# List of nonlinear ordinary differential equations

Differential equations :
Scope :
Fields
Natural sciencesEngineering
Astronomy
Physics
Chemistry
Biology
Geology
Applied mathematics
Continuum mechanics
Chaos theory
Dynamical systems
Social sciences
Economics
Population dynamics
List of named differential equations :
Classification :
Types
Ordinary
Partial
Differential-algebraic
Integro-differential
Fractional
Linear
Non-linear
By variable type
Dependent and independent variables
Autonomous
Coupled / Decoupled
Exact
Homogeneous / Nonhomogeneous
Features
Order
Operator
Notation :
Relation to processes
Difference (discrete analogue)
Stochastic
Stochastic partial
Delay :
Solution :
Existence and uniqueness
Picard–Lindelöf theorem
Peano existence theorem
Carathéodory's existence theorem
Cauchy–Kowalevski theorem :
General topics
Initial conditions
Boundary values
Dirichlet
Neumann
Robin
Cauchy problem
Wronskian
Phase portrait
Lyapunov / Asymptotic / Exponential stability
Rate of convergence
Series / Integral solutions
Numerical integration
Dirac delta function :
Solution methods
Inspection
Method of characteristics
Euler
Exponential response formula
Finite difference (Crank–Nicolson)
Finite element
Infinite element
Finite volume
Galerkin
Petrov–Galerkin
Green's function
Integrating factor
Integral transforms
Perturbation theory
Runge–Kutta
Separation of variables
Undetermined coefficients
Variation of parameters :
People :
List
Isaac Newton
Gottfried Leibniz
Jacob Bernoulli
Leonhard Euler
Joseph-Louis Lagrange
Józef Maria Hoene-Wroński
Joseph Fourier
Augustin-Louis Cauchy
George Green
Carl David Tolmé Runge
Martin Kutta
Rudolf Lipschitz
Ernst Lindelöf
Émile Picard
Phyllis Nicolson
John Crank :
vte :
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 ${\frac {dy}{dx}}={\frac {P(x,y)+yR(x,y)}{Q(x,y)+xR(x,y)}}$ A generalization of the Jacobi equation. Can be solved without quadratures, if certain amount of distinct particular solutions are found. In case $P\left(x,y\right)$ and $Q\left(x,y\right)$ are homogenous polynomials of the same degree, this equation is either homogenous itself, or can be reduced to a Bernoulli differential 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 ${\frac {d}{dx}}f(x)=f(x){\big1-f(x){\big )}$ 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 ${\begin{aligned}{\dot {a}}(n,t)&=a(n,t){\Bigb(n+1,t)-b(n,t){\Big )}\\{\dot {b}}(n,t)&=2{\Biga(n,t)^{2}-a(n-1,t)^{2}{\Big )}\end{aligned}}$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 Trachenko-Zaccone 1 ${\frac {dy}{dt}}=-{\frac {y}{\tau }}\exp(Ky)$ Condensed matter physics; relaxation in amorphous solids and glass transition

## 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 |
| 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 |
| 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 |