 Solution

Substitute $\hat{u}=e^{-b(t)s^2}$ into the transformed partial differential equation. Comparing coefficients gives $b^{\prime }=1-2b$, $b(0)=0$, whenceThe stated Gaussian Fourier transform pair, with its inverse factor $1/(2\pi )$, givesThis is a normal distribution density with mean zero and variance $1-e^{-2t}$. Its total integral is one. As $t\downarrow 0$ it converges to $\delta$ in the sense of distributions, while as $t\to \infty$ it tends to the standard normal distribution density. This is the transition density from zero of the Ornstein-Uhlenbeck process with drift $-x$ and diffusion coefficient $\sqrt{2}$.