Homogeneous Linear ODEs of Second Order

TL;DR

• Standard form of a second-order linear ODE: $y^{\prime\prime} + p(x)y^{\prime} + q(x)y = r(x)$
  • Coefficients: Functions $p$, $q$
  • Input: $r(x)$
  • Output or response: $y(x)$
• Homogeneous and Nonhomogeneous
  • Homogeneous: When $r(x)\equiv0$ in the standard form.
  • Nonhomogeneous: When $r(x)\not\equiv 0$ in the standard form.
• Superposition principle: For a homogeneous linear ODE $y^{\prime\prime} + p(x)y^{\prime} + q(x)y = 0$, any linear combination of two of its solutions on an open interval $I$ is also a solution of the given equation. That is, the sum and constant multiples of any solutions to the given homogeneous linear ODE are also solutions.
• Basis or fundamental system: A pair of linearly independent solutions $(y_1, y_2)$ of a homogeneous linear ODE on an interval $I$.
• Reduction of order: If one solution to a second-order homogeneous ODE is known, a second, linearly independent solution (i.e., a basis) can be found by solving a first-order ODE. This method is called reduction of order.
• Applications of reduction of order: A general second-order ODE $F(x, y, y^\prime, y^{\prime\prime})=0$, whether linear or nonlinear, can be reduced to a first-order ODE using reduction of order in the following cases:
  • $y$ does not appear explicitly.
  • $x$ does not appear explicitly.
  • The equation is homogeneous linear and one solution is already known.

Prerequisites

• Basic Concepts of Modeling
• Separation of Variables
• Solution of First-Order Linear ODEs

Second-Order Linear ODEs

A second-order ordinary differential equation is called linear if it can be written in the form

\[y^{\prime\prime} + p(x)y^{\prime} + q(x)y = r(x) \label{eqn:standard_form}\tag{1}\]

and nonlinear otherwise.

When $p$, $q$, and $r$ are functions of any $x$, this equation is linear with respect to $y$ and its derivatives.

The form of Eq. ($\ref{eqn:standard_form}$) is called the standard form of a second-order linear ODE. If the first term of a given second-order linear ODE is $f(x)y^{\prime\prime}$, we can obtain the standard form by dividing both sides of the equation by $f(x)$.

The functions $p$ and $q$ are called coefficients, $r(x)$ is the input, and $y(x)$ is the output or the response to the input and initial conditions.

Homogeneous Second-Order Linear ODEs

Let $J$ be an interval $a<x<b$ where we want to solve Eq. ($\ref{eqn:standard_form}$). If $r(x)\equiv 0$ for the interval $J$ in Eq. ($\ref{eqn:standard_form}$), then

\[y^{\prime\prime} + p(x)y^{\prime} + q(x)y = 0 \label{eqn:homogeneous_linear_ode}\tag{2}\]

and this is called homogeneous.

Nonhomogeneous Linear ODEs

If $r(x)\not\equiv 0$ in the interval $J$, the equation is called nonhomogeneous.

Superposition Principle

A function of the form \(y = c_1y_1 + c_2y_2 \quad \text{(where }c_1, c_2\text{ are arbitrary constants)}\tag{3}\) is called a linear combination of $y_1$ and $y_2$.

The following holds true.

Superposition principle
For the homogeneous linear ODE ($\ref{eqn:homogeneous_linear_ode}$), any linear combination of two of its solutions on an open interval $I$ is also a solution of Eq. ($\ref{eqn:homogeneous_linear_ode}$). That is, the sum and constant multiples of any solutions to the given homogeneous linear ODE are also solutions.

Proof

Let $y_1$ and $y_2$ be solutions of Eq. ($\ref{eqn:homogeneous_linear_ode}$) on an interval $I$. Substituting $y=c_1y_1+c_2y_2$ into Eq. ($\ref{eqn:homogeneous_linear_ode}$) gives

\[\begin{align*} y^{\prime\prime} + py^{\prime} + qy &= (c_1y_1+c_2y_2)^{\prime\prime} + p(c_1y_1+c_2y_2)^{\prime} + q(c_1y_1+c_2y_2) \\ &= c_1y_1^{\prime\prime} + c_2y_2^{\prime\prime} + p(c_1y_1^{\prime} + c_2y_2^{\prime}) + q(c_1y_1+c_2y_2) \\ &= c_1(y_1^{\prime\prime} + py_1^{\prime} + qy_1) + c_2(y_2^{\prime\prime} + py_2^{\prime} + qy_2) \\ &= 0 \end{align*}\]

which becomes an identity. Therefore, $y$ is a solution of Eq. ($\ref{eqn:homogeneous_linear_ode}$) on the interval $I$. $\blacksquare$

Note that the superposition principle holds only for homogeneous linear ODEs and not for nonhomogeneous linear or nonlinear ODEs.

Basis and General Solution

Review of Key Concepts from First-Order ODEs

As we saw previously in Basic Concepts of Modeling, an Initial Value Problem for a first-order ODE consists of the ODE and an initial condition $y(x_0)=y_0$. The initial condition is necessary to determine the arbitrary constant $c$ in the general solution of the given ODE, and the resulting solution is called a particular solution. Let’s now extend these concepts to second-order ODEs.

Initial Value Problem and Initial Conditions

An initial value problem for the second-order homogeneous ODE ($\ref{eqn:homogeneous_linear_ode}$) consists of the given ODE ($\ref{eqn:homogeneous_linear_ode}$) and two initial conditions

\[y(x_0) = K_0, \quad y^{\prime}(x_0)=K_1 \label{eqn:init_conditions}\tag{4}\]

These conditions are needed to determine the two arbitrary constants $c_1$ and $c_2$ in the general solution of the ODE

\[y = c_1y_1 + c_2y_2 \label{eqn:general_sol}\tag{5}\]

Linear Independence and Dependence

Let’s briefly discuss the concepts of linear independence and dependence. This is necessary to define a basis later.
Two functions $y_1$ and $y_2$ are said to be linearly independent on an interval $I$ where they are defined if for all points in $I$,

\[k_1y_1(x) + k_2y_2(x) = 0 \Leftrightarrow k_1=0\text{ and }k_2=0 \label{eqn:linearly_independent}\tag{6}\]

Otherwise, $y_1$ and $y_2$ are said to be linearly dependent.

If $y_1$ and $y_2$ are linearly dependent (i.e., statement ($\ref{eqn:linearly_independent}$) is not true), then with $k_1 \neq 0$ or $k_2 \neq 0$, we can divide both sides of the equation in ($\ref{eqn:linearly_independent}$) to write

\[y_1 = - \frac{k_2}{k_1}y_2 \quad \text{or} \quad y_2 = - \frac{k_1}{k_2}y_2\]

which shows that $y_1$ and $y_2$ are proportional.

Basis, General Solution, and Particular Solution

Returning to our discussion, for Eq. ($\ref{eqn:general_sol}$) to be a general solution, $y_1$ and $y_2$ must be solutions to Eq. ($\ref{eqn:homogeneous_linear_ode}$) and also be linearly independent (not proportional to each other) on the interval $I$. A pair of solutions $(y_1, y_2)$ of Eq. ($\ref{eqn:homogeneous_linear_ode}$) that are linearly independent on an interval $I$ is called a basis or a fundamental system of solutions for Eq. ($\ref{eqn:homogeneous_linear_ode}$) on $I$.

By using the initial conditions to determine the two constants $c_1$ and $c_2$ in the general solution ($\ref{eqn:general_sol}$), we obtain a unique solution that passes through the point $(x_0, K_0)$ and has a slope of $K_1$ at that point. This is called a particular solution of the ODE ($\ref{eqn:homogeneous_linear_ode}$).

If Eq. ($\ref{eqn:homogeneous_linear_ode}$) is continuous on an open interval $I$, it is guaranteed to have a general solution, and this general solution includes all possible particular solutions. In this case, Eq. ($\ref{eqn:homogeneous_linear_ode}$) does not have a singular solution that cannot be obtained from the general solution.

Reduction of Order

If we can find one solution to a second-order homogeneous ODE, we can find a second, linearly independent solution—that is, a basis—by solving a first-order ODE as follows. This method is called reduction of order.

For a second-order homogeneous ODE in standard form with $y^{\prime\prime}$, not $f(x)y^{\prime\prime}$,

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

let’s assume we know one solution $y_1$ on an open interval $I$.

Now, let’s set the second solution we are looking for as $y_2 = uy_1$, and substitute

\[\begin{align*} y &= y_2 = uy_1, \\ y^{\prime} &= y_2^{\prime} = u^{\prime}y_1 + uy_1^{\prime}, \\ y^{\prime\prime} &= y_2^{\prime\prime} = u^{\prime\prime}y_1 + 2u^{\prime}y_1^{\prime} + uy_1^{\prime\prime} \end{align*}\]

into the equation to get

\[(u^{\prime\prime}y_1 + 2u^{\prime}y_1^{\prime} + uy_1^{\prime\prime}) + p(u^{\prime}y_1 + uy_1^{\prime}) + quy_1 = 0 \tag{7}\]

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

\[y_1u^{\prime\prime} + (py_1+2y_1^{\prime})u^{\prime} + (y_1^{\prime\prime} + py_1^{\prime} + qy_1)u = 0\]

However, since $y_1$ is a solution to the given equation, the expression in the last parenthesis is $0$. Thus, the term with $u$ disappears, leaving an ODE in terms of $u^{\prime}$ and $u^{\prime\prime}$. Dividing the remaining ODE by $y_1$ and setting $u^{\prime}=U$ and $u^{\prime\prime}=U^{\prime}$, we obtain the following first-order ODE.

\[U^{\prime} + \left(\frac{2y_1^{\prime}}{y_1} + p \right) U = 0.\]

Using Separation of Variables and integrating,

\[\begin{align*} \frac{dU}{U} &= - \left(\frac{2y_1^{\prime}}{y_1} + p \right) dx \\ \ln|U| &= -2\ln|y_1| - \int p dx \end{align*}\]

and taking the exponential of both sides, we finally get

\[U = \frac{1}{y_1^2}e^{-\int p dx} \tag{8}\]

Since we set $U=u^{\prime}$, we have $u=\int U dx$. The second solution $y_2$ we are looking for is

\[y_2 = uy_1 = y_1 \int U dx\]

Since $\cfrac{y_2}{y_1} = u = \int U dx$ cannot be a constant as long as $U>0$, $y_1$ and $y_2$ form a basis of solutions.

Applications of Reduction of Order

A general second-order ODE $F(x, y, y^\prime, y^{\prime\prime})=0$, whether linear or nonlinear, can be reduced to a first-order ODE using reduction of order when $y$ does not appear explicitly, when $x$ does not appear explicitly, or, as seen before, when the equation is homogeneous linear and one solution is already known.

Case where $y$ does not appear explicitly

In $F(x, y^\prime, y^{\prime\prime})=0$, setting $z=y^{\prime}$ reduces the equation to a first-order ODE in $z$, $F(x, z, z^{\prime})$.

Case where $x$ does not appear explicitly

In $F(y, y^\prime, y^{\prime\prime})=0$, setting $z=y^{\prime}$ gives $y^{\prime\prime} = \cfrac{d y^{\prime}}{dx} = \cfrac{d y^{\prime}}{dy}\cfrac{dy}{dx} = \cfrac{dz}{dy}z$. This reduces the equation to a first-order ODE in $z$, $F(y,z,z^\prime)$, where $y$ takes the role of the independent variable $x$.