Hello(adsbygoogle = window.adsbygoogle || []).push({});

In our math course, we encountered the following elliptic PDE:

[tex]

y^{2}u_{xx} + u_{yy} = 0

[/tex]

In order to solve it, we converted it to the characteristic equation,

[tex]

y^{2}\left(\frac{dy}{dx}\right)^{2} + 1 = 0

[/tex]

Next, we wrote:

[tex]\frac{dy}{dx} = \frac{i}{y}[/tex]

My question is: the characteristic equation has no solution in [itex]\mathbb{R}[/itex] but we went ahead and mechanically wrote the expression for [itex]dy/dx[/itex]. Does this mean that we should regard [itex]x[/itex] and [itex]y[/itex] as complex variables? If so, how does one reconcile with the fact that some solution to the PDE as [itex]u(x,y) = c[/itex] is a surface in [itex](x,y,u)[/itex] space? Perhaps this is a trivial question, but I'm just starting to learn PDEs. Does this also mean that we should not ascribe a physical significance to x and y as coordinates in [itex]\mathbb{R}^2[/itex]?

Thanks.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Help needed with an elliptic PDE

**Physics Forums | Science Articles, Homework Help, Discussion**