Theory

Introduction

The solution method for homogeneous second-order linear equations is part of the solution method for inhomogeneous ones. The homogeneous theory provides the parameters, while a single “particular” solution to the inhomogeneous equation provides an “offset.” This offset shifts the vector space of solutions to the homogeneous equation over to the affine space of solutions to the inhomogeneous equation.

The situation is exactly akin to that of solutions to homogeneous and inhomogeneous matrix equations, $A\mathbf{x}=0$ and $A\mathbf{x}=\mathbf{b}$. The solutions $\mathbf{x}$ to $A\mathbf{x}=0$ form a vector space, namely the null space of $A$. The solutions to $A\mathbf{x}=\mathbf{b}$ form an affine space that is given by adding a single particular solution ${\mathbf{x}}_0$ satisfying $A{\mathbf{x}}_0=\mathbf{b}$ to the null space of $A$. The reason: any two solutions to $A\mathbf{x}=\mathbf{b}$ differ by a solution to $A\mathbf{x}=0$. Therefore the affine space of all solutions to $A\mathbf{x}=\mathbf{b}$ can be obtained by shifting the vector space of solutions to $A\mathbf{x}=0$ over by an offset which is one particular solution $A\mathbf{x}=\mathbf{b}$.

A generic second-order linear inhomogeneous ODE is given as follows:

$$y^{\prime \prime }+p(t)y^{\prime }+q(t)y=f(t).$$

This ODE determines a corresponding homogeneous ODE:

$$y^{\prime \prime }+p(t)y^{\prime }+q(t)y=0.$$

Given two “independent” solutions $y_1(t)$ and $y_2(t)$ to the homogeneous ODE, any linear combination of them is also a solution, and a complete family of solutions is determined as their span:

$$C_1{y}_1(t)+C_2{y}_2(t).$$

Given a single particular solution $y_I(t)$ to the original inhomogeneous equation, the complete family of solutions to the inhomogeneous equation is the span for the corresponding homogeneous equation shifted by that particular solution as an offset:

$$C_1{y}_1(t)+C_2{y}_2(t)+y_I(t).$$