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saching

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**[SOLVED] transforming a parabolic pde to normal form**

## Homework Statement

The problem is to transform the PDE to normal form.

The PDE in question is parabolic: U[tex]_{xx}[/tex] - 2U[tex]_{xy}[/tex] + U[tex]_{yy}[/tex] = 0 but I also need to solve other problems for hyperbolic pde's so general advice would be appreciated.

## Homework Equations

The characteristic equation is: Ay'[tex]^{2}[/tex] - 2By' + C = 0

The new variables should be v=x, w=psi, and the normal form is U[tex]_{ww}[/tex]=F[tex]_{2}[/tex]

## The Attempt at a Solution

The solutions manual provides:

I get lost right after we solve the characteristic equation. I don't understand how the variable substitution works or what is going on after that. My textbook only offers 1 example similar to this problem with no explanation of how it goes from step to step...so I'm completely lost. I looked online for information but the limited amount of stuff I did find is too technical(I read through all of them).