An applied problem in ODE leading to Interval of Existence issue ODE books have a section on interval of existence of a solution with a standard set of problems, such as what is the interval of existence for $y'+(\tan t) y = t/(t-2), y(3)=5$, or $y'=y^3, y(0)=2$. I am looking for a "modelling" problem leading to such an issue. That is, a natural situation where the issue comes up and has a physically meaningful answer. 
 A: My research involves the shapes made by bent semirigid materials like paper. The shapes made by paper when it is bent are solutions to differential equations which are guaranteed to exist (obviously, since the paper must do something) as long as the "springiness" or rigidity of the paper is a continuous function of the length of the paper. However, if we introduce a crease in the paper, we treat this as a discontinuity in its rigidity, and we are not guaranteed the same solution on one side of the crease as the other. In fact, engineers and origami artists take advantage of this fact in their craft.
Another example of when the existence theorem comes into play in a physical situation is in wave mechanics. The wave equation (in one dimension) is: $$\frac{\partial^2 f}{\partial t^2}=v(x)\frac{\partial ^2 f}{\partial x^2},$$
where $v(x)$ may not be continuous. The wave equation has to be solved separately on each interval over which $v(x)$ is continuous, and the different sections are matched together by matching boundary conditions. Unfortunately, this is a PDE, and doesn't exactly answer your question. 
However, a similar equation in quantum mechanics is an O.D.E. This is known as the time-independant Schrodinger equation, and is used to solve for the wavefunction $\psi(x)$ of a particle:
$$-\frac{\hbar^2}{2m} \frac{d^2\psi(x)}{dx^2}+V(x)\psi(x)=E\psi(x).$$ Here, $V(x)$ represents the potential energy the particle must have in order to be at the position $x$, and $E$ is the total energy of the particle. The wavefunction is related to where we can expect to observe the particle (its physical position). Again, $V(x)$ may not be continuous, so the Schrodinger equation must be solved separately on each interval over which $V(x)$ is continuous. The applications of this equation are numerous. A simple one is known as the "finite square well problem." In this problem, we want to simulate a particle confined to a "box." In this case, the potential, $V(x)$, is zero on a bounded interval (the inside of the "box"), and $V(x)$ is a positive constant $U$ outside this interval (the outside of the "box"). If $\psi(x)$ is a solution to the Schrodinger equation for this potential, with $0<E<U$, then the solution inside the interval is sinusoidal, and the solution outside the interval is exponential, decaying to $0$ at $\pm\infty$. The most difficult part of solving this problem is using the constants of integration to guarantee that $\psi(x)$ is both continuous and differentiable everywhere.
Another interesting aspect of the Schrodinger equation is that we are really looking for solutions not just on the reals, but on the extended reals, with points at $\pm\infty$. Since physically, we expect that the particle cannot exist at $\infty$, we expect that the wave function approaches zero as $x\to\pm\infty$. Thus, we have to consider that even though potentials like $V(x)=x^2$ are continuous over the reals, they may not be continuous at $\infty$. For this reason, we cannot choose an arbitrary energy to solve this problem. Only some values of $E$ allow us to guarantee that $$\lim_{x\to\pm\infty}\psi(x)=0.$$
Because $E$ can only take on some discrete values, we say that it is quantized, which is why we call it quantum mechanics.
