What you're trying to do is introduce a new coordinate system to tranform you pde to one that is easier (i.e. missing a derivative) you'll find in general, this will take you to trying to solve part of the original problem.

For your first problem, it's separable so

so

For your second, you can use your change of variables (a=1, b=1) to get

- again, separable.

Have you learned the method of characteristics yet? It's really much quicker.