List of nonlinear ordinary differential equations

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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.

Contents

Mathematics

NameOrderEquationApplicationReference
Abel's differential equation of the first kind 1Class of differential equation which may be solved implicitly [1]
Abel's differential equation of the second kind 1Class of differential equation which may be solved implicitly [1]
Bernoulli equation 1Class of differential equation which may be solved exactly [2]
Binomial differential equation Class of differential equation which may sometimes be solved exactly [3]
Briot-Bouquet Equation 1Class of differential equation which may sometimes be solved exactly [4]
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 [5] [6] [7]
Chrystal's equation 1Generalization of Clairaut's equation with a singular solution [8]
Clairaut's equation 1Particular case of d'Alembert's equation which may be solved exactly [9]
d'Alembert's equation or Lagrange's equation1May be solved exactly [10]
Darboux equation 1Can be reduced to a Bernoulli differential equation; a general case of the Jacobi equation [11]
Elliptic function 1Equation for which the elliptic functions are solutions [12]
Euler's differential equation 1A separable differential equation [13]
Euler's differential equation1A differential equation which may be solved with Bessel functions [13]
Jacobi equation 1Special case of the Darboux equation, integrable in closed form [14]
Loewner differential equation 1Important in complex analysis and geometric function theory [15]
Logistic differential equation (sometimes known as the Verhulst model)2Special case of the Bernoulli differential equation; many applications including in population dynamics [16]
Lorenz attractor 1 Chaos theory, dynamical systems, meteorology [17]
Nahm equations 1 Differential geometry, gauge theory, mathematical physics, magnetic monopoles [18]
Painlevé I transcendent 2One of fifty classes of differential equation of the form ; the six Painlevé transcendents required new special functions to solve [19]
Painlevé II transcendent 2One of fifty classes of differential equation of the form ; the six Painlevé transcendents required new special functions to solve [19]
Painlevé III transcendent 2One of fifty classes of differential equation of the form ; the six Painlevé transcendents required new special functions to solve [19]
Painlevé IV transcendent 2One of fifty classes of differential equation of the form ; the six Painlevé transcendents required new special functions to solve [19]
Painlevé V transcendent 2One of fifty classes of differential equation of the form ; the six Painlevé transcendents required new special functions to solve [19]
Painlevé VI transcendent 2All of the other Painlevé transcendents are degenerations of the sixth [19]
Rabinovich–Fabrikant equations 1 Chaos theory, dynamical systems [20]
Riccati equation 1Class of first order differential equations that is quadratic in the unknown. Can reduce to Bernoulli differential equation or linear differential equation in certain cases [21]
Rössler attractor 1 Chaos theory, dynamical systems [22]

Physics

NameOrderEquationApplicationsReference
Bellman's equation or Emden-Fowler's equation 2 (Emden-Fowler) which reduces to if (Bellman) Diffusion in a slab [23]
Besant-Rayleigh-Plesset equation 2Spherical bubble in fluid dynamics [24]
Blasius equation 3 Blasius boundary layer [25]
Chandrasekhar's white dwarf equation 2 Gravitational potential of white dwarf in astrophysics [26]
De Boer-Ludford equation 2 Plasma physics [27]
Emden–Chandrasekhar equation 2 Astrophysics [26]
Ermakov-Pinney equation 2 Electromagnetism, oscillation, scalar field cosmologies [28] [29]
Falkner–Skan equation 3 Falkner–Skan boundary layer [30]
Friedmann equations 2 and Physical cosmology [31]
Heisenberg equation of motion 1 Quantum mechanics [32]
Ivey's equation 2 Space charge theory [33]
Kidder equation 2 Flow through porous medium [34]
Krogdahl equation 2 Stellar pulsation in astrophysics [35]
Lagerstrom equation 2One dimensional viscous flow at low Reynolds numbers [36]
Lane–Emden equation or polytropic differential equation2 Astrophysics [37]
Liñán's equation 2 Combustion [38]
Pendulum equation2 Mechanics [39]
Poisson–Boltzmann equation (1d case)2 Inflammability and the theory of thermal explosions [40]
Stuart–Landau equation 1 Hydrodynamic stability [41]
Taylor–Maccoll equation 2 where Flow behind a conical shock wave [42]
Thomas–Fermi equation 2 Quantum mechanics [43] [44]
Toda lattice 1where Model of one-dimensional crystal in solid-state physics, Langmuir oscillations in plasma, quantum cohomology; notable for being a completely integrable system [45]
Trachenko-Zaccone 1 Condensed matter physics; relaxation in amorphous solids and glass transition [46] [47]

Engineering

NameOrderEquationApplicationsReference
Duffing equation 2 Oscillators, hysteresis, chaotic dynamical systems [48]
Lewis regulator 2 Oscillators [49]
Liénard equation 2 with odd and even Oscillators, electrical engineering, dynamical systems [50]
Rayleigh equation 2 Oscillators (especially auto-oscillation), acoustics; the Van der Pol equation is a Rayleigh equation [51]
Van der Pol equation 2 Oscillators, electrical engineering, chaotic dynamical systems [52]

Chemistry

NameOrderEquationApplicationsReference
Brusselator 1A type of autocatalytic reaction modelled at constant concentration [53]
Oregonator 1A type of autocatalytic reaction modelled at constant concentration [54]

Biology and medicine

NameOrderEquationApplicationsReference
Allee effect 1 Population biology [55]
Arditi–Ginzburg equations 1 Population dynamics [56]
FitzHugh–Nagumo model or Bonhoeffer-van der Pol model1 Action potentials in neurons, oscillators [57]
Hodgkin-Huxley equations 1 Action potentials in neurons [58]
Kuramoto model 1 Synchronization, coupled oscillators [59]
Lotka–Volterra equations 1 Population dynamics [60]
Price equation 1 Evolution and change in allele frequency over time [61]
SIR model 1 Epidemiology [62]

Economics and finance

NameOrderEquationApplicationsReference
Bass diffusion model 1A Riccati equation used in marketing to describe product adoption [63]
Ramsey–Cass–Koopmans model 1 Neoclassical economics model of economic growth [64] [65]
Solow–Swan model 1Model of long run economic growth [66]

See also

References

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