A metal rod of length $L = 1$ is kept at constant temperatures at the two ends $u(x = 0, t) = u(x = 1, t) = 0$ and is heated with a constant heat source S(x) concentrated in the middle $(S(x) = 1$ for $0.4 ≤ x < 0.6$ and $S(x) = 0$ elsewhere). The temperature distribution u(x, t) satisfies the heat equation: $$ \frac{\partial u}{\partial t}=D\frac{\partial^2 u}{\partial x^2} + S(x) $$
where D = 1 is the heat diffusion coefficient. The initial condition is $u(x, t = 0) = 0$ (i.e. there is a constant temperature before the heat source is switched on). After a certain time the solution $u(x, t)$ converges towards a time-independent stationary temperature distribution $u_s(x)$. Find an expression for $u_s(x)$.
I solved $u(x) = c_1\cdot x$ for the first domain [0,0.4] and I solved $u(x) = c_3(1-x) $ for the third domain [0.6,1]. Then I integrate $u'' +1=0$ and get $u=-\frac12 x^2 +c_5\cdot x +c_6$, however, I don't know how to find the unknown integration variable.