Mass Diffusion Equation (Fick's Second Law)

Question

Solving the Mass Diffusion Equation $$\frac{\partial \rho}{\partial t} = D_m\frac{\partial ^2 \rho}{\partial x^2}$$ for $$\rho (x,t) = \frac{1}{\sqrt{D_mt}}f\left(\frac{x}{\sqrt{D_mt}}\right)$$ and $$\int_{-\infty}^{\infty} \rho(x,t) dx = M$$ where M is the total mass of the diffusing particles, I obtained a differential equation of the form $$2f''(u) + uf'(u) + f(u) = 0$$ Above, $u = \frac{x}{\sqrt{D_mt}}$. How do I analytically solve this differential equation to find the functional form of $f(u)$?

Answer 1

$$2f''(u) + uf'(u) + f(u) = 2f''+(uf)'=0$$so $$2f'+uf=C$$From the boundary conditions, C = 0. So, $$\frac{f'}{f}=-\frac{u}{2}$$The solution to this differential equation is: $$f=Ae^{-\frac{u^2}{4}}$$where A is a constant of integration. So, $$\rho=\frac{A}{\sqrt{D_mt}}e^{-\frac{u^2}{4}}$$To determine A, all you need to do is satisfy the condition $$\int_{-\infty}^{+\infty}{\rho du}=\frac{M}{\sqrt{D_mt}}$$or$$A\int_{-\infty}^{+\infty}e^{-\frac{u^2}{4}}du=M$$