# I Second order DE with Sine function

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1. Apr 7, 2017

### Figaro

I have this second order differential equation but I'm stumped as to how to solve this since the zeroth order term has a Sine function in it and the variable is embedded.

$\ddot y(t) + 3H (1+Q) \dot y(t) -m^2 f \sin(\frac{y(t)}{f}) = 0$

$H~$, $~Q~$, $~m~$, and $~f~$ are just constants.

I even tried to use DSolve in mathematica but there is an error. How do I solve this? Can anyone guide me with this problem?

Last edited: Apr 7, 2017
2. Apr 7, 2017

### BvU

 and there's always this

3. Apr 8, 2017

### Figaro

Are you pertaining to the small angle approximation? I know that is a possibility but the problem is that $\theta = \frac{y[t]}{f} \approx 1$.

4. Apr 8, 2017

### BvU

Small angle is the first approach. The article in the link discusses the full equation in section 3.

5. Apr 8, 2017

### aheight

If you're interested in non-linear equations, I recommend this book which includes a step-by step solution of the non-linear pendulum:

http://store.doverpublications.com/0486609715.html

6. Apr 9, 2017

### Figaro

I still can't find the solution to my given equation, can you kindly give me a hint/note onto where I should look?

7. Apr 9, 2017

### aheight

I was referring to the equation $y''=k\sin(y)$. That's not yours but if you first study how this one is solved exactly in terms of elliptic functions, then perhaps you can adapt the method to yours. It's been a while that I've studied it (using the book I quoted) and I no longer have the book and don't recall exactly how it's done.