How to solve: ((1/k)')^2 + T^2(1/k)^2 - T^2 = 0

[Pwantar]

That equation again is: ((1/k)')^2 + T^2(1/k)^2 - T^2 = 0
where k is a function of s, T is a constant, and ' denotes differentiation with respect to s.

If you're wondering, it arises when you try to describe the family of curves with constant torsion that lie on the unit sphere. Here, T is the torsion, and k is the curvature. And boy do I wish I remembered my differential equations class a little better.


My attempt:

Let 1/k = R.

Then (R')^2 + T^2*R^2 -T^2 = 0

Suppose R = A*cos(Ts) + B*sin(Ts)

Plugging it in, it is found that A^2+B^2=1

So, R = cos(a)*sin(Ts)+sin(a)+cos(Ts)

So K = R^-1...

Which does not satisfy the equation :(


I also tried setting it up like this:

K' +Sqrt[T^2*K^2*(K^4-K^2)]

 

[tiny-tim]

Hi Pwantar!

Then (R')^2 + T^2*R^2 -T^2 = 0

Suppose R = A*cos(Ts) + B*sin(Ts)

Nooo …

what are the roots of the characteristic equation? ​

 

[HallsofIvy]

That's a non-linear equation. Neither Acos(Ts)+ Bsin(Ts), nor the concept of "characteristic equation" apply.

 

[tiny-tim]

oops!

That's a non-linear equation. Neither Acos(Ts)+ Bsin(Ts), nor the concept of "characteristic equation" apply.

oops! thanks, HallsofIvy!

(all those ^2s confusing me)

better write it R'/√(1 - R²) = ±T ​

 

[Pwantar]

As I'm too lazy to recheck what I've written now that I'm already at this fancy text box thing that I don't understand, but love, I would like to point out an error that may or may not be present, as well as a possible solution.

((1/k)')^2 + T^2*(1/k)^2 - T^2 = 0

(-k'/k^2)^2 + T^2*(1/k)^2 - T^2 = 0

(k')^2/k^4 + T^2*(1/k)^2 - T^2 = 0

(k')^2 = T^2*k^2*(k^2-1)

dk/ds = T*k*Sqrt(k^2-1)

-atan(1/sqrt(k^2-1) = Ts + C, where C is an arbitrary constant

k = sqrt(1/tan(-Ts-C)^2 + 1)

k = 1/cos(Ts + C)

which happily satisfies my equation... the thing I don't like is taking the square root. Perhaps I should consider both the negative and positive sqrts... however, just looking at it, I don't think the solution will change.One more thing...
That Solution to the equation for R that I mentioned earlier actually does work... I guessed it based off of me forseeing a few sin^2 + cos^2 dealies cancelling out.

the knowledge that k=1/R does not appear to be sufficient information to find k given R... however, should a = Pi/2, it works, save for the phase angle C. It could be that I made a mistake somewhere. Perhaps I'll actually try to find my work for it, examine it, and notify you of anything interesting.Whoops, I made some mistakes in what I typed, but got the answer... I'll leave it as an exercise to you to figure out what they are