Examples of nonlinear ordinary differential equations with elementary solutions.
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.
A very simple non-linear system to analyze is what I like to call the "Parachute Equation" which is essentially
$$\ddot{x}+k\dot{x}^2-g=0 \tag{1}$$
With initial conditions $x(0)=0$ and $\dot{x}(0)=0$.
where $\displaystyle k=\frac{\pi \rho C_d D^2}{8m}$
such that:
$m$ is the mass of the body and parachute,
$\rho$ is the density of the fluid in which the body moves,
$C_d$ is the drag coefficient for the parachute ($1.5$ for parabolic profile and $0.75$ for flat),
$D$ is the effective diameter of the parachute.
$(1)$ admits the solution:
$$x=\frac{1}{k}\left(\log\left(\frac{e^{2\sqrt{gk}t}+1}{2}\right)-\sqrt{gk}{t}\right)$$
$(1)$ can also be converted into velocity form considering $\dot{x}=v$ into
$$\dot{v}+k{v}^2-g=0 \tag{2}$$
Solution of $(2)$ is
$$ v=\sqrt{\frac{g}{k}}\left(1-\frac{2}{e^{2\sqrt{gk}t}+1}\right) $$
For the first order non linear OE, you certainly are aware of Clairaut's equation. I see you are searching for higher order, so considering the following: $$y^{'''}=(x-1)^2+y^2+y'-2\\\\ y(1)=1,~y'(1)=0,~y''(1)=2$$ We can find a particular solution, of course by using series method and the undetermined coefficient method simultaneously. W e will see that $$y_p(x)=1+(x-1)^2-1/6(x-1)^3+1/12(x-1)^4+...$$