Let $I$ be an open interval in $\mathbb R$ and $c\in C^0(I; \mathbb R_+^\ast)$ a continuous function on it taking strictly positive values. Is there a simple argument showing that the ODE $u''(t) = c(t) u(t)$ admits a fundamental system $(v,w)$ such that $v$ is increasing and positive, $w$ is decreasing and positive, in the strict sense, respectively? How explicitly can they be represented?
The question arised while studying linear diffusions where this seems (to me) to be a standard result derived by probabilistic means.
Motivated by the case of constant $c$, an ansatz $u(t) = \exp(f(t))$ yields the non-linear equation $$ f''(t) + (f'(t))^2 = c(t).$$