Assume a solution u of the form f(x) g(y) to get
(f')^2 g^2 / 4 + f^2 g g' = f g.
Divide by f^2 g to get
1/4 (f'/f)^2 g + g' = 1/f, so
g' = 1/f - 1/4 (f'/f)^2 g.
Think of y as the independent variable and x being held constant. This last equation is a simple differential equation in g which is easily solved. In fact the solution to
g' = b - a g is g = b/a + c exp(- a y), where c is any constant.
Apply this to the formula above to get a general solution to the original pde. Then substitute for the specific condition.
Justin: When I look back at this I see that there is a problem with it, namely g is assumed to be a function only of y, not of x, but it is clearly a function of x. This solution is flawed. Sorry. If I see how to fix it I will write more.