Integrating Quasilinear Equations

[iris_m]

(urgent) help with integrals needed

I was doing an exercise on quasilinear equations, and have come to the point here I have to solve the following:

$$dy=\frac{du}{x-u}$$, where $$u=u(x, y)$$.

How do I integrate this?

Thank you!

 

[HallsofIvy]

Saying that u is a function of x and y means that x depends upon u. The integral should depend strongly on just what that dependence is. How did you get that?

 

[iris_m]

I'll write everything from the beginning then, I probably got something wrong.

I have to solve the following quasilinear equation:

$$x u_x + y u_y= xy-yu$$

I'm trying to find the general soultion so I do the standard procedure from my textbook:

$$\frac{dx}{x}=\frac{dy}{y}=\frac{du}{y(x-u)}$$


$$\frac{dx}{x}=\frac{dy}{y}$$ gives me
$$ln x=ln y + ln \overline{c}$$, $$\phi (x, y)=c=\frac{x}{y}$$

And now I try to do the same with $$\frac{dy}{y}=\frac{du}{y(x-u)}$$ to get $$\psi(x, y, u)$$, but don't know how.

Please, help me.