I had figured this to be three problems, not one?

They don't separate but I can't believe that your instructor would just give you a set of problems that haven't (and won't) be discussed either in class or the reading.

Skim the chapter and see if you find this:

Method of Characteristics.

-Dan

Addundum:

Actually, the second two equations do separate. Just remember than when integrating over, say x, we take y to be a constant.

For example, in the second one we have

Let u(x, y) = X(x)Y(y)

\(\displaystyle u_{xx} = 4y^2u \implies X''Y = 4y^2 XY \implies \dfrac{X''}{X} = 4y^2 \text{ and } Y \neq 0\)

You would treat the y^2 as a constant.

Since these last two are separable and you haven't gone further than that(?) is it possible that the first has a typo? Possibly \(\displaystyle u_y + u = e^{x + y}\) ? This does separate.

-Dan