## anonymous one year ago I was studying the differential equation of the pendulum, you know, y'' + k sin (y) = 0, with k constant, and I know it can be approssimated, for little y angles, to y'' + ky = 0. My question is: how do I solve the equation without estimating sin (y) = y?

1. IrishBoy123

you can note that $$y'' + ky = 0$$ $$(D^2 + k) y = 0$$ where D is the differential operator $$D = \pm i \sqrt{k}$$ so characteristic equation $$y = A e^{i \ \sqrt{k}x} + B e^{-i \ sqrt{k}x}$$ and try the suggested solution to see if it works in this differential equation and expanding that solution out gives you sinusoids

2. IrishBoy123

but the mere fact you have a restorative force [-kx] means that you have simple harmonic motion and so the solution is sinusoid

3. anonymous

I'm sorry, what do you mean with "expanding"? Taylor? And there's a thing I don't understand, did you do that approximation, sin (y) = y, in the first equation you wrote?

4. IrishBoy123

maybe there is some ambiguity there yes, without the linearisation you have to solve something else $$y'' + k \sin y = 0$$ and that's way above my pay grade!! here's why :p

5. anonymous

Well, now I have understood what my mechanics book meant with "analytically difficult"!!!!! Thank you IrishBoy, you've been very kind!

6. IrishBoy123

and thank you for this insight! i feel very humble again :p

7. beginnersmind

Is the equation y'' + P(x) = 0, solvable for a general polynomial P(x)? It looks like it should be, but the answer wolfram gives me for y'' + k(x-(1/6)*x^3) = 0 is giberrish.

8. beginnersmind

Silly me, it should be y'' + P(y) = 0

9. anonymous

I had the same idea, expanding with Taylor, it looks like a Bernoulli equation, even though it's not a first order equation, so I'll try to divide and change y variables. Let's see if something sensed can come out of this!