When ODE tells more than the explicit solution. "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."

There are examples where the qualitative approach is very helpful.
Newton's Law  of Cooling  $$ T'=-k(T-A)$$ is a good example.
The temperature of an object changes proportional to the difference of the object temperature and the room temperature.
The heat equation, $$ \frac {\partial u}{\partial t}= -k \frac {\partial^2 u}{\partial x^2}$$ is another example.
The concavity of heat with respect the state indicates, the rate of change of  the temperature with respect the time. 
That explains the effect of the temperature of the neighboring points on the object.  
