I have to solve the following:
$\displaystyle
dy=\frac{du}{x-u}$, where $\displaystyle u=u(x, y)$.
How do I solve this?
Thank you so much for all your help!
No. I have to solve a quasilinear equation:
$\displaystyle x u_x + y u_y= xy-yu$
I'm trying to find the general soultion so I do the standard procedure from my textbook:
$\displaystyle \frac{dx}{x}=\frac{dy}{y}=\frac{du}{y(x-u)}$
$\displaystyle \frac{dx}{x}=\frac{dy}{y}$ gives me
$\displaystyle ln x=ln y + ln c$, $\displaystyle \phi (x, y)=c=\frac{x}{y}$
And now I try to do the same with $\displaystyle \frac{dy}{y}=\frac{du}{y(x-u)}$ to get $\displaystyle \psi(x, y, u)$, but don't know how.
I've had a similar problem:
$\displaystyle \frac{dy}{y+u}=\frac{du}{y-u}$, and this is what I did:
$\displaystyle ydy-udy=ydu+udu$
$\displaystyle (y-u)dy-(y-u)du=2udu$
Then I substituted $\displaystyle z=y-u$ and got $\displaystyle zdz=2udu$.
But I don't know what to do with the problem I stated above..
Please, could anyone help?
I'm stuck and unfortunately, I'm running out of time to solve it.