I have the following PDE:

$$x^2u_{xx} - y^2u_{yy}-2yu_y = 0 $$

I noted that this looks very similar to the wave equation,

$$u_{yy} = C^2u_{xx}$$

I am, however, unable to proceed from here.

  • 2
    $\begingroup$ A change of variables to $\hat{x} = \log(x)$ might be useful as $x^2u_{xx} = u_{\hat{x}\hat{x}} - u_{\hat{x}}$. If you do this for both $x$ and $y$ you should get a linear PDE without any explicit coordinate dependence. $\endgroup$ – Winther Feb 16 '16 at 19:53
  • $\begingroup$ It's already linear. You can use separation of variables straightaway no? $\endgroup$ – Chinny84 Feb 16 '16 at 20:16
  • $\begingroup$ after seperating variables, I obtain $\frac{x^2}{\phi} \phi '' = - \lambda $ and $\frac{y^2}{g} g '' -\frac{2y}{g} g' = -\lambda$ where $u(x,y) = \phi(x)g(y)$ and $\lambda$ is the separation constant. How do I proceed? $\endgroup$ – Dr. John A Zoidberg Feb 17 '16 at 14:50

A method of solving is shown below. The general solution is : $$u(x,y)=\sqrt{\frac{x}{y}}\:\Phi(xy)+\Psi\left(\frac{x}{y}\right)$$ where $\Phi$ and $\Psi$ are any derivable functions.

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