Find the equilibrium solution of
$$ u_t(t,x) = u_{xx} (t,x) + x^2, \ 0<x<L\\ u(t,0) = 0,\ u(t,L) = 0 $$
I know that the equilibrium solution must satisfy: $[u_e(x)]' = 0 \ \forall t$. It must also satisfies the boundary conditions. I'm not sure what the first step is though.
At equilibrium, $u_t(t,x) = 0$ for $x \in [0,L]$.
The steady state solution $U(x)$ satisfies
$$U_{xx} (x) + x^2=0, \ 0<x<L\\ U(0) = 0,\ U(L) = 0.$$
Integrating we get
$$U(x) = -\frac1{12}x^4 + C_1x + C_2.$$
Applying the boundary conditions we find
$$C_1 = \frac1{12}L^3,\ C_2 = 0,$$
and
$$U(x) = \frac1{12}[L^3x - x^4].$$
