I have the following PDE:

$\delta_\epsilon(S(\epsilon)\phi(x,\epsilon))+\delta_x\phi(x,\epsilon) = -T(\epsilon)\phi(x,\epsilon)$

Deltas represent partial derivatives, for ease of notation. Does it have an analytical solution? Or at least a semi-analytical solution dependent on T and S? I tried to do separation of variables, but the $S(\epsilon)$ is throwing me off. Any ideas?


1 Answer 1


Write it as $$ S\,\partial_\epsilon\phi+\partial_x\phi=-(S'+T)\phi. $$ This is a first order linear equation that can be soved by the method of characteristics.

  • $\begingroup$ And I'm assuming you arrived at this by just expanding the $\epsilon$ derivative with Chain Rule? $\endgroup$
    – Louis
    Jul 14, 2015 at 10:42
  • $\begingroup$ Not the chain rule but the product rule. $\endgroup$ Jul 14, 2015 at 10:44
  • $\begingroup$ RIght. Thank you very much $\endgroup$
    – Louis
    Jul 14, 2015 at 11:36

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