I'm in trouble with this problem
$$\begin{cases} u_{t}(x,t)-u_{xx}(x,t)-2u_{x}(x,t)=0\qquad \mathrm {for}\;(x,t)\in(0,\ell)\times\mathbb R^{+} \\ u(x,0)=2e^{-x}\sin(\frac{3\pi x}{\ell})\qquad\qquad\qquad\;\;\; \mathrm{for}\;x \in (0,\ell) \\ u(0,t)=u(\ell,t)=0\qquad\qquad\qquad\qquad\;\mathrm{for} \;t\in\mathbb{R^{+}} \end{cases} $$ where $\ell \in \mathbb{R^{+}}$.
It'd a PDE whit boundary and initial conditions, but I see it a little bit strange for my knowledge. It has both first and second derivatives for spatial ($x$) part and a first order derivative for the temporal part. I start the study as always:
$$U(x,t)=X(x)T(t)$$ $$T'X=TX''+2TX'$$ $$\frac{T'}{T}(t)=\frac{X''+2X'}{X}(x)$$ $$\frac{T'}{T}=\frac{X''+2X'}{X}=\lambda$$ for some constant $\lambda \in \mathbb{R}$. I have to solve, considering the boundary conditions: $$\begin{cases} X''(x)+2X'(x)=\lambda X\\ X(0)=X(\ell)=0 \end{cases} $$ and, for the temporal part: $$T(t)=\lambda T(t)$$ with $t>0$.
Ok, but now I'm stucked. I don't know how to treat the spatial component in order to select some value of $\lambda$ to see what happens putting the boundary conditions to find the costants. Could someone give me a hint to let me start and solve the problem? I really want to learn this stuff! Thank you!.
Edit:
My progress (My idea)
$$X(x)=c_{1}e^{-(\sqrt{\lambda + 1}+1)x}+c_{2}e^{(\sqrt{\lambda + 1}-1)x}$$
I rewrite as $$X(x)=c_{1}e^{-(\sqrt{\lambda + 1})x}e^{x}+c_{2}e^{(\sqrt{\lambda + 1})x}e^{-x}$$
Now I suppose $\lambda+1=\alpha^{2}<0$ in order to have the solution like as:
$$X(x)=a\cos(\alpha x)+b\sin(\alpha x)$$ Using the boundary conditions: $$X(0)=a=0$$ $$X(\ell)=b\sin(\alpha\ell)=0$$ So $$\alpha=\frac{n \pi}{l}$$ I have finally found $\lambda$ and a value for $c_{1}=0$ and $c_{2} \in \mathbb{R}\neq0.$
$$\lambda=-\alpha^{2}-1$$ Note the minus sign for $\alpha^{2}$: I placed it because of the hypothesis for $a^{2}<0$. So $$U(x,t)=c_{2}\sin(\frac{n\pi}{\ell}x)e^{-x}e^{-(1+\frac{n^{2}\pi^{2}}{\ell ^{2}})t}$$
Including the temporal part as $T'(t)=c_{n}e^{-(\alpha^{2}+1)t}$.