Solution of the differential equation of a pendulum with a block (air resistance) The differential equation for a pendulum with air resistance is
$$
\ddot{y} + (b/m) \dot{y}^2+\frac{D}{m}y = 0
$$
What is a solution of the differential equation? I had problems to solve it.
 A: There are (as far as I know) no closed form solutions, and certainly none tht are simple, to this differential equation. As @G.Smith points out, there is no reason to assume every differential equation has a closed form solution, and certainly not a non-linear equation like this one.
You may obtain a solution using something like WolframAlpha, or integrate it numerically.
As @bolbteppa mentioned in the comment above, there is a way to get $y'(t)$ as a function of $y(t)$. If this is what you're interested in, you can define $v = y'$ and rewrite your equation as $$v \frac{\text{d}v}{\text{d}y} + \frac{b}{m}v^2 + \frac{D}{m}y =0 \quad \implies \quad \frac{1}{2}\frac{\text{d}}{\text{d}y} (v^2) + \frac{b}{m} v^2 = - \frac{D}{m} y,$$ which is a linear non-homogeneous first order differential equation in $v^2(y)$, and as such can be solved very easily using standard methods. This gives you $v$ as a function of $y$, but sadly can't be easily integrated to give either in terms of $t$!
A: Tho solve the quadratically damped harmonic oscillator, you need two equations:
$$ \ddot y + \frac b m \dot y^2 + \frac D m = 0\ \ \ \ \dot y < 0 $$
$$ \ddot y - \frac b m \dot y^2 + \frac D m = 0\ \ \ \ \dot y > 0 $$
so that the damping force is always opposing the velocity.
A standard brute force method is to use a power series:
$$ y(t) = \sum_{n=0}^{\infty}a_nt^n $$
$$ \dot y(t) = \sum_{n=1}^{\infty}na_nt^{n-1} =  \sum_{n=0}^{\infty}(n+1)a_{n+1}t^n$$
$$ \ddot y(t) = \sum_{n=2}^{\infty}n(n-1)a_nt^{n-2} =  \sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2}t^n$$
So you can try that.
A clever or novel method is worthy of publication, e.g.:

