Mathematical Methods for Physics and Engineering : A Comprehensive Guide

15.2 LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS

Use Green’s functions to solve

d^2 y

dx^2

+y=f(x), (15.69)

subject to the one-point boundary conditionsy(0) =y′(0) = 0.

We again require (15.67) to hold and so again we assume a Green’s function of the form

G(x, z)=

{

A(z)sinx+B(z)cosx forx<z,

C(z)sinx+D(z)cosx forx>z.

However, we now requireG(x, z) to obey the boundary conditionsG(0,z)=G′(0,z)=0,
which implyA(z)=B(z) = 0. Therefore we have

G(x, z)=

{

0forx<z,

C(z)sinx+D(z)cosx forx>z.

Applying the continuity conditions onG(x, z) as before now gives

C(z)sinz+D(z)cosz=0,

C(z)cosz−D(z)sinz=1,

which are solved to give

C(z)=cosz, D(z)=−sinz.

So finally the Green’s function is given by

G(x, z)=

{

0forx<z,

sin(x−z)forx>z,

and the general solution to (15.69) that obeys the boundary conditionsy(0) =y′(0) = 0 is

y(x)=

∫∞

0

G(x, z)f(z)dz

=

∫x

0

sin(x−z)f(z)dz.

Finally, we consider how to deal with inhomogeneous boundary conditions

such asy(a)=α,y(b)=βory(0) =y′(0) =γ,whereα, β, γare non-zero. The

simplest method of solution in this case is to make a change of variable such that

the boundary conditions in the new variable,usay, are homogeneous, i.e.u(a)=

u(b)=0oru(0) =u′(0) = 0 etc. Fornth-order equations we generally require

nboundary conditions to fix the solution, but thesenboundary conditions can

be of various types: we could have then-point boundary conditionsy(xm)=ym

form=1ton, or the one-point boundary conditionsy(x 0 )=y′(x 0 )=···=

y(n−1)(x 0 )=y 0 , or something in between. In all cases a suitable change of variable

is

u=y−h(x),

whereh(x)isan(n−1)th-order polynomial that obeys the boundary conditions.