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: $D \in \mathbb{R}_{>0}$

This answer was for the original version of the question asked
I would divide through by $y=f(\phi)$ first, and integrate w.r.t $y$:
$$\int\frac{y''}{y}dy=C\int 2\phi - 1 dy=Cy(2\phi-1)+G$$
Where $G \in \mathbb{R}$.
Then use a substitution, however I am not sure if the LHS integral has a closed form (exact) solution.
