A very simple heat problem

Solve the heat problem
$$
\begin{array}{ll}
u_t=u_{xx} & u(x,0) = \sin(x)-\sin(5x)/3 \\
u(0,t) = 0 & u(\pi,t) = 0.
\end{array}
$$

Recall that the situation: $$u_t=ku_{xx}, u(0,t)=u(L,t)=0, u(x,0)=f(x)$$ has a general solution of $$u_n(x,t)=e^{\frac{-kn^2\pi^2t}{L^2}}sin(\frac{n\pi x}{L})$$. In this particular case, $k=1$ and $L=\pi$, so for an initial condition of $u_n(x,0)=sin(nx)$ the solution is $u_n(x,t)=e^{-n^2t}sin(nx)$. However the initial condition in this case is not that simple. This is where it is useful to know that the nature of these solutions is such that any scalar multiple and linear combination of a solution(s) is also a solution. Thus using this formula we get that $$u(x,t)=e^{-t}sin(x)-\frac{1}{3}e^{-25t}sin(5x)$$