Properties of Solutions of $u'' + c(t)f(u)h(u') = 0$ with Explicit Initial Conditions

Conditions are stated, determining the behavior of a solution of the nonlinear equation mentioned in the title, which are expressed wholly and explicitly in terms of the initial conditions and the given functions $h(s)$, $f(s)$ and $c(t)$. Avoiding a priori assumptions that a solution is proper or possesses some other property, the results illuminate the variety of behavior which can exist for a single equation under various initial conditions. The basic restrictions imposed here on the defining functions are that $h(s)$ and $c(t)$ have constant sign and $sf(s) > 0$ for $s \ne 0$. The method of proof for some of the results involves the introduction of two Lyapunov functions which do not require that $c(t)$ be monotone.