"ODE is not just about 'solving' an equation and spitting out a (probably nasty) formula" -- this is what I want my (undergraduate) students to learn from my course this summer.
One example I am looking for is a scenario where one extracts info about a function from the ODE it satisfies much more easily than from the solution's explicit formula. There are easy example to see this. For instance, the IVP $y'=y; y(0)=1$ tells us that the function will be increasing on zero to infinity, by taking another derivative that it will be concave up, etc. However, one may rightly argue that $e^x$ which is the solution easily gives these properties.
So, I am looking for a less trivial, yet, interesting example where it is much easier to understand a function from its ODE than from its explicit formula.
Do you have such examples? I will appreciate them.
**The example may be important from computational/numerical point of view."