Extensions of Second-Order Linear Constant-Coefficient Equations

1Introduction

This lecture gives a unified account of characteristic-root types and nonhomogeneous terms for second-order linear equations with constant coefficients.

2Theoretical Context

The results below refine the constant-coefficient theory. The preceding lecture develops the general theory that also permits variable coefficients.

3Classification of Homogeneous Solutions

For

the characteristic equation is $$a{r}^2+br+c=0$$ar^2+br+c=0. If its roots are $$\alpha \pm i\beta$$\alpha\pm i\beta, the real-valued general solution is

If $$r$$r is a repeated root, $$e^{rx}$$e^{rx} and $$x{e}^{rx}$$xe^{rx} provide two linearly independent solutions and hence the required two dimensions of the solution space.

4Decomposition of Nonhomogeneous Equations

For

linearity gives the decomposition

where $$y_h$$y_h is the general homogeneous solution and $$y_p$$y_p is one particular solution. For constant coefficients, the method of undetermined coefficients applies when a finite-dimensional function space containing the forcing term is invariant under differentiation. If the forcing term resonates with a homogeneous solution, multiply the trial functions by a sufficient power of $$x$$x to remove the overlap. For a general continuous forcing term, variation of parameters is the appropriate candidate.

5Examples

The characteristic roots of

are $$-1\pm 2i$$-1\pm2i. Therefore,

For

the forcing term $$\cos x$$\cos x belongs to the homogeneous solution space. The usual trial function $$A\cos x+B\sin x$$A\cos x+B\sin x therefore fails, and it must be replaced by

Substitution gives $$A=0$$A=0 and $$B=\text{[PARSE ERROR: Undefined("Command("tfrac")")]}12$$B=\tfrac12, so $$y_p=\text{[PARSE ERROR: Undefined("Command("tfrac")")]}x2\sin x$$y_p=\tfrac{x}{2}\sin x.

6Scope and Limitations

The classifications by complex and repeated roots and the method of undetermined coefficients operate within the constant-coefficient setting. Variable-coefficient equations may instead require power-series methods or special functions.

7Exercises

8Related Lectures

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