I am teaching single-variable real analysis and I want to give the students a concrete example of application of the Picard--Lindelöf theorem for a first-order ODE $$ \frac{dx}{dt}=f(t,x),$$ where $t$ and $x$ are real variables.
The example should be explicit and come from a simple problem in the sciences (e.g., motion of a particle, population growth). The example should not be solvable by standard methods of integration such as separation of variables or variation of constants.
Do you have any suggestion for such an example?
