Name
|
Order
|
Equation
|
Application
|
Reference
|
Abel's differential equation of the first kind
|
1
|
|
Class of differential equation which may be solved implicitly
|
[1]
|
Abel's differential equation of the second kind
|
1
|
|
Class of differential equation which may be solved implicitly
|
[1]
|
Bernoulli equation
|
1
|
|
Class 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
|
1
|
|
Class 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
|
1
|
|
Generalization of Clairaut's equation with a singular solution
|
[8]
|
Clairaut's equation
|
1
|
|
Particular case of d'Alembert's equation which may be solved exactly
|
[9]
|
d'Alembert's equation or Lagrange's equation
|
1
|
|
May be solved exactly
|
[10]
|
Darboux equation
|
1
|
|
Can be reduced to a Bernoulli differential equation; a general case of the Jacobi equation
|
[11]
|
Elliptic function
|
1
|
|
Equation for which the elliptic functions are solutions
|
[12]
|
Euler's differential equation
|
1
|
|
A separable differential equation
|
[13]
|
Euler's differential equation
|
1
|
|
A differential equation which may be solved with Bessel functions
|
[13]
|
Jacobi equation
|
1
|
|
Special case of the Darboux equation, integrable in closed form
|
[14]
|
Loewner differential equation
|
1
|
|
Important in complex analysis and geometric function theory
|
[15]
|
Logistic differential equation (sometimes known as the Verhulst model)
|
2
|
|
Special 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
|
2
|
|
One of fifty classes of differential equation of the form ; the six Painlevé transcendents required new special functions to solve
|
[19]
|
Painlevé II transcendent
|
2
|
|
One of fifty classes of differential equation of the form ; the six Painlevé transcendents required new special functions to solve
|
[19]
|
Painlevé III transcendent
|
2
|
|
One of fifty classes of differential equation of the form ; the six Painlevé transcendents required new special functions to solve
|
[19]
|
Painlevé IV transcendent
|
2
|
|
One of fifty classes of differential equation of the form ; the six Painlevé transcendents required new special functions to solve
|
[19]
|
Painlevé V transcendent
|
2
|
|
One of fifty classes of differential equation of the form ; the six Painlevé transcendents required new special functions to solve
|
[19]
|
Painlevé VI transcendent
|
2
|
|
All of the other Painlevé transcendents are degenerations of the sixth
|
[19]
|
Rabinovich–Fabrikant equations
|
1
|
|
Chaos theory, dynamical systems
|
[20]
|
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
|
[21]
|
Rössler attractor
|
1
|
|
Chaos theory, dynamical systems
|
[22]
|