Re: [Phys-l] a nonlinear ODE

[John Denker <jsd@av8n.com>]

On 12/28/2008 06:30 PM, Carl Mungan wrote:
> Consider the following differential equation for motion in a single
> variable x:
>
> x*a=k*t                     [1]
>
> where x=position, a=acceleration, t=time, and k=real positive
> constant.
>
> What is the general solution for x(t)?
>
> By trying x=c*t^n, I get x=sqrt(4k/3)*t^1.5. But isn't that just a
> particular solution? Shouldn't I expect two arbitrary constants in
> the general solution?

The question is directly on target.  There is in fact a
two-parameter family of solutions.

There are few if any general techniques for dealing with
nonlinear differential equations.  (This stands in contrast
with linear DEs, for which some quite powerful general
techniques are known.)

There is one rule to live by:  Start by drawing a picture.
I plotted some solutions to equation [1] for k=1 and
various initial conditions:
  http://www.av8n.com/physics/img48/xddx.png

Doing it took me less time than it takes to tell about it.
For nonzero x, rewrite the equation as 
         (d/dt) (d/dt) x = k t / x          [2]
and now it looks like every other equation of motion you've
ever seen, i.e. acceleration in response to a known force.
You get to choose the initial x and the initial x_dot.
Then just turn the crank, time-stepping the equation of 
motion.  Equation [2] gives you x_dot_dot, which tells
you the new x_dot, which tells you the new x, and we're
off to the races.

Remember:  Draw the picture.

================

Also, ask yourself how badly you need a closed-form solution
to equation [1].  Oftentimes you can understand more about
the solution from looking at the differential equation itself
and/or looking at the picture than you could from looking
at the closed-form solution.

... and then there's the distinct possibility that no closed-
form solution exists (in terms of known functions).