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## Homework Statement

Find the solution of

[tex]yu_x + xu_y = (y-x)e^{x-y}[/tex]

that satisfies the auxiliary condition

[tex]u(x,0) = x^4 + e^x[/tex]

## Homework Equations

Given in question

## The Attempt at a Solution

The general solution to this is [tex]u(x,y) = f(y^2-x^2)[/tex]

Applying the auxiliary condition I get

[tex]x^4 + e^x = u(x,0) = f(0^2-x^2)[/tex]

This results in

[tex]x^4 + e^x = f(-x^2)[/tex]

This is where I'm getting stuck. I need to "make" something on the left side that resembles what is shown in the parenthesis.

For example:

[tex]x^4 = f(-x^2)[/tex]

Re-writing this would give

[tex](-x^2)^2 = f(-x^2)[/tex]