Section 14.3 Partial Derivatives

The diffusion equation
\[\frac{\partial c}{\partial t} = D \frac{\partial^2 c}{\partial x^2}\]
where \(D\) is a positive constant, describes the diffusion of heat through a solid, or the concentration of a pollutant at time \(t\) at a distance \(x\) from the source of the pollution, or the invasion of alien species into a new habitat.

Verify that the function
\[c(x, t) = \frac{1}{\sqrt{4\pi Dt}} e^{-x^2/(4Dt)}\]
is a solution of the diffusion equation.