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# 2nd ord ODE, non-const cffts watch

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1. Now let

so

ERRORS from here on down, please ignore!

So we need to find K(x) s.t.

where K(x) = k'(x).

But this ones 'easy' ...

So...

2. (Original post by Mathmoid)

Oopsie.

3. (Original post by generalebriety)
Oopsie.

Oh carp!
4. (Original post by Mathmoid)
Oh carp!
In conclusion (as my brain is rapidly switching off) to find p(x) we need to solve:

As this is non-linear I think it will very much depend on r(x) as to whether we can find solutions for K and what these will look like.

If it is possible to find solutions, I think my method works in principle, but there is no guarantee that the integrals will be easy / possible even from this point onwards.
5. Here's what I did.

I thought I could maybe simplify this by finding a substitution that would turn it into an ode with constant coeffts.

let

Then,

Substituting for into the original ode, and rearranging, gives,

Now, for a const-coefft ode, I need

For , I ended up with

Even if I ignored the const of integration, this solution would only give a const-coefft ode for r(x) = an exponential function - not too much of an improvement beyond standard methods.

For ... well, this is where I stopped and came here looking for help.

It was starting to get a bit hairy now. Even for a simple poynomial such as r(x) = ax² + bx + c, solving for f wasn't going to be easy. And then there was no guarantee that would end up a constant!!

I am sooo glad there was a simpler solution to my prob
6. That's some brave work in the face of sheer hopelessness.
7. I don't think you're going to get nice solutions most of the time.

After all when r(x) = -x this is Airy's equation, which has special function solutions.

I think your best approach is with power series given that r is a polynomial.

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Updated: August 3, 2007
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