(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Solve the equation [tex]u_{x}+2xy^{2}u_{y}=0[/tex] with [tex]u(x,0)=\phi(x)[/tex]

2. Relevant equations

Implicit function theorem

[tex]\frac{dy}{dx}=-\frac{\partial u/\partial x}{\partial u/\partial y}[/tex]

3. The attempt at a solution

[tex]-\frac{u_x}{u_y}=\frac{dy}{dx}=2xy^2[/tex]

Separating variables

[tex]\frac{dy}{y^2}=2xdx[/tex]

[tex]\frac{-1}{y}=x^2+c[/tex]

[tex]C=x^2+\frac{1}{y}[/tex]

So [tex]u(x,y)=f(x^2+\frac{1}{y})[/tex]

The boundary condition is given as evaluating at [tex]y=0[/tex] which doesn't seem to make sense. Any thoughts? Thanks!

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# Homework Help: Linear 1st order PDE (boundary conditions)

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