I am solving a differential problem with boundary conditions and stuck at a point and looking for helps.Here is what I did.
The problem is: Solve the differential equation $$\frac{\partial u}{\partial t} = \frac{\partial^2u}{\partial x^2} + u$$ with boundary conditions$$ u(0,t)=u(1,t)=0$$
I tried using separating variable method where $$ u(x,t)=X(x)T(t)$$ and got $$XT'=X''T+1$$ and lost how to proceed. How do I proceed to solve this?