solving $u_t = u_{xx} +2u_x +u+1$

Question

I have this problem : $$u_t = u_{xx} +2u_x +u+1$$ with boundaries of $$0<x<1$$ $$ 0<t$$
and we also have $$ u(x,0) = x$$ $$u_x (0,t) = sin t$$ $$ u(1,t) + u_x(1,t) =2$$

I need to solve this problem but I have no idea about the way I can use the last condition. can someone please help me solve this problem?

Answer 1

Multiplying by $e^x$ we arrive at \begin{align} (e^x(u+1))_t=(e^x(u+1))_{xx}. \end{align} Taking $e^x(u+1)=w$ we arrive at the heat equation \begin{align} w_t=w_{xx}. \end{align} Can you solve it from here?