Mathematical Methods for Physics and Engineering : A Comprehensive Guide

HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS

f(x)=0:

an(x)

dny

dxn

+an− 1 (x)

dn−^1 y

dxn−^1

+···+a 1 (x)

dy

dx

+a 0 (x)y=0. (15.2)

To determine the general solution of (15.2), we must findnlinearly independent

functions that satisfy it. Once we have found these solutions, the general solution

is given by a linear superposition of thesenfunctions. In other words, if then

solutions of (15.2) arey 1 (x),y 2 (x),...,yn(x), then the general solution is given by

the linear superposition

yc(x)=c 1 y 1 (x)+c 2 y 2 (x)+···+cnyn(x), (15.3)

where thecmare arbitrary constants that may be determined ifnboundary

conditions are provided. The linear combinationyc(x) is called thecomplementary

functionof (15.1).

The question naturally arises how we establish that anynindividual solutions to

(15.2) are indeed linearly independent. Fornfunctions to be linearly independent

over an interval, there must not existanyset of constantsc 1 ,c 2 ,...,cnsuch that

c 1 y 1 (x)+c 2 y 2 (x)+···+cnyn(x) = 0 (15.4)

over the interval in question, except for the trivial casec 1 =c 2 =···=cn=0.

A statement equivalent to (15.4), which is perhaps more useful for the practical

determination of linear independence, can be found by repeatedly differentiating

(15.4),n−1 times in all, to obtainnsimultaneous equations forc 1 ,c 2 ,...,cn:

c 1 y 1 (x)+c 2 y 2 (x)+···+cnyn(x)=0

c 1 y 1 ′(x)+c 2 y 2 ′(x)+···+cnyn′(x)=0

..

.

c 1 y( 1 n−1)(x)+c 2 y 2 (n−1)+···+cny(nn−1)(x)=0,

(15.5)

where the primes denote differentiation with respect tox. Referring to the

discussion of simultaneous linear equations given in chapter 8, if the determinant

of the coefficients ofc 1 ,c 2 ,...,cnis non-zero then the only solution to equations

(15.5) is the trivial solutionc 1 =c 2 =···=cn= 0. In other words, thenfunctions

y 1 (x),y 2 (x),...,yn(x) are linearly independent over an interval if

W(y 1 ,y 2 ,...,yn)=

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

y 1 y 2 ... yn

y 1 ′ y 2 ′

..

.

..

.

..

.

..

.

y 1 (n−1) ... ... yn(n−1)

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

= 0 (15.6)

over that interval;W(y 1 ,y 2 ,...,yn) is called theWronskianof the set of functions.

It should be noted, however, that the vanishing of the Wronskian does not

guarantee that the functions are linearly dependent.