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

The DE can be rearranged to:
$$Df''(\phi)+\frac{1}{2}\phi f'(\phi)+\frac{1}{4}f(\phi)=0$$
Or in alternate notation:
$$Dy''+\frac{1}{2}\phi y'+\frac{1}{4}y=0$$
This has an Integrating Factor of 
$$e^{\int \frac{1}{2D}\phi d\phi}$$
Evaluating the Integral, $I$;
$$I=\int \frac{1}{2D}\phi d\phi=\frac{\phi^2}{4D}+C$$
Hence I.F. is $Ae^{\frac{\phi^2}{4D}}$, where $A=e^C$
Multiplying the DE by the IF, and cancelling the $A$'s:
$$De^{\frac{\phi^2}{4D}}y''+\frac{1}{2}\phi e^{\frac{\phi^2}{4D}}y'+\frac{1}{4}e^{\frac{\phi^2}{4D}}y=0$$
Hence
$$(De^{\frac{\phi^2}{4D}}y')'=-\frac{1}{4}e^{\frac{\phi^2}{4D}}y$$
Integrating w.r.t. $\phi$, and taking terms without $\phi$ outside integral:
$$De^{\frac{\phi^2}{4D}}y'=-\frac{1}{4}y\int e^{\frac{\phi^2}{4D}} d\phi$$
$$\frac{y'}{y}=-\frac{1}{4De^{\frac{\phi^2}{4D}}}\int e^{\frac{\phi^2}{4D}} d\phi$$
Hence
$$\int\frac{y'}{y}dy=-\int[\frac{1}{4De^{\frac{\phi^2}{4D}}}\int e^{\frac{\phi^2}{4D}} d\phi] dy$$
$$\ln y=-[\frac{1}{4De^{\frac{\phi^2}{4D}}}\int e^{\frac{\phi^2}{4D}} d\phi] \int dy=[\frac{y+C}{4De^{\frac{\phi^2}{4D}}}\int e^{\frac{\phi^2}{4D}} d\phi] $$
The Integral has no exact solution afaik.
