List of nonlinear ordinary differential equations explained

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
	1	(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	^[1]
Abel's differential equation of the second kind	1	(g _{o(x)} + g _{1(x)} y) (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
	1	(dy)/(dx) + P(x) y = Q(x) y ^{n}	Class of differential equation which may be solved exactly	^[2]
Binomial differential equation	m	\left( y' \right) ^{m} = f(x,y)	Class of differential equation which may sometimes be solved exactly	^[3]
Briot-Bouquet Equation	1	x ^{m} y' = f(x, y)	Class of differential equation which may sometimes be solved exactly	^[4]
Cherwell-Wright differential equation	1	(dx)/(dt) = (a - x(t -1 ))x(t) or the related form f'(x) = (- f(x)f(\sqrt{x)/()}{ 2 x})	An example of a nonlinear delay differential equation; applications in number theory, distribution of primes, and control theory	^[5] ^[6] ^[7]
Chrystal's equation	1	\left( (dy)/(dx) \right) ^{2} + Ax (dy)/(dx) + By + Cx ^{2} =0	Generalization of Clairaut's equation with a singular solution	^[8]
Clairaut's equation	1	y = x (dy)/(dx) + f\left( (dy)/(dx) \right)	Particular case of d'Alembert's equation which may be solved exactly	^[9]
d'Alembert's equation or Lagrange's equation	1	y = x f\left( (dy)/(dx) \right) + g\left( (dy)/(dx) \right)	May be solved exactly	^[10]
Darboux equation	1	(dy)/(dx) = (P(x,y) + y R(x,y))/(Q(x,y) + xR(x,y))	Can be reduced to a Bernoulli differential equation; a general case of the Jacobi equation	^[11]
Elliptic function	1	y' = \sqrt{\left( 1- y ^{2\right)\left(1-k} ^{2y} ^{2\right)}}	Equation for which the elliptic functions are solutions	^[12]
Euler's differential equation	1	(dy)/(dx) + (\sqrt{a _{0+a} _{1y} + a _{2} y ^{2} + a _{3} y ^{3} + a _{4} y ^{4})/() } = 0	A separable differential equation	^[13]
Euler's differential equation	1	y' + y' ^{2} = \alpha x ^{m}	A differential equation which may be solved with Bessel functions
Jacobi equation	1	(dy)/(dx) = (Axy + By ^{2} + ax + by + c)/(Ax ^{2} + Bxy + \alpha x + \beta y + \gamma)	Special case of the Darboux equation, integrable in closed form	^[14]
Loewner differential equation	1	(dw)/(dt) = - wp _{t(w)}	Important in complex analysis and geometric function theory	^[15]
Logistic differential equation (sometimes known as the Verhulst model)	2	(d)/(dx) f(x) = f(x) ( 1 - f(x) )	Special case of the Bernoulli differential equation; many applications including in population dynamics	^[16]
Lorenz attractor	1	\begin{align} (d x)/(d t) & = \sigma (y - x) \\ (d y)/(d t) & = x (\rho - z) - y \\ (d z)/(d t) & = x y - \beta z \end{align}	Chaos theory, dynamical systems, meteorology	^[17]
Nahm equations	1	\begin{align} (dT _{1})/(dz) & = [T _{2,T} ^{3 ]\\ (dT _{2})/(dz)} & = [T _{3,T} ^{1 ]\\ (dT _{3})/(dz)} & = [T _{1,T} _{2] \end{align}}	Differential geometry, gauge theory, mathematical physics, magnetic monopoles	^[18]
Painlevé I transcendent	2	(d ^{2y})/(dt ^{2}) = 6 y ^{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	^[19]
Painlevé II transcendent	2	(d ^{2y})/(dt ^{2}) = 2 y ^{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 (d ^{2y})/(dt ^{2}) = t \left( (dy)/(dt) \right) ^{2- y (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 (d ^{2y})/(dt ^{2}) = \tfrac 12 \left( (dy)/(dt) \right) ^{2+\beta+2(t} ^{2-\alpha)y} ^{2+4ty} ^{3+\tfrac32y} ^{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	(d ^{2y})/(= \left( dt ^{2}) (1)/(+ 2 y) (1)/(y -1) \right) \left( (dy)/(dt) \right) ^{2 - (1)/(t)} (dy)/(+ dt) (( y -1 ) ^{2})/(t ^{2}) \left(\alpha y + (\beta)/(y) \right) + \gamma (y)/(+ \delta t) (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{align} (d ^{2y})/(& = dt ^{2}) (1)/(\left( 2) (1)/(+ y) (1)/(+ y -1) (1)/(y - t) \right)\left( (dy)/(dt) \right) ^{2- \left( (1)/(t)} + (1)/(+ t -1) (1)/(\right) y - t) (dy)/(\\ & + dt) (y(y -1 )(y - t))/(\left(\alpha + \beta t ^{2(t-1)} ^{2}) (t)/(+ \gamma y ^{2}) (t -1)/(+ \delta (y -1 ) ^{2}) (t(t -1 ))/((y - t) ^{2}) \right)\\ \end{align}	All of the other Painlevé transcendents are degenerations of the sixth
Rabinovich–Fabrikant equations	1	• \begin{align} x & = y (z - 1 + x ^{2)} + \gamma x \\ • y & = x ( 3 z + 1 - x ^{2)} + \gamma y \\ • z & = -2 z (\alpha + xy) \end{align}	Chaos theory, dynamical systems	^[20]
Riccati equation	1	(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	^[21]
Rössler attractor	1	\begin{align} (dx)/(dt) & = - y - z \ (dy)/(dt) & = x + ay \ (dz)/(dt) & = b + z(x - c) \end{align}	Chaos theory, dynamical systems

Physics

Name	Order	Equation	Applications	Reference
Bellman's equation or Emden-Fowler's equation	2	(d)/(dt) \left(t ^{\rho} (du)/(dt) \right) = t ^{\sigma} u ^{\rho} (Emden-Fowler) which reduces to (d ^{2u})/(dx ^{2}) = \phi ^{2} u ^{p} if \sigma + \rho = 0 (Bellman)	Diffusion in a slab	^[22]
	2	R (d ^{2R})/(dt ^{2}) + (3)/(\left( 2) (dR)/(dt) \right) ^{2} + (4 \nu _{L})/(R) (dR)/(dt) + (2 \sigma)/(\rho _{L} R) + (\Delta P(t))/(\rho) = 0	Spherical bubble in fluid dynamics	^[23]
	3	(d ^{3y})/(dx ^{3}) + y (d ^{2y})/(dx ^{2}) =0	Blasius boundary layer	^[24]
	2	(1)/(x ^{2}) (d)/(dx) \left(x ^{2} (dy)/(dx) \right) + (y ^{2} - c) ^{3/2} =0	Gravitational potential of white dwarf in astrophysics	^[25]
	2	(d ^{2y})/(dx ^{2}) - xy = y	y	^\alpha, \ \alpha>0	Plasma physics	^[26]
Emden–Chandrasekhar equation	2	(1)/(\xi ^{2}) (d)/(d\xi) \left(\xi ^{2} (d\psi)/(d\xi) \right) = e ^{-\psi}	Astrophysics
Ermakov-Pinney equation	2	\ddot{x} + \omega ^{2} x = (h ^{2})/(x ^{3})	Electromagnetism, oscillation, scalar field cosmologies	^[27] ^[28]
Falkner–Skan equation	3	(d ^{3y})/(dx ^{3}) + y (d ^{2y})/(dx ^{2}) + \beta \left[ 1- \left( (dy)/(dx) \right) ^{2\right]=0}	Falkner–Skan boundary layer	^[29]
Friedmann equations	2	(\ddot{a)/() } = -\frac\left(\rho+\frac\right) + \frac and (• a ^{2} + kc ^{2})/(a ^{2}) = (8 \pi G \rho +Λ c ^{2})/(3)	Physical cosmology	^[30]
Heisenberg equation of motion	1	(d\hat{A(t))/() } = \frac[\hat{\mathcal{H}}, \hat{A}]	Quantum mechanics	^[31]
Ivey's equation	2	(d ^{2y})/(dx ^{2}) - (1)/(\left( y) (dy)/(dx) \right) ^{2} + (2)/(x) (dy)/(dx) + ky ^{2=0}	Space charge theory	^[32]
Kidder equation	2	\sqrt{ 1- \alpha y} (d ^{2y})/(dx ^{2}) + 2 x (dy)/(dx) =0 , 0 \leq\alpha\leq 1	Flow through porous medium	^[33]
Krogdahl equation	2	(d ^{2} Q)/(d\tau ^{2}) = - Q + (2)/(3) λ Q ^{2} + (14)/(27) λ ^{2} Q ^{3} + \mu( 1- Q ^{2 ) (dQ)/(d\tau)} + (2)/(3) λ ( 1-λ Q) \left( (dQ)/(d\tau) \right) ^{2+ …}	Stellar pulsation in astrophysics	^[34]
Lagerstrom equation	2	y'' + (k)/(r) y' + \epsilon y' y = 0	One dimensional viscous flow at low Reynolds numbers	^[35]
Lane–Emden equation or polytropic differential equation	2	(1)/(\xi ^{2}) (d)/(d\xi) \left({\xi ^{2} (d\theta)/(d\xi) }\right) + \theta^n = 0	Astrophysics	^[36]
Liñán's equation	2	(d ^{2y})/(d\zeta ^{2}) = (y ^{2-\zeta} ^{2)e} ^{- \delta ^{1/3} (y + \gamma \zeta)}	Combustion	^[37]
Pendulum equation	2	(d ^{2\theta})/(+ dt ^{2}) (g)/(\ell) \sin\theta =0	Mechanics	^[38]
Poisson–Boltzmann equation (1d case)	2	(d ^{2\theta})/(dz ^{2}) + (k)/(z) (d\theta)/(dz) = - \delta e ^{\theta}	Inflammability and the theory of thermal explosions	^[39]
Stuart–Landau equation	1	(dA)/(dt) = \sigma A - (l)/(2) A	A	^2	Hydrodynamic stability
Taylor–Maccoll equation	2	(c ^{2-f'} ^{2)} f'' + c ^{2} \cot\theta f' + ( 2 c ^{2-f'} ^{2)} f = 0 , c = c(f ^{2+f'} ^{2)} where f' = (df)/(d\theta)	Flow behind a conical shock wave	^[40]
Thomas–Fermi equation	2	(d ^{2y})/(dx ^{2}) = (1)/(\sqrt x) y ^{3/2}	Quantum mechanics^[41]	^[42]
Toda lattice	1	• \begin{align} a (n,t) & = a(n,t) ( b(n +1 ,t) - b(n,t) ) \\ • b (n,t) & = 2 ( a(n,t) ^{2-a(n-1,t)} ^{2) \end{align}} where \begin{align} a(n,t) & = (1)/(2) {\rm e} ^{-(q(n+1,t)} \\ b(n,t) & =- (1)/(2) p(n,t) \end{align}	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	(d ^{2x})/(dt ^{2}) + \mu (dx)/(dt) + \alpha x + \beta x ^{3} = \gamma \cos \omega t	Oscillators, hysteresis, chaotic dynamical systems	^[43]
Lewis regulator	2	y'' + ( 1 -	y	)y' + y = 0	Oscillators	^[44]
Liénard equation	2	{d ^{2x} \over dt ^{2}} + f(x){dx \over dt} + g(x) = 0 with f odd and g even	Oscillators, electrical engineering, dynamical systems	^[45]
Rayleigh equation	2	y'' + F(y') + y = 0	Oscillators (especially auto-oscillation), acoustics; the Van der Pol equation is a Rayleigh equation	^[46]
Van der Pol equation	2	{d ^{2x} \over dt ^{2}-\mu(1-x} ^{2){dx} \over dt} + x = 0	Oscillators, electrical engineering, chaotic dynamical systems	^[47]

Chemistry

Name	Order	Equation	Applications	Reference
Brusselator	1	\begin{align} {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{align}	A type of autocatalytic reaction modelled at constant concentration	^[48]
Oregonator	1	\begin{align} (d [X])/(dt) & = k _{I} [A] [Y] - k _{II} [X] [Y] + k _{III} [A] [X] - 2 k _{IV} [X] ^{2} \\ (d [Y])/(dt) & = - k _{I} [A] [Y] - k _{II} [X] [Y] + (1)/(2) f k _{V} [B] [Z] \\ (d [Z])/(dt) & = 2 k _{III} [A] [X] - k _{V} [B] [Z] \end{align}	A type of autocatalytic reaction modelled at constant concentration	^[49]

Biology and medicine

Name	Order	Equation	Applications	Reference
Allee effect	1	(dN)/(dt) = - r N \left( 1 - (N)/(A) \right) \left( 1 - (N)/(K) \right)	Population biology	^[50]
Arditi–Ginzburg equations	1	\begin{align} (dN)/(dt) & = f(N)N - g{\left(\tfrac N P \right)}P\\ (dP)/(dt) & = e g{\left(\tfrac N P \right)}P - uP \end{align}	Population dynamics	^[51]
FitzHugh–Nagumo model or Bonhoeffer-van der Pol model	1	• \begin{align} v & = v - (v ^{3})/(3) - w + RI _{\rm} \\ \tau • w & = v + a - b w \end{align}	Action potentials in neurons, oscillators
Hodgkin-Huxley equations	1	\begin{align} I & = C _{m ({ d)/(V)} _{m}{{d}t}} + \bar{g} ^{4 (V} _{m} - V _{K)} + \bar{g} ^{3 h(V} _{m} - V _{Na} ) + \bar{g} _{l(V} _{m} - V _{l)\\ (dn)/(dt)} & = \alpha _{n(V} _{m)(1} - n) - \beta _{n(V} _{m)} n\\ (dm)/(dt) & = \alpha _{m(V} _{m)(1} - m) - \beta _{m(V} _{m)} m\\ (dh)/(dt) & = \alpha _{h(V} _{m)(1} - h) - \beta _{h(V} _{m)} h \end{align}	Action potentials in neurons	^[52]
Kuramoto model	1	(d \theta _{i})/(d t) = \omega _{i} + (1)/(N) \sum ^{N} _{j =1} K _{ij} \sin(\theta _{j} - \theta _{i),} i = 1 \ldots N	Synchronization, coupled oscillators	^[53]
	1	\begin{align} (dx)/(dt) & = \alpha x - \beta xy\\ (dy)/(dt) & = \delta xy - \gamma y \end{align}	Population dynamics	^[54]
Price equation	1	{d\over{dt}} E (x) = \underbrace{ Cov (x,f)} _{Selectioneffect} + \underbrace{E( • x )} _{Dynamiceffect}	Evolution and change in allele frequency over time	^[55]
SIR model	1	\begin{align} (dS)/(dt) & = - \beta S I\\ (dI)/(dt) & = \beta S I - \gamma I\\ (dR)/(dt) & = \gamma I \end{align}	Epidemiology	^[56]

Economics and finance

Name	Order	Equation	Applications	Reference
Bass diffusion model	1	(dF)/(dt) = ( 1- F)(p + qF)	A Riccati equation used in marketing to describe product adoption	^[57]
Ramsey–Cass–Koopmans model	1	• \begin{align} k & = f(k) - (n + \delta)k - c\\ • c & = \sigma(c) \left[ f _{k(k)} - \delta - \rho \right] ⋅ c \end{align}	Neoclassical economics model of economic growth	^[58] ^[59]
Solow–Swan model	1	• k (t) = sk(t) ^{\alpha} - (n + g + \delta)k(t)	Model of long run economic growth	^[60]

See also

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

Notes and References

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