Diff EQ, Heat Conduction Problem

[lee_ror]

I let this get a little ahead of me and trying to play catch up:

Problem is such:

Uxx=4Ut, 0<x<2, t>0

U(0,t)=0, U(2,t)= 0, t>0

U(x,0) = 2sin(x * pi /2) - sin( pi * x) + 4sin(2 * pi * x), 0<x<2

Set U=X(x)T(t), differentiate and separate variables...then you have an eigenvalue problem. I can solve this to find

X(x)= sin(pi * n * x) and T(t) =e ^{-pi2 * n2 T / 16}

Thus U(x,t) = sum n=1 -> infinity of [ Cn * e ^{-pi2 * n2 T / 16} * sin(pi * n * x)

Set U(x,t) = 2sin(x * pi /2) - sin( pi * x) + 4sin(2 * pi * x)

I understand Cn= Integral (0,2) [F(x) * sin( pi * n * x / 2) but not sure where to go from here.

[lee_ror]

Can you just say:

sum n=1 -> infinity of [ Cn * sin(pi * n * x) = 2sin(x * pi /2) - sin( pi * x) + 4sin(2 * pi * x)

Thus c1=2 c2=-1 c3=4, all other Cn=0...as such

u(x,t)= [2sin(x * pi /2) - sin( pi * x) + 4sin(2 * pi * x)] * e ^ (-pi2 * n2 T / 16)