Solving the heat equation

 P: 59 1. The problem statement, all variables and given/known data Let $s = x/\sqrt{t}$ and look for a solution to the heat equation $u_{t} = u_{xx}$ which is of the form u(x,t) = f(s) and satisfies the IC u(x, 0) = 0 and the BC u(0, t) = 1 and u(∞, t) = 0. 2. Relevant equations $∫e^{x^{2}} = \sqrt{\pi}$ 3. The attempt at a solution Let f(s) = u(x, t) Substituting f(s) into the heat equation: $\frac{-xf'(s)}{2t^{3/2}} = \frac{f''(s)}{t}$ $s = x/\sqrt{t}$ $-0.5sf'(s) = f''(s)$ It's now just an ODE. $f'(s) = g(s)$ $-0.5sg(s) = g'(s)$ $ln(g(s)) = -0.25s^2 + constant$ $g(s) = Ke^{-0.25s^{2}} = f'(s)$ $f(s) = ∫f'(s)ds = Ke^{0.25}∫e^{-s^{2}}ds = Ke^{0.25}\pi^{1/2}$ So now I'm left with a trivial solution of f(s) = u(x,t) = constant. I don't think this is what u(x,t) is supposed to be. Which part did I do incorrectly?