Weird ODE containing cos.

1. Reid

36
1. The problem statement, all variables and given/known data

I can not determine the solution to the diff. eq.

2. Relevant equations

$$\ddot{x}+k \cos{x}=0$$.

The constant k is postive.

3. The attempt at a solution

I tried solving it with the methods I know and it all ended up in a big mess. I am just not used to the second term, \cos. Doe's anyone know what to do?

2. Office_Shredder

4,487
Staff Emeritus
It's GOT to be something of the form

$$x=arccos(y)$$ for some y a function of t, based on an argument of 'this is really fricking hard otherwise'. Try the substitution and see what you get

3. HallsofIvy

41,265
Staff Emeritus
Since the independent variable, which I will call "t", does not appear explicitely in the equation, this is a candidate for "quadrature". Let $y= dx/dt$. Then $d^2x/dt^2= (dy/dx)(dx/dt)$ by the chain rule. But since $y= dx/dt$, that is $d^2x/dt^2= y dy/dx$ and the equation converts to the first order equation, for y as a function of x, $y dy/dx= -cos(x)$ which is $y dy= -cos(x)dx$. Integrating, $(1/2)y^2= sin(x)+ C$. Now we have $y^2= -2 sin(x)+ 2C$ or $y= dx/dt= \sqrt{2C- 2sin(x)}$ so $dx= \sqrt{2C- 2sin(x)}dx$.

I am not sure that is going to be easy to integrate, but that gives the solution.