Thread: PDE Separation of variables

I'm a little lost and need a little help figuring out where to start.

ut = uxx, 0 < x < 1, t > 0
u(x,0) = f(x)
ux(1,t) = u(1,t)
ux(0,t) = 0

The fact that u(x,0) is equal to f(x) is throwing me off and I'm not sure How I go about solving for lambda = 0, lambda > 0 and lambda < 0.

Thanks!

ut = uxx = K implies a solution of the form

u = K(x^2/2 + t) + Bx + C

First condition : Kx^2/2 + Bx + C = f(x)
Second condition: K + B = K/2 + Kt + B + C
or Kt + C = K/2
or C = K/2 - Kt
Third condition: B = 0

Thus u(x,t) = K/2(1+x^2)

K is found from condition one.
K = 2f(x)/(x^2 + 1 - 2t)

Thus u(x,t) = (1+x^2)f(x)/(1-2t+x^2)