678 CHAPTER 34. SOME PARTIAL DIFFERENTIAL EQUATIONS

Then, as above, bn = 0 and an must be chosen such that

f (x) =∞

∑n=1

an sin(nπx

4

)Thus

an =24

∫ 3

1

(1− (x−2)2

)sin(nπx

4

)dx

Then after doing the hard work, you end up with

an =−164cos 3

4 nπ +nπ sin 34 nπ−4cos 1

4 nπ +nπ sin 14 nπ

n3π3

Then the solution is

u(x, t) =∞

∑n=1

(−16

4cos 34 nπ +nπ sin 3

4 nπ−4cos 14 nπ +nπ sin 1

4 nπ

n3π3

)·

cos(

αnπ

4t)

sin(nπx

4

)Let α = .5 to give a specific example. Here is a graph of the function of two variablesin which the sum is taken up to n = 6. The t axis goes from 0 to 10 and if you fix t andimagine a cross section, it will be x→ u(x, t).

-1

0

0

1

5 4210 0

34.3 Nonhomogeneous ProblemsFor the sake of completeness, here is a brief discussion of what can be done if you have anonhomogeneous equation of the form ut = auxx + f along with an initial condition

u(x,0) = g(x)

and boundary conditions. As before, there are eigenfunctions yn satisfying the boundaryconditions and

y′′n =−λ2nyn, lim

n→∞λ n = ∞

such that also ∫ L

0yn (x)ym (x)dx = δ nm =

{1 if n = m0 if n ̸= m