Given the initial boundary value problem for the heat equation: $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial t^2}$$ $$u(0,t) = 0, t>0$$ $$u(x,0) = f(x), 0<x<\infty$$ Where $u=u(x,t)$ and $f(x)$ is a given smooth function. Solve this problem by taking Fourier sine transforms and Fourier inverse sine transforms in the $x$ variable
So I've been able to use the Fourier sine transforms to transform the whole equation to get: $$\frac{\partial }{\partial t}F_s(u)+\alpha^2F_s(u)=0$$ and using the integrating factor $e^{\alpha^2t}$, I was able to show that $$\frac{\partial }{\partial t}e^{\alpha^2t}F_s(u)=0$$ What's the next step in the methods? How do the initial boundary values apply here?