I really don't know how to start to solve it: $$\left\{\begin{array}{ll} u_{t}=ku_{xx}-\lambda^{2}u, & x\in(0,\ell), t>0 \\ u(0,t)=u(\ell,t)=0, & t>0 \\ u(x,0)=h(x), & x\in(0,\ell) \end{array}\right.$$
I know there are some tips, like using some function $v(x,t)$ to solve the "original" heat equation (the homogeneous one), but I didn't find anything about how to start solving a problem like this one.
I just need to know what are the first steps.
Using the integrating factor method borrowed from first order linear ODEs, your equation becomes $\frac{\partial}{\partial t}(e^{\lambda^2 t}u)=k \frac{\partial^2}{\partial x^2} \left ( e^{\lambda^2 t} u \right )$. Thus $e^{\lambda^2 t} u$ solves the ordinary heat equation.