Characteristic curves of this PDE

[WannaBe22]

Homework Statement

Let [tex] f(x,y) [/tex] be the soloution of [tex]xu_x +yu_y = u^4 [/tex] that is defined in the whole plane. Prove that [tex] f = 0 [/tex] .
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:
[tex] \frac{dx}{x} = \frac{dy}{y} = \frac{du}{u^4} [/tex] . From these equations we will receive: [tex] y=c_1 \cdot x [/tex] and [tex] u^3 = \frac{1}{-3ln(x)-3c_s} [/tex] ... 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 [itex]u^3[/itex]. 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 [tex] u^3 [/tex] - f(y) for some function f? That is: [tex] u^3 = \frac{1}{-3ln(x) - 3c_2 +f(y)} [/tex]
If so, then because we have a singularity in [tex] ln(x) = -c_2 [/tex] , I don't think we have any restrictions on this f... We'll have a singularity anyway...

Am I right?

Thanks