I have an exercise as homework and I am stuck.
If $k\in R$ the heat equation is
$\frac{\partial f}{\partial t} - k \left(\frac{\partial ^2f}{\partial x^2}+\frac{\partial ^2f}{\partial y^2} \right) =0$
the first question is to find if function $f(x,y,t) = (\cos x + \cos y)e^{-kt}$ is a solution. Which is fairly easy to do.
The second question asks to find functions of type $f(x,y,t)= x^at^b$ that are solutions to the equations too.
I tried to find the derivatives and the put them back into the heat function but I end up with 3 unknown variables. I guess that I must use the result of the first question but I don't know what to do.
Any help?