dirichlet boundary condition and first derivatives

Question

I am interested in solving an ODE of the form: $$y''[x]+y'[x]+y[x]=g[x],$$ with the boundary condition $y[0]=y[L]=0$. Is expressing $y[x]$ in terms of its Fourier sine series of any use in this context, given the presence of the first derivative term? Analogously, is the Laplace transform of any use, given that the boundary condition does not involve the first derivative of y[x]? Are there alternative approaches that I may consider? Thank you.