# Partial Differential Equation with initial condition on a curve

Solve $u_t + 3t u_x = u$ with the initial condition $u(x,t)=1+\cos x$ on the curve $x + 3t = 0$.
• Michael Lee : If the PDE is $u_t+3tu_x)=u$ , what you wrote is false : $x+3t$ is NOT a characteristic curve. – JJacquelin Aug 15 '17 at 16:24
• 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. – JJacquelin Aug 16 '17 at 5:59