Do the partial derivatives, plug into the equations and then the coefficients of like powers of x and y (x and t in the second question) must be equal. That will give you a set of simultaneous equations for the unknowns independent of x and y which you should solve.

In fact for question 1, you will obtain a solution for a and b from the first of the C-R equations, and you will just need to show that this is also a solution for the second

RonL