# Solving a PDE through separation of variables

I have the following PDE:

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

after seperating variables, I obtain after separating 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 should I proceed?

It is more powerful to use the method of characteristics (MoC) than separation of variables (SoV) for OP's problem. In this answer we give a sketched derivation of the full solution using the MoC.

1. Notice that OP's linear 2nd order PDE $$x^2 u_{xx}-y^2 u_{yy}-2yu_y~=~0$$ is form-invariant under scalings $(x,y)\to (\lambda z, \mu y)$. This suggests that we should define new variables $$X~=~\ln|x|\quad\text{and}\quad Y~=~\ln|y|.$$ Then OP's linear 2nd order PDE becomes a linear 2nd order PDE $$u_{XX}-u_X-u_{YY}-u_Y~=~0$$ with constant coefficient functions.

2. If we define $$X^{\pm}~=~X\pm Y,$$ the PDE factorizes $$(4\partial_--2)\partial_+u~=~0,$$ or equivalently, $$\partial_- e^{-X^-/2}\partial_+u~=~0,$$ with full solution $$u~=~ F(X^-)+ e^{X^-/2}G(X^+),$$ where $F,G$ are arbitrary functions.

3. Going back to the original variables, the solution reads $$u~=~ f(x/y) + \sqrt{\left| \frac{x}{y}\right|} g(xy),$$ where $f,g$ are arbitrary functions. This can in turn be beautified to$^1$ $$u~=~ f(x/y) + x g(xy)$$ for arbitrary functions $f,g$. The above full solution would have been more difficult to derive using SoV.

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$^1$ Hat tip: Robert Israel.

• How did you know to choose $X= ln |x|$ and $Y=ln|y|$? – Dr. John A Zoidberg Feb 17 '16 at 21:52
• From the form-invariance under scaling. I updated the answer. – Qmechanic Feb 17 '16 at 22:10
• I don't understand. I can see why you chose to find new variables, but how did you find $X=ln|x|$? – Dr. John A Zoidberg Feb 17 '16 at 22:17
• So that $x\partial_x=\partial_X$. – Qmechanic Feb 17 '16 at 22:23
• A slightly nicer form, avoiding apparent problems at $x=0$, is $u = f(x/y) + x g(xy)$. – Robert Israel Feb 17 '16 at 22:25