Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2008 / Paper 2 / 8

The proposed function has partial derivatives $u_t=1$ and $u_{xx}=2\alpha$. Substitution in the heat equation requires $1=\alpha$, henceThe coefficient $\beta$ remains arbitrary.

Within this polynomial family, $u(0,0)=0$ already holds and $u(1,0)=1+\beta =0$ gives $\beta =-1$. Thus one solution isTwo specified point values alone do not give uniqueness among all solutions of the heat equation.