Quasi linear PDE 1st order

Question

I have a problem with the following PDE:

I tried to solve it, but I get stuck at this point:

Answer 1

The general solution must satisfy the following equation (Mathematica tells this): $$f_1\left(x u(x,y),\frac{y^2 u(x,y)-x}{2 u(x,y)}\right)=0$$ with $f_1(x,y)$ an arbitrary function.

We choose by trial $f_1(x,y)=x\cdot y$, hence

$$\text{f1}\left(x u(x,y),\frac{y^2 u(x,y)-x}{2 u(x,y)}\right)=\frac{1}{2} x \left(y^2 u(x,y)-x\right)=0$$

The solution which satisfies the PDE is then: $\mathbf{u(x,y)=\frac{x}{y^2}}$

And also $u(x^2,x)=1$ satisfies the boundary condition.