Heat Equation Problem

Find the solution u(x,t) of the heat equation:
ut = 1/2 uxx

(a) with initial data u(x,0) = x
(b) with initial data u(x,0) = x^2
(c) with initial data u(x,0) = sinx
(d) with initial data u(x,0) = 0 x < 0 and u(x,0) = 1 or x >/= 0

I know the solution of the heat equation with given initial data is unique. So if you happen to guess a function that satisfies the heat equation and satisfies initial condition, then it is a unique solution.