What are the Boundary Conditions for Solving Uxx - Uy - Ux =0?

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The discussion focuses on solving the second-order partial differential equation (PDE) Uxx - Uy - Ux = 0 without employing separation of variables. The method of characteristics is highlighted as a potential approach, although the original poster lacks familiarity with it. The conversation also touches on the importance of boundary conditions (BCs) and suggests using weight functions and eigenfunctions related to Fourier series for a more general solution.

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I'm stuck on this problem. I need to get the most general solution to:

Uxx - Uy - Ux =0

without using separation of variables.

I have no clue how to use the method of characteristics for second order PDE's. My prof hasn't taught us anything about that. All we learned how to do is factor operators.

Help would be appreciated!
 
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Does the equation look familiar if you put x+y=t ?
 
There are a couple of ways to solve this

One them to specify weight functions based on boundary conditions and apply these eigenfunctions, as so, it is related to Fourier series ...

Tell me what are the BC's?
 

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