APM346--2020S > Chapter 2

S2.1 online textbook problem #23

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wuyuning:
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.

Victor Ivrii:
Since solving $x,y$ you get a circle of the constant radius $r$, you can parametrize it $x=r\cos(t)$, $y=r\sin(t)$; then integration will be easy. Don't forget in the end to get rid of $t,r$, leaving only $x,y$