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.
Thanks in advance.