Solve: $\frac{\partial u}{\partial t}=\frac{\partial^2u}{\partial x^2}+4\frac{\partial u}{\partial x}+2u$ for $0
The way I am solving it is by reducing it to one involving the standard heat equation by setting $u(x,t)=(e^{\alpha x+\beta t})U(x,t)$. Then substituting this into the problem and choosing $\alpha$ and $\beta$ to obtain a standard problem for $U(x,t)$ for a bar with ends kept at zero temperature. I have gotten to the point where I differentiated the equation and plugged it back into the initial PDE and took out $e^{\alpha x+\beta t}$. It's after here that I am unsure of what to do. What I dont understand is choosing alpha and beta or the following steps to finish the overall problem. Also, I apologize for the syntax, I don't have Latex or anything on my computer and the derivative notation is supposed to be partial derivatives.