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 {|class="wikitable" style="text-align: left" |-style="background: #eee" !Name !Order !Equation !Application !Reference |- |Abel's differential equation of the first kind |1 | |Class of differential equation which may be solved implicitly
|Abel's differential equation of the second kind |1 | |Class of differential equation which may be solved implicitly
|Bernoulli equation |1 | |Class of differential equation which may be solved exactly
|Binomial differential equation
|Class of differential equation which may sometimes be solved exactly
|Briot-Bouquet Equation |1 | |Class of differential equation which may sometimes be solved exactly
|Cherwell-Wright differential equation |1 | or the related form |An example of a nonlinear delay differential equation; applications in number theory, distribution of primes, and control theory
|Chrystal's equation |1 | |Generalization of Clairaut's equation with a singular solution
|Clairaut's equation |1 | |Particular case of d'Alembert's equation which may be solved exactly
|d'Alembert's equation or Lagrange's equation |1 | |May be solved exactly
|Darboux equation |1 | |A generalization of the Jacobi equation. Can be solved without quadratures, if certain amount of distinct particular solutions are found. In case and 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 | |Equation for which the elliptic functions are solutions
|Euler's differential equation |1 | |A separable differential equation
|Euler's differential equation |1 | |A differential equation which may be solved with Bessel functions
|Jacobi equation |1 | |Special case of the Darboux equation, integrable in closed form
|Loewner differential equation |1 | |Important in complex analysis and geometric function theory
|Logistic differential equation (sometimes known as the Verhulst model) |2 | |Special case of the Bernoulli differential equation; many applications including in population dynamics
|Lorenz attractor |1 | |Chaos theory, dynamical systems, meteorology
|Nahm equations |1 | |Differential geometry, gauge theory, mathematical physics, magnetic monopoles
|Painlevé I transcendent |2 | |One of fifty classes of differential equation of the form ; the six Painlevé transcendents required new special functions to solve
|Painlevé II transcendent |2 | |One of fifty classes of differential equation of the form ; the six Painlevé transcendents required new special functions to solve
|Painlevé III transcendent |2 | |One of fifty classes of differential equation of the form ; the six Painlevé transcendents required new special functions to solve
|Painlevé IV transcendent |2 | |One of fifty classes of differential equation of the form ; the six Painlevé transcendents required new special functions to solve
|Painlevé V transcendent |2 | |One of fifty classes of differential equation of the form ; the six Painlevé transcendents required new special functions to solve
|Painlevé VI transcendent |2 | |All of the other Painlevé transcendents are degenerations of the sixth
|Rabinovich–Fabrikant equations |1 | |Chaos theory, dynamical systems
|Riccati equation |1 | |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 | |Chaos theory, dynamical systems
Physics {|class="wikitable" style="text-align: left" |-style="background: #eee" !Name !Order !Equation !Applications !Reference |- |Bellman's equation or Emden-Fowler's equation |2 | (Emden-Fowler) which reduces to if (Bellman) |Diffusion in a slab
|Besant-Rayleigh-Plesset equation |2 | |Spherical bubble in fluid dynamics
|Blasius equation |3 | |Blasius boundary layer
|Chandrasekhar's white dwarf equation |2 | |Gravitational potential of white dwarf in astrophysics
|De Boer-Ludford equation |2 | |Plasma physics
|Emden–Chandrasekhar equation |2 | |Astrophysics
|Ermakov-Pinney equation |2 | |Electromagnetism, oscillation, scalar field cosmologies
|Falkner–Skan equation |3 | |Falkner–Skan boundary layer
|Friedmann equations |2 | and |Physical cosmology
|Heisenberg equation of motion |1 | |Quantum mechanics
|Ivey's equation |2 | |Space charge theory
|Kidder equation |2 | |Flow through porous medium
|Krogdahl equation |2 | |Stellar pulsation in astrophysics
|Lagerstrom equation |2 | |One dimensional viscous flow at low Reynolds numbers
|Lane–Emden equation or polytropic differential equation |2 | |Astrophysics
|Liñán's equation |2 | |Combustion
|Pendulum equation |2 | |Mechanics
|Poisson–Boltzmann equation (1d case) |2 | |Inflammability and the theory of thermal explosions
|Stuart–Landau equation |1 | |Hydrodynamic stability
|Taylor–Maccoll equation |2 | where |Flow behind a conical shock wave
|Thomas–Fermi equation |2 | |Quantum mechanicsDavis, Harold Thayer. Introduction to nonlinear differential and integral equations. Courier Corporation, 1962.
|Toda lattice |1 |where |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 | |Condensed matter physics; relaxation in amorphous solids and glass transition
Engineering {|class="wikitable" style="text-align: left" |-style="background: #eee" !Name !Order !Equation !Applications !Reference |- |Duffing equation |2 | |Oscillators, hysteresis, chaotic dynamical systems
|Lewis regulator |2 | |Oscillators
|Liénard equation |2 | with odd and even |Oscillators, electrical engineering, dynamical systems
|Rayleigh equation |2 | |Oscillators (especially auto-oscillation), acoustics; the Van der Pol equation is a Rayleigh equation
|Van der Pol equation |2 | |Oscillators, electrical engineering, chaotic dynamical systems
Chemistry NameOrderEquationApplicationsReferenceBrusselator1A type of autocatalytic reaction modelled at constant concentrationOregonator1A type of autocatalytic reaction modelled at constant concentration
Biology and medicine NameOrderEquationApplicationsReferenceAllee effect1Population biologyArditi–Ginzburg equations1 Population dynamicsFitzHugh–Nagumo model or Bonhoeffer-van der Pol model1Action potentials in neurons, oscillatorsHodgkin-Huxley equations1Action potentials in neuronsKuramoto model1Synchronization, coupled oscillatorsLotka–Volterra equations 1Population dynamicsPrice equation1Evolution and change in allele frequency over timeSIR model1Epidemiology
Economics and finance NameOrderEquationApplicationsReferenceBass diffusion model1A Riccati equation used in marketing to describe product adoptionRamsey–Cass–Koopmans model1Neoclassical economics model of economic growthSolow–Swan model1Model 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
References