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I recently had an assignment question asking to solve a PDE.
The question is as follows:

Solve $u_t + 3t u_x = u$ with the initial condition $u(x,t)=1+\cos x$ on the curve $x + 3t = 0$.

I know how to solve without the initial conditions, but I would like to know how to solve for the full solution with the initial conditions. Thanks!

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  • $\begingroup$ If you edit the question to include what you have done in the case with no initial condition, then maybe you can receive help from there. Also, the question title mentions 'Non-linear PDE', but the equation you have stated is linear. Are you sure that you have typed up the question properly? $\endgroup$ – ekkilop Aug 15 '17 at 11:25
  • $\begingroup$ Michael Lee : If the PDE is $u_t+3tu_x)=u$ , what you wrote is false : $x+3t$ is NOT a characteristic curve. $\endgroup$ – JJacquelin Aug 15 '17 at 16:24
  • $\begingroup$ You changed the wording of the question, but the typo is still in it. The general solution of $u_t+3tu_x=u$ is : $$u(x,t)=e^tF(x-\frac{3}{2}t^2)$$ where $F$ is an arbitrary function. Of course, it is possible to determine $F$ according to the condition $u=1+\cos(x)$ on the curve $x+3t=0$. But this is too complicated for a scolar exercise. Probably you made a mistake in copying $u_t+3tu_x=u$. Sorry, I will be away for some days, So I will not be avalable to help you more. $\endgroup$ – JJacquelin Aug 16 '17 at 5:59
  • $\begingroup$ See the duplicate (answered) question: math.stackexchange.com/questions/2395039/… $\endgroup$ – Merkh Aug 16 '17 at 21:48

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