Characteristic curves of this PDE

[WannaBe22]

Homework Statement

Let $$f(x,y)$$ be the soloution of $$xu_x +yu_y = u^4$$ that is defined in the whole plane. Prove that $$f = 0$$ .
Hint: Think of the characteristic curves of this PDE.

HOPE You'll be able to help me

Thanks in advance!

Homework Equations

The Attempt at a Solution

By trying to solve this problem, I've got this subidinary equations:
$$\frac{dx}{x} = \frac{dy}{y} = \frac{du}{u^4}$$ . From these equations we will receive: $$y=c_1 \cdot x$$ and $$u^3 = \frac{1}{-3ln(x)-3c_s}$$ ... But can it help us? I think we are missing this way a few other soloutions...

Help is needed!
Thanks !

 

[HallsofIvy]

You are missing one undetermined constant in your last formula for $u^3$. What must that constant be in order that u is "defined in the whole plane"?

 

[WannaBe22]

You mean that we need to add to the result for $$u^3$$ - f(y) for some function f? That is: $$u^3 = \frac{1}{-3ln(x) - 3c_2 +f(y)}$$
If so, then because we have a singularity in $$ln(x) = -c_2$$ , I don't think we have any restrictions on this f... We'll have a singularity anyway...

Am I right?

Thanks