# Examples of parameter dependent ODEs

I would like to have some examples of simple parameter dependent ODEs. I would like the solutions to have some physical meaning. I'll give one example so it clear what I am after:

Example 1: The Lane-Emden equation from Astrophysics can be rewritten so that the first root of the solution becomes an unknown. Let $v$ be the unknown first root then the parameter-dependent ODE becomes,

$u'' +(2/x)u' + v^2u =0, u(0)=0, u'(0)=1, u(1)=0.$

The Lane-Emden equation model stellar formation and the parameter $v$ determines the polytropic region.

I would like to have perhaps two/three interesting examples.

$x''=-kx$ models simple harmonic motion. The parameter $k$ determines the period.

Here is a blog post by John Baez about a two dimensional system with a Hopf bifurcation and applications in quantitative ecology and climate science.

Two ideas:

(1) Consider a beam of uniform cross section and length L, with endpoints fixed at $x = 0$ and $x = L$. If the beam is subject to equal and opposite forces $P$ at the endpoints, then the displacement of the beam can be modeled by the eigenvalue problem

$\frac{d^{2}y}{dx^{2}} + \mu^{2}y = 0$, with $y(0) = y(L) = 0$ and $\mu^{2} = \frac{P}{EI}$.

($E$ and $I$ are both constants. They represent Young's modulus and the moment of inertia of the cross section of the beam, respectively.)

(2) One variant of the logistic equation is

$\frac{dP}{dt} = rP(1 - \frac{P}{K})$,

which describes the growth of a single population $P$ over time. The parameter $r$ is the growth rate, and $K$ is the carrying capacity.

Most of the usual test sets for testing ODE solvers feature ODEs obtained from real-life applications. There's the CWI-Bari test set, test sets from Ernst Hairer and Jeff Cash, as well as the DETEST suite from Toronto (see these two articles for details).

One of my favorites is the restricted three-body problem (this question displays a few relevant links); it is not at once obvious that for certain preset positions of the "spaceship" that one would get a periodic orbit ("Arenstorf" orbits).