Variation of Parameters and the Wronskian

1Introduction

This lecture develops variation of parameters, which constructs a particular solution of a nonhomogeneous equation from a fundamental pair of homogeneous solutions. It formulates nonvanishing of the Wronskian as an interval-wide condition and explains the role of constants of integration.

2Hypotheses and a Fundamental Pair

On an interval $$I$$I, consider

where $$P,Q,R$$P,Q,R are continuous on $$I$$I. Let $$y_1,y_2$$y_1,y_2 be a fundamental pair for the corresponding homogeneous equation; that is, they are linearly independent solutions on $$I$$I.

Define the {Wronskian} by

Abel's identity states that

Thus, if $$W\left(x_0\right)\ne 0$$W(x_0)\ne0 at one point of $$I$$I, then $$W\left(x\right)\ne 0$$W(x)\ne0 throughout $$I$$I. Conversely, vanishing at one point implies vanishing throughout the interval. For a homogeneous equation with continuous coefficients, nonvanishing need not be assumed separately at every point.

3Derivation of Variation of Parameters

Set

and impose

Then $$y_{p^{\prime }}=u_1{y}_{1^{\prime }}+u_2{y}_{2^{\prime }}$$y_p'=u_1y_1'+u_2y_2'. Differentiating once more and substituting into the original equation leaves

Therefore,

Since $$W\ne 0$$W\ne0 on $$I$$I, Cramer's rule gives

4Definite-Integral Formula

Fix a base point $$x_0\in I$$x_0\in I. Then

so a particular solution is

This choice satisfies $$u_1\left(x_0\right)=u_2\left(x_0\right)=0$$u_1(x_0)=u_2(x_0)=0. If indefinite integrals are used instead, their arbitrary constants contribute $$C_1{y}_1+C_2{y}_2$$C_1y_1+C_2y_2 and are absorbed into the homogeneous solution. Hence the general solution is

5Example: $$y^{\prime }\text{'}+y=\tan x$$y''+y=\tan x

Fix any interval on which $$\tan x$$\tan x is continuous,

The functions $$y_1=\cos x$$y_1=\cos x and $$y_2=\sin x$$y_2=\sin x form a fundamental pair on $$I_k$$I_k, with $$W=1$$W=1. Thus,

One choice of antiderivatives is

The cross terms cancel in $$u_1{y}_1+u_2{y}_2$$u_1y_1+u_2y_2, yielding

This expression is defined on the selected interval $$I_k$$I_k, and direct substitution verifies $$y_{p^{\prime }}\text{'}+y_p=\tan x$$y_p''+y_p=\tan x. Different integration constants merely add a homogeneous solution.

6Method Selection and Limitations

Undetermined coefficients has lower computational cost when its hypotheses hold. Variation of parameters is less dependent on the form of $$R$$R, but a fundamental pair must already be known, and the resulting integrals need not be elementary. The method must be applied separately on each interval where $$P,Q,R$$P,Q,R are simultaneously continuous.

An analogous formula using a fundamental matrix exists for first-order linear systems. Its details are deferred to the subsequent lectures on systems.

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