I had to solve the equation $u_t(t,x)=u_{xx}(t,x)$ with $x\in (0,\pi), t>0$ s.t. $u(t,0)=u(t,\pi)=0$ for $t>0$ and $u(0,x)=\sin(3x)-\sin(x)$ for $x\in (0,\pi)$. So I tried to find solutions with separable variable, i.e. of the form $$u(x,t)=h(x)f(t).$$
Then I have to solve the equation $h''(x)=\alpha h(x)$ with $h(0)=h(\pi)=0$. I found $h_n(x)=\sin(nx)$ for all $n$. Then, I solved $f'(t)=-n^2f(t)$ what $f_n(t)=\alpha _ne^{-n^2t}$. At the end, $$u_n(t,x)=\alpha _n f_n(t)h_n(x)$$ solve $u_{t}=u_{xx}$ with $u_n(t,0)=u_n(t,\pi)=0$. To use the last condition, I proved that $u(t,x)=\sum_{k=0}^\infty u_n(t,x)$ solve the system and is s.t. $u(t,0)=u(t,\pi)=0$. Then I found the $\alpha _n$ to get $u(0,x)=\sin(3x)-\sin(x)$, and I derived $\alpha _1=-1$, $\alpha _3=1$ and $\alpha _n=0$ for all $n\neq 1,3$. So finally, $$u(t,x)=e^{-9t}\sin(3x)-e^{-t}\sin(x).$$
Question : But is it with separable variable ? What are $h$ and $f$ s.t. $u(t,x)=h(x)f(t)$ ? More generaly, does $f_1(t)g_1(x)+f_2(t)g_2(x)$ can be written as $f(t)g(x)$ ?
If a sum of separated heat equation solutions of the form $T(t)X(x)$ were again such a solution, then the technique of separation of variables would not produce general solutions of heat equation. So don't worry about trying to find a final solution that is separated. Your final solution is correct because it satisfies the heat equation and the boundary conditions, and the solution is unique.