Given positive scalars $k_1,k_2, x_1, u_1$ and continuous non-decreasing function $f: R\to R$ with $f(0) = 0$ how to prove the existence of unique $C^2([0,x_1])$ solution of the problem:
$$u'' = k_1u'+k_2f(u)\quad \hbox{ on }\ \langle0,x_0\rangle$$ $$u(0) = 0$$ $$u(x_1) = u_1.$$
Does it imply that $u\geq 0$?
I know this is standard result, but I have difficulties with references to this result.