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Let u be a solution of the PDE $$ u_{xy}+au_x+b u_y+cu=0,~~~~~~~~~~~~~~a,b,c=const.~~~~~(*) $$ Consider $$ v(x,y):=u(x,y)\exp(bx+ay). $$ Find the PDE which $v$ fullfills.

Could you please give me a hint how to find that PDE?

(*) is a linear PDE of degree 2, right? Do I have to find the normal form?

As you see: I do not really know how to solve this question.

1 Answers1

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$u(x,y)=v(x,y)\exp(-bx-ay)$. Now differentiate that, and combine the results of $u_{xy}$ and so on, to add up to 0 on the right-hand side

Empy2
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  • You mean I have to determine $u_{x}, u_y,u_{xx},u_{yy},u_{xy},u_{yx}$ for $u(x,y)=v(x,y)\exp(-bx-ay)$ ? –  Oct 14 '13 at 15:35
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    No, just $u_{xy}, u_x, u_y$. Then feed your results into the original equation. $\exp(-bx-ay)$ should be a common factor – Empy2 Oct 14 '13 at 15:44
  • If I did not make a mistake, then $v$ fullfills the PDE $v_{xy}+(c-ab)\cdot v=0$. –  Oct 14 '13 at 15:59