List of nonlinear ordinary differential equations

See also List of nonlinear partial differential equations.

A–F[]

Name	Order	Equation	Applications
Abel's differential equation of the first kind	1	${\displaystyle {\frac {dy}{dx}}=f_{o}(x)+f_{1}(x)y+f_{2}(x)y^{2}+f_{3}(x)y^{3}}$	Mathematics
Abel's differential equation of the second kind	1	${\displaystyle (g_{o}(x)+g_{1}(x)y){\frac {dy}{dx}}=f_{o}(x)+f_{1}(x)y+f_{2}(x)y^{2}+f_{3}(x)y^{3}}$	Mathematics
or	2	${\displaystyle {\frac {d^{2}y}{dx^{2}}}=kx^{a}y^{b}}$	Mathematics
Bernoulli equation	1	${\displaystyle {\frac {dy}{dx}}+P(x)y=Q(x)y^{n}}$	Mathematics
Besant-Rayleigh-Plesset equation	2	${\displaystyle R{\frac {d^{2}R}{dt^{2}}}+{\frac {3}{2}}\left({\frac {dR}{dt}}\right)^{2}+{\frac {4\nu }{R}}{\frac {dR}{dt}}+{\frac {2\gamma }{\rho R}}+{\frac {\Delta P(t)}{\rho }}=0}$	Fluid dynamics
Blasius equation	3	${\displaystyle {\frac {d^{3}y}{dx^{3}}}+y{\frac {d^{2}y}{dx^{2}}}=0}$	Blasius boundary layer
Chandrasekhar equation	2	${\displaystyle {\frac {1}{\xi ^{2}}}{\frac {d}{d\xi }}\left(\xi ^{2}{\frac {d\psi }{d\xi }}\right)=e^{-\psi }}$	Astrophysics
Chandrasekhar's white dwarf equation	2	${\displaystyle {\frac {1}{x^{2}}}{\frac {d}{dx}}\left(x^{2}{\frac {dy}{dx}}\right)+(y^{2}-c)^{3/2}=0}$	Astrophysics
Chrystal's equation	1	${\displaystyle \left({\frac {dy}{dx}}\right)^{2}+Ax{\frac {dy}{dx}}+By+Cx^{2}=0}$	Mathematics
Clairaut's equation	1	${\displaystyle y=x{\frac {dy}{dx}}+f\left({\frac {dy}{dx}}\right)}$	Mathematics
D'Alembert's equation	1	${\displaystyle y=xf\left({\frac {dy}{dx}}\right)+g\left({\frac {dy}{dx}}\right)}$	Mathematics
	1	${\displaystyle {\frac {dy}{dx}}={\frac {P(x,y)+yR(x,y)}{Q(x,y)+xR(x,y)}}}$	Mathematics
	2	${\displaystyle {\frac {d^{2}y}{dx^{2}}}-xy=2y|y|^{\alpha },\ \alpha >0}$	Plasma physics
Duffing equation	2	${\displaystyle {\frac {d^{2}x}{dt^{2}}}+\mu {\frac {dx}{dt}}+\alpha x+\beta x^{3}=\gamma \cos \omega t}$	Oscillators
Emden equation	2	${\displaystyle {\frac {1}{x^{2}}}{\frac {d}{dx}}\left(x^{2}{\frac {dy}{dx}}\right)=f(y)}$	Astrophysics
Euler's differential equation	1	${\displaystyle {\frac {dy}{dx}}+{\frac {\sqrt {a_{0}+a_{1}y+a_{2}y^{2}+a_{3}y^{3}+a_{4}y^{4}}}{\sqrt {a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}}}}=0}$	Mathematics
Falkner–Skan equation	3	${\displaystyle {\frac {d^{3}y}{dx^{3}}}+y{\frac {d^{2}y}{dx^{2}}}+\beta \left[1-\left({\frac {dy}{dx}}\right)^{2}\right]=0}$	Falkner–Skan boundary layer

G–K[]

Name	Order	Equation	Applications
	2	${\displaystyle {\frac {d^{2}y}{dx^{2}}}-{\frac {1}{y}}\left({\frac {dy}{dx}}\right)^{2}+{\frac {2}{x}}{\frac {dy}{dx}}+ky^{2}=0}$
	1	${\displaystyle {\frac {dy}{dx}}={\frac {Axy+By^{2}+ax+by+c}{Ax^{2}+Bxy+\alpha x+\beta y+\gamma }}}$	Mathematics
	2	${\displaystyle {\sqrt {1-\alpha y}}{\frac {d^{2}y}{dx^{2}}}+2x{\frac {dy}{dx}}=0,\ 0<\alpha <1}$	Flow through porous medium
	2	${\displaystyle {\frac {d^{2}Q}{d\tau ^{2}}}=-Q+{\frac {2}{3}}\lambda Q^{2}-{\frac {14}{27}}\lambda ^{2}Q^{3}+\mu (1-Q^{2}){\frac {dQ}{d\tau }}+{\frac {2}{3}}\lambda (1-\lambda Q)\left({\frac {dQ}{d\tau }}\right)^{2}}$	Stellar pulsation

L–Q[]

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

R–Z[]

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

References[]