Laplace's Equation using similarity variable and others

I've really spent a long time struggling with these.

The first one is finding the similarity solutions of

$\displaystyle \frac{\partial u}{\partial t} = x\frac{\partial^2 u}{\partial x^2}$

using the similarity variable $\displaystyle n = xt^a$, where a is a constant I have to determine all possible values of

The other problem is solving Laplace's equation

$\displaystyle \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}=0$

using the similarity variable $\displaystyle n=xy^a$, where a is a constant I have to determine all possible values of

my attempts at answering the questions

for the second one I managed to solve by the normal method of writing u as a product of a function of x only and a function of y only and working through to get:

$\displaystyle u = (c_1e^{kx} + c_2e^{-kx})(c_3cos(ky) + c_4sin(ky))$

for constants k and c subscript 1-4

but i'm not sure how to write up a solution using their similarity variable, or how to be sure how i've found all the possible constants a