How to perform the following change of variables?

Question

Suppose I have a function $f(t,x,y)$ such that $$f_t = \nabla^26f(t,x,y).$$ I want to perform a change of variables so that for $f(t',x',y')$ we have $$f_{t'} = \nabla^2f(t',x',y').$$ Expanding the original expression we get $$f_t = 6\left(\frac{\partial^2 f}{\partial t^2} + \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}\right)$$ and I know that $x$ is related to $x'$ by $$\frac{\partial f}{\partial x} = \frac{\partial f}{\partial x'} \frac{\partial x'}{\partial x}.$$ Also, $y$ is related to $y'$ by a similar relation as above. I am trying to figure out where to go from here. I tried taking the derivative of the expression for $\frac{\partial f}{\partial x}$ but without any success. I am just looking for a hint on where to go next, not a full solution.

Answer 1

Try $(x',y',t') = (\alpha x,\alpha y,\alpha t)$ for some fixed number $\alpha$, to be determined.