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.

-
Thanks. That looks good. –  alext87 Aug 23 '11 at 14:40

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.

-