I have this equation to solve \begin{equation} \partial_t F = \partial^2_y F \end{equation} and got two independent solutions through some trial and error.
\begin{equation} F_1 = erf\left(\frac{y}{2\sqrt{t}}\right) \\ F_2 = \frac{1}{\sqrt{t}}\exp\left( \frac{-y^2}{4{t}}\right) \end{equation} I'm wondering how many other possible solutions exist to this equation. I'll be glad to be directed to a chapter in some mathematics text that deals with the number of solutions for a linear partial differential equation.
There are whole books on this. For the Diffusion Equation, you can't do better than The Mathematics of Diffusion, by J. Crank. Another classic is Conduction of Heat in Solids, by Carslaw and Jaeger.