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I want to solve the PDE $$u_x-6u_y=u$$ by the method of characteristic curves.

Cauchy data is $u(x,y)=e^x$ on the line $y=-6x+2$.

I found the characteristics as $c_1=6x+y$ and $c_2=ue^{-x}$ and when I apply $c_2=f(c_1)$, I get $f(2)=1.$

I can' t finish the rest of the question...

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  • $\begingroup$ The characteristic curves $y=-6x+c$ are parallel with the line on which initial data is given, in which case we are not able to generate a unique solution from the initial data. $\endgroup$ – shrinklemma Apr 13 '17 at 6:53
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    $\begingroup$ It has infinitely many solutions. Every function $f$ for wich $f(2)=1$ satisfies the conditions. $\endgroup$ – Rafa Budría Apr 13 '17 at 8:51
  • $\begingroup$ Could you offer some texts about the question? I will study from it. $\endgroup$ – user384789 Apr 13 '17 at 15:03
  • $\begingroup$ I don' t exactly understand, unfortunately. $\endgroup$ – user384789 Apr 13 '17 at 18:03
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General solution of equation is

$$u=f(6x+y)e^x$$

From Cauchy data we get $$f(2)e^x=e^x$$ or $$f(2)=1$$

Then solution of our problem is $u=f(6x+y)e^x$, where $f(x)$ is any function satisfying the condition $f(2)=1$.

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