PDE Transport Equation(?) with Decay and Forcing Term So I am kind of lost on how to solve this PDE IVP.
$${\mathrm du \over \mathrm dt}+2{\mathrm du \over \mathrm dx}+4u=x$$ where $t>0$ and $x$ is in $R$, with the initial condition $u(x,0)=uo(x)=e^x$
Then consider the case for $x$ is contained in $[-1,1]$ boundary. Specify which boundary point is the inflow and solve the problem with the boundary condition $\phi (inflow,t)=e^t$.
So I normally know how to solve ${\mathrm du \over \mathrm dt}+2{\mathrm du \over \mathrm dx}+4u=x$ but I'm kinda confused on how to do it when it equals a nonconstant. Do I try ${\mathrm du \over \mathrm dt}+2{\mathrm du \over \mathrm dx}=x-4u$$?
Please help!!!

To make the problem simpler, let $v\left(x,t\right)=e^{2x}u\left(x,t\right)$
so that
$$
u_{t}+2u_{x}+4u=e^{-2x}v_{t}+2\left(-2e^{-2x}v+e^{-2x}v_{x}\right)+4e^{-2x}v=e^{-2x}\left(v_{t}+2v_{x}\right)
$$
and hence we look for a function $v$ s.t.
$$
\left\langle 1,2\right\rangle \cdot\nabla v=e^{2x}x.
$$
You should now be able to apply the method of characteristics for a first order PDE and use the relationship between $v$ and $u$ to find an expression for $u$.
