What exactly is the maximal solution of an ODE and why do we care?

Question

I am reading these notes on the definition of a maximal solution of an ODE i.e. http://www.math.lmu.de/~philip/publications/lectureNotes/ODE.pdf

But the definition is so abstract and no example is provided!

From what I can gather, the maximal solution of an ODE is the solution to an ODE that exists for the longest time.

But why do we care about this?

For example, given $\dot x = -x$, the solution is $x(t) = K\exp(-t)$ where $K$ is some constant. Solution exists for all times. And?

Can someone provide a concrete example of a differential equation where we need to care about maximal solution and why it matters?

Answer 1

Check the equation $\dot x = x^2$. All the nontrivial solutions have "blowup in finite time" (i.e., a vertical asymptote).

Answer 2

The idea of maximal solution might come from three body problem. People have very strong interests to know if planets in solar system may eventually collapse onto each other? Without considering the effect of chemical reaction from the sun, the question is roughly equivallent to prove the maximal solution to three body problem exists for $ 0 < t < \infty$. The existence of solution of ODE, except trivial ones, are ususally proved locally, so finding a way to extend the solution to a possible maximal interval is of interests to see when the disaster can happen or never happen. This is one person's view.

Answer 3

When trying to find a solution of ODE $$ \frac{dy}{dx} = \frac{1}{\sqrt{x^2-1}} $$ as

$$ y = C+ \cosh^{-1}x $$

just has no real existence at all in the interval $ x < 1. $ The nature of the ODE, the phenomenon it represents, influence its solution as well.