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Yesterday I asked a question asking how a system of 1st order PDEs is solved. They can be written as

$$u_x+v_y+w_z=c(u, v, w, x, y, z)$$ $$u_y=v_x, \quad v_z=w_y, \quad u_z=w_x$$

In the particular case I'm interested in, $c(u, v, w, x, y, z)$ is linear in $u, v,$ and $w$. In the previous question, someone suggested to solve the equations using the method of characteristics, but all of the literature I can find, as well as other questions on this forum, always solve for a single unknown $u$ (cf., Chapter 3 of Evans' book, A, B, C, and D).

Maybe I'm missing something obvious, but I don't see how to generalize the method to three unknowns $u, v, w$.

Chaotic
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    You can use the method of characteristics by following the solution I gave here, as well as solutions by others here and here, provided the PDE system is diagonalisable. Also, are you sure it isn't $u_{y} = v_{x}$? And the system is overdetermined as is. – Matthew Cassell May 03 '22 at 13:45
  • Thank you, I'll look at the solution. You're right, I did indeed make a mistake in the second equation. – Chaotic May 03 '22 at 17:41

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