Solving Nonlinear ODEs with Trigonometric Functions in Physics

[Mindscrape]

I have a couple ODEs that I need to solve. I was probably just going to put them into mathematica, but I like finding the analytical way also. The first one is

$$\frac{d}{dx}\left( \frac{(y + \lambda)y'}{\sqrt{1+y'^2}} \right) = \sqrt{1+y'^2}$$

Lambda is a constant and y' is dy/dx. I suppose that after all the quotients and products are evaluated it could be separated, but that is a lot of work if there is a nice trick to employ.

The other one, the one I am actually curious about since it is nonlinear, is actually a pair of ODEs

$$ma^2(sin^2\theta \ddot{\phi}+ 2\phi sin\theta cos\theta) = 0$$
and
$$ma^2 \ddot{\theta} = -mga sin\theta + 2a^2 sin\theta cos\theta \dot{\phi}^2$$

In this problem, m, g, and a are constants. I need to solve the coupled equations for both phi and theta. I was thinking that maybe I could convert the equations into first order ODEs, and then solve the system of equations, but I'm not sure how to deal with the nonlinearity of the $$\dot{\phi}^2$$ and trig functions.

These came from assigned physics HW problems (Legrangians/Hamiltonians), so I assume they can be solved without mathematica.

*Nevermind about the first one, I solved it with separation and integration tables. The solution was $$y = c cosh((x-b)/c) - \lambda$$ in case anyone guessed.

 

[quasar987]

I had to do a similar problem earlier this semester and it was implicit in the question that we could do a small angle approximation.

Also, you might want to check out arildno's complicated answer (post #7) about how we can drop the squares angular velocity also in the small angle aprox:

https://www.physicsforums.com/showthread.php?t=152090