Change of variable for differential equation

Without thinking too hard about this, if you really have a
differential functional equation as in your 'ugly' example, I would
not try to make such an equation with 't' as an independent variable.

The only apparent way to do that is to write
  x = a*t - b, plug in and differentiate.  Having the
  atan(diff(h(x),x)) in there will produce a more complicated
  differential equation which you probably can't solve and will have to
  resort to numerical techniques.  This will probably be much more
  numerically unstable than the original equation.

I would solve the first equation as best you can, getting h = h(x).

You can then plug in x = a*t -b and get your function in terms of t.

Just my 0.02 !

-sen

On Sun, 17 Jun 2007, Daniel Lakeland wrote:

> On Sun, Jun 17, 2007 at 03:07:12PM -0400, sen1 at math.msu.edu wrote:
>> There is no 'h' on the rhs, so all you have to do is:
>>
>> dh/dt = dh/dx *dx/dt = dh/dt * a = g(a*t + b)
>>
>> So,
>>   dh/dt = (1/a)* g(a*t + b)
>>
>> Once you have this, higher derivatives are no problem.
>
> Unfortunately, the example operation I gave was mostly just
> example... The real situations might involve a nonlinear differential
> equation or what have you. The goal being to be able to transform from
> a fairly general differential equation on an x coordinate to a related
> equation on a t coordinate that is a simple scaling and translation of
> x.
>
> so for example something ugly like
>
> diff(h(x),x) - diff(h(x),x,2)*atan(diff(h(x),x)) = g(x)
>
> where I'd prefer to have maxima calculate the appropriate expression
> for the change of variable on the left because I'm lazy :-)
>
>
>

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