Searching for a second order ODE whose solution is bell shape (Gaussian function)

Question

I'm studying a nonequilibrium dynamics of a stochastic system. I found that in mean-field approximation the numerical solution resembles a bell shaped function (Gaussian function) with is zero at initial time, then reaches its maximum and finally decays to zero. I was wondering if there exist a second order ODE whose solutions are smooth and resemble Gaussian curves. I know that one can get the ODE satisfies by a Gaussian function just deriving it twice. But this is not the point. I'm interested in a general ODE which exhibits solutions which have similar behaviour of Gaussians. Thank you in advance.

Answer 1

Well, start with a solution that "resembles a Gaussian" and find an ODE that has that solution.

For example, if $y = \exp(-f(x)^2)$, then $y$ satisfies the second order differential equation $$y'' + 2 f(x) f'(x) y' + 2 (f'(x)^2 + f(x) f''(x)) y = 0$$

Answer 2

The SDE (Stochastic Differential Equation) $dx_t = dW_t$ has a probability density function $p(x,t)$ that satisfies the PDE (partial differential equation) $$ \frac{\partial{p(x,t)}}{\partial{t}} = \frac{1}{2}\frac{\partial^2{p(x,t)}}{\partial{x^2}} $$ (aka the heat equation), whose solution is the Gaussian function you described (with variance being time $t$), $$ p(t,x) = \frac{1}{\sqrt{2\pi t}}e^{-\frac{x^2}{2t}} $$ Note that the intial condition is a delta distribution, e.g. $\infty$ at origin and 0 at rest of the $x$ axis.

More generally, there is a class of stochastic processes (or SDEs) called Ornstein–Uhlenbeck process Maybe you can start from the Wiki page Wave Equation/Brownian Motion and go from there.