Consider IVP $$y’’-q(x)y=0, y(0)=a, y’(0)=1$$ where $q(x)$ is continuous and positive on $\mathbb R$. Then my observation is as
If $a>0$, every solution is convex and hence strictly increasing on $[0,\infty)$, because solutions start with positive value and remains convex as long as $y>0$. Again, if $a<0$, then every solution is concave and hence strictly decreasing on $(-\infty, 0]$, because solutions start with negative value and remains concave as long as $y<0$.
Please comment or correct my concept . Thank you.