TITLE: Examples of nonlinear ordinary differential equations with elementary solutions.
QUESTION [15 upvotes]: I am looking for nice examples of nonlinear ordinary differential equations that have simple solutions in terms of elementary functions. (But are not trivial to find, like, for example, with separation of variables).
A perfect example of what I am looking for is the Lane-Emden equation of index 5:
$$y''+\frac{2}{x}y'+y^5=0,\qquad y(0)=1,y'(0)=0$$
which admits the solution
$$y(x)=\frac{1}{\sqrt{1+x^2/3}}$$
Another very good example is
$$4y^2y'''-18yy'y''+15y'^3=0$$
from which we find
$$y(x)=\frac{1}{(ax^2+bx+c)^2}$$
for some constants $a,b,c$.
Do you have more examples like these? The higher the order, the better. (But I consider, for example, $H(x,y'',y''')=0$ to be of order 1). And it is a plus if it comes from a physics problem.

REPLY [2 votes]: Just for fun I came up with a DEQ that has an unspecified order.
$$
\frac{\mathrm{d}^n y}{\mathrm{d} x^n} = \frac{1}{x^n + \mathrm{e}^y}
$$
It's solution is
$$
y = \ln \left(\left[\frac{(-1)^{n-1}}{n!}-1\right]x^n\right).
$$
Here's another one:
$$
\frac{\mathrm{d}^n y}{\mathrm{d} x^n} = y+e^{nx}
$$
It's solution for n>1 is
$$
y=\frac{e^{nx}}{n^n -1}.
$$
If you eliminate the requirement $y(0)=1$, the generalized Lane-Emden equation
$$
y''+\frac{2}{x}y'+y^N=0
$$
has the solution
$$
y=\left(\frac{xi}{\sqrt{\beta^2+\beta}}\right)^\beta
$$
where
$$
\beta=\frac{2}{1-N}.
$$
This solution fails at $N=1,3$.
Last but not least, the one-dimensional fluid drag equation $F=-\alpha v^2$ corresponds to the DEQ
$$
m\ddot{y}=-\alpha \dot{y}^2
$$
where $m$ is mass in kilograms and $\alpha$ is a unitless drag coefficient which is a function of shape, density and viscosity. It also has the conditions that $y(t=0)=0$ and $\dot{y}(t=0)=v_o$ where $v_o$ is initial velocity. Its solution is
$$
y=\frac{m}{\alpha} \ln\bigg(1+\frac{\alpha}{m}v_ot\bigg).
$$