2. First show that for the completely insulated bar

ut = c2uxx; 0 < x < L; t > 0:
ux(0; t) = 0; t > 0
ux(L; t) = 0; t > 0
u(x; 0) = f(x); 0 < x < L

the separation of variables gives the solution

u(x; t) = A0 + 1 Ancos n x e ( cn )2t
L

n=1 L
X
where
2 Z0 L n x
An = f(x)cos dx; n = 1; 2; 3:::;

L L
then nd the temperature in the bar with L = ; c = 1; and f(x) = x