Nonhomogeneous Linear ODEs of Second Order

TL;DR

• The general solution of a second-order nonhomogeneous linear differential equation $y^{\prime\prime} + p(x)y^{\prime} + q(x)y = r(x)$:
  • $y(x) = y_h(x) + y_p(x)$
  • $y_h$: general solution of the homogeneous equation $y^{\prime\prime} + p(x)y^{\prime} + q(x)y = 0$, where $y_h = c_1y_1 + c_2y_2$
  • $y_p$: particular solution of the nonhomogeneous equation
• The response term $y_p$ is determined solely by the input $r(x)$, and remains unchanged for the same nonhomogeneous equation regardless of initial conditions. The difference between two particular solutions of a nonhomogeneous equation is a solution of the corresponding homogeneous equation.
• Existence of general solution: A general solution always exists when the coefficients $p(x)$, $q(x)$ and the input function $r(x)$ are continuous
• Non-existence of singular solutions: The general solution includes all solutions of the equation (i.e., singular solutions do not exist)

Prerequisites

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

General Solution and Particular Solution of Second-Order Nonhomogeneous Linear Differential Equations

Consider the second-order nonhomogeneous linear differential equation

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

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 differential equation

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

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

\[y(x) = y_h(x) + y_p(x) \label{eqn:general_sol}\tag{3}\]

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

In other words, when we add an input $r(x)$ that depends only on the independent variable $x$ to the homogeneous differential equation ($\ref{eqn:homogeneous_linear_ode}$), a corresponding term $y_p$ is added to the response, and this additional response term $y_p$ is determined solely by the input $r(x)$, independent 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 initial conditions), the $y_p$ term that is independent of initial conditions cancels out, leaving only the difference between ${y_h}_1$ and ${y_h}_2$, which by the superposition principle is a solution to equation ($\ref{eqn:homogeneous_linear_ode}$).

Relationship Between Solutions of Nonhomogeneous and Homogeneous Differential Equations

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

Proof

(a)

Let’s denote the left 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 $I$, we have:

\[L[y + \tilde{y}] = L[y] + L[\tilde{y}] = r + 0 = r.\]

(b)

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

\[L[y - y^*] = L[y] - L[y^*] = r - r = 0.\ \blacksquare\]

The General Solution Includes All Solutions

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

Theorem 2: The General Solution of a Nonhomogeneous Differential Equation 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 an open interval $I$, then any solution of equation ($\ref{eqn:nonhomogeneous_linear_ode}$) on interval $I$ can be obtained from the general solution ($\ref{eqn:general_sol}$) by assigning appropriate 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 interval $I$, and let $x_0$ be any point in interval $I$. By the existence theorem for general solutions of homogeneous equations with continuous coefficients, $y_h = c_1y_1 + c_2y_2$ exists, and by the method of variation of parameters (which we will explore later), $y_p$ also exists, so the general solution ($\ref{eqn:general_sol}$) of equation ($\ref{eqn:nonhomogeneous_linear_ode}$) exists on interval $I$. Now, by Theorem 1(b), $Y = y^* - y_p$ is a solution of the homogeneous differential equation ($\ref{eqn:homogeneous_linear_ode}$) on interval $I$, and at $x_0$:

\[\begin{gather*} Y(x_0) = y^*(x_0) - y_p(x_0) \\ Y^{\prime}(x_0) = {y^*}^{\prime}(x_0) - y_p^{\prime}(x_0) \end{gather*}\]

By the existence and uniqueness theorem for initial value problems, there exists a unique solution $Y$ of the homogeneous differential equation ($\ref{eqn:homogeneous_linear_ode}$) on interval $I$ satisfying these initial conditions, which can be obtained by assigning appropriate 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 differential equation ($\ref{eqn:nonhomogeneous_linear_ode}$) can be obtained from the general solution ($\ref{eqn:general_sol}$). $\blacksquare$