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OpenStudy (anonymous):

System of second-order ODEs?

OpenStudy (anonymous):

\[\frac{\text{d}^2\theta}{\text{d}t^2}=-g\frac{\sin\theta}{r}\]\[\frac{\text{d}^2r}{\text{d}t^2}=g\cos\theta-\frac{k}{m}(r-\ell)\]\[\theta(0)=\theta_0,~~\theta'(0)=0\]\[r(0)=\ell,~~r'(0)=0\]

OpenStudy (anonymous):

oh dear lord, a polar system of second order DE's I don't think I can do it, but I'll be checking my notes in the meantime, bookmark*

OpenStudy (experimentx):

Here is a general solution generated by Maple without IC what kind of system is that? the first looks like large angle pendulum, the second one looks like elastic harmonic oscillation. so, in general, this is a elastic pendulum?

OpenStudy (anonymous):

Yes, this is the differential equation for an elastic (spring) pendulum. http://en.wikipedia.org/wiki/Spring_pendulum I don't really understand the solution... I guess I just don't know how to read maple. I was almost certain that a closed-form solution would have to involve elliptic integrals, right? Or no closed-form solution? I was really just looking for a simplification of the system so I could form a numerical solution... But here's something cool: http://www.wolframalpha.com/input/?i=spring+pendulum&a=*C.spring+pendulum-_*Formula- I wonder how W|A solves it...

OpenStudy (experimentx):

probably ... numerical solution by changing into first order system [4x4]. even considering r to be constant ... the first DE is non linear. the solution looks like \[ t = {2 \over \omega_0} \log (\tan \left( \theta + \pi \over 4 \right))\]

OpenStudy (experimentx):

probably numerical ...

OpenStudy (anonymous):

How did you get that solution? Actually, maybe tell me about conversion from 2x2 second-order to 4x4 first-order? I don't know how to do this...

OpenStudy (anonymous):

And since \(\omega_0=0\), your solution doesn't work...

OpenStudy (experimentx):

no ... that's just for large angle pendulum. I don't think I could solve the system analytically.

OpenStudy (experimentx):

any second order system can be converted to first order system by letting a new variable. it is given here http://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-iv-first-order-systems/ just the basics .... but it allows your to convert your system from second order to first order ... and you can apply numerical approach to solve the system.

OpenStudy (experimentx):

sorry ... the \( \omega_0 \) is the angular frequency of the system.

OpenStudy (experimentx):

Woops ... even in my book, it assumes IC such that integral Constant doesn't make it elliptical anymore.

OpenStudy (experimentx):

let's see .. it's interesting system.

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