I have the following problem of the heat equation:
$w_t - w_{xx}-4w_x+w=0; x\in(0,1),t>0$
$w(0,x)=e^x(\sin(2\pi x)-\sin(5\pi x));x\in(0,1)$
$w(t,0)=w(t,1)=0;t>0$
My attempt:
Doing the change of variable $u(t,x)=w(t,x)e^{5t+2x}$
We have now the problem:
$u_t - u_{xx}=0; x\in(0,1),t>0$
$u(0,x)=e^{3x}(\sin(2\pi x)-\sin(5\pi x));x\in(0,1)$
$u(t,0)=u(t,1)=0;t>0$
But doing with separation of variables $u(t,x)=V(t)X(x)$ I have to solve the ODE $X''+\lambda X=0 $ but then the initial condition of $X(0)=X(1)=0$ could not happen that $X(x)$ is the form of $c_n \cos(n \pi x)+b_n \sin(n\pi x)$ for any number n. I cant follow from here...
Any help would be appreciated. Thanks