Characteristic curves of this PDE

In summary, the solution to the given partial differential equation is f(x,y) = 0, as shown by the characteristic curves of the PDE. The undetermined constant in the formula for u^3 must be chosen in such a way that u is defined in the whole plane. Adding f(y) to the result for u^3 does not impose any restrictions, as there will be a singularity regardless.
  • #1
WannaBe22
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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 !
 
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  • #2


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"?
 
  • #3


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
 
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FAQ: Characteristic curves of this PDE

What is a characteristic curve of a PDE?

A characteristic curve of a PDE (partial differential equation) is a curve in the solution space that satisfies the PDE. It is a method used to visualize and analyze the behavior of a PDE by tracking the evolution of its solution along these curves.

How are characteristic curves determined?

The characteristic curves of a PDE are determined by the coefficients of the PDE and the initial/boundary conditions. These curves can be found by solving the characteristic equations, which are derived from the PDE.

What information can be obtained from characteristic curves?

Characteristic curves provide important information about the behavior of a PDE, such as the type of solution (e.g. smooth or discontinuous), the existence and uniqueness of solutions, and the stability of the solution.

Can characteristic curves be used to solve a PDE?

Characteristic curves can be used as a tool to aid in the solution of a PDE, but they do not provide an exact solution. They can help determine the form of the solution and assist in finding the appropriate boundary/initial conditions for a given problem.

Are characteristic curves unique for each PDE?

Yes, the characteristic curves for a specific PDE are unique and specific to that particular equation. They depend on the specific coefficients and conditions of the PDE, and can vary greatly from one equation to another.

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