- #1
maverick280857
- 1,789
- 4
Hello
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.
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.