Separating variables in the PDE $u_{tt}+2u_t+u=u_{xx}$

Question

Separating variables in the PDE $u_{tt}+2u_t+u=u_{xx}$

In the ODE of $T(t)$ (Second last equation) shouldn't it be the one stay outside the bracket?

Answer 1

It looks like the error may be earlier. The separation of variables should give:

$\dfrac{\ddot{T} +2\dot{T}}{T}+1 = \dfrac{\ddot{X}}{X} = \lambda$

The next lines look okay:

$\ddot{T} +2\dot{T}+T = \lambda T$

$\ddot{T} +2\dot{T}+(1-\lambda)T = 0$

$\ddot{T} +2\dot{T}+(1+n^2)T = 0$