Can You Tackle This Week's Challenging PDE?

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Here is this week's POTW:

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Solve the PDE $$3\frac{\partial u}{\partial x} + 4 \frac{\partial u}{\partial y} = f(x,y)$$

where $f$ is a smooth function of $x$ and $y$.
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Remember to read the https://mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to https://mathhelpboards.com/forms.php?do=form&fid=2!

 

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No one answered this week's problem. You can read my solution below.

Spoiler

Let $\xi = 3x + 4y$ and $\eta = 4x - 3y$. Then $$3\frac{\partial u}{\partial x} + 4\frac{\partial u}{\partial y} = (3^2 + 4^2) \frac{\partial u}{\partial \xi} = 25\frac{\partial u}{\partial \xi}$$ Thus $$u(\xi, \eta) = \frac{1}{25}\int f(\xi, \eta)\, d\xi + g(\eta)$$ for some smooth function $g$. Therefore $$u(x,y) = \frac{1}{5}\int_C f\, ds + g(4x - 3y)$$ where $\int_C f\, ds$ is the line integral of $f$ (with respect to arclength) over the characteristic segment from the $y$-axis to $(x,y)$.