The following problem is a homework and I am not asking for a solution but simply for a hint on how to approach it. The task is given as follows:
Find all second-order polynomial solutions to the heat equation: $u_{t}-u_{xx}=0$
This is what I have done so far:
Given that $u(t, x)$ is a second-order polynomial, the equation will be the following:
$u(t, x) = ax^2+bxt+ct^2+dx+et+f$ (where $a, b, c, d, e, f$ are constants)
$u_{t}(t, x)=bx+2ct+e$
$u_{x}(t, x)=2ax+bt+d$
$u_{xx}(t, x)=2a$
From what follows that $bx+2ct+e-2a=0$
Is there anything else that I'm missing? How should one approach this problem without any previous knowledge of PDEs?
Thanks.