List_of_nonlinear_ordinary_differential_equations

List of nonlinear ordinary differential equations

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A–F

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

G–K

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

L–Q

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

R–Z

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

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This article uses material from the Wikipedia article List_of_nonlinear_ordinary_differential_equations, and is written by contributors. Text is available under a CC BY-SA 4.0 International License; additional terms may apply. Images, videos and audio are available under their respective licenses.

[1] [1]
Davis, Harold Thayer. Introduction to nonlinear differential and integral equations. Courier Corporation, 1962.

[1]

List_of_nonlinear_ordinary_differential_equations

A–F

G–K

L–Q

R–Z

References

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