| Posted on Wednesday, 05 February, 2003 - 10:21 am: |
If we solve the equation dy/dx=y, we get ex (ignoring the constant factor, which is uninteresting as far as this question goes.)
Similarly, d2 y/dx2 =y yields ex and e-x
If we take fourth derivatives, we get sin(x) and cos(x) as well.
All these functions have neat and fundamental properties, quite apart from the ones I've listed.
Now suppose we had y = d8 y/dx8 . That would give us four new linearly independent functions. I think these new ones would have interesting properties as well. Probably not periodicity, but maybe something analogous to sin{2}(x)+cos2 (x)=1.
Can anyone suggest an approach for investigating this?
| Posted on Wednesday, 05 February, 2003 - 11:58 am: |
Hauke, for any differential equation of the form y=dn y/d xn we can try a solution proportional to ew x, for some unknown w. Then substituting in gives the equation wn=1, i.e. w is any of the nth roots of unity: e2pi/n, e4pi/n, ..., 1. You could therefore get expressions for the solutions as combinations of cos's and sin's (using Euler's formula ei k=cos k+i sin k), and see what happens in terms of linear independence etc. Does that help at all?
| Posted on Thursday, 06 February, 2003 - 12:23 am: |
Yeah, I went down that track already. I tried y=esqrt(i)x , and (after some algebra) got the following functions:
y=1-x4 /4!+x8 /8! + ...
y=x-x5 /5!+x9 /9! + ...
y=x2 /2!-x6 /6!+x10 /10! + ...
y=x3 /3!-x7 /7!+x11 /11! + ...
Those functions are easy to derive, and seem to confirm my suspicions that they are sort of 'extensions' of sine and cosine because they have alternating positive and negative terms in their power series expansion, and because two of themm are odd functions and two even.
How to discover if there are any more interesting properties of these functions is still unclear to me.
| Posted on Thursday, 06 February, 2003 - 10:26 am: |
Alex's point is that these are just combinations of exp, cos and sin. Your first function, 1-x4/4!+x8/8!+..., is just 1/2(ex/Ö2+e-x/Ö2)cos x/Ö2. The second is
| 1/(2Ö2)((ex/Ö2+e-x/Ö2 )sin x/Ö2+ex/Ö2-e-x/Ö2cos x/ | _______ Ö 2 | ) |
. One can find similar expressions for the other two. David
| Posted on Friday, 07 February, 2003 - 07:54 pm: |
Hauke, the general solution to the DEQ dn y/dxn = y is given by
,where aj are arbitrary constants.
To prove this, consider the series expansion for y in terms of x, and then try to evaluate
|
¥ å m=0 | xm n+j/ (m n+j)! |
,
for constant (integer) n and j. Consider j = 0 first.
With regards to your original question, I'd imagine that there are some functions to be found in the above solution that obey special laws, like the pythagorean identity, but I'm not sure whether these laws would be special enough to merit much interest, still less since the functions, as said above, are only combinations of other standard ones.
Brad