Let's say we have a second-order differential equation of the form: $$ y''(t)= a(t)y'(t)+b(t)y(t). $$ Assume also that $a(t)$ and $b(t)$ are bounded, positive for all time, and with any regularity we wish.
QUESTION: If I have now an inequality of the form $$ f''(t)\leq a(t)f'(t)+b(t)f(t), $$ is there any relation between $y(t)$ and $f(t)$? Something like Gronwall's inequality or Petrovitsch theorem for a second-order ODE?