I am trying to solve the following boundary value problem involving the heat equation: $$ \frac{\partial{u}}{\partial{t}} - \frac{1}{4} \frac{\partial^2{u}}{\partial{x}^2} = 0,\;\; t > 0 \text{ and } 0 < x < 1, \\ u(0,t) = u(1,t) = 0,\;\; t > 0, \\ u(x,0) = \sin{(2 \pi x)} - \frac{1}{3} \sin{(4 \pi x)}, \;\; 0 < x < 1. $$ I used separation of variables $u(x,t) = X(x) T(t)$ and got down to $u(x,t) = \sum_{k=1}^{\infty} b_k e^{-\frac{1}{4} (k \pi)^2 t} \sin{(k \pi x)}$ for some $b_k$.
My question is, how do I use the initial condition $u(x, 0)$ and finish this problem off by finding $b_k$? I assume I would have to use Fourier series on sine, but I'm not sure how to proceed with this. Any help is appreciated.
Buy uniqueness of Fourier series expansions the expression for $u(x,0)$ gives $b_2=1,b_4=-\frac 1 3$ and $b_k=0$ for all other $k$.