- #1
kingwinner
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Homework Statement
Find the general solution to the PDE and solve the initial value problem:
y2 (ux) + x2 (uy) = 2 y2, initial condition u(x, y) = -2y on y3 = x3 - 2
2. Homework Equations /concepts
First order linear non-homogeneous PDEs
The Attempt at a Solution
I know that the general solution to the non-homogeneous PDE = a particular soltuion to it + the general solution to the assoicated homogenous PDE, so I first consider to assocatied homogeneous equation:
y2 (ux) + x2 (uy) = 0
The characteristic equation is dy/dx = x2/y2
And the solution to this ODE is y3 - x3 = C where C is an arbitrary constant.
So the general solution to the homogeneous equation is u = f(y3 - x3) where f is arbitrary differentiable function of one variable.
But then I am stuck. How can I find a particular solution to the original non-homogeneous PDE? Is there a systematic way to find one such solution?
Any help is appreciated! :)