LVK has already answered existence and the non-negativity of the solution. For existence, "shooting" works. For each parameter $\alpha \ge 0$, consider the solution to the IVP \[ \begin{split} &u'' = k_1 u' + k_2 f(u) \\ &u(0) = 0, \\ &u'(0) = \alpha. \end{split} \] Denote the unique solution to this IVP by $u_\alpha$. Then, $u_\alpha(x_1)$ is a continuous function of $\alpha$. Observe that $u_0$ is identically $0$, so $u_0(x_1) = 0$. Furthermore, by rewriting the ODE, we have \[ ( e^{-k_1 t} u' ) ' = e^{-k_1 t} k_2 f(u). \] Integrating both sides and using $f(u) \ge 0$, we get that $u_\alpha'(t) \ge \alpha$ for all $t \in [0, x_1]$. It then follows that $u_\alpha(x_1) \ge \alpha\, x_1$, so for $\alpha$ large enough, $u_\alpha(x_1) > u_1$.
The IVT gives the existence of an $\alpha$ for which a solution exists.