I am trying to solve problem #23$$yu_{x} - xu_{y} = x^2$$ using characteristic line, first I have \begin{align*}

\frac{dx}{y} &= -\frac{dy}{x} = \frac{du}{x^2}

\end{align*}

Then I get the characteristic line is given by $$C = \frac{1}{2}y^2 + \frac{1}{2}x^2$$

Next, I solve the first term and third term, I have \begin{align*}

\frac{du}{x^2} &= \frac{dx}{y} \\

du &= \frac{x^2}{y}dx

\end{align*}

Here is my problem, that is we can not integrate right-hand side without eliminating the variable $y$, but if we try to replace $y$ in terms of $x$ and $C$, the result does not look integrable and really messy.