List of nonlinear ordinary differential equations

Differential equations are prominent in many scientific areas. Nonlinear ones are of particular interest for their commonality in describing real-world systems and how much more difficult they are to solve compared to linear differential equations. This list presents nonlinear ordinary differential equations that have been named, sorted by area of interest.

Mathematics

Name	Order	Equation	Application	Reference
Abel's differential equation of the first kind	1	${\frac {dy}{dx}}=f_{o}(x)+f_{1}(x)y+f_{2}(x)y^{2}+f_{3}(x)y^{3}$	Class of differential equation which may be solved implicitly
Abel's differential equation of the second kind	1	$(g_{o}(x)+g_{1}(x)y){\frac {dy}{dx}}=f_{o}(x)+f_{1}(x)y+f_{2}(x)y^{2}+f_{3}(x)y^{3}$	Class of differential equation which may be solved implicitly
Bernoulli equation	1	${\frac {dy}{dx}}+P(x)y=Q(x)y^{n}$	Class of differential equation which may be solved exactly
Binomial differential equation	$m$	$\left(y'\right)^{m}=f(x,y)$	Class of differential equation which may sometimes be solved exactly
Briot-Bouquet Equation	1	$x^{m}y'=f(x,y)$	Class of differential equation which may sometimes be solved exactly
Cherwell-Wright differential equation	1	${\frac {dx}{dt}}=(a-x(t-1))x(t)$ or the related form $f'(x)={\frac {-f(x)f({\sqrt {x}})}{2x}}$	An example of a nonlinear delay differential equation; applications in number theory, distribution of primes, and control theory
Chrystal's equation	1	$\left({\frac {dy}{dx}}\right)^{2}+Ax{\frac {dy}{dx}}+By+Cx^{2}=0$	Generalization of Clairaut's equation with a singular solution
Clairaut's equation	1	$y=x{\frac {dy}{dx}}+f\left({\frac {dy}{dx}}\right)$	Particular case of d'Alembert's equation which may be solved exactly
d'Alembert's equation or Lagrange's equation	1	$y=xf\left({\frac {dy}{dx}}\right)+g\left({\frac {dy}{dx}}\right)$	May be solved exactly
Darboux equation	1	${\frac {dy}{dx}}={\frac {P(x,y)+yR(x,y)}{Q(x,y)+xR(x,y)}}$	Can be reduced to a Bernoulli differential equation; a general case of the Jacobi equation
Elliptic function	1	$y'={\sqrt {\left(1-y^{2}\right)\left(1-k^{2}y^{2}\right)}}$	Equation for which the elliptic functions are solutions
Euler's differential equation	1	${\frac {dy}{dx}}+{\frac {\sqrt {a_{0}+a_{1}y+a_{2}y^{2}+a_{3}y^{3}+a_{4}y^{4}}}{\sqrt {a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}}}}=0$	A separable differential equation
Euler's differential equation	1	$y'+y'^{2}=\alpha x^{m}$	A differential equation which may be solved with Bessel functions
Jacobi equation	1	${\frac {dy}{dx}}={\frac {Axy+By^{2}+ax+by+c}{Ax^{2}+Bxy+\alpha x+\beta y+\gamma }}$	Special case of the Darboux equation, integrable in closed form
Loewner differential equation	1	${\frac {dw}{dt}}=-wp_{t}(w)$	Important in complex analysis and geometric function theory
Logistic differential equation (sometimes known as the Verhulst model)	2	${\frac {d}{dx}}f(x)=f(x){\big (}1-f(x){\big )}$	Special case of the Bernoulli differential equation; many applications including in population dynamics
Lorenz attractor	1	${\begin{aligned}{\frac {\mathrm {d} x}{\mathrm {d} t}}&=\sigma (y-x)\\{\frac {\mathrm {d} y}{\mathrm {d} t}}&=x(\rho -z)-y\\{\frac {\mathrm {d} z}{\mathrm {d} t}}&=xy-\beta z\end{aligned}}$	Chaos theory, dynamical systems, meteorology
Nahm equations	1	${\begin{aligned}{\frac {dT_{1}}{dz}}&=[T_{2},T_{3}]\\{\frac {dT_{2}}{dz}}&=[T_{3},T_{1}]\\{\frac {dT_{3}}{dz}}&=[T_{1},T_{2}]\end{aligned}}$	Differential geometry, gauge theory, mathematical physics, magnetic monopoles
Painlevé I transcendent	2	${\frac {d^{2}y}{dt^{2}}}=6y^{2}+t$	One of fifty classes of differential equation of the form $y''=R(y',y,t)$; the six Painlevé transcendents required new special functions to solve
Painlevé II transcendent	2	${\frac {d^{2}y}{dt^{2}}}=2y^{3}+ty+\alpha$	One of fifty classes of differential equation of the form $y''=R(y',y,t)$; the six Painlevé transcendents required new special functions to solve
Painlevé III transcendent	2	$ty{\frac {d^{2}y}{dt^{2}}}=t\left({\frac {dy}{dt}}\right)^{2}-y{\frac {dy}{dt}}+\delta t+\beta y+\alpha y^{3}+\gamma ty^{4}$	One of fifty classes of differential equation of the form $y''=R(y',y,t)$; the six Painlevé transcendents required new special functions to solve
Painlevé IV transcendent	2	$y{\frac {d^{2}y}{dt^{2}}}={\tfrac {1}{2}}\left({\frac {dy}{dt}}\right)^{2}+\beta +2(t^{2}-\alpha )y^{2}+4ty^{3}+{\tfrac {3}{2}}y^{4}$	One of fifty classes of differential equation of the form $y''=R(y',y,t)$; the six Painlevé transcendents required new special functions to solve
Painlevé V transcendent	2	${\frac {d^{2}y}{dt^{2}}}=\left({\frac {1}{2y}}+{\frac {1}{y-1}}\right)\left({\frac {dy}{dt}}\right)^{2}-{\frac {1}{t}}{\frac {dy}{dt}}+{\frac {(y-1)^{2}}{t^{2}}}\left(\alpha y+{\frac {\beta }{y}}\right)+\gamma {\frac {y}{t}}+\delta {\frac {y(y+1)}{y-1}}$	One of fifty classes of differential equation of the form $y''=R(y',y,t)$; the six Painlevé transcendents required new special functions to solve
Painlevé VI transcendent	2	${\begin{aligned}{\frac {d^{2}y}{dt^{2}}}&={\frac {1}{2}}\left({\frac {1}{y}}+{\frac {1}{y-1}}+{\frac {1}{y-t}}\right)\left({\frac {dy}{dt}}\right)^{2}-\left({\frac {1}{t}}+{\frac {1}{t-1}}+{\frac {1}{y-t}}\right){\frac {dy}{dt}}\\&+{\frac {y(y-1)(y-t)}{t^{2}(t-1)^{2}}}\left(\alpha +\beta {\frac {t}{y^{2}}}+\gamma {\frac {t-1}{(y-1)^{2}}}+\delta {\frac {t(t-1)}{(y-t)^{2}}}\right)\\\end{aligned}}$	All of the other Painlevé transcendents are degenerations of the sixth
Rabinovich–Fabrikant equations	1	${\begin{aligned}{\dot {x}}&=y(z-1+x^{2})+\gamma x\\{\dot {y}}&=x(3z+1-x^{2})+\gamma y\\{\dot {z}}&=-2z(\alpha +xy)\end{aligned}}$	Chaos theory, dynamical systems
Riccati equation	1	${\frac {dy}{dx}}+Q(x)y+R(x)y^{2}=P(x)$	Class of first order differential equations that is quadratic in the unknown. Can reduce to Bernoulli differential equation or linear differential equation in certain cases
Rössler attractor	1	${\begin{aligned}{\frac {dx}{dt}}&=-y-z\\{\frac {dy}{dt}}&=x+ay\\{\frac {dz}{dt}}&=b+z(x-c)\end{aligned}}$	Chaos theory, dynamical systems

Physics

Name	Order	Equation	Applications	Reference
Bellman's equation or Emden-Fowler's equation	2	${\frac {d}{dt}}\left(t^{\rho }{\frac {du}{dt}}\right)=t^{\sigma }u^{\rho }$ (Emden-Fowler) which reduces to ${\frac {d^{2}u}{dx^{2}}}=\phi ^{2}u^{p}$ if $\sigma +\rho =0$ (Bellman)	Diffusion in a slab
Besant-Rayleigh-Plesset equation	2	$R{\frac {d^{2}R}{dt^{2}}}+{\frac {3}{2}}\left({\frac {dR}{dt}}\right)^{2}+{\frac {4\nu _{L}}{R}}{\frac {dR}{dt}}+{\frac {2\sigma }{\rho _{L}R}}+{\frac {\Delta P(t)}{\rho }}=0$	Spherical bubble in 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	${\frac {1}{x^{2}}}{\frac {d}{dx}}\left(x^{2}{\frac {dy}{dx}}\right)+(y^{2}-c)^{3/2}=0$	Gravitational potential of white dwarf in astrophysics
De Boer-Ludford equation	2	${\frac {d^{2}y}{dx^{2}}}-xy=y|y|^{\alpha },\ \alpha >0$	Plasma physics
Emden–Chandrasekhar equation	2	${\frac {1}{\xi ^{2}}}{\frac {d}{d\xi }}\left(\xi ^{2}{\frac {d\psi }{d\xi }}\right)=e^{-\psi }$	Astrophysics
Ermakov-Pinney equation	2	${\ddot {x}}+\omega ^{2}x={\frac {h^{2}}{x^{3}}}$	Electromagnetism, oscillation, scalar field cosmologies
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
Friedmann equations	2	${\frac {\ddot {a}}{a}}=-{\frac {4\pi G}{3}}\left(\rho +{\frac {3p}{c^{2}}}\right)+{\frac {\Lambda c^{2}}{3}}$ and ${\frac {{\dot {a}}^{2}+kc^{2}}{a^{2}}}={\frac {8\pi G\rho +\Lambda c^{2}}{3}}$	Physical cosmology
Heisenberg equation of motion	1	${\frac {d{\hat {A(t)}}}{dt}}={\frac {i}{\hbar }}[{\hat {\mathcal {H}}},{\hat {A}}]$	Quantum mechanics
Ivey's equation	2	${\frac {d^{2}y}{dx^{2}}}-{\frac {1}{y}}\left({\frac {dy}{dx}}\right)^{2}+{\frac {2}{x}}{\frac {dy}{dx}}+ky^{2}=0$	Space charge theory
Kidder equation	2	${\sqrt {1-\alpha y}}{\frac {d^{2}y}{dx^{2}}}+2x{\frac {dy}{dx}}=0,\ 0\leq \alpha \leq 1$	Flow through porous medium
Krogdahl equation	2	${\frac {d^{2}Q}{d\tau ^{2}}}=-Q+{\frac {2}{3}}\lambda Q^{2}+{\frac {14}{27}}\lambda ^{2}Q^{3}+\mu (1-Q^{2}){\frac {dQ}{d\tau }}+{\frac {2}{3}}\lambda (1-\lambda Q)\left({\frac {dQ}{d\tau }}\right)^{2}+\cdots$	Stellar pulsation in astrophysics
Lagerstrom equation	2	$y''+{\frac {k}{r}}y'+\epsilon y'y=0$	One dimensional viscous flow at low Reynolds numbers
Lane–Emden equation or polytropic differential equation	2	${\frac {1}{\xi ^{2}}}{\frac {d}{d\xi }}\left({\xi ^{2}{\frac {d\theta }{d\xi }}}\right)+\theta ^{n}=0$	Astrophysics
Liñán's equation	2	${\frac {d^{2}y}{d\zeta ^{2}}}=(y^{2}-\zeta ^{2})e^{-\delta ^{1/3}(y+\gamma \zeta )}$	Combustion
Pendulum equation	2	${\frac {d^{2}\theta }{dt^{2}}}+{\frac {g}{\ell }}\sin \theta =0$	Mechanics
Poisson–Boltzmann equation (1d case)	2	${\frac {d^{2}\theta }{dz^{2}}}+{\frac {k}{z}}{\frac {d\theta }{dz}}=-\delta e^{\theta }$	Inflammability and the theory of thermal explosions
Stuart–Landau equation	1	${\frac {dA}{dt}}=\sigma A-{\frac {l}{2}}A|A|^{2}$	Hydrodynamic stability
Taylor–Maccoll equation	2	$(c^{2}-f'^{2})f''+c^{2}\cot \theta f'+(2c^{2}-f'^{2})f=0,\quad c=c(f^{2}+f'^{2})$ where $f'={\frac {df}{d\theta }}$	Flow behind a conical shock wave
Thomas–Fermi equation	2	${\frac {d^{2}y}{dx^{2}}}={\frac {1}{\sqrt {x}}}y^{3/2}$	Quantum mechanics
Toda lattice	1	${\begin{aligned}{\dot {a}}(n,t)&=a(n,t){\Big (}b(n+1,t)-b(n,t){\Big )}\\{\dot {b}}(n,t)&=2{\Big (}a(n,t)^{2}-a(n-1,t)^{2}{\Big )}\end{aligned}}$where ${\begin{aligned}a(n,t)&={\frac {1}{2}}{\rm {e}}^{-(q(n+1,t)-q(n,t))/2}\\b(n,t)&=-{\frac {1}{2}}p(n,t)\end{aligned}}$	Model of one-dimensional crystal in solid-state physics, Langmuir oscillations in plasma, quantum cohomology; notable for being a completely integrable system

Engineering

Name	Order	Equation	Applications	Reference
Duffing equation	2	${\frac {d^{2}x}{dt^{2}}}+\mu {\frac {dx}{dt}}+\alpha x+\beta x^{3}=\gamma \cos \omega t$	Oscillators, hysteresis, chaotic dynamical systems
Lewis regulator	2	$y''+(1-|y|)y'+y=0$	Oscillators
Liénard equation	2	${d^{2}x \over dt^{2}}+f(x){dx \over dt}+g(x)=0$ with $f$ odd and $g$ even	Oscillators, electrical engineering, dynamical systems
Rayleigh equation	2	$y''+F(y')+y=0$	Oscillators (especially auto-oscillation), acoustics; the Van der Pol equation is a Rayleigh equation
Van der Pol equation	2	${d^{2}x \over dt^{2}}-\mu (1-x^{2}){dx \over dt}+x=0$	Oscillators, electrical engineering, chaotic dynamical systems

Chemistry

Name	Order	Equation	Applications	Reference
Brusselator	1	${\begin{aligned}{d \over dt}\left\{X\right\}&=\left\{A\right\}+\left\{X\right\}^{2}\left\{Y\right\}-\left\{B\right\}\left\{X\right\}-\left\{X\right\}\\{d \over dt}\left\{Y\right\}&=\left\{B\right\}\left\{X\right\}-\left\{X\right\}^{2}\left\{Y\right\}\end{aligned}}$	A type of autocatalytic reaction modelled at constant concentration
Oregonator	1	${\begin{aligned}{\frac {d[X]}{dt}}&=k_{I}[A][Y]-k_{II}[X][Y]+k_{III}[A][X]-2k_{IV}[X]^{2}\\{\frac {d[Y]}{dt}}&=-k_{I}[A][Y]-k_{II}[X][Y]+{\frac {1}{2}}fk_{V}[B][Z]\\{\frac {d[Z]}{dt}}&=2k_{III}[A][X]-k_{V}[B][Z]\end{aligned}}$	A type of autocatalytic reaction modelled at constant concentration

Biology and medicine

Name	Order	Equation	Applications	Reference
Allee effect	1	${\frac {dN}{dt}}=-rN\left(1-{\frac {N}{A}}\right)\left(1-{\frac {N}{K}}\right)$	Population biology
Arditi–Ginzburg equations	1	${\begin{aligned}{\frac {dN}{dt}}&=f(N)\,N-g{\left(\!{\tfrac {N}{P}}\!\right)}P\\{\frac {dP}{dt}}&=e\,g{\left(\!{\tfrac {N}{P}}\!\right)}P-uP\end{aligned}}$	Population dynamics
FitzHugh–Nagumo model or Bonhoeffer-van der Pol model	1	${\begin{aligned}{\dot {v}}&=v-{\frac {v^{3}}{3}}-w+RI_{\rm {ext}}\\\tau {\dot {w}}&=v+a-bw\end{aligned}}$	Action potentials in neurons, oscillators
Hodgkin-Huxley equations	1	${\begin{aligned}I&=C_{m}{\frac {{\mathrm {d} }V_{m}}{{\mathrm {d} }t}}+{\bar {g}}_{\text{K}}n^{4}(V_{m}-V_{K})+{\bar {g}}_{\text{Na}}m^{3}h(V_{m}-V_{Na})+{\bar {g}}_{l}(V_{m}-V_{l})\\{\frac {dn}{dt}}&=\alpha _{n}(V_{m})(1-n)-\beta _{n}(V_{m})n\\{\frac {dm}{dt}}&=\alpha _{m}(V_{m})(1-m)-\beta _{m}(V_{m})m\\{\frac {dh}{dt}}&=\alpha _{h}(V_{m})(1-h)-\beta _{h}(V_{m})h\end{aligned}}$	Action potentials in neurons
Kuramoto model	1	${\frac {d\theta _{i}}{dt}}=\omega _{i}+{\frac {1}{N}}\sum _{j=1}^{N}K_{ij}\sin(\theta _{j}-\theta _{i}),\qquad i=1\ldots N$	Synchronization, coupled oscillators
Lotka–Volterra equations	1	${\begin{aligned}{\frac {dx}{dt}}&=\alpha x-\beta xy\\{\frac {dy}{dt}}&=\delta xy-\gamma y\end{aligned}}$	Population dynamics
Price equation	1	${d \over {dt}}\mathbb {E} (x)=\underbrace {{\text{Cov}}(x,f)} _{\text{Selection effect}}+\underbrace {\mathbb {E} ({\dot {x}})} _{\text{Dynamic effect}}$	Evolution and change in allele frequency over time
SIR model	1	${\begin{aligned}{\frac {dS}{dt}}&=-\beta SI\\{\frac {dI}{dt}}&=\beta SI-\gamma I\\{\frac {dR}{dt}}&=\gamma I\end{aligned}}$	Epidemiology

Economics and finance

Name	Order	Equation	Applications	Reference
Bass diffusion model	1	${\frac {dF}{dt}}=(1-F)(p+qF)$	A Riccati equation used in marketing to describe product adoption
Ramsey–Cass–Koopmans model	1	${\begin{aligned}{\dot {k}}&=f(k)-(n+\delta )k-c\\{\dot {c}}&=\sigma (c)\left[f_{k}(k)-\delta -\rho \right]\cdot c\end{aligned}}$	Neoclassical economics model of economic growth
Solow–Swan model	1	${\dot {k}}(t)=sk(t)^{\alpha }-(n+g+\delta )k(t)$	Model of long run economic growth

See also

• List of linear ordinary differential equations
• List of nonlinear partial differential equations
• List of named differential equations
• List of stochastic differential equations