Is it possible to solve this 2nd order non linear ODE analytically? I am a high school student working on an exploration assessment. My aim is to find a mathematical model for the angular displacement of a damped simple pendulum over time. I derived this ODE using Euler-Lagrange equation with a term for dissipative energy Euler-Lagrange Eq. Clearly, this is a 2nd order non-linear ODE and I was just wondering whether an analytical solution is possible.
$$\frac{d^2\theta}{dt^2}+\frac cm\frac{d\theta}{dt}+\frac gl\sin\theta=0$$
where c, m, g and l are constants, and theta is angular displacement.
Thanks in advance.
 A: Unfortunately, this system does not have an analytic solution. For the case where there is no damping force, you have the system
$$\frac{d^2 \theta}{dt^2}+\frac{g}{l}\sin{\theta} = 0 $$
Even this, however, unfortunately does not have an analytic solution. However, you can see over here for a solution in terms of elliptic integrals. Elliptic integrals are a class special functions that are defined by an integral (special functions are basically functions that cannot be written in terms of elementary functions like sin, cos, exponentials, polynomials, etc. or combinations of those). If you see in the paper I linked, it will have them defined. As you can see in the paper, it is possible to write $\theta(t)$ in terms of these special integrals and so you can maybe continue with your assignment by doing some numerical simulations with this using stuff like python (the numpy library seems to have some special functions capabilities as seen here. 
Of course for the more general with the damping forces, you can use general numerical methods like Euler's method (see here for more info). Applications like Mathematica or Matlab even can do all the numerical solving also if you have access to that. 
That being said, if you are OK with making the assumption that the pendulum doesn't move too far from equilibrium, we can use a small angles approximation to say that $\sin{\theta} \approx \theta$ for small $\theta$, then your equation does have an analytic solution. Using this, your equation is now
$$\frac{d^2 \theta}{dt^2}+\frac{c}{m}\frac{d\theta}{dt}+\frac{g}{l}\theta = 0$$
This is a second order linear differential equation that you can solve using a characteristic equation. The solution to get by doing this, however, will only apply for when the oscillations are small. 
