Solving inhomogeneous wave equation I am studying wave equation for exam 
$$u_{tt}=u_{xx} +\theta(x,t)$$ 
and I can't solve the next inhomogeneous wave equation at all (although it seems really simple):
$$
\begin{cases}
u_{tt}=u_{xx} +1 \\ 
u(0,t)=u(L,t)=0
\end{cases}
$$
We used superposition to get  $$u=\sum_1^k X_k(x)*T_k(t)=\sum_1^k \sin(k\pi t/L)*T_k(t)$$
where $2L$ is the period, then we calculated some sums and got DE 
$$T_k''(t)+(k\pi /L)^2T_k(t)=C_k(t)$$ 
where $C_k(t)=2/L*\int_0^L\theta(x,t)\sin(k\pi x/L)\,dx$.  
If $\theta(x,t)=1$, then $C_k(t)=4/k\pi$ for all even $k$. So my question is how do I solve this DE if right side doesn't include variable $t$? 
$$T_k''(t)+(k\pi /L)^2T_k(t)=4/k\pi$$ 
I know there are other possibilities for solving this (Laplace, D'alembert...) but the professor explained only this one (sloppy). If anybody can solve this I would be really happy. Thank you for answers.
 A: A differential equation of the type
$$y'' + k y=c$$
is a linear inhomogeneous second order ordinary differential equation. 
I assume you know that, in order to find the general solution, you need to solve the homogenous equation associated to it, and then find a particular solution. 
There are standard methods to find a general solution for the homogeneous equation (look at the roots of the associated equation $\lambda^2+k\lambda=0$, etc) and then we need to find a particular solution. 
To find a particular solution, we need a good guess, which should depend on the inhomogeneous term. In order to do that you should try functions of the type $y_{PS}(t)=\text{constant}$ (as your "inhomogeneous term" is also a constant).
Here you can find a good guide for solutions, together with many examples.
A: Let me suggest a different method. Since in the problem $\theta(x,t)=1$ dos not depend on $t$, find a function $v(x)$ such that $0=v_{xx}+1$, $v(0)=v(L)=0$. Then let $u=v+w$ and see what equation you obtain for $w$.
