Examples of nonlinear ordinary differential equations with elementary solutions.

Question

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.

Answer 1

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

$$ v=\sqrt{\frac{g}{k}}\left(1-\frac{2}{e^{2\sqrt{gk}t}+1}\right) $$

Answer 2

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 $$y_p(x)=1+(x-1)^2-1/6(x-1)^3+1/12(x-1)^4+...$$

Answer 3

I would certainly like to mention six Painlevé equations. These are the only nonlinear 2nd order ODEs of the form $w''=F(t,w,w')$ whose solutions cannot have movable critical points. Painlevé equations have a lot of applications in various areas of mathematics, including integrable models, random matrices, algebraic and differential geometry and combinatorics.

It is known (proven rigorously) that the general solutions of Painlevé equations, in a sense, cannot be expressed in terms of classical functions. However, many particular solutions are known for special values of parameters and integration constants. You can check, for instance, NIST Library for explicit examples.

Such solutions include classical special function solutions (for example, for Painlevé II they are expressed in terms of Airy functions, and for Painlevé VI in terms of Gauss hypergeometric functions). Also, e.g. Painlevé VI has a family of elliptic solutions and a lot of algebraic solutions with polynomial degree going up to $72$ and genus up to $7$.

Concerning physics applications, one could look at the long list on p.19 here to get an idea.

Finally, to give an illustration: Painlevé VI equation \begin{align} \nonumber\frac{d^2w}{dt^2}=\;\frac{1}{2}\left(\frac{1}{w}+\frac{1}{w-1} +\frac{1}{w-t}\right)\left(\frac{dw}{dt}\right)^2 - \left(\frac{1}{t}+\frac{1}{t-1}+\frac{1}{w-t}\right)\frac{dw}{dt}\,+\\ \nonumber +\, \frac{w(w-1)(w-t)}{2t^2(t-1)^2}\left((\theta_{\infty}-1)^2-\frac{\theta_x^2 t}{w^2}+ \frac{\theta_y^2(t-1)}{(w-1)^2}+\frac{(1-\theta_z^2)t(t-1)}{(w-t)^2}\right) \end{align} with parameters $(\theta_x,\theta_y,\theta_z,\theta_{\infty})=\left(\frac25,\frac15,\frac13,\frac23\right)$ has a $5$-branch genus $0$ algebraic solution given by parametric equations \begin{align}&w=\frac{2(s^2+s+7)(5s-2)}{s(s+5)(4s^2-5s+10)}, \\ & t=\frac{27(5s-2)^2}{(s+5)(4s^2-5s+10)^2}. \end{align}