user input such as mouse clicks, touches, and key presses
• Among the tasks between FCP and TTI (Time to Interactive)*, tasks that run for ≥ 50 ms are considered long tasks. For each long task, the time beyond 50 ms is called the blocking portion, and TBT is the sum of all blocking portions.

* TTI itself is overly sensitive to outliers in network responses and long tasks, leading to low consistency and high variance, so it was removed from Lighthouse scoring starting with Lighthouse 10.

The most common causes of long tasks are unnecessary or inefficient JavaScript loading, parsing, and execution. The Chrome Developers docs and Google’s web.dev recommend reducing JavaScript payload via code splitting so each chunk runs within 50 ms, and, if needed, offloading work from the main thread to a separate Service Worker to run in multiple threads.

Lighthouse scoring thresholds

According to the Chrome Developers docs, Lighthouse uses the following thresholds:

CLS (Cumulative Layout Shift)

An example of an unexpected layout shift

Video source: Cumulative Layout Shift (CLS) | Articles | web.dev

I sense deep rage in that cursor movement

• Unexpected layout shifts degrade UX in many ways, such as suddenly moving text that causes readers to lose their place, or misclicks on links and buttons
• The exact method for calculating the CLS score is described on Google’s web.dev
• As shown in the image below, you should target ≤ 0.1

Image source: Cumulative Layout Shift (CLS) | Articles | web.dev

SI (Speed Index)

• Measures how quickly content is visually displayed during page load
• Lighthouse records a video of the page loading in the browser, analyzes it to compute frame-by-frame progression, and then uses the Speedline Node.js module to compute the SI score

Any improvement that speeds up page loading—including what we covered for FCP, LCP, and TBT—will generally improve the SI score as well. Rather than representing a single stage of loading, SI reflects the overall loading process to some extent.

Lighthouse scoring thresholds

According to the Chrome Developers docs, Lighthouse uses the following thresholds:

TL;DR

• Newton’s law of universal gravitation: $\mathbf{F} = -G\cfrac{mM}{r^2}\mathbf{e}_r$
• For objects with continuous mass distribution and finite size: $\mathbf{F} = -Gm\int_V \cfrac{dM}{r^2}\mathbf{e}_r = -Gm\int_V \cfrac{\rho(\mathbf{r^\prime})\mathbf{e}_r}{r^2} dv^{\prime}$
  • $\rho(\mathbf{r^{\prime}})$: mass density at a point located at position vector $\mathbf{r^{\prime}}$ from an arbitrary origin
  • $dv^{\prime}$: volume element at a point located at position vector $\mathbf{r^{\prime}}$ from an arbitrary origin
• Gravitational field vector:
  • A vector representing the force per unit mass experienced by a particle in the field created by an object of mass $M$
  • $\mathbf{g} = \cfrac{\mathbf{F}}{m} = - G \cfrac{M}{r^2}\mathbf{e}_r = - G \int_V \cfrac{\rho(\mathbf{r^\prime})\mathbf{e}_r}{r^2}dv^\prime$
  • Has dimensions of force per unit mass or acceleration
• Gravitational potential:
  • $\mathbf{g} \equiv -\nabla \Phi$
  • Has dimensions of (force per unit mass) × (distance) or energy per unit mass
  • $\Phi = -G\cfrac{M}{r}$
  • Only the relative difference in gravitational potential has meaning; the specific value itself is meaningless
  • Usually the condition $\Phi \to 0$ as $r \to \infty$ is arbitrarily set to remove ambiguity
  • $U = m\Phi, \quad \mathbf{F} = -\nabla U$
• Gravitational potential inside and outside a spherical shell (Shell theorem)
  • When $R>a$:
    • $\Phi(R>a) = -\cfrac{GM}{R}$
    • When calculating the gravitational potential at any external point due to a spherically symmetric mass distribution, the object can be treated as a point mass
  • When $R<b$:
    • $\Phi(R<b) = -2\pi\rho G(a^2 - b^2)$
    • Inside a spherically symmetric mass shell, the gravitational potential is constant regardless of position, and the gravitational force is $0$
  • When $b<R<a$: $\Phi(b<R<a) = -4\pi\rho G \left( \cfrac{a^2}{2} - \cfrac{b^3}{3R} - \cfrac{R^2}{6} \right)$

Gravitational Field

Newton’s Law of Universal Gravitation

Newton had already systematized and numerically verified the law of universal gravitation before 11666 HE. Nevertheless, it took another 20 years until he published his results in his book Principia in 11687 HE, because he could not justify the calculation method that assumed the Earth and Moon as point masses without size. Fortunately, using the calculus that Newton himself invented later, we can prove that problem, which was not easy for Newton in the 1600s, much more easily.

According to Newton’s law of universal gravitation, every mass particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. Mathematically, this is expressed as:

Image source

• Author: Wikimedia user Dennis Nilsson
• License: CC BY 3.0

The unit vector $\mathbf{e}_r$ points from $M$ toward $m$, and the negative sign indicates that the force is attractive. That is, $m$ is pulled toward $M$.

Cavendish’s Experiment

The experimental verification of this law and the determination of the value of $G$ was accomplished by British physicist Henry Cavendish in 11798 HE. Cavendish’s experiment used a torsion balance consisting of two small spheres fixed to the ends of a light rod. These two spheres were each attracted toward two other large spheres positioned nearby. The currently accepted value of $G$ is $6.673 \pm 0.010 \times 10^{-11} \mathrm{N\cdot m^2/kg^2}$.

Despite $G$ being one of the oldest known fundamental constants, it is known with lower precision than most other fundamental constants such as $e$, $c$, and $\hbar$. Even today, much research is being conducted to determine the value of $G$ with higher precision.

For Objects with Finite Size

The law in equation ($\ref{eqn:law_of_gravitation}$) can strictly only be applied to point particles. If one or both objects have finite size, we need the additional assumption that the gravitational force field is a linear field to calculate the force. That is, we assume that the total gravitational force on a particle of mass $m$ from several other particles can be found by vector addition of each force. For objects with continuous mass distribution, the sum is replaced by an integral:

• $\rho(\mathbf{r^{\prime}})$: mass density at a point located at position vector $\mathbf{r^{\prime}}$ from an arbitrary origin
• $dv^{\prime}$: volume element at a point located at position vector $\mathbf{r^{\prime}}$ from an arbitrary origin

If both objects of mass $M$ and mass $m$ have finite size, a second volume integral over $m$ is also needed to find the total gravitational force.

Gravitational Field Vector

The gravitational field vector $\mathbf{g}$ is defined as the vector representing the force per unit mass experienced by a particle in the field created by an object of mass $M$:

or

Here, the direction of $\mathbf{e}_r$ varies with $\mathbf{r^\prime}$.

This quantity $\mathbf{g}$ has dimensions of force per unit mass or acceleration. The magnitude of the gravitational field vector $\mathbf{g}$ near the Earth’s surface is equal to what we call the gravitational acceleration constant, with $|\mathbf{g}| \approx 9.80\mathrm{m/s^2}$.

Gravitational Potential

Definition

The gravitational field vector $\mathbf{g}$ varies as $1/r^2$, and therefore satisfies the condition ($\nabla \times \mathbf{g} \equiv 0$) for being expressible as the gradient of some scalar function (potential). Thus we can write:

where $\Phi$ is called the gravitational potential, and has dimensions of (force per unit mass) × (distance) or energy per unit mass.

Since $\mathbf{g}$ depends only on the radius, $\Phi$ also varies with $r$. From equations ($\ref{eqn:g_vector}$) and ($\ref{eqn:gradient_phi}$):

Integrating this gives:

Since only the relative difference in gravitational potential has meaning and the absolute magnitude of the value is meaningless, we can omit the integration constant. Usually the condition $\Phi \to 0$ as $r \to \infty$ is arbitrarily set to remove ambiguity, and equation ($\ref{eqn:g_potential}$) satisfies this condition.

For continuous mass distributions, the gravitational potential is:

For mass distributed on a thin shell surface:

And for a linear mass source with linear density $\rho_l$:

Physical Meaning

Consider the work per unit mass $dW^\prime$ done by an object when it moves by $d\mathbf{r}$ in a gravitational field.

In this equation, $\Phi$ is a function of position coordinates only, expressed as $\Phi=\Phi(x_1, x_2, x_3) = \Phi(x_i)$. Therefore, the work per unit mass done by an object when moved from one point to another in a gravitational field equals the potential difference between those two points.

If we define the gravitational potential at infinity to be $0$, then $\Phi$ at any point can be interpreted as the work per unit mass required to move the object from infinity to that point. Since the potential energy of an object equals the product of its mass and the gravitational potential $\Phi$, if $U$ is the potential energy:

Therefore, the gravitational force on an object is obtained by taking the negative gradient of its potential energy:

When an object is placed in a gravitational field created by some mass, there is always some potential energy. Strictly speaking, this potential energy resides in the field itself, but it is conventionally expressed as the potential energy of the object.

Example: Gravitational Potential Inside and Outside a Spherical Shell (Shell Theorem)

Coordinate Setup & Expressing Gravitational Potential as an Integral

Let’s find the gravitational potential inside and outside a uniform spherical shell with inner radius $b$ and outer radius $a$. While the gravitational force due to a spherical shell can be obtained by directly calculating the force components acting on a unit mass in the field, using the potential method is simpler.

In the figure above, let’s calculate the potential at point $P$ at distance $R$ from the center. Assuming uniform mass distribution in the shell, $\rho(r^\prime)=\rho$, and due to symmetry about the azimuthal angle $\phi$ with respect to the line connecting the sphere’s center and point $P$:

By the law of cosines:

Since $R$ is constant, differentiating this equation with respect to $r^\prime$:

Substituting this into equation ($\ref{eqn:spherical_shell_1}$):

Here, $r_\mathrm{max}$ and $r_\mathrm{min}$ are determined by the position of point $P$.

When $R>a$

The mass $M$ of the spherical shell is:

Therefore, the potential is:

Comparing the gravitational potential due to a point mass of mass $M$ in equation ($\ref{eqn:g_potential}$) with the result just obtained ($\ref{eqn:spherical_shell_outside_2}$), we see they are identical. This means that when calculating the gravitational potential at any external point due to a spherically symmetric mass distribution, we can assume all mass is concentrated at the center. Most spherical celestial bodies of a certain size or larger, such as Earth or the Moon, fall into this category, as they can be considered as countless spherical shells with the same center but different diameters nested like Matryoshka dolls. This provides the justification for assuming celestial bodies like Earth or the Moon as point masses without size in calculations mentioned at the beginning of this post.

When $R<b$

Inside a spherically symmetric mass shell, the gravitational potential is constant regardless of position, and the gravitational force is $0$.

This is also a major reason why the “Hollow Earth theory,” one of the representative pseudosciences, is nonsense. If Earth were a spherical shell with a hollow interior as claimed by the Hollow Earth theory, no gravitational force would act on any object inside that cavity. Considering Earth’s mass and volume, such a hollow cannot exist, and even if it did, life forms there would not live with the inner surface of the spherical shell as ground, but would float in a weightless state like in a space station.
Microorganisms may live several kilometers deep underground, but at least not in the form claimed by the Hollow Earth theory. I also really enjoy Jules Verne’s novel “Journey to the Center of the Earth” and the movie “Journey to the Center of the Earth,” but we should enjoy fiction as fiction and not seriously believe it.

When $b<R<a$

Results

The gravitational potential $\Phi$ in the three regions obtained above, and the corresponding magnitude of the gravitational field vector $|\mathbf{g}|$ as functions of distance $R$, are shown graphically as follows:

• Python visualization code: yunseo-kim/physics-visualizations repository
• License: See here

We can see that both the gravitational potential and the magnitude of the gravitational field vector are continuous. If the gravitational potential were discontinuous at any point, the gradient of the potential at that point, i.e., the magnitude of gravity, would become infinite, which is not physically reasonable, so the potential function must be continuous at all points. However, the derivative of the gravitational field vector is discontinuous at the inner and outer surfaces of the shell.

Example: Galactic Rotation Curves

According to astronomical observations, in many spiral galaxies that rotate about their centers, such as the Milky Way and Andromeda Galaxy, most of the observable mass is concentrated near the center. However, the orbital velocities of masses in these spiral galaxies greatly disagree with theoretically predicted values based on the observable mass distribution, as can be seen in the following graph, and remain nearly constant beyond a certain distance.

Image source

• Author: Wikipedia user PhilHibbs
• License: Public Domain

Left: Predicted galactic rotation from observable mass | Right: Actual observed galactic rotation.

Video source

• Original file (Ogg Theora video) link: https://commons.wikimedia.org/wiki/File:Galaxy_rotation_under_the_influence_of_dark_matter.ogv
• Author: Ingo Berg
• License: CC BY-SA 3.0
• Simulation method and code used: https://beltoforion.de/en/spiral_galaxy_renderer/

The image file previously embedded on this page, Rotation curve of spiral galaxy Messier 33 (Triangulum).png, was deleted from Wikimedia Commons after it was determined to be a derivative work by Wikimedia user Mario De Leo that plagiarized Prof. Mark Whittle of the University of Virginia’s non-free work without proper citation. Accordingly, it has also been removed from this page.

Let’s predict the orbital velocity as a function of distance when the galaxy’s mass is concentrated at the center, confirm that this prediction does not match the observational results, and show that the mass $M(R)$ distributed within distance $R$ from the galactic center must be proportional to $R$ to explain the observations.

First, if the galactic mass $M$ is concentrated at the center, the orbital velocity at distance $R$ is:

In this case, an orbital velocity decreasing as $1/\sqrt{R}$ is predicted, as shown by the dotted lines in the two graphs above. However, according to observational results, the orbital velocity $v$ is nearly constant regardless of distance $R$, so the prediction and observations do not match. These observational results can only be explained if $M(R)\propto R$.

Setting $M(R) = kR$ using proportionality constant $k$:

From this, astrophysicists conclude that many galaxies must contain undiscovered “dark matter,” and this dark matter must account for more than 90% of the universe’s mass. However, the identity of dark matter has not yet been clearly revealed, and while not mainstream theory, attempts like Modified Newtonian Dynamics (MOND) exist to explain observational results without assuming the existence of dark matter. Today, these research fields are at the forefront of astrophysics.

TL;DR

• Method of Undetermined Coefficients is applicable to:
  • Linear ODEs $y^{\prime\prime} + ay^{\prime} + by = r(x)$
  • with constant coefficients $a$ and $b$,
  • and where the input $r(x)$ is an exponential function, a power of $x$, a cosine or sine, or sums and products of such functions.
• Choice Rules for the Method of Undetermined Coefficients
  • (a) Basic Rule: If $r(x)$ in Eq. ($\ref{eqn:linear_ode_with_constant_coefficients}$) is one of the functions in the first column of the table, choose the corresponding $y_p$ from the same row and determine its undetermined coefficients by substituting $y_p$ and its derivatives into Eq. ($\ref{eqn:linear_ode_with_constant_coefficients}$).
  • (b) Modification Rule: If a term in your choice for $y_p$ is a solution of the corresponding homogeneous ODE $y^{\prime\prime} + ay^{\prime} + by = 0$, multiply this term by $x$ (or by $x^2$ if this solution corresponds to a double root of the characteristic equation of the homogeneous ODE).
  • (c) Sum Rule: If $r(x)$ is a sum of functions in the first column of the table, choose for $y_p$ the sum of the functions in the corresponding rows of the second column.

Term in $r(x)$	Choice for $y_p(x)$
$ke^{\gamma x}$	$Ce^{\gamma x}$
$kx^n\ (n=0,1,\cdots)$	$K_nx^n + K_{n-1}x^{n-1} + \cdots + K_1x + K_0$
$k\cos{\omega x}$ $k\sin{\omega x}$	$K\cos{\omega x} + M\sin{\omega x}$
$ke^{\alpha x}\cos{\omega x}$ $ke^{\alpha x}\sin{\omega x}$	$e^{\alpha x}(K\cos{\omega x} + M\sin{\omega x})$

Prerequisites

• Homogeneous Linear ODEs of Second Order
• Homogeneous Linear ODEs with Constant Coefficients
• Euler-Cauchy Equation
• Wronskian, Existence and Uniqueness of Solutions
• Nonhomogeneous Linear ODEs of Second Order
• Vector Spaces, Linear Span (Linear Algebra)

Method of Undetermined Coefficients

Consider a second-order nonhomogeneous linear ordinary differential equation where $r(x) \not\equiv 0$

and its corresponding homogeneous ordinary differential equation

As we saw in Nonhomogeneous Linear ODEs of Second Order, to solve an initial value problem for the nonhomogeneous linear ODE ($\ref{eqn:nonhomogeneous_linear_ode}$), we must first solve the homogeneous ODE ($\ref{eqn:homogeneous_linear_ode}$) to find $y_h$, then find a particular solution $y_p$ of Eq. ($\ref{eqn:nonhomogeneous_linear_ode}$) to obtain the general solution

So, how can we find $y_p$? A general method for finding $y_p$ is the method of variation of parameters, but in some cases, a much simpler method, the method of undetermined coefficients, can be applied. It is a frequently used method in engineering, especially as it can be applied to models of vibrating systems and RLC electrical circuits.

The method of undetermined coefficients is suitable for linear ODEs with constant coefficients $a$ and $b$, and where the input $r(x)$ is an exponential function, a power of $x$, a cosine or sine, or sums and products of such functions:

The key to the method of undetermined coefficients is that an $r(x)$ of this form has derivatives that are similar in form to itself. To apply this method, we choose a $y_p$ that is similar in form to $r(x)$ but has unknown coefficients, which are determined by substituting $y_p$ and its derivatives into the given ODE. For forms of $r(x)$ that are practically important in engineering, the rules for choosing an appropriate $y_p$ are as follows.

Choice Rules for the Method of Undetermined Coefficients
(a) Basic Rule: If $r(x)$ in Eq. ($\ref{eqn:linear_ode_with_constant_coefficients}$) is one of the functions in the first column of the table, choose the corresponding $y_p$ from the same row and determine its undetermined coefficients by substituting $y_p$ and its derivatives into Eq. ($\ref{eqn:linear_ode_with_constant_coefficients}$).
(b) Modification Rule: If a term in your choice for $y_p$ is a solution of the corresponding homogeneous ODE $y^{\prime\prime} + ay^{\prime} + by = 0$, multiply this term by $x$ (or by $x^2$ if this solution corresponds to a double root of the characteristic equation of the homogeneous ODE).
(c) Sum Rule: If $r(x)$ is a sum of functions in the first column of the table, choose for $y_p$ the sum of the functions in the corresponding rows of the second column.

Term in $r(x)$	Choice for $y_p(x)$
$ke^{\gamma x}$	$Ce^{\gamma x}$
$kx^n\ (n=0,1,\cdots)$	$K_nx^n + K_{n-1}x^{n-1} + \cdots + K_1x + K_0$
$k\cos{\omega x}$ $k\sin{\omega x}$	$K\cos{\omega x} + M\sin{\omega x}$
$ke^{\alpha x}\cos{\omega x}$ $ke^{\alpha x}\sin{\omega x}$	$e^{\alpha x}(K\cos{\omega x} + M\sin{\omega x})$

This method has the advantage of being not only simple but also self-correcting. If you choose $y_p$ incorrectly or with too few terms, you will arrive at a contradiction. If you choose too many terms, the coefficients of the unnecessary terms will turn out to be $0$, leading to the correct result. Even if something goes wrong while applying the method, you will naturally notice it during the solution process. Therefore, as long as you choose a reasonably appropriate $y_p$ according to the choice rules above, you can try it without much hesitation.

Proof of the Sum Rule

Consider a nonhomogeneous linear ODE of the form $r(x) = r_1(x) + r_2(x)$:

Now, let’s assume that the following two equations, with the same left-hand side but with inputs $r_1$ and $r_2$, have particular solutions ${y_p}_1$ and ${y_p}_2$, respectively.

If we denote the left-hand side of the given equation as $L[y]$, then due to the linearity of $L[y]$, the sum rule holds because the following is satisfied for $y_p = {y_p}_1 + {y_p}_2$.

Example: $y^{\prime\prime} + ay^{\prime} + by = ke^{\gamma x}$

According to the basic rule (a), we set $y_p = Ce^{\gamma x}$ and substitute it into the given equation $y^{\prime\prime} + ay^{\prime} + by = ke^{\gamma x}$:

Case where $\gamma^2 + a\gamma + b \neq 0$

We can determine the undetermined coefficient $C$ and find $y_p$ as follows.

Case where $\gamma^2 + a\gamma + b = 0$

In this case, we must apply the modification rule (b). First, let’s find the roots of the characteristic equation of the homogeneous ODE $y^{\prime\prime} + ay^{\prime} + by = 0$ by using the fact that $b = -\gamma^2 - a\gamma = -\gamma(a + \gamma)$.

From this, we obtain the basis for the homogeneous ODE:

Case where $\gamma \neq -a-\gamma$

Since the chosen $y_p = Ce^{\gamma x}$ is a solution of the corresponding homogeneous ODE but not a double root, we multiply this term by $x$ according to the modification rule (b) and set $y_p = Cxe^{\gamma x}$.

Now, substituting the modified $y_p$ back into the given equation $y^{\prime\prime} + ay^{\prime} - \gamma(a + \gamma)y = ke^{\gamma x}$:

Case where $\gamma = -a-\gamma$

In this case, the chosen $y_p = Ce^{\gamma x}$ corresponds to a double root of the characteristic equation of the homogeneous ODE. Therefore, according to the modification rule (b), we multiply this term by $x^2$ and set $y_p = Cx^2 e^{\gamma x}$.

Now, substituting the modified $y_p$ back into the given equation $y^{\prime\prime} - 2\gamma y^{\prime} + \gamma^2 y = ke^{\gamma x}$:

Extension of the Method: $r(x)$ as a Product of Functions

Consider a nonhomogeneous linear ODE where $r(x)$ is of the form $k x^n e^{\alpha x}\cos(\omega x)$:

If we assume $r(x)$ is a product of functions like an exponential function $e^{\alpha x}$, a power of $x$ like $x^m$, and a cosine or sine function like $\cos{\omega x}$ or $\sin{\omega x}$ (here we assume cosine without loss of generality), or a sum of such products (i.e., it can be expressed as a sum and product of functions from the first column of the previous table), we will show that a solution $y_p$ exists which is a sum and product of functions from the second column of the same table.

For a rigorous proof, some parts are described using linear algebra and are marked with an asterisk (*). You can skip these parts and still get a general understanding.

Defining the Vector Space $V$*

For an $r(x)$ of the form \(\begin{align*} r(x) &= C_1x^{n_1}e^{\alpha_1 x} \times C_2x^{n_2}e^{\alpha_2 x}\cos(\omega x) \times \cdots \\ &= C x^n e^{\alpha x}\cos(\omega x) \end{align*}\)

we can define a vector space $V$ such that $r(x) \in V$ as follows:

Derivative Forms of Exponential, Polynomial, and Trigonometric Functions

The derivative forms of the basic functions presented in the first column of the previous table are as follows.

• Exponential function: $\cfrac{d}{dx}e^{\alpha x} = \alpha e^{\alpha x}$
• Polynomial function: $\cfrac{d}{dx}x^m = mx^{m-1}$
• Trigonometric functions: $\cfrac{d}{dx}\cos\omega x = -\omega\sin\omega x, \quad \cfrac{d}{dx}\sin\omega x = \omega\cos\omega x$

The derivatives obtained by differentiating these functions are also expressed as a sum of the same kinds of functions.

Therefore, if functions $f$ and $g$ are the functions above or their sums, applying the product rule to $r(x) = f(x)g(x)$ gives

and here, $f$, $f^{\prime}$, $f^{\prime\prime}$ and $g$, $g^{\prime}$, $g^{\prime\prime}$ can all be written as sums or constant multiples of exponential, polynomial, and trigonometric functions. Thus, $r^{\prime}(x) = (fg)^{\prime}$ and $r^{\prime\prime}(x) = (fg)^{\prime\prime}$, like $r(x)$, can also be expressed as sums and products of these functions.

Invariance of $V$ under the Differential Operator $D$ and Linear Transformation $L$*

That is, not only $r(x)$ itself, but also $r^{\prime}(x)$ and $r^{\prime\prime}(x)$ are linear combinations of terms of the form $x^k e^{\alpha x}\cos(\omega x)$ and $x^k e^{\alpha x}\sin(\omega x)$, so

Not limiting this to just $r(x)$, if we introduce the differential operator $D$ for all elements of the previously defined vector space $V$ to express this more generally, the vector space $V$ is closed under the differentiation operation $D$. Therefore, if we denote the left-hand side of the given equation, $y^{\prime\prime} + ay^{\prime} + by$, as $L[y]$, then $V$ is invariant under $L$.

Since $r(x) \in V$ and $V$ is invariant under $L$, there exists another element $y_p \in V$ that satisfies $L[y_p] = r$.

Ansatz

Therefore, if we choose an appropriate $y_p$ as a sum of all possible product terms using undetermined coefficients $A_0, A_1, \dots, A_n$ and $K, M$ as follows, we can determine the undetermined coefficients by substituting $y_p$ (or $xy_p$, $x^2y_p$) and its derivatives into the given equation, according to the basic rule (a) and the modification rule (b). Here, $n$ should be determined according to the degree of $x$ in $r(x)$.

$\blacksquare$

If the given input $r(x)$ includes several different values of $\alpha_i$ and $\omega_j$, you must choose $y_p$ to include all possible terms of the form $x^{k}e^{\alpha_i x}\cos(\omega_j x)$ and $x^{k}e^{\alpha_i x}\sin(\omega_j x)$ for each $\alpha_i$ and $\omega_j$ value.
The advantage of the method of undetermined coefficients is its simplicity. If the ansatz becomes too complicated and this advantage is lost, it might be better to use the method of variation of parameters, which will be discussed later.

Extension of the Method: Euler-Cauchy Equation

The method of undetermined coefficients can be utilized not only for homogeneous linear ODEs with constant coefficients but also for the Euler-Cauchy equation:

Change of Variables

By substituting $x = e^t$ to transform it into a homogeneous linear ODE with constant coefficients, we get

which, as we have seen before, allows us to convert the Euler-Cauchy equation into the following homogeneous linear ODE with constant coefficients in terms of $t$.

Now, we can apply the previously discussed method of undetermined coefficients to Eq. ($\ref{eqn:substituted}$) to solve for $t$, and finally, use $t = \ln x$ to find the solution in terms of $x$.

Case where $r(x)$ is a power of $x$, a natural logarithm, or a sum/product of such functions

In particular, if the input $r(x)$ consists of powers of $x$, natural logarithms, or sums and products of such functions, an appropriate $y_p$ can be chosen directly according to the following choice rules for the Euler-Cauchy equation.

Choice Rules for the Method of Undetermined Coefficients: For Euler-Cauchy Equations
(a) Basic Rule: If $r(x)$ in Eq. ($\ref{eqn:euler_cauchy}$) is one of the functions in the first column of the table, choose the corresponding $y_p$ from the same row and determine its undetermined coefficients by substituting $y_p$ and its derivatives into Eq. ($\ref{eqn:euler_cauchy}$).
(b) Modification Rule: If a term in your choice for $y_p$ is a solution of the corresponding homogeneous ODE $x^2y^{\prime\prime} + axy^{\prime} + by = 0$, multiply this term by $\ln{x}$ (or by $(\ln{x})^2$ if this solution corresponds to a double root of the characteristic equation of the homogeneous ODE).
(c) Sum Rule: If $r(x)$ is a sum of functions in the first column of the table, choose for $y_p$ the sum of the functions in the corresponding rows of the second column.

Term in $r(x)$	Choice for $y_p(x)$
$kx^m\ (m=0,1,\cdots)$	$Ax^m$
$kx^m \ln{x}\ (m=0,1,\cdots)$	$x^m(B\ln x + C)$
$k(\ln{x})^s\ (s=0,1,\cdots)$	$D_0 + D_1\ln{x} + \cdots + D_{s-1}(\ln{x})^{s-1} + D_s(\ln{x})^s$
$kx^m (\ln{x})^s$ $(m=0,1,\cdots ;\; s=0,1,\cdots)$	$x^m \left( D_0 + D_1\ln{x} + \cdots + D_{s-1}(\ln{x})^{s-1} + D_s(\ln{x})^s \right)$

This way, for practically important forms of the input $r(x)$, we can find the same $y_p$ as obtained through the change of variables more quickly and easily. You can derive these choice rules for the Euler-Cauchy equation by substituting $\ln{x}$ for $x$ in the original choice rules we looked at earlier.

TL;DR

• General solution of a second-order nonhomogeneous linear ODE $y^{\prime\prime} + p(x)y^{\prime} + q(x)y = r(x)$:
  • $y(x) = y_h(x) + y_p(x)$
  • $y_h$: The general solution of the homogeneous ODE $y^{\prime\prime} + p(x)y^{\prime} + q(x)y = 0$, which is $y_h = c_1y_1 + c_2y_2$
  • $y_p$: A particular solution of the given nonhomogeneous ODE
• The response term $y_p$ is determined solely by the input $r(x)$. For the same nonhomogeneous ODE, $y_p$ does not change even if the initial conditions change. The difference between two particular solutions of a nonhomogeneous ODE is a solution of the corresponding homogeneous ODE.
• Existence of a general solution: If the coefficients $p(x)$, $q(x)$, and the input function $r(x)$ of a nonhomogeneous ODE are continuous, a general solution always exists.
• Non-existence of singular solutions: The general solution includes all solutions of the equation (i.e., no singular solutions exist).

Prerequisites

• Homogeneous Linear ODEs of Second Order
• The Wronskian, Existence and Uniqueness of Solutions

General and Particular Solutions of Second-Order Nonhomogeneous Linear ODEs

Consider the second-order nonhomogeneous linear ordinary differential equation

where $r(x) \not\equiv 0$. The general solution of equation ($\ref{eqn:nonhomogeneous_linear_ode}$) on an open interval $I$ is the sum of the general solution $y_h = c_1y_1 + c_2y_2$ of the corresponding homogeneous ODE

and a particular solution $y_p$ of equation ($\ref{eqn:nonhomogeneous_linear_ode}$), in the form

Furthermore, a particular solution of equation ($\ref{eqn:nonhomogeneous_linear_ode}$) on the interval $I$ is a solution obtained from equation ($\ref{eqn:general_sol}$) by assigning specific values to the arbitrary constants $c_1$ and $c_2$ in $y_h$.

In other words, adding an input $r(x)$, which depends only on the independent variable $x$, to the homogeneous ODE ($\ref{eqn:homogeneous_linear_ode}$) adds a corresponding term $y_p$ to the response. This added response term $y_p$ is determined solely by the input $r(x)$, regardless of the initial conditions. As we will see later, if we take the difference between any two solutions $y_1$ and $y_2$ of equation ($\ref{eqn:nonhomogeneous_linear_ode}$) (i.e., the difference between particular solutions for two different sets of initial conditions), the term $y_p$, which is independent of the initial conditions, cancels out, leaving only the difference between ${y_h}_1$ and ${y_h}_2$. By the Superposition Principle, this difference is a solution of equation ($\ref{eqn:homogeneous_linear_ode}$).

Relationship Between Solutions of Nonhomogeneous and Corresponding Homogeneous ODEs

Theorem 1: Relationship Between Solutions of Nonhomogeneous ODE ($\ref{eqn:nonhomogeneous_linear_ode}$) and Homogeneous ODE ($\ref{eqn:homogeneous_linear_ode}$)
(a) The sum of a solution $y$ of the nonhomogeneous ODE ($\ref{eqn:nonhomogeneous_linear_ode}$) and a solution $\tilde{y}$ of the homogeneous ODE ($\ref{eqn:homogeneous_linear_ode}$) on some open interval $I$ is a solution of equation ($\ref{eqn:nonhomogeneous_linear_ode}$) on $I$. In particular, equation ($\ref{eqn:general_sol}$) is a solution of equation ($\ref{eqn:nonhomogeneous_linear_ode}$) on $I$.
(b) The difference between two solutions of the nonhomogeneous ODE ($\ref{eqn:nonhomogeneous_linear_ode}$) on an interval $I$ is a solution of the homogeneous ODE ($\ref{eqn:homogeneous_linear_ode}$) on $I$.

Proof

(a)

Let’s denote the left-hand side of equations ($\ref{eqn:nonhomogeneous_linear_ode}$) and ($\ref{eqn:homogeneous_linear_ode}$) as $L[y]$. Then, for any solution $y$ of equation ($\ref{eqn:nonhomogeneous_linear_ode}$) and any solution $\tilde{y}$ of equation ($\ref{eqn:homogeneous_linear_ode}$) on interval $I$, the following holds:

(b)

For any two solutions $y$ and $y^*$ of equation ($\ref{eqn:nonhomogeneous_linear_ode}$) on interval $I$, the following holds:

The General Solution of a Nonhomogeneous ODE Includes All Solutions

For a homogeneous ODE ($\ref{eqn:homogeneous_linear_ode}$), we know that the general solution includes all solutions. Let’s show that the same holds for the nonhomogeneous ODE ($\ref{eqn:nonhomogeneous_linear_ode}$).

Theorem 2: The General Solution of a Nonhomogeneous ODE Includes All Solutions
If the coefficients $p(x)$, $q(x)$, and the input function $r(x)$ of equation ($\ref{eqn:nonhomogeneous_linear_ode}$) are continuous on some open interval $I$, then every solution of equation ($\ref{eqn:nonhomogeneous_linear_ode}$) on $I$ can be obtained from the general solution ($\ref{eqn:general_sol}$) of equation ($\ref{eqn:nonhomogeneous_linear_ode}$) on $I$ by assigning suitable values to the arbitrary constants $c_1$ and $c_2$ in $y_h$.

Proof

Let $y^*$ be any solution of equation ($\ref{eqn:nonhomogeneous_linear_ode}$) on $I$, and let $x_0$ be any $x$ in the interval $I$. By the theorem on the Existence of a General Solution for homogeneous ODEs with continuous variable coefficients, $y_h = c_1y_1 + c_2y_2$ exists. Also, by the method of variation of parameters, which we will discuss later, $y_p$ also exists. Therefore, the general solution ($\ref{eqn:general_sol}$) of the nonhomogeneous ODE ($\ref{eqn:nonhomogeneous_linear_ode}$) exists on the interval $I$. Now, by Theorem 1(b) which we proved earlier, $Y = y^* - y_p$ is a solution of the homogeneous ODE ($\ref{eqn:homogeneous_linear_ode}$) on interval $I$, and at $x_0$,

According to the Existence and Uniqueness Theorem for Initial Value Problems, for the initial conditions above, there exists a unique particular solution $Y$ of the homogeneous ODE ($\ref{eqn:homogeneous_linear_ode}$) on interval $I$, which can be obtained by assigning suitable values to $c_1$ and $c_2$ in $y_h$. Since $y^* = Y + y_p$, we have shown that any particular solution $y^*$ of the nonhomogeneous ODE ($\ref{eqn:nonhomogeneous_linear_ode}$) can be obtained from the general solution ($\ref{eqn:general_sol}$). $\blacksquare$

TL;DR

For a second-order homogeneous linear ordinary differential equation with continuous variable coefficients $p$ and $q$ on an interval $I$

\[y^{\prime\prime} + p(x)y^{\prime} + q(x)y = 0\]

and initial conditions

\[y(x_0)=K_0, \qquad y^{\prime}(x_0)=K_1\]

the following four theorems hold.

1. Existence and Uniqueness Theorem for Initial Value Problems: The initial value problem consisting of the given equation and initial conditions has a unique solution $y(x)$ on the interval $I$.
2. Test for Linear Dependence/Independence using the Wronskian: For two solutions $y_1$ and $y_2$ of the equation, if there exists an $x_0$ in the interval $I$ where the Wronskian $W(y_1, y_2) = y_1y_2^{\prime} - y_2y_1^{\prime}$ is $0$, then the two solutions are linearly dependent. Furthermore, if there exists an $x_1$ in the interval $I$ where $W\neq 0$, then the two solutions are linearly independent.
3. Existence of a General Solution: The given equation has a general solution on the interval $I$.
4. Nonexistence of Singular Solutions: This general solution includes all solutions of the equation (i.e., no singular solutions exist).

Prerequisites

• Solution of First-Order Linear ODEs
• Homogeneous Linear ODEs of Second Order
• Homogeneous Linear ODEs with Constant Coefficients
• Euler-Cauchy Equation
• Inverse Matrix, Singular Matrix, and Determinant

Homogeneous Linear ODEs with Continuous Variable Coefficients

Previously, we examined the general solutions of Homogeneous Linear ODEs with Constant Coefficients and the Euler-Cauchy Equation. In this article, we extend the discussion to a more general case: a second-order homogeneous linear ordinary differential equation with arbitrary continuous variable coefficients $p$ and $q$.

We will investigate the existence and form of the general solution for this equation. Additionally, we will explore the uniqueness of the solution to the Initial Value Problem composed of the ODE ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) and the following two initial conditions:

To state the conclusion upfront, the core of this discussion is that a linear ordinary differential equation with continuous coefficients does not have a singular solution (a solution that cannot be obtained from the general solution).

Existence and Uniqueness Theorem for Initial Value Problems

Existence and Uniqueness Theorem for Initial Value Problems
If $p(x)$ and $q(x)$ are continuous functions on some open interval $I$, and $x_0$ is in $I$, then the initial value problem consisting of Eqs. ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) and ($\ref{eqn:initial_conditions}$) has a unique solution $y(x)$ on the interval $I$.

The proof of existence will not be covered here; we will only look at the proof of uniqueness. Proving uniqueness is typically simpler than proving existence.
If you are not interested in the proof, you may skip this section and proceed to Linear Dependence and Independence of Solutions.

Proof of Uniqueness

Let’s assume that the initial value problem consisting of the ODE ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) and initial conditions ($\ref{eqn:initial_conditions}$) has two solutions, $y_1(x)$ and $y_2(x)$, on the interval $I$. If we can show that their difference

is identically zero on the interval $I$, this implies that $y_1 \equiv y_2$ on $I$, which means the solution is unique.

Since Eq. ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) is a homogeneous linear ODE, the linear combination $y$ of $y_1$ and $y_2$ is also a solution to the equation on $I$. Since $y_1$ and $y_2$ satisfy the same initial conditions ($\ref{eqn:initial_conditions}$), $y$ satisfies the conditions

Now, consider the function

and its derivative

From the ODE, we have

Substituting this into the expression for $z^{\prime}$ gives

Now, since $y$ and $y^{\prime}$ are real,

From this and the definition of $z$, we can derive two inequalities:

Since ${y^{\prime}}^2 \leq y^2 + {y^{\prime}}^2 = z$, this leads to

In the same way, from Eqs. ($\ref{eqn:z_prime}$) and ($\ref{eqn:inequalities}$), we get

These two inequalities, ($\ref{eqn:inequality_6a}$) and ($\ref{eqn:inequality_6b}$), are equivalent to the following inequalities:

The integrating factors for the left-hand sides of these two expressions are

Since $h$ is continuous, the indefinite integral $\int h(x)\ dx$ exists. As $F_1$ and $F_2$ are positive, from ($\ref{eqn:inequalities_7}$) we obtain

This means that on the interval $I$, $F_1 z$ is non-increasing and $F_2 z$ is non-decreasing. By Eq. ($\ref{eqn:initial_conditions_*}$), we have $z(x_0) = 0$, so

Finally, dividing both sides of the inequalities by the positive functions $F_1$ and $F_2$, we can show the uniqueness of the solution as follows:

Linear Dependence and Independence of Solutions

Let’s briefly recall what we covered in Second-Order Homogeneous Linear ODEs. The general solution on an open interval $I$ is constructed from a basis $y_1$, $y_2$ on $I$, which is a pair of linearly independent solutions. Here, $y_1$ and $y_2$ being linearly independent on an interval $I$ means that for all $x$ in the interval, the following holds:

If the above is not satisfied, and $k_1y_1(x) + k_2y_2(x) = 0$ holds for at least one non-zero $k_1$ or $k_2$, then $y_1$ and $y_2$ are linearly dependent on the interval $I$. In this case, for all $x$ in the interval $I$,

which means $y_1$ and $y_2$ are proportional.

Now let’s look at the following test for linear independence/dependence of solutions.

Test for Linear Dependence/Independence using the Wronskian
i. If the ODE ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) has continuous coefficients $p(x)$ and $q(x)$ on an open interval $I$, then a necessary and sufficient condition for two solutions $y_1$ and $y_2$ of Eq. ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) to be linearly dependent on $I$ is that their Wronski determinant, or simply Wronskian, which is the following determinant,

\[W(y_1, y_2) = \begin{vmatrix} y_1 & y_2 \\ y_1^{\prime} & y_2^{\prime} \\ \end{vmatrix} = y_1y_2^{\prime} - y_2y_1^{\prime} \label{eqn:wronskian}\tag{10}\]

is zero at some $x_0$ in the interval $I$.

\[\exists x_0 \in I: W(x_0)=0 \iff y_1 \text{ and } y_2 \text{ are linearly dependent}\]

ii. If $W=0$ at a point $x=x_0$ in the interval $I$, then $W=0$ for all $x$ in the interval $I$.

\[\exists x_0 \in I: W(x_0)=0 \implies \forall x \in I: W(x)=0\]

In other words, if there exists an $x_1$ in the interval $I$ such that $W\neq 0$, then $y_1$ and $y_2$ are linearly independent on that interval $I$.

\[\begin{align*} \exists x_1 \in I: W(x_1)\neq 0 &\implies \forall x \in I: W(x)\neq 0 \\ &\implies y_1 \text{ and } y_2 \text{ are linearly independent} \end{align*}\]

The Wronskian was first introduced by the Polish mathematician Józef Maria Hoene-Wroński and was named after him posthumously in 11882 HE by the Scottish mathematician Sir Thomas Muir.

Proof

i. (a)

Let $y_1$ and $y_2$ be linearly dependent on the interval $I$. Then, either Eq. ($\ref{eqn:linearly_dependent}$a) or ($\ref{eqn:linearly_dependent}$b) holds on $I$. If Eq. ($\ref{eqn:linearly_dependent}$a) holds, then

Similarly, if Eq. ($\ref{eqn:linearly_dependent}$b) holds, then

Thus, we can confirm that the Wronskian $W(y_1, y_2)=0$ for all $x$ in the interval $I$.

i. (b)

Conversely, suppose that $W(y_1, y_2)=0$ for some $x = x_0$. We will show that $y_1$ and $y_2$ are linearly dependent on the interval $I$. Consider the system of linear equations for the unknowns $k_1$, $k_2$:

This can be expressed in the form of a vector equation:

The coefficient matrix of this vector equation is

and the determinant of this matrix is $W(y_1(x_0), y_2(x_0))$. Since $\det(A) = W=0$, $A$ is a singular matrix that does not have an inverse matrix. Therefore, the system of equations ($\ref{eqn:linear_system}$) has a non-trivial solution $(c_1, c_2)$ other than the zero vector $(0,0)$, where at least one of $k_1$ and $k_2$ is not zero. Now, let’s introduce the function

Since Eq. ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) is homogeneous and linear, by the Superposition Principle, this function is a solution of ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) on the interval $I$. From Eq. ($\ref{eqn:linear_system}$), we can see that this solution satisfies the initial conditions $y(x_0)=0$, $y^{\prime}(x_0)=0$.

Meanwhile, there exists a trivial solution $y^* \equiv 0$ that satisfies the same initial conditions $y^*(x_0)=0$, ${y^*}^{\prime}(x_0)=0$. Since the coefficients $p$ and $q$ of Eq. ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) are continuous, the uniqueness of the solution is guaranteed by the Existence and Uniqueness Theorem for Initial Value Problems. Therefore, $y \equiv y^*$. That is, on the interval $I$,

Since at least one of $c_1$ and $c_2$ is not zero, this does not satisfy ($\ref{eqn:linearly_independent}$), which means that $y_1$ and $y_2$ are linearly dependent on the interval $I$.

ii.

If $W(x_0)=0$ at some point $x_0$ in the interval $I$, then by i.(b), $y_1$ and $y_2$ are linearly dependent on the interval $I$. Then, by i.(a), $W\equiv 0$. Therefore, if there is even one point $x_1$ in the interval $I$ where $W(x_1)\neq 0$, then $y_1$ and $y_2$ are linearly independent. $\blacksquare$

The General Solution Includes All Solutions

Existence of a General Solution

If $p(x)$ and $q(x)$ are continuous on an open interval $I$, then the equation ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) has a general solution on the interval $I$.

Proof

By the Existence and Uniqueness Theorem for Initial Value Problems, the ODE ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) has a solution $y_1(x)$ on the interval $I$ that satisfies the initial conditions

and a solution $y_2(x)$ on the interval $I$ that satisfies the initial conditions

The Wronskian of these two solutions at $x=x_0$ has a non-zero value:

Therefore, by the Test for Linear Dependence/Independence using the Wronskian, $y_1$ and $y_2$ are linearly independent on the interval $I$. Thus, these two solutions form a basis of solutions for Eq. ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) on the interval $I$, and a general solution $y = c_1y_1 + c_2y_2$ with arbitrary constants $c_1$, $c_2$ must exist on the interval $I$. $\blacksquare$

Nonexistence of Singular Solutions

If the ODE ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) has continuous coefficients $p(x)$ and $q(x)$ on some open interval $I$, then every solution $y=Y(x)$ of Eq. ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) on the interval $I$ is of the form

\[Y(x) = C_1y_1(x) + C_2y_2(x) \label{eqn:particular_solution}\tag{13}\]

where $y_1$, $y_2$ form a basis of solutions for Eq. ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) on the interval $I$, and $C_1$, $C_2$ are suitable constants.
That is, Eq. ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) does not have a singular solution, which is a solution that cannot be obtained from the general solution.

Proof

Let $y=Y(x)$ be any solution of Eq. ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) on the interval $I$. Now, by the Existence of a General Solution theorem, the ODE ($\ref{eqn:homogeneous_linear_ode_with_var_coefficients}$) has a general solution on the interval $I$:

Now we must show that for any $Y(x)$, there exist constants $c_1$, $c_2$ such that $y(x)=Y(x)$ on the interval $I$. Let’s first show that we can find values for $c_1$, $c_2$ such that for an arbitrary $x_0$ in $I$, we have $y(x_0)=Y(x_0)$ and $y^{\prime}(x_0)=Y^{\prime}(x_0)$. From Eq. ($\ref{eqn:general_solution}$), we get

Since $y_1$ and $y_2$ form a basis, the determinant of the coefficient matrix, which is the Wronskian $W(y_1(x_0), y_2(x_0))$, is non-zero. Therefore, Eq. ($\ref{eqn:vector_equation_2}$) can be solved for $c_1$ and $c_2$. Let the solution be $(c_1, c_2) = (C_1, C_2)$. Substituting this into Eq. ($\ref{eqn:general_solution}$) gives the following particular solution:

Since $C_1$, $C_2$ are the solution to Eq. ($\ref{eqn:vector_equation_2}$),

By the uniqueness part of the Existence and Uniqueness Theorem for Initial Value Problems, we have $y^* \equiv Y$ for all $x$ in the interval $I$. $\blacksquare$

TL;DR

• Euler-Cauchy equation: $x^2y^{\prime\prime} + axy^{\prime} + by = 0$
• Auxiliary equation: $m^2 + (a-1)m + b = 0$
• The form of the general solution can be divided into three cases, as shown in the table, depending on the sign of the discriminant $(1-a)^2 - 4b$ of the auxiliary equation.

Case	Roots of Auxiliary Equation	Basis of Solutions for Euler-Cauchy Equation	General Solution of Euler-Cauchy Equation
I	Distinct real roots $m_1$, $m_2$	$x^{m_1}$, $x^{m_2}$	$y = c_1 x^{m_1} + c_2 x^{m_2}$
II	Real double root $m = \cfrac{1-a}{2}$	$x^{(1-a)/2}$, $x^{(1-a)/2}\ln{x}$	$y = (c_1 + c_2 \ln x)x^m$
III	Complex conjugate roots $m_1 = \cfrac{1}{2}(1-a) + i\omega$, $m_2 = \cfrac{1}{2}(1-a) - i\omega$	$x^{(1-a)/2}\cos{(\omega \ln{x})}$, $x^{(1-a)/2}\sin{(\omega \ln{x})}$	$y = x^{(1-a)/2}[A\cos{(\omega \ln{x})} + B\sin{(\omega \ln{x})}]$

Prerequisites

• Homogeneous Linear ODEs of Second Order
• Homogeneous Linear ODEs with Constant Coefficients
• Euler’s Formula

Auxiliary Equation

The Euler-Cauchy equation is an ordinary differential equation of the form

with given constants $a$ and $b$, and an unknown function $y(x)$. Substituting

into Eq. ($\ref{eqn:euler_cauchy_eqn}$) gives

which simplifies to

From this, we obtain the auxiliary equation

and the necessary and sufficient condition for $y=x^m$ to be a solution of the Euler-Cauchy equation ($\ref{eqn:euler_cauchy_eqn}$) is that $m$ is a root of the auxiliary equation ($\ref{eqn:auxiliary_eqn}$).

Solving the quadratic equation ($\ref{eqn:auxiliary_eqn}$) gives the roots

and from this, the two functions

are solutions to equation ($\ref{eqn:euler_cauchy_eqn}$).

As with Homogeneous Linear ODEs with Constant Coefficients, we can divide this into three cases based on the sign of the discriminant $(1-a)^2 - 4b$ of the auxiliary equation ($\ref{eqn:auxiliary_eqn}$).

• $(1-a)^2 - 4b > 0$: Distinct real roots
• $(1-a)^2 - 4b = 0$: Real double root
• $(1-a)^2 - 4b < 0$: Complex conjugate roots

Forms of the General Solution Based on the Sign of the Discriminant

I. Distinct Real Roots $m_1$ and $m_2$

In this case, a basis of solutions for equation ($\ref{eqn:euler_cauchy_eqn}$) on any interval is

and the corresponding general solution is

II. Real Double Root $m = \cfrac{1-a}{2}$

When $(1-a)^2 - 4b = 0$, i.e., $b=\cfrac{(1-a)^2}{4}$, the quadratic equation ($\ref{eqn:auxiliary_eqn}$) has only one root $m = m_1 = m_2 = \cfrac{1-a}{2}$. Therefore, the one solution of the form $y = x^m$ we can obtain is

and the Euler-Cauchy equation ($\ref{eqn:euler_cauchy_eqn}$) takes the form

Now, let’s find another linearly independent solution $y_2$ using reduction of order.

If we set the second solution we are looking for as $y_2=uy_1$, we get

Since $\exp \left(-\int \cfrac{a}{x}\ dx \right) = \exp (-a\ln x) = \exp(\ln{x^{-a}}) = x^{-a}$,

and integrating gives $u = \ln x$.

Therefore, $y_2 = uy_1 = y_1 \ln x$, and since their quotient is not a constant, $y_1$ and $y_2$ are linearly independent. The general solution corresponding to the basis $y_1$ and $y_2$ is

III. Complex Conjugate Roots

In this case, the roots of the auxiliary equation ($\ref{eqn:auxiliary_eqn}$) are $m = \cfrac{1}{2}(1-a) \pm i\sqrt{b - \frac{1}{4}(1-a)^2}$, and the corresponding two complex solutions of the Euler-Cauchy equation ($\ref{eqn:euler_cauchy_eqn}$) can be written as follows, using the fact that $x=e^{\ln x}$.

By setting $t=\sqrt{b - \frac{1}{4}(1-a)^2}\ln x$ and using Euler’s formula $e^{it} = \cos{t} + i\sin{t}$, we can see that

and from this, we obtain the following two real solutions

Since their quotient $\cos\left(\sqrt{b - \frac{1}{4}(1-a)^2}\ln x \right)$ is not a constant, the two solutions above are linearly independent and thus form a basis for the Euler-Cauchy equation ($\ref{eqn:euler_cauchy_eqn}$) by the superposition principle. From this, we obtain the following real general solution.

However, the case where the auxiliary equation of an Euler-Cauchy equation has complex conjugate roots is not of great practical importance.

Transformation to a Homogeneous Linear ODE with Constant Coefficients

The Euler-Cauchy equation can be transformed into a homogeneous linear ODE with constant coefficients through a change of variables.

By substituting $x = e^t$, we get

and the Euler-Cauchy equation ($\ref{eqn:euler_cauchy_eqn}$) is transformed into the following homogeneous linear ODE with constant coefficients in terms of $t$.

If we solve equation ($\ref{eqn:substituted}$) for $t$ by applying the solution method for homogeneous linear ODEs with constant coefficients, and then transform the resulting solution back into a solution in terms of $x$ using $t = \ln{x}$, we obtain the same results as seen before.

TL;DR

• $n$th-term test for divergence: $\lim_{n\to\infty} a_n \neq 0 \Rightarrow \text{series }\sum a_n \text{ diverges}$
• Convergence/divergence of geometric series: The geometric series $\sum ar^{n-1}$
  • converges if $|r| < 1$
  • diverges if $|r| \geq 1$
• Convergence/divergence of $p$-series: The $p$-series $\sum \cfrac{1}{n^p}$
  • converges if $p>1$
  • diverges if $p\leq 1$
• Comparison Test: If $0 \leq a_n \leq b_n$, then
  • $\sum b_n < \infty \ \Rightarrow \ \sum a_n < \infty$
  • $\sum a_n = \infty \ \Rightarrow \ \sum b_n = \infty$
• Limit Comparison Test: If $\lim_{n\to\infty} \frac{a_n}{b_n} = c \text{ (}c\text{ is a finite positive number)}$, then both series $\sum a_n$ and $\sum b_n$ either both converge or both diverge
• For a series of positive terms $\sum a_n$ and a positive number $\epsilon < 1$
  • If $\sqrt[n]{a_n}< 1-\epsilon$ for all $n$, then the series $\sum a_n$ converges
  • If $\sqrt[n]{a_n}> 1+\epsilon$ for all $n$, then the series $\sum a_n$ diverges
• Root Test: For a series of positive terms $\sum a_n$, if the limit $\lim_{n\to\infty} \sqrt[n]{a_n} =: r$ exists, then
  • the series $\sum a_n$ converges if $r<1$
  • the series $\sum a_n$ diverges if $r>1$
• Ratio Test: For a sequence of positive terms $(a_n)$ and $0 < r < 1$
  • If $a_{n+1}/a_n \leq r$ for all $n$, then the series $\sum a_n$ converges
  • If $a_{n+1}/a_n \geq 1$ for all $n$, then the series $\sum a_n$ diverges
• For a sequence of positive numbers $(a_n)$, if the limit $\rho := \lim_{n\to\infty} \cfrac{a_{n+1}}{a_n}$ exists, then
  • the series $\sum a_n$ converges if $\rho < 1$
  • the series $\sum a_n$ diverges if $\rho > 1$
• Integral Test: For a continuous, decreasing function $f: \left[1,\infty \right) \rightarrow \mathbb{R}$ with $f(x)>0$ for all $x$, the series $\sum f(n)$ converges if and only if the integral $\int_1^\infty f(x)\ dx := \lim_{b\to\infty} \int_1^b f(x)\ dx$ converges
• Alternating Series Test: An alternating series $\sum a_n$ converges if the following conditions are satisfied:
  1. $a_n$ and $a_{n+1}$ have opposite signs for all $n$
  2. $|a_n| \geq |a_{n+1}|$ for all $n$
  3. $\lim_{n\to\infty} a_n = 0$
• A series that converges absolutely also converges. The converse is not true.

Prerequisites

• Sequences and Series

Introduction

In the previous post on Sequences and Series, we covered the definitions of convergence and divergence of series. In this post, we will summarize various methods for determining whether a series converges or diverges. Generally, testing for convergence or divergence of a series is much easier than finding the exact sum of the series.

The $n$th-Term Test

For a series $\sum a_n$, we call $a_n$ the general term of the series.

The following theorem allows us to easily identify some obviously divergent series, making it a wise first step when testing for convergence or divergence to avoid wasting time.

$n$th-term test for divergence
If a series $\sum a_n$ converges, then

\[\lim_{n\to\infty} a_n=0\]

That is,

\[\lim_{n\to\infty} a_n \neq 0 \Rightarrow \text{series }\sum a_n \text{ diverges}\]

Proof

Let $l$ be the sum of a convergent series $\sum a_n$ and define the partial sum of the first $n$ terms as

Then,

Therefore, for sufficiently large $n$ (where $n > N$),

From the definition of convergence of a sequence,

Caution

The converse of this theorem is generally not true. A classic example that demonstrates this is the harmonic series.

The harmonic series is a series whose terms are the reciprocals of an arithmetic sequence, forming a harmonic sequence. The most well-known harmonic series is

This series diverges, as can be shown by:

Thus, despite the fact that the harmonic series $H_n$ diverges, its general term $1/n$ converges to $0$.

If $\lim_{n\to\infty} a_n \neq 0$, then the series $\sum a_n$ must diverge, but assuming that a series $\sum a_n$ converges just because $\lim_{n\to\infty} a_n = 0$ is dangerous. In such cases, other methods must be used to determine convergence or divergence.

Geometric Series

The geometric series derived from a geometric sequence with first term 1 and common ratio $r$,

is the most important and fundamental series. From the equation

we get

Meanwhile,

Therefore, we know that the necessary and sufficient condition for the geometric series ($\ref{eqn:geometric_series}$) to converge is $|r| < 1$.

Convergence/divergence of geometric series
The geometric series $\sum ar^{n-1}$

• converges if $|r| < 1$
• diverges if $|r| \geq 1$

From this, we obtain

Geometric Series and Approximations

The identity ($\ref{eqn:sum_of_geometric_series}$) is useful for finding approximations of $\cfrac{1}{1-r}$ when $|r| < 1$.

Substituting $r=-\epsilon$ and $n=2$ into this equation, we get

Therefore, if $0 < \epsilon < 1$, then

which gives us

From this, we can see that for sufficiently small positive $\epsilon$, $\cfrac{1}{1 + \epsilon}$ can be approximated by $1 - \epsilon$.

$p$-Series Test

For a positive real number $p$, a series of the following form is called a $p$-series:

Convergence/divergence of $p$-series
The $p$-series $\sum \cfrac{1}{n^p}$

• converges if $p>1$
• diverges if $p\leq 1$

When $p=1$ in a $p$-series, we get the harmonic series, which we’ve already shown diverges.
The problem of finding the value of the $p$-series when $p=2$, i.e., $\sum \cfrac{1}{n^2}$, is known as the ‘Basel problem’, named after the hometown of the Bernoulli family, which produced several famous mathematicians over multiple generations and first proved that this series converges. The answer to this problem is known to be $\cfrac{\pi^2}{6}$.

More generally, the $p$-series where $p>1$ is called the zeta function. This is a special function introduced by Leonhard Euler in 11740 HE and later named by Riemann, defined as

This topic somewhat deviates from the main subject of this post, and frankly, as an engineering student rather than a mathematician, I don’t know much about it, so I won’t cover it here. However, Leonhard Euler showed that the zeta function can also be expressed as an infinite product of primes, known as the Euler Product, and subsequently, the zeta function has occupied a central position in various fields under analytic number theory. The Riemann zeta function, which extends the domain of the zeta function to complex numbers, and the important unsolved problem related to it, the Riemann hypothesis, are among these.

Returning to our original topic, the proof of the $p$-series test requires the Comparison Test and the Integral Test, which will be discussed later. However, the convergence/divergence of $p$-series can be usefully applied in the Comparison Test along with geometric series, which is why I’ve intentionally placed it earlier in this post.

Proof

i) When $p>1$

The integral

converges, so by the Integral Test, the series $\sum \cfrac{1}{n^p}$ also converges.

ii) When $p\leq 1$

In this case,

Since we know that the harmonic series $\sum \cfrac{1}{n}$ diverges, by the Comparison Test, $\sum \cfrac{1}{n^p}$ also diverges.

Conclusion

By i) and ii), the $p$-series $\sum \cfrac{1}{n^p}$ converges if $p>1$ and diverges if $p \leq 1$. $\blacksquare$

Comparison Test

Jakob Bernoulli’s Comparison Test is useful for determining the convergence/divergence of a series of positive terms, where each term is a non-negative real number.

Since a series of positive terms forms an increasing sequence, it must converge unless it diverges to infinity ($\sum a_n = \infty$). Therefore, in a series of positive terms, the expression

means that the series converges.

Comparison Test
If $0 \leq a_n \leq b_n$, then

• $\sum b_n < \infty \ \Rightarrow \ \sum a_n < \infty$
• $\sum a_n = \infty \ \Rightarrow \ \sum b_n = \infty$

In particular, for series of positive terms that have forms similar to the geometric series $\sum ar^{n-1}$ or the $p$-series $\sum \cfrac{1}{n^p}$ that we’ve examined earlier, such as $\sum \cfrac{1}{n^2 + n}$, $\sum \cfrac{\log n}{n^3}$, $\sum \cfrac{1}{2^n + 3^n}$, $\sum \cfrac{1}{\sqrt{n}}$, $\sum \sin{\cfrac{1}{n}}$, it’s a good idea to actively try the Comparison Test.

All the other convergence/divergence tests that will be discussed later can be derived from this Comparison Test, making it arguably the most important test.

Limit Comparison Test

For series of positive terms $\sum a_n$ and $\sum b_n$, if the dominant terms in the numerator and denominator of the ratio $a_n/b_n$ cancel out, resulting in $\lim_{n\to\infty} \cfrac{a_n}{b_n}=c \text{ (}c\text{ is a finite positive number)}$, and if we know whether the series $\sum b_n$ converges or diverges, then we can use the following Limit Comparison Test.

Limit Comparison Test
If

\[\lim_{n\to\infty} \frac{a_n}{b_n} = c \text{ (}c\text{ is a finite positive number)}\]

then both series $\sum a_n$ and $\sum b_n$ either both converge or both diverge. That is, $ \sum a_n < \infty \ \Leftrightarrow \ \sum b_n < \infty$.

Root Test

Theorem
For a series of positive terms $\sum a_n$ and a positive number $\epsilon < 1$

• If $\sqrt[n]{a_n}< 1-\epsilon$ for all $n$, then the series $\sum a_n$ converges
• If $\sqrt[n]{a_n}> 1+\epsilon$ for all $n$, then the series $\sum a_n$ diverges

Corollary: Root Test
For a series of positive terms $\sum a_n$, if the limit

\[\lim_{n\to\infty} \sqrt[n]{a_n} =: r\]

exists, then

• the series $\sum a_n$ converges if $r<1$
• the series $\sum a_n$ diverges if $r>1$

In the corollary above, if $r=1$, the test is inconclusive, and other methods must be used to determine convergence or divergence.

Ratio Test

Ratio Test
For a sequence of positive terms $(a_n)$ and $0 < r < 1$

• If $a_{n+1}/a_n \leq r$ for all $n$, then the series $\sum a_n$ converges
• If $a_{n+1}/a_n \geq 1$ for all $n$, then the series $\sum a_n$ diverges

Corollary
For a sequence of positive terms $(a_n)$, if the limit $\rho := \lim_{n\to\infty} \cfrac{a_{n+1}}{a_n}$ exists, then

• the series $\sum a_n$ converges if $\rho < 1$
• the series $\sum a_n$ diverges if $\rho > 1$

Integral Test

Integration can be used to determine the convergence/divergence of a series composed of a decreasing sequence of positive terms.

Integral Test
For a continuous, decreasing function $f: \left[1,\infty \right) \rightarrow \mathbb{R}$ with $f(x)>0$ for all $x$, the series $\sum f(n)$ converges if and only if the integral

\[\int_1^\infty f(x)\ dx := \lim_{b\to\infty} \int_1^b f(x)\ dx\]

converges.

Proof

Since the function $f(x)$ is continuous, decreasing, and always positive, the inequality

holds. Adding these inequalities from $n=1$ to the general term, we get

Now, using the Comparison Test, we obtain the desired result. $\blacksquare$

Alternating Series

A series $\sum a_n$ where each term $a_n$ is non-zero and has a sign opposite to that of the next term $a_{n+1}$, i.e., where positive and negative terms alternate, is called an alternating series.

For alternating series, the following theorem discovered by the German mathematician Gottfried Wilhelm Leibniz can be usefully applied to determine convergence/divergence.

Alternating Series Test
If

1. $a_n$ and $a_{n+1}$ have opposite signs for all $n$,
2. $|a_n| \geq |a_{n+1}|$ for all $n$, and
3. $\lim_{n\to\infty} a_n = 0$,

then the alternating series $\sum a_n$ converges.

Absolute Convergence

For a series $\sum a_n$, if the series $\sum |a_n|$ converges, we say that “the series $\sum a_n$ converges absolutely.”

The following theorem holds:

Theorem
A series that converges absolutely also converges.

The converse of the above theorem is not true.
If a series converges but does not converge absolutely, we say it “converges conditionally.”

Proof

For a real number $a$, define

Then,

Since $0 \leq a^\pm \leq |a|$, by the Comparison Test, if the series $\sum |a_n|$ converges, then both series $\sum a_n^+$ and $\sum a_n^-$ also converge, and therefore, by the basic properties of convergent series,

also converges. $\blacksquare$

Sequences

In calculus, a sequence primarily refers to an infinite sequence. That is, a sequence is a function defined on the set of all natural numbers

• If the values of this function are real numbers, we call it a ‘real sequence’; if complex numbers, a ‘complex sequence’; if points, a ‘point sequence’; if matrices, a ‘matrix sequence’; if functions, a ‘function sequence’; if sets, a ‘set sequence’. However, all of these can be simply referred to as ‘sequences’.

Usually, for the field of real numbers $\mathbb{R}$, in a sequence $\mathbf{a}: \mathbb{N} \to \mathbb{R}$, we denote

and represent this sequence as

or

*In the process of defining a sequence, instead of using the set of all natural numbers $\mathbb{N}$ as the domain, we can also use the set of non-negative integers

\[\mathbb{N}_0 := \{0\} \cup \mathbb{N} = \{0,1,2,\dots\}\]

or

\[\{2,3,4,\dots \}\]

For example, when dealing with power series theory, it’s more natural to have $\mathbb{N}_0$ as the domain.

Convergence and Divergence

If a sequence $(a_n)$ converges to a real number $l$, we write

and call $l$ the limit of the sequence $(a_n)$.

The rigorous definition using the epsilon-delta argument is as follows:

\[\lim_{n\to \infty} a_n = l \overset{def}\Longleftrightarrow \forall \epsilon > 0,\, \exists N \in \mathbb{N}\ (n > N \Rightarrow |a_n - l| < \epsilon)\]

In other words, if for any positive $\epsilon$, there always exists a natural number $N$ such that $|a_n - l | < \epsilon$ when $n>N$, it means that the difference between $a_n$ and $l$ becomes infinitely small for sufficiently large $n$. Therefore, we define that a sequence $(a_n)$ satisfying this condition converges to the real number $l$.

A sequence that does not converge is said to diverge. The convergence or divergence of a sequence does not change even if a finite number of its terms are altered.

If each term of the sequence $(a_n)$ grows infinitely large, we write

and say that it diverges to positive infinity. Similarly, if each term of the sequence $(a_n)$ becomes infinitely small, we write

and say that it diverges to negative infinity.

Basic Properties of Convergent Sequences

If sequences $(a_n)$ and $(b_n)$ both converge (i.e., have limits), then the sequences $(a_n + b_n)$ and $(a_n \cdot b_n)$ also converge, and

Also, for any real number $t$,

These properties are called the basic properties of convergent sequences or basic properties of limits.

$e$, the Base of Natural Logarithm

The base of natural logarithm is defined as

This is considered one of the most important constants in mathematics.

The term ‘natural constant’ is widely used only in Korea, but this is not a standard term. The official term registered in the mathematics terminology dictionary by the Korean Mathematical Society is ‘base of natural logarithm’, and the expression ‘natural constant’ cannot be found in this dictionary. Even in the Standard Korean Language Dictionary of the National Institute of Korean Language, the word ‘natural constant’ cannot be found, and in the dictionary definition of ‘natural logarithm’, it only mentions “a specific number usually denoted as e”.
In English-speaking countries and Japan, there is no corresponding term, and in English, it’s mainly referred to as ‘the base of the natural logarithm’ or shortened to ‘natural base’, or ‘Euler’s number’ or ‘the number $e$’.
Since the origin is unclear and it has never been recognized as an official term by the Korean Mathematical Society, and it’s not used anywhere else in the world except Korea, there’s no reason to insist on using such a term. Therefore, from now on, I will refer to it as ‘the base of natural logarithm’ or simply denote it as $e$.

Series

For a sequence

the sequence of partial sums

is called the series of the sequence $\mathbf{a}$. The series of the sequence $(a_n)$ is denoted as

Convergence and Divergence of Series

If the series obtained from the sequence $(a_n)$

converges to some real number $l$, we write

The limit value $l$ is called the sum of the series $\sum a_n$. The symbol

can represent either the series or the sum of the series, depending on the context.

A series that does not converge is said to diverge.

Basic Properties of Convergent Series

From the basic properties of convergent sequences, we obtain the following basic properties of convergent series. For a real number $t$ and two convergent series $\sum a_n$, $\sum b_n$,

The convergence of a series is not affected by changes in a finite number of terms. That is, if $a_n=b_n$ for all but finitely many $n$ in two sequences $(a_n)$, $(b_n)$, the series $\sum a_n$ converges if and only if the series $\sum b_n$ converges.

TL;DR

Newton’s Laws of Motion

1. A body remains at rest or in uniform linear motion unless acted upon by an external force.
2. The rate of change of momentum of a body is equal to the force applied to it.
   • $\vec{F} = \cfrac{d\vec{p}}{dt} = \cfrac{d}{dt}(m\vec{v}) = m\vec{a}$
3. When two bodies exert forces on each other, these forces are equal in magnitude and opposite in direction.
   • $\vec{F_1} = -\vec{F_2}$

Principle of Equivalence

• Inertial mass: The mass that determines a body’s acceleration when a given force is applied
• Gravitational mass: The mass that determines the gravitational force between a body and other bodies
• Currently, inertial mass and gravitational mass are known to clearly agree within an error range of about $10^{-12}$
• The assertion that inertial mass and gravitational mass are exactly equal is called the principle of equivalence

Newton’s Laws of Motion

Newton’s laws of motion are three laws published by Isaac Newton in his work Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy, abbreviated as ‘Principia’) in the year 11687 of the Holocene calendar. These laws form the foundation of Newtonian mechanics.

1. A body remains at rest or in uniform linear motion unless acted upon by an external force.
2. The rate of change of momentum of a body is equal to the force applied to it.
3. When two bodies exert forces on each other, these forces are equal in magnitude and opposite in direction.

Newton’s First Law

I. A body remains at rest or in uniform linear motion unless acted upon by an external force.

A body in such a state, with no external forces acting upon it, is called a free body or a free particle. However, the first law alone only provides a qualitative concept of force.

Newton’s Second Law

II. The rate of change of momentum of a body is equal to the force applied to it.

Newton defined momentum as the product of mass and velocity:

From this, Newton’s second law can be expressed as:

Despite their names, Newton’s first and second laws are actually closer to ‘definitions’ of force rather than ‘laws’. Also, we can see that the definition of force depends on the definition of ‘mass’.

Newton’s Third Law

III. When two bodies exert forces on each other, these forces are equal in magnitude and opposite in direction.

This is also known as the ‘law of action and reaction’ and applies when the force exerted by one body on another is directed along the line connecting the two points of action. Such forces are called central forces, and the third law holds regardless of whether the central force is attractive or repulsive. Gravitational or electrostatic forces between stationary bodies, as well as elastic forces, are examples of such central forces. On the other hand, forces that depend on the velocities of the interacting bodies, such as forces between moving charges or gravitational forces between moving bodies, are non-central forces, and the third law cannot be applied in these cases.

Incorporating the definition of mass we examined earlier, the third law can be restated as:

III$^\prime$. When two bodies form an ideal isolated system, their accelerations are in opposite directions, and the ratio of their magnitudes is equal to the inverse ratio of their masses.

By Newton’s third law:

Substituting the second law ($\ref{eqn:2nd_law}$) into this:

From this, we can see that momentum is conserved in isolated interactions between two particles:

Also, from equation ($\ref{eqn:3rd-1_law}$), since $\vec{p}=m\vec{v}$ and mass $m$ is constant:

This gives us:

Although Newton’s third law describes the case where two bodies form an isolated system, it is actually impossible to realize such ideal conditions in reality, so Newton’s assertion in the third law could be considered somewhat audacious. Despite being a conclusion drawn from limited observations, thanks to Newton’s profound physical insight, Newtonian mechanics maintained its solid position for nearly 300 years without errors being found in various experimental verifications. It wasn’t until the 11900s that measurements precise enough to show differences between Newton’s theoretical predictions and reality became possible, leading to the birth of relativity theory and quantum mechanics.

Inertial Mass and Gravitational Mass

One method of determining the mass of an object is to compare its weight with a standard weight using a tool like a balance. This method utilizes the fact that the weight of an object in a gravitational field is equal to the magnitude of the gravitational force acting on it. In this case, the second law $\vec{F}=m\vec{a}$ takes the form $\vec{W}=m\vec{g}$. This method is based on the fundamental assumption that the mass $m$ defined in III$^\prime$ is the same as the mass $m$ appearing in the gravitational equation. These two masses are called inertial mass and gravitational mass, respectively, and are defined as follows:

• Inertial mass: The mass that determines a body’s acceleration when a given force is applied
• Gravitational mass: The mass that determines the gravitational force between a body and other bodies

Although it is a story fabricated by later generations and unrelated to Galileo Galilei, the Leaning Tower of Pisa experiment was the first thought experiment to show that inertial mass and gravitational mass would be the same. Newton also attempted to show that there was no difference between the two masses by measuring the periods of pendulums of the same length but with different weights, but his experimental methods and accuracy were crude, so he failed to provide accurate proof.

Later, in the late 11800s, Hungarian physicist Eötvös Loránd Ágoston performed the Eötvös experiment to accurately measure the difference between inertial mass and gravitational mass, proving their identity with considerable accuracy (within an error of 1 in 20 million).

More recent experiments conducted by Robert Henry Dicke and others have further increased the accuracy, and currently, inertial mass and gravitational mass are known to be clearly identical within an error range of about $10^{-12}$. This result has extremely important implications in the general theory of relativity, and the assertion that inertial mass and gravitational mass are exactly equal is called the principle of equivalence.

TL;DR

• Second-order homogeneous linear ODE with constant coefficients: $y^{\prime\prime} + ay^{\prime} + by = 0$
• Characteristic equation: $\lambda^2 + a\lambda + b = 0$
• Depending on the sign of the discriminant $a^2 - 4b$ of the characteristic equation, the form of the general solution can be divided into three cases as shown in the table:

Case	Roots of Characteristic Equation	Basis of the ODE’s Solution	General Solution of the ODE
I	Distinct real roots $\lambda_1$, $\lambda_2$	$e^{\lambda_1 x}$, $e^{\lambda_2 x}$	$y = c_1e^{\lambda_1 x} + c_2e^{\lambda_2 x}$
II	Real double root $\lambda = -\cfrac{1}{2}a$	$e^{-ax/2}$, $xe^{-ax/2}$	$y = (c_1 + c_2 x)e^{-ax/2}$
III	Complex conjugate roots $\lambda_1 = -\cfrac{1}{2}a + i\omega$, $\lambda_2 = -\cfrac{1}{2}a - i\omega$	$e^{-ax/2}\cos{\omega x}$, $e^{-ax/2}\sin{\omega x}$	$y = e^{-ax/2}(A\cos{\omega x} + B\sin{\omega x})$

Prerequisites

• Bernoulli Equation
• Homogeneous Linear ODEs of Second Order
• Euler’s formula

Characteristic Equation

Let’s consider a second-order homogeneous linear ordinary differential equation with constant coefficients $a$ and $b$:

This type of equation has important applications in mechanical and electrical vibrations.

We have previously found the general solution of the logistic equation in Bernoulli Equation, and according to it, the solution to the first-order linear ODE with a constant coefficient $k$,

is the exponential function $y = ce^{-kx}$ (the case where $A=-k$ and $B=0$ in equation (4) of that post).

Therefore, for a similarly shaped equation like ($\ref{eqn:ode_with_constant_coefficients}$), we can first try a solution of the form

Of course, this is merely a guess, and there is no guarantee that the general solution will actually have this form. However, if we can find any two linearly independent solutions, we can obtain the general solution by the superposition principle, as we saw in Homogeneous Linear ODEs of Second Order.
As we will see shortly, there are also cases where we need to find a different form of solution.

Substituting Eq. ($\ref{eqn:general_sol}$) and its derivatives

into Eq. ($\ref{eqn:ode_with_constant_coefficients}$) gives

Therefore, if $\lambda$ is a root of the characteristic equation

then the exponential function ($\ref{eqn:general_sol}$) is a solution to the ordinary differential equation ($\ref{eqn:ode_with_constant_coefficients}$). Solving the quadratic equation ($\ref{eqn:characteristic_eqn}$) gives

and from this, the two functions

become solutions to equation ($\ref{eqn:ode_with_constant_coefficients}$).

The terms characteristic equation and auxiliary equation are often used interchangeably; they mean exactly the same thing. You can use either term.

Now, we can divide the problem into three cases depending on the sign of the discriminant $a^2 - 4b$ of the characteristic equation ($\ref{eqn:characteristic_eqn}$).

• $a^2 - 4b > 0$: Distinct real roots
• $a^2 - 4b = 0$: Real double root
• $a^2 - 4b < 0$: Complex conjugate roots

Form of the General Solution based on the Sign of the Discriminant

I. Distinct Real Roots $\lambda_1$ and $\lambda_2$

In this case, a basis of solutions for equation ($\ref{eqn:ode_with_constant_coefficients}$) on any interval is

and the corresponding general solution is

II. Real Double Root $\lambda = -\cfrac{a}{2}$

If $a^2 - 4b = 0$, the quadratic equation ($\ref{eqn:characteristic_eqn}$) yields only one root $\lambda = \lambda_1 = \lambda_2 = -\cfrac{a}{2}$. Therefore, the only solution of the form $y = e^{\lambda x}$ we can obtain is

To obtain a basis, we need to find a second solution $y_2$ that is linearly independent of $y_1$.

In this situation, we can use reduction of order, which we have discussed before. We set the second solution we are looking for as $y_2=uy_1$, and substitute

into equation ($\ref{eqn:ode_with_constant_coefficients}$) to get

Grouping the terms by $u^{\prime\prime}$, $u^\prime$, and $u$ gives

Here, since $y_1$ is a solution to equation ($\ref{eqn:ode_with_constant_coefficients}$), the expression in the last parenthesis is $0$. Also, since

the expression in the first parenthesis is also $0$. Thus, only $u^{\prime\prime}y_1 = 0$ remains, which implies $u^{\prime\prime}=0$. Integrating twice gives $u = c_1x + c_2$. Since the integration constants $c_1$ and $c_2$ can be any value, we can simply choose $c_1=1$ and $c_2=0$ to set $u=x$. Then we have $y_2 = uy_1 = xy_1$. Since $y_1$ and $y_2$ are linearly independent, they form a basis. Therefore, when the characteristic equation ($\ref{eqn:characteristic_eqn}$) has a double root, a basis of solutions for equation ($\ref{eqn:ode_with_constant_coefficients}$) on any interval is

and the corresponding general solution is

III. Complex Conjugate Roots $-\cfrac{1}{2}a + i\omega$ and $-\cfrac{1}{2}a - i\omega$

In this case, $a^2 - 4b < 0$, and since $\sqrt{-1} = i$, from Eq. ($\ref{eqn:lambdas}$) we have

Here, let’s define the real number $\omega = \sqrt{b-\cfrac{1}{4}a^2}$.

With $\omega$ defined as above, the roots of the characteristic equation ($\ref{eqn:characteristic_eqn}$) are the complex conjugate roots $\lambda = -\cfrac{1}{2}a \pm i\omega$. The corresponding two complex solutions to equation ($\ref{eqn:ode_with_constant_coefficients}$) are

However, in this case, we can obtain a basis of real solutions as follows.

From Euler’s formula

and by substituting $-t$ for $t$ in the above equation to get

we can add and subtract these two equations to obtain:

The complex exponential function $e^z$ of a complex variable $z = r + it$ with real part $r$ and imaginary part $it$ can be defined using the real functions $e^r$, $\cos t$, and $\sin t$ as follows.

Here, setting $r=-\cfrac{1}{2}ax$ and $t=\omega x$, we can write:

By the superposition principle, the sum and constant multiples of these complex solutions are also solutions. Therefore, by adding the two equations side by side and multiplying both sides by $\cfrac{1}{2}$, we can obtain the first real solution $y_1$ as follows.

Similarly, by subtracting the second equation from the first and multiplying both sides by $\cfrac{1}{2i}$, we can obtain the second real solution $y_2$.

Since $\cfrac{y_1}{y_2} = \cot{\omega x}$ is not a constant, $y_1$ and $y_2$ are linearly independent on any interval and thus form a basis of real solutions for equation ($\ref{eqn:ode_with_constant_coefficients}$). From this, we obtain the general solution

Overview

In early July 12024, I added multi-language support to this blog, which is hosted on GitHub Pages with Jekyll, by applying the Polyglot plugin. This series shares the bugs encountered while applying the Polyglot plugin to the Chirpy theme, their solutions, and how to write the HTML header and sitemap.xml with SEO in mind. The series consists of 3 posts, and this is the third post in the series.

• Part 1: Applying Polyglot Plugin & Modifying HTML Header and Sitemap
• Part 2: Implementing Language Selection Button & Localizing Layout Language
• Part 3: Troubleshooting Chirpy Theme Build Failures and Search Function Errors (this post)

This series was originally planned as two parts, but as I added more content over several revisions, the length increased significantly, so it has been reorganized into three parts.

Requirements

• The built result (web pages) must be served under language-specific paths (e.g., /posts/ko/, /posts/ja/).
• To minimize the additional time and effort for multi-language support, the build process should automatically recognize the language based on the local file path (e.g., /_posts/ko/, /_posts/ja/) without needing to manually specify ‘lang’ and ‘permalink’ tags in the YAML front matter of each Markdown file.
• The header of each page on the site must meet Google’s SEO guidelines for multilingual search by including appropriate Content-Language meta tags, hreflang alternate tags, and canonical links.
• The site must provide all language-specific page links in a single sitemap.xml file without omissions, and this sitemap.xml file must exist only at the root path without duplication.
• All features provided by the Chirpy theme must function correctly on each language page. If not, they must be modified to work properly.
  • ‘Recently Updated’ and ‘Trending Tags’ features work correctly.
  • No errors during the build process using GitHub Actions.
  • The post search function in the top-right corner of the blog works correctly.

Before We Begin

This post is a continuation of Part 1 and Part 2, so if you haven’t read them yet, I recommend reading the previous posts first.

Troubleshooting (‘relative_url_regex’: target of repeat operator is not specified)

(+ 12025.10.08. Update) This bug was fixed in Polyglot version 1.11.

After completing the previous steps, when I ran the bundle exec jekyll serve command to test the build, it failed with the error 'relative_url_regex': target of repeat operator is not specified.

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...(omitted)
                    ------------------------------------------------
      Jekyll 4.3.4   Please append `--trace` to the `serve` command 
                     for any additional information or backtrace. 
                    ------------------------------------------------
/Users/yunseo/.gem/ruby/3.2.2/gems/jekyll-polyglot-1.8.1/lib/jekyll/polyglot/
patches/jekyll/site.rb:234:in `relative_url_regex': target of repeat operator 
is not specified: /href="?\/((?:(?!*.gem)(?!*.gemspec)(?!tools)(?!README.md)(
?!LICENSE)(?!*.config.js)(?!rollup.config.js)(?!package*.json)(?!.sass-cache)
(?!.jekyll-cache)(?!gemfiles)(?!Gemfile)(?!Gemfile.lock)(?!node_modules)(?!ve
ndor\/bundle\/)(?!vendor\/cache\/)(?!vendor\/gems\/)(?!vendor\/ruby\/)(?!en\/
)(?!ko\/)(?!es\/)(?!pt-BR\/)(?!ja\/)(?!fr\/)(?!de\/)[^,'"\s\/?.]+\.?)*(?:\/[^
\]\[)("'\s]*)?)"/ (RegexpError)

...(omitted)

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exclude:
  - "*.gem"
  - "*.gemspec"
  - docs
  - tools
  - README.md
  - LICENSE
  - "*.config.js"
  - package*.json

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    # a regex that matches relative urls in a html document
    # matches href="baseurl/foo/bar-baz" href="/foo/bar-baz" and others like it
    # avoids matching excluded files.  prepare makes sure
    # that all @exclude dirs have a trailing slash.
    def relative_url_regex(disabled = false)
      regex = ''
      unless disabled
        @exclude.each do |x|
          regex += "(?!#{x})"
        end
        @languages.each do |x|
          regex += "(?!#{x}\/)"
        end
      end
      start = disabled ? 'ferh' : 'href'
      %r{#{start}="?#{@baseurl}/((?:#{regex}[^,'"\s/?.]+\.?)*(?:/[^\]\[)("'\s]*)?)"}
    end

    # a regex that matches absolute urls in a html document
    # matches href="http://baseurl/foo/bar-baz" and others like it
    # avoids matching excluded files.  prepare makes sure
    # that all @exclude dirs have a trailing slash.
    def absolute_url_regex(url, disabled = false)
      regex = ''
      unless disabled
        @exclude.each do |x|
          regex += "(?!#{x})"
        end
        @languages.each do |x|
          regex += "(?!#{x}\/)"
        end
      end
      start = disabled ? 'ferh' : 'href'
      %r{(?<!hreflang="#{@default_lang}" )#{start}="?#{url}#{@baseurl}/((?:#{regex}[^,'"\s/?.]+\.?)*(?:/[^\]\[)("'\s]*)?)"}
    end

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    def relative_url_regex(disabled = false)
      regex = ''
      unless disabled
        @exclude.each do |x|
          escaped_x = Regexp.escape(x)
          regex += "(?!#{escaped_x})"
        end
        @languages.each do |x|
          escaped_x = Regexp.escape(x)
          regex += "(?!#{escaped_x}\/)"
        end
      end
      start = disabled ? 'ferh' : 'href'
      %r{#{start}="?#{@baseurl}/((?:#{regex}[^,'"\s/?.]+\.?)*(?:/[^\]\[)("'\s]*)?)"}
    end

    def absolute_url_regex(url, disabled = false)
      regex = ''
      unless disabled
        @exclude.each do |x|
          escaped_x = Regexp.escape(x)
          regex += "(?!#{escaped_x})"
        end
        @languages.each do |x|
          escaped_x = Regexp.escape(x)
          regex += "(?!#{escaped_x}\/)"
        end
      end
      start = disabled ? 'ferh' : 'href'
      %r{(?<!hreflang="#{@default_lang}" )#{start}="?#{url}#{@baseurl}/((?:#{regex}[^,'"\s/?.]+\.?)*(?:/[^\]\[)("'\s]*)?)"}
    end

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exclude: # Modified with reference to https://github.com/untra/polyglot/issues/204
  # - "*.gem"
  - jekyll-theme-chirpy.gemspec # - "*.gemspec"
  - tools
  - README.md
  - LICENSE
  - purgecss.config.js # - "*.config.js"
  - rollup.config.js
  - package*.json

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  <body>
    {% include sidebar.html lang=lang %}

    <div id="main-wrapper" class="d-flex justify-content-center">
      <div class="container d-flex flex-column px-xxl-5">
        
        (...omitted...)

        {% include_cached search-results.html lang=lang %}
      </div>

      <aside aria-label="Scroll to Top">
        <button id="back-to-top" type="button" class="btn btn-lg btn-box-shadow">
          <i class="fas fa-angle-up"></i>
        </button>
      </aside>
    </div>

    (...omitted...)

    {% include_cached search-loader.html lang=lang %}
  </body>

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<!-- The Search results -->

<div id="search-result-wrapper" class="d-flex justify-content-center d-none">
  <div class="col-11 content">
    <div id="search-hints">
      {% include_cached trending-tags.html %}
    </div>
    <div id="search-results" class="d-flex flex-wrap justify-content-center text-muted mt-3"></div>
  </div>
</div>

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{% capture result_elem %}
  <article class="px-1 px-sm-2 px-lg-4 px-xl-0">
    <header>
      <h2><a href="{url}">{title}</a></h2>
      <div class="post-meta d-flex flex-column flex-sm-row text-muted mt-1 mb-1">
        {categories}
        {tags}
      </div>
    </header>
    <p>{snippet}</p>
  </article>
{% endcapture %}

{% capture not_found %}<p class="mt-5">{{ site.data.locales[include.lang].search.no_results }}</p>{% endcapture %}

<script>
  {% comment %} Note: dependent library will be loaded in `js-selector.html` {% endcomment %}
  document.addEventListener('DOMContentLoaded', () => {
    SimpleJekyllSearch({
      searchInput: document.getElementById('search-input'),
      resultsContainer: document.getElementById('search-results'),
      json: '{{ '/assets/js/data/search.json' | relative_url }}',
      searchResultTemplate: '{{ result_elem | strip_newlines }}',
      noResultsText: '{{ not_found }}',
      templateMiddleware: function(prop, value, template) {
        if (prop === 'categories') {
          if (value === '') {
            return `${value}`;
          } else {
            return `<div class="me-sm-4"><i class="far fa-folder fa-fw"></i>${value}</div>`;
          }
        }

        if (prop === 'tags') {
          if (value === '') {
            return `${value}`;
          } else {
            return `<div><i class="fa fa-tag fa-fw"></i>${value}</div>`;
          }
        }
      }
    });
  });
</script>

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---
layout: compress
swcache: true
---

[
  {% for post in site.posts %}
  {
    "title": {{ post.title | jsonify }},
    "url": {{ post.url | relative_url | jsonify }},
    "categories": {{ post.categories | join: ', ' | jsonify }},
    "tags": {{ post.tags | join: ', ' | jsonify }},
    "date": "{{ post.date }}",
    {% include no-linenos.html content=post.content %}
    {% assign _content = content | strip_html | strip_newlines %}
    "snippet": {{ _content | truncate: 200 | jsonify }},
    "content": {{ _content | jsonify }}
  }{% unless forloop.last %},{% endunless %}
  {% endfor %}
]

stateDiagram
  state "Changes" as CH
  state "Build start" as BLD
  state "Create search.json" as IDX
  state "Static Website" as DEP
  state "In Test" as TST
  state "Search Loader" as SCH
  state "Results" as R
    
  [*] --> CH: Make Changes
  CH --> BLD: Commit & Push origin
  BLD --> IDX: jekyll build
  IDX --> TST: Build Complete
  TST --> CH: Error Detected
  TST --> DEP: Deploy
  DEP --> SCH: Search Input
  SCH --> R: Return Results
  R --> [*]

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{% capture result_elem %}
  <article class="px-1 px-sm-2 px-lg-4 px-xl-0">
    <header>
      {% if site.active_lang != site.default_lang %}
      <h2><a {% static_href %}href="/{{ site.active_lang }}{url}"{% endstatic_href %}>{title}</a></h2>
      {% else %}
      <h2><a href="{url}">{title}</a></h2>
      {% endif %}

(...omitted...)

<script>
  {% comment %} Note: dependent library will be loaded in `js-selector.html` {% endcomment %}
  document.addEventListener('DOMContentLoaded', () => {
    {% assign search_path = '/assets/js/data/search.json' %}
    {% if site.active_lang != site.default_lang %}
      {% assign search_path = '/' | append: site.active_lang | append: search_path %}
    {% endif %}
    
    SimpleJekyllSearch({
      searchInput: document.getElementById('search-input'),
      resultsContainer: document.getElementById('search-results'),
      json: '{{ search_path | relative_url }}',
      searchResultTemplate: '{{ result_elem | strip_newlines }}',

(...omitted)