Solving the homogeneous heat equation using the method of fundamental solutions I'm trying to solve the PDE $u_t=Du_{xx}$ using the method of fundamental solutions. I've used the ansatz $u(x,t)=t^{\alpha/2}f(x/\sqrt{t})$ and this has given me the following ODE: $$\frac{\alpha}{2}f(\phi)-\frac{1}{2}\phi f'(\phi)=Df''(\phi)$$ where $\phi=x/\sqrt{t}$. 
I'm then told to let $\alpha=-1/2$, leaving me with the ODE $$-\frac{1}{4}f(\phi)-\frac{1}{2}\phi f'(\phi)=Df''(\phi)$$
I'm not sure how to solve this though. 
Note: $D \in \mathbb{R}_{>0}$

Let's start from the top. Make the ansatz
$$
u(x,t) = t^\alpha f\left(\frac{x}{2\sqrt{Dt}}\right)
$$
Then after some massaging of the differential equation and substituting $\phi = x/(2\sqrt{Dt})$, you end up with
$$
f''(\phi) + 2\phi f'(\phi) - 4\alpha f(\phi) = 0
$$
The form of the differential operator suggests the substitution $f(\phi) = e^{-\phi^2}h(\phi)$, which then gives
$$
h''(\phi) - 2\phi h'(\phi) - 2(1+2\alpha)h(\phi) = 0 
$$
This is a well-known differential equation called Hermite's equation. These solutions normally blow up extremely quickly for $\phi \rightarrow\pm \infty$, but for certain special values of $\alpha$ it will have polynomial solutions. Those values are $\alpha = -n-1/2, n\in \mathbb N$, and the polynomials are denoted by $H_n(\phi)$. These will be the only solutions for which $f(\phi)$ remains finite for all $\phi$. So, substituting back into the original expression, we find the fundamental solutions are 
$$
u(x,t) = A_n t^{-n-1/2}H_n\left(\frac{x}{2\sqrt{Dt}}\right)\exp\left(\frac{x^2}{4Dt}\right)
$$
for some normalization constant $A_n$.
