Are there real numbers a and b such that $f(x,y,t) = x^a t^b$ satisfies the heat equation? The question is in the title. The heat equation is as follows:
$$
\frac{\partial f}{\partial t} = k \left( \frac{\partial^2 f}{\partial x^2} +  \frac{\partial^2 f}{\partial y^2} \right),\quad k\in\mathbb{R}
$$

Attempt at solution
Plugging the requested form into the above equation yields:
$$
\frac{\partial f}{\partial t} = b\ t^{b-1} x^a\\
\frac{\partial^2 f}{\partial x^2} = a (a-1)\ t^b x^{a-2}\\
\frac{\partial^2 f}{\partial y^2} = 0
$$
Which leads to showing that:
$$
b\ x^2 = k\ a(a-1)\ t
$$
I'm not sure how to proceed from this point. Is this the correct procedure to solve this problem? Thanks!

From your derivations, it seems to me that $b=0$ and $a=1$ would work fine for instance...
