By the method of separation of variables, solve the equation

$\displaystyle u_{tt}+4u_{t}-u_{xx}=5\sin(2\pi x)$

for $\displaystyle 0\leq x\leq1 $ with boundary conditions $\displaystyle u(0,\, t)=u(1,\, t)=0$ and initial conditions $\displaystyle u(x,\,0)=u_{t}(x,\,0)=0.$