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

using the similarity variable , where a is a constant I have to determine all possible values of

The other problem is solving Laplace's equation

using the similarity variable , 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:

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