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I am given the following equation (Mathieu's equation) in my subject of Numerical Analysis :

$$ \frac{d^2 x}{dt^2}=-\omega^2(1+\epsilon\cos(t))x $$

I am supposed to find those frequencies $\omega$ for which the system becomes unstable. And then draw a graph of $\epsilon (<<1) $ as a function of $\omega$ pointing out regions where the solution becomes unstable. I will have to really scan different values of $\omega$ for a fixed $\epsilon$, then change $\epsilon$ and again scan $\omega$.

This all has to be solved using Runge-Kutta order 4 method. The initial values of $x(0)$ and $x'(0)$ have to be taken as some non-zero values.

I am neither a physics student, nor a mathematics student. Please explain me in layman's terms where should I search for the required frequencies.

PS : The professor has suggested V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd edition, Springer-Verlag, pages 113 − 120. There I came across a theorem stating : All points on the $\omega$-axis except the integers and half integers $\omega=k/2,k=0,1,2 ....$ correspond to strongly stable systems. I am having a little knowledge of SHM but I am not able to get the significance of $k$ here.

EDIT - I have come to know that for the equation of the type :

$$ \frac{d^2 x}{dt^2} + (a-2q\cos(2t))x=0 $$

The stable and unstable regions are given by the following graph:

enter image description here

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    $\begingroup$ Not a student of mathematics, nor physics, and your professor suggests Arnold? He/she must be a sadist! $\endgroup$
    – Danu
    Feb 20, 2015 at 18:28
  • $\begingroup$ Not his fault. I am a computer science student, taking the course of NA offered by the Dept. of Physics. $\endgroup$
    – Gaurav
    Feb 20, 2015 at 18:53
  • $\begingroup$ @Gaurav I suggest you to look first in Wikipedia at the article with the Mathieu equation $\endgroup$
    – Sofia
    Feb 20, 2015 at 21:31
  • $\begingroup$ In order to understand an answer to your question you must know linear algebra and differential equations. Do you? $\endgroup$
    – Ali Moh
    Feb 20, 2015 at 22:22
  • $\begingroup$ @AliMoh : I have read the article from Wikipedia. I have basic knowledge of LA and DE. Please have a look at the EDIT done and proceed. The complete theory is not required, just the final results would suffice. $\endgroup$
    – Gaurav
    Mar 11, 2015 at 8:42

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