Consider the problem of finding the solution of the heat equation for an infinite rod. The heat equation is $$u_{t} = \frac{1}{2} u_{xx}$$ and we want to find a solution $u(x,t)$. So taking the Fourier Transform of both sides with respect to $x$, we get $$ \int_{-\infty}^{\infty} u_{t}(s,t) e^{-2 \pi i s x} \ dx = (2 \pi i s)^{2} \mathcal{F} u(s,t)$$
$$ \int_{-\infty}^{\infty}\frac{\partial u(s,t)}{\partial t} e^{-2 \pi i s x} \ dx = (2 \pi i s)^{2} \mathcal{F} u(s,t)$$
$$ \frac{\partial}{\partial t} \int_{-\infty}^{\infty} u(s,t)e^{-2 \pi i s x} \ dx = (2 \pi i s)^{2} \mathcal{F} u(s,t)$$
$$ \frac{\partial \mathcal{F} u(s,t)}{\partial t} = (2 \pi i s)^{2} \mathcal{F} u(s,t)$$
What is a good way of solving for $\mathcal{F} u(s,t)$?
