Question

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?

Answer 1

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

Answer 2

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

Answer 3

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?

Answer 4

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

Answer 5

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

Answer 6

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

Answer 7

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.

Answer 8

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

Answer 9

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!