Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS

wherey 1 (x)andy 2 (x)arelinearly independentsolutions of (16.1), andc 1 andc 2

are constants that are fixed by the boundary conditions (if supplied).

A full discussion of the linear independence of sets of functions was given

at the beginning of the previous chapter, but for just two functionsy 1 andy 2

to be linearly independent we simply require thaty 2 is not a multiple ofy 1.

Equivalently,y 1 andy 2 must be such that the equation

c 1 y 1 (x)+c 2 y 2 (x)=0

isonlysatisfied forc 1 =c 2 = 0. Therefore the linear independence ofy 1 (x)and

y 2 (x) can usually be deduced by inspection but in any case can always be verified

by the evaluation of the Wronskian of the two solutions,

W(x)=

∣

∣

∣

∣

y 1 y 2

y′ 1 y′ 2

∣

∣

∣

∣=y^1 y

′

2 −y^2 y

′

1. (16.3)

IfW(x)= 0 anywhere in a given interval theny 1 andy 2 are linearly independent

in that interval.

An alternative expression forW(x), of which we will make use later, may be

derived by differentiating (16.3) with respect toxto give

W′=y 1 y′′ 2 +y′ 1 y′ 2 −y 2 y′′ 1 −y′ 2 y′ 1 =y 1 y′′ 2 −y′′ 1 y 2.

Since bothy 1 andy 2 satisfy (16.1), we may substitute fory′′ 1 andy′′ 2 to obtain

W′=−y 1 (py′ 2 +qy 2 )+(py′ 1 +qy 1 )y 2 =−p(y 1 y′ 2 −y′ 1 y 2 )=−pW.

Integrating, we find

W(x)=Cexp

{

−

∫x

p(u)du

}

, (16.4)

whereCis a constant. We note further that in the special casep(x)≡0weobtain

W=constant.

The functionsy 1 =sinxandy 2 =cosxare both solutions of the equationy′′+y=

0. Evaluate the Wronskian of these two solutions, and hence show that they are linearly

independent.

The Wronskian ofy 1 andy 2 is given by

W=y 1 y 2 ′−y 2 y′ 1 =−sin^2 x−cos^2 x=− 1.

SinceW= 0 the two solutions are linearly independent. We also note thaty′′+y=0is
a special case of (16.1) withp(x) = 0. We therefore expect, from (16.4), thatWwill be a
constant, as is indeed the case.

From the previous chapter we recall that, once we have obtained the general

solution to the homogeneous second-order ODE (16.1) in the form (16.2), the

general solution to theinhomogeneousequation

y′′+p(x)y′+q(x)y=f(x) (16.5)