List of nonlinear ordinary differential equations

See also List of nonlinear partial differential equations and List of linear ordinary differential equations.

A–F

Name	Order	Equation	Applications
Abel's differential equation of the first kind	1	[math]\displaystyle{ \frac{dy}{dx} = f_o(x) + f_1(x) y + f_2(x) y^2 + f_3(x) y^3 }[/math]	Mathematics
Abel's differential equation of the second kind	1	[math]\displaystyle{ (g_o(x) + g_1(x) y)\frac{dy}{dx} = f_o(x) + f_1(x) y + f_2(x) y^2 + f_3(x) y^3 }[/math]	Mathematics
Bellman's equation or Emden-Fowler's equation	2	[math]\displaystyle{ \frac{d^2y}{dx^2} = k x^a y^b }[/math]	Mathematics
Bernoulli equation	1	[math]\displaystyle{ \frac{dy}{dx} + P(x) y = Q(x) y^n }[/math]	Mathematics
Besant-Rayleigh-Plesset equation	2	[math]\displaystyle{ R\frac{d^2R}{dt^2} + \frac{3}{2}\left(\frac{dR}{dt}\right)^2 + \frac{4\nu}{R}\frac{dR}{dt} + \frac{2\gamma}{\rho R} + \frac{\Delta P(t)}{\rho} = 0 }[/math]	Fluid dynamics
Blasius equation	3	[math]\displaystyle{ \frac{d^3y}{dx^3} + y \frac{d^2y}{dx^2} =0 }[/math]	Blasius boundary layer
Chandrasekhar equation	2	[math]\displaystyle{ \frac{1}{\xi^2} \frac{d}{d\xi}\left(\xi^2 \frac{d\psi}{d\xi}\right)= e^{-\psi} }[/math]	Astrophysics
Chandrasekhar's white dwarf equation	2	[math]\displaystyle{ \frac{1}{x^2} \frac{d}{dx}\left(x^2 \frac{dy}{dx}\right) + (y^2 - c)^{3/2}=0 }[/math]	Astrophysics
Chrystal's equation	1	[math]\displaystyle{ \left(\frac{dy}{dx}\right)^2 + Ax \frac{dy}{dx} + By + Cx^2 =0 }[/math]	Mathematics
Clairaut's equation	1	[math]\displaystyle{ y= x\frac{dy}{dx} + f\left(\frac{dy}{dx}\right) }[/math]	Mathematics
D'Alembert's equation	1	[math]\displaystyle{ y = x f\left(\frac{dy}{dx}\right) + g\left(\frac{dy}{dx}\right) }[/math]	Mathematics
Darboux equation	1	[math]\displaystyle{ \frac{dy}{dx} = \frac{P(x,y) + y R(x,y)}{Q(x,y) + xR(x,y)} }[/math]	Mathematics
De Boer-Ludford equation	2	[math]\displaystyle{ \frac{d^2y}{dx^2} -xy =2y |y|^\alpha, \ \alpha\gt 0 }[/math]	Plasma physics
Duffing equation	2	[math]\displaystyle{ \frac{d^2x}{dt^2} + \mu \frac{dx}{dt} + \alpha x + \beta x^3 = \gamma \cos \omega t }[/math]	Oscillators
Emden equation	2	[math]\displaystyle{ \frac{1}{x^2} \frac{d}{dx}\left(x^2 \frac{dy}{dx}\right)= f(y) }[/math]	Astrophysics
Euler's differential equation	1	[math]\displaystyle{ \frac{dy}{dx} + \frac{\sqrt{a_0+a_1y +a_2 y^2 + a_3 y^3 + a_4 y^4}}{\sqrt{a_0+a_1x +a_2 x^2 + a_3 x^3 + a_4 x^4}} = 0 }[/math]	Mathematics
Falkner–Skan equation	3	[math]\displaystyle{ \frac{d^3y}{dx^3} + y \frac{d^2y}{dx^2} + \beta \left[1-\left(\frac{dy}{dx}\right)^2\right]=0 }[/math]	Falkner–Skan boundary layer

G–K

Name	Order	Equation	Applications
Ivey's equation	2	[math]\displaystyle{ \frac{d^2y}{dx^2} - \frac{1}{y} \left(\frac{dy}{dx}\right)^2 + \frac{2}{x}\frac{dy}{dx} + ky^2=0 }[/math]
Jacobi's differential equation	1	[math]\displaystyle{ \frac{dy}{dx} = \frac{Axy + By^2 + ax + by + c}{Ax^2 + Bxy +\alpha x +\beta y + \gamma} }[/math]	Mathematics
Kidder equation	2	[math]\displaystyle{ \sqrt{1-\alpha y} \frac{d^2y}{dx^2} + 2x \frac{dy}{dx}=0,\ 0\lt \alpha\lt 1 }[/math]	Flow through porous medium
Krogdahl equation	2	[math]\displaystyle{ \frac{d^2 Q}{d\tau^2} = -Q + \frac{2}{3}\lambda Q^2 - \frac{14}{27} \lambda^2 Q^3 + \mu(1-Q^2)\frac{dQ}{d\tau} + \frac{2}{3}\lambda(1-\lambda Q) \left(\frac{dQ}{d\tau}\right)^2 }[/math]	Stellar pulsation

L–Q

Name	Order	Equation	Applications
Lane–Emden equation	2	[math]\displaystyle{ \frac{1}{\xi^2} \frac{d}{d\xi} \left({\xi^2 \frac{d\theta}{d\xi}}\right) + \theta^n = 0 }[/math]	Astrophysics
Langmuir equation	2	[math]\displaystyle{ 3y\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 + 4y \frac{dy}{dx} + y^2 =1 }[/math]	Environmental Engineering
Langmuir-Blodgett equation	2	[math]\displaystyle{ \sqrt{y}\frac{d^2y}{dx^2}=e^x }[/math]
Langmuir-Boguslavski equation	2	[math]\displaystyle{ \frac{d}{dx}\left(x^n\frac{dy}{dx}\right) = \frac{1}{\sqrt{y}} }[/math]
Liñán's equation	2	[math]\displaystyle{ \frac{d^2y}{d\zeta^2} =(y^2-\zeta^2)e^{-\delta^{1/3}(y+\gamma \zeta)} }[/math]	Combustion
Painlevé I transcendent	2	[math]\displaystyle{ \frac{d^2y}{dt^2} = 6 y^2 + t }[/math]	Mathematics
Painlevé II transcendent	2	[math]\displaystyle{ \frac{d^2y}{dt^2} = 2 y^3 + ty + \alpha }[/math]	Mathematics
Painlevé III transcendent	2	[math]\displaystyle{ ty\frac{d^2y}{dt^2} = t \left(\frac{dy}{dt} \right)^2-y\frac{dy}{dt} + \delta t + \beta y + \alpha y^3 + \gamma ty^4 }[/math]	Mathematics
Painlevé IV transcendent	2	[math]\displaystyle{ y\frac{d^2y}{dt^2}=\tfrac12 \left(\frac{dy}{dt} \right)^2+\beta+2(t^2-\alpha)y^2+4ty^3+\tfrac32y^4 }[/math]	Mathematics
Painlevé V transcendent	2	[math]\displaystyle{ \frac{d^2y}{dt^2}=\left(\frac{1}{2 y }+\frac{1}{ y -1}\right) \left( \frac{dy}{dt} \right)^2 -\frac{1}{t} \frac{dy}{dt}+\frac{( y -1)^2}{t^2}\left(\alpha y +\frac{\beta}{ y }\right) +\gamma\frac{ y }{t}+\delta\frac{ y ( y +1)}{ y -1} }[/math]	Mathematics
Painlevé VI transcendent	2	[math]\displaystyle{ \frac{d^2y}{dt^2}=\frac{1}{2}\left(\frac{1}{y}+\frac{1}{y-1}+\frac{1}{y-t}\right)\left( \frac{dy}{dt} \right)^2-\left(\frac{1}{t}+\frac{1}{t-1}+\frac{1}{y-t}\right)\frac{dy}{dt} +\frac{y(y-1)(y-t)}{t^2(t-1)^2} \left(\alpha+\beta\frac{t}{y^2}+\gamma\frac{t-1}{(y-1)^2}+\delta\frac{t(t-1)}{(y-t)^2}\right) }[/math]	Mathematics
Poisson-Boltzmann equation	2	[math]\displaystyle{ \frac{d^2y}{dx^2} + \frac{\alpha}{x} \frac{dy}{dx} = e^y }[/math]	Statistical Physics

R–Z

Name	Order	Equation	Applications
Rayleigh equation	2	[math]\displaystyle{ \frac{d^2y}{dx^2} + k \frac{dy}{dx} + m \left(\frac{dy}{dx}\right)^3 + n^2 y =0 }[/math]	Hydrodynamic stability
Riccati equation	1	[math]\displaystyle{ \frac{dy}{dx} + Q(x) y + R(x) y^2 = P(x) }[/math]	Mathematics
Stuart–Landau equation	1	[math]\displaystyle{ \frac{dA}{dt} = \gamma A - \alpha A |A|^2 }[/math]	Hydrodynamic stability
Thomas–Fermi equation	2	[math]\displaystyle{ \frac{d^2y}{dx^2} = \frac{1}{\sqrt x}y^{3/2} }[/math]	Quantum mechanics
Van der Pol equation	2	[math]\displaystyle{ {d^2x \over dt^2}-\mu(1-x^2){dx \over dt}+x= 0 }[/math]	Oscillators