# A question about a PDE

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...

• 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. – shrinklemma Apr 13 '17 at 6:53
• It has infinitely many solutions. Every function $f$ for wich $f(2)=1$ satisfies the conditions. – Rafa Budría Apr 13 '17 at 8:51
• Could you offer some texts about the question? I will study from it. – user384789 Apr 13 '17 at 15:03
• I don' t exactly understand, unfortunately. – user384789 Apr 13 '17 at 18:03

$$u=f(6x+y)e^x$$
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 $f(2)=1$.