List of nonlinear ordinary differential equations

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	${\displaystyle {\frac {dy}{dx}}=f_{o}(x)+f_{1}(x)y+f_{2}(x)y^{2}+f_{3}(x)y^{3}}$	Class of differential equation which may be solved implicitly	[1]
Abel's differential equation of the second kind	1	${\displaystyle (g_{o}(x)+g_{1}(x)y){\frac {dy}{dx}}=f_{o}(x)+f_{1}(x)y+f_{2}(x)y^{2}+f_{3}(x)y^{3}}$	Class of differential equation which may be solved implicitly	[1]
Bernoulli equation	1	${\displaystyle {\frac {dy}{dx}}+P(x)y=Q(x)y^{n}}$	Class of differential equation which may be solved exactly	[2]
Binomial differential equation	${\displaystyle m}$	${\displaystyle \left(y'\right)^{m}=f(x,y)}$	Class of differential equation which may sometimes be solved exactly	[3]
Briot-Bouquet Equation	1	${\displaystyle x^{m}y'=f(x,y)}$	Class of differential equation which may sometimes be solved exactly	[4]
Cherwell-Wright differential equation	1	${\displaystyle {\frac {dx}{dt}}=(a-x(t-1))x(t)}$ or the related form ${\displaystyle f'(x)={\frac {-f(x)f({\sqrt {x}})}{2x}}}$	An example of a nonlinear delay differential equation; applications in number theory, distribution of primes, and control theory	[5] [6] [7]
Chrystal's equation	1	${\displaystyle \left({\frac {dy}{dx}}\right)^{2}+Ax{\frac {dy}{dx}}+By+Cx^{2}=0}$	Generalization of Clairaut's equation with a singular solution	[8]
Clairaut's equation	1	${\displaystyle y=x{\frac {dy}{dx}}+f\left({\frac {dy}{dx}}\right)}$	Particular case of d'Alembert's equation which may be solved exactly	[9]
d'Alembert's equation or Lagrange's equation	1	${\displaystyle y=xf\left({\frac {dy}{dx}}\right)+g\left({\frac {dy}{dx}}\right)}$	May be solved exactly	[10]
Darboux equation	1	${\displaystyle {\frac {dy}{dx}}={\frac {P(x,y)+yR(x,y)}{Q(x,y)+xR(x,y)}}}$	A generalization of the Jacobi equation. Can be solved without quadratures, if certain amount of distinct particular solutions are found. In case ${\displaystyle P\left(x,y\right)}$ and ${\displaystyle Q\left(x,y\right)}$ are homogenous polynomials of the same degree, this equation is either homogenous itself, or can be reduced to a Bernoulli differential equation.	[11]
Elliptic function	1	${\displaystyle y'={\sqrt {\left(1-y^{2}\right)\left(1-k^{2}y^{2}\right)}}}$	Equation for which the elliptic functions are solutions	[12]
Euler's differential equation	1	${\displaystyle {\frac {dy}{dx}}+{\frac {\sqrt {a_{0}+a_{1}y+a_{2}y^{2}+a_{3}y^{3}+a_{4}y^{4}}}{\sqrt {a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}}}}=0}$	A separable differential equation	[13]
Euler's differential equation	1	${\displaystyle y'+y'^{2}=\alpha x^{m}}$	A differential equation which may be solved with Bessel functions	[13]
Jacobi equation	1	${\displaystyle {\frac {dy}{dx}}={\frac {Axy+By^{2}+ax+by+c}{Ax^{2}+Bxy+\alpha x+\beta y+\gamma }}}$	Special case of the Darboux equation, integrable in closed form	[14]
Loewner differential equation	1	${\displaystyle {\frac {dw}{dt}}=-wp_{t}(w)}$	Important in complex analysis and geometric function theory	[15]
Logistic differential equation (sometimes known as the Verhulst model)	2	${\displaystyle {\frac {d}{dx}}f(x)=f(x){\big (}1-f(x){\big )}}$	Special case of the Bernoulli differential equation; many applications including in population dynamics	[16]
Lorenz attractor	1	${\displaystyle {\begin{aligned}{\frac {\mathrm {d} x}{\mathrm {d} t}}&=\sigma (y-x)\\{\frac {\mathrm {d} y}{\mathrm {d} t}}&=x(\rho -z)-y\\{\frac {\mathrm {d} z}{\mathrm {d} t}}&=xy-\beta z\end{aligned}}}$	Chaos theory, dynamical systems, meteorology	[17]
Nahm equations	1	${\displaystyle {\begin{aligned}{\frac {dT_{1}}{dz}}&=[T_{2},T_{3}]\\{\frac {dT_{2}}{dz}}&=[T_{3},T_{1}]\\{\frac {dT_{3}}{dz}}&=[T_{1},T_{2}]\end{aligned}}}$	Differential geometry, gauge theory, mathematical physics, magnetic monopoles	[18]
Painlevé I transcendent	2	${\displaystyle {\frac {d^{2}y}{dt^{2}}}=6y^{2}+t}$	One of fifty classes of differential equation of the form ${\displaystyle y''=R(y',y,t)}$; the six Painlevé transcendents required new special functions to solve	[19]
Painlevé II transcendent	2	${\displaystyle {\frac {d^{2}y}{dt^{2}}}=2y^{3}+ty+\alpha }$	One of fifty classes of differential equation of the form ${\displaystyle y''=R(y',y,t)}$; the six Painlevé transcendents required new special functions to solve	[19]
Painlevé III transcendent	2	${\displaystyle ty{\frac {d^{2}y}{dt^{2}}}=t\left({\frac {dy}{dt}}\right)^{2}-y{\frac {dy}{dt}}+\delta t+\beta y+\alpha y^{3}+\gamma ty^{4}}$	One of fifty classes of differential equation of the form ${\displaystyle y''=R(y',y,t)}$; the six Painlevé transcendents required new special functions to solve	[19]
Painlevé IV transcendent	2	${\displaystyle y{\frac {d^{2}y}{dt^{2}}}={\tfrac {1}{2}}\left({\frac {dy}{dt}}\right)^{2}+\beta +2(t^{2}-\alpha )y^{2}+4ty^{3}+{\tfrac {3}{2}}y^{4}}$	One of fifty classes of differential equation of the form ${\displaystyle y''=R(y',y,t)}$; the six Painlevé transcendents required new special functions to solve	[19]
Painlevé V transcendent	2	${\displaystyle {\frac {d^{2}y}{dt^{2}}}=\left({\frac {1}{2y}}+{\frac {1}{y-1}}\right)\left({\frac {dy}{dt}}\right)^{2}-{\frac {1}{t}}{\frac {dy}{dt}}+{\frac {(y-1)^{2}}{t^{2}}}\left(\alpha y+{\frac {\beta }{y}}\right)+\gamma {\frac {y}{t}}+\delta {\frac {y(y+1)}{y-1}}}$	One of fifty classes of differential equation of the form ${\displaystyle y''=R(y',y,t)}$; the six Painlevé transcendents required new special functions to solve	[19]
Painlevé VI transcendent	2	${\displaystyle {\begin{aligned}{\frac {d^{2}y}{dt^{2}}}&={\frac {1}{2}}\left({\frac {1}{y}}+{\frac {1}{y-1}}+{\frac {1}{y-t}}\right)\left({\frac {dy}{dt}}\right)^{2}-\left({\frac {1}{t}}+{\frac {1}{t-1}}+{\frac {1}{y-t}}\right){\frac {dy}{dt}}\\&+{\frac {y(y-1)(y-t)}{t^{2}(t-1)^{2}}}\left(\alpha +\beta {\frac {t}{y^{2}}}+\gamma {\frac {t-1}{(y-1)^{2}}}+\delta {\frac {t(t-1)}{(y-t)^{2}}}\right)\\\end{aligned}}}$	All of the other Painlevé transcendents are degenerations of the sixth	[19]
Rabinovich–Fabrikant equations	1	${\displaystyle {\begin{aligned}{\dot {x}}&=y(z-1+x^{2})+\gamma x\\{\dot {y}}&=x(3z+1-x^{2})+\gamma y\\{\dot {z}}&=-2z(\alpha +xy)\end{aligned}}}$	Chaos theory, dynamical systems	[20]
Riccati equation	1	${\displaystyle {\frac {dy}{dx}}+Q(x)y+R(x)y^{2}=P(x)}$	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	[21]
Rössler attractor	1	${\displaystyle {\begin{aligned}{\frac {dx}{dt}}&=-y-z\\{\frac {dy}{dt}}&=x+ay\\{\frac {dz}{dt}}&=b+z(x-c)\end{aligned}}}$	Chaos theory, dynamical systems	[22]

Physics

Name	Order	Equation	Applications	Reference
Bellman's equation or Emden-Fowler's equation	2	${\displaystyle {\frac {d}{dt}}\left(t^{\rho }{\frac {du}{dt}}\right)=t^{\sigma }u^{\rho }}$ (Emden-Fowler) which reduces to ${\displaystyle {\frac {d^{2}u}{dx^{2}}}=\phi ^{2}u^{p}}$ if ${\displaystyle \sigma +\rho =0}$ (Bellman)	Diffusion in a slab	[23]
Besant-Rayleigh-Plesset equation	2	${\displaystyle R{\frac {d^{2}R}{dt^{2}}}+{\frac {3}{2}}\left({\frac {dR}{dt}}\right)^{2}+{\frac {4\nu _{L}}{R}}{\frac {dR}{dt}}+{\frac {2\sigma }{\rho _{L}R}}+{\frac {\Delta P(t)}{\rho }}=0}$	Spherical bubble in fluid dynamics	[24]
Blasius equation	3	${\displaystyle {\frac {d^{3}y}{dx^{3}}}+y{\frac {d^{2}y}{dx^{2}}}=0}$	Blasius boundary layer	[25]
Chandrasekhar's white dwarf equation	2	${\displaystyle {\frac {1}{x^{2}}}{\frac {d}{dx}}\left(x^{2}{\frac {dy}{dx}}\right)+(y^{2}-c)^{3/2}=0}$	Gravitational potential of white dwarf in astrophysics	[26]
De Boer-Ludford equation	2	${\displaystyle {\frac {d^{2}y}{dx^{2}}}-xy=y|y|^{\alpha },\ \alpha >0}$	Plasma physics	[27]
Emden–Chandrasekhar equation	2	${\displaystyle {\frac {1}{\xi ^{2}}}{\frac {d}{d\xi }}\left(\xi ^{2}{\frac {d\psi }{d\xi }}\right)=e^{-\psi }}$	Astrophysics	[26]
Ermakov-Pinney equation	2	${\displaystyle {\ddot {x}}+\omega ^{2}x={\frac {h^{2}}{x^{3}}}}$	Electromagnetism, oscillation, scalar field cosmologies	[28] [29]
Falkner–Skan equation	3	${\displaystyle {\frac {d^{3}y}{dx^{3}}}+y{\frac {d^{2}y}{dx^{2}}}+\beta \left[1-\left({\frac {dy}{dx}}\right)^{2}\right]=0}$	Falkner–Skan boundary layer	[30]
Friedmann equations	2	${\displaystyle {\frac {\ddot {a}}{a}}=-{\frac {4\pi G}{3}}\left(\rho +{\frac {3p}{c^{2}}}\right)+{\frac {\Lambda c^{2}}{3}}}$ and ${\displaystyle {\frac {{\dot {a}}^{2}+kc^{2}}{a^{2}}}={\frac {8\pi G\rho +\Lambda c^{2}}{3}}}$	Physical cosmology	[31]
Heisenberg equation of motion	1	${\displaystyle {\frac {d{\hat {A(t)}}}{dt}}={\frac {i}{\hbar }}[{\hat {\mathcal {H}}},{\hat {A}}]}$	Quantum mechanics	[32]
Ivey's equation	2	${\displaystyle {\frac {d^{2}y}{dx^{2}}}-{\frac {1}{y}}\left({\frac {dy}{dx}}\right)^{2}+{\frac {2}{x}}{\frac {dy}{dx}}+ky^{2}=0}$	Space charge theory	[33]
Kidder equation	2	${\displaystyle {\sqrt {1-\alpha y}}{\frac {d^{2}y}{dx^{2}}}+2x{\frac {dy}{dx}}=0,\ 0\leq \alpha \leq 1}$	Flow through porous medium	[34]
Krogdahl equation	2	${\displaystyle {\frac {d^{2}Q}{d\tau ^{2}}}=-Q+{\frac {2}{3}}\lambda Q^{2}+{\frac {14}{27}}\lambda ^{2}Q^{3}+\mu (1-Q^{2}){\frac {dQ}{d\tau }}+{\frac {2}{3}}\lambda (1-\lambda Q)\left({\frac {dQ}{d\tau }}\right)^{2}+\cdots }$	Stellar pulsation in astrophysics	[35]
Lagerstrom equation	2	${\displaystyle y''+{\frac {k}{r}}y'+\epsilon y'y=0}$	One dimensional viscous flow at low Reynolds numbers	[36]
Lane–Emden equation or polytropic differential equation	2	${\displaystyle {\frac {1}{\xi ^{2}}}{\frac {d}{d\xi }}\left({\xi ^{2}{\frac {d\theta }{d\xi }}}\right)+\theta ^{n}=0}$	Astrophysics	[37]
Liñán's equation	2	${\displaystyle {\frac {d^{2}y}{d\zeta ^{2}}}=(y^{2}-\zeta ^{2})e^{-\delta ^{1/3}(y+\gamma \zeta )}}$	Combustion	[38]
Pendulum equation	2	${\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+{\frac {g}{\ell }}\sin \theta =0}$	Mechanics	[39]
Poisson–Boltzmann equation (1d case)	2	${\displaystyle {\frac {d^{2}\theta }{dz^{2}}}+{\frac {k}{z}}{\frac {d\theta }{dz}}=-\delta e^{\theta }}$	Inflammability and the theory of thermal explosions	[40]
Stuart–Landau equation	1	${\displaystyle {\frac {dA}{dt}}=\sigma A-{\frac {l}{2}}A|A|^{2}}$	Hydrodynamic stability	[41]
Taylor–Maccoll equation	2	${\displaystyle (c^{2}-f'^{2})f''+c^{2}\cot \theta f'+(2c^{2}-f'^{2})f=0,\quad c=c(f^{2}+f'^{2})}$ where ${\displaystyle f'={\frac {df}{d\theta }}}$	Flow behind a conical shock wave	[42]
Thomas–Fermi equation	2	${\displaystyle {\frac {d^{2}y}{dx^{2}}}={\frac {1}{\sqrt {x}}}y^{3/2}}$	Quantum mechanics [43]	[44]
Toda lattice	1	${\displaystyle {\begin{aligned}{\dot {a}}(n,t)&=a(n,t){\Big (}b(n+1,t)-b(n,t){\Big )}\\{\dot {b}}(n,t)&=2{\Big (}a(n,t)^{2}-a(n-1,t)^{2}{\Big )}\end{aligned}}}$where ${\displaystyle {\begin{aligned}a(n,t)&={\frac {1}{2}}{\rm {e}}^{-(q(n+1,t)-q(n,t))/2}\\b(n,t)&=-{\frac {1}{2}}p(n,t)\end{aligned}}}$	Model of one-dimensional crystal in solid-state physics, Langmuir oscillations in plasma, quantum cohomology; notable for being a completely integrable system	[45]
Trachenko-Zaccone	1	${\displaystyle {\frac {dy}{dt}}=-{\frac {y}{\tau }}\exp(Ky)}$	Condensed matter physics; relaxation in amorphous solids and glass transition	[46] [47]

Engineering

Name	Order	Equation	Applications	Reference
Duffing equation	2	${\displaystyle {\frac {d^{2}x}{dt^{2}}}+\mu {\frac {dx}{dt}}+\alpha x+\beta x^{3}=\gamma \cos \omega t}$	Oscillators, hysteresis, chaotic dynamical systems	[48]
Lewis regulator	2	${\displaystyle y''+(1-|y|)y'+y=0}$	Oscillators	[49]
Liénard equation	2	${\displaystyle {d^{2}x \over dt^{2}}+f(x){dx \over dt}+g(x)=0}$ with ${\displaystyle f}$ odd and ${\displaystyle g}$ even	Oscillators, electrical engineering, dynamical systems	[50]
Rayleigh equation	2	${\displaystyle y''+F(y')+y=0}$	Oscillators (especially auto-oscillation), acoustics; the Van der Pol equation is a Rayleigh equation	[51]
Van der Pol equation	2	${\displaystyle {d^{2}x \over dt^{2}}-\mu (1-x^{2}){dx \over dt}+x=0}$	Oscillators, electrical engineering, chaotic dynamical systems	[52]

Chemistry

Name	Order	Equation	Applications	Reference
Brusselator	1	${\displaystyle {\begin{aligned}{d \over dt}\left\{X\right\}&=\left\{A\right\}+\left\{X\right\}^{2}\left\{Y\right\}-\left\{B\right\}\left\{X\right\}-\left\{X\right\}\\{d \over dt}\left\{Y\right\}&=\left\{B\right\}\left\{X\right\}-\left\{X\right\}^{2}\left\{Y\right\}\end{aligned}}}$	A type of autocatalytic reaction modelled at constant concentration	[53]
Oregonator	1	${\displaystyle {\begin{aligned}{\frac {d[X]}{dt}}&=k_{I}[A][Y]-k_{II}[X][Y]+k_{III}[A][X]-2k_{IV}[X]^{2}\\{\frac {d[Y]}{dt}}&=-k_{I}[A][Y]-k_{II}[X][Y]+{\frac {1}{2}}fk_{V}[B][Z]\\{\frac {d[Z]}{dt}}&=2k_{III}[A][X]-k_{V}[B][Z]\end{aligned}}}$	A type of autocatalytic reaction modelled at constant concentration	[54]

Biology and medicine

Name	Order	Equation	Applications	Reference
Allee effect	1	${\displaystyle {\frac {dN}{dt}}=-rN\left(1-{\frac {N}{A}}\right)\left(1-{\frac {N}{K}}\right)}$	Population biology	[55]
Arditi–Ginzburg equations	1	${\displaystyle {\begin{aligned}{\frac {dN}{dt}}&=f(N)\,N-g{\left(\!{\tfrac {N}{P}}\!\right)}P\\{\frac {dP}{dt}}&=e\,g{\left(\!{\tfrac {N}{P}}\!\right)}P-uP\end{aligned}}}$	Population dynamics	[56]
FitzHugh–Nagumo model or Bonhoeffer-van der Pol model	1	${\displaystyle {\begin{aligned}{\dot {v}}&=v-{\frac {v^{3}}{3}}-w+RI_{\rm {ext}}\\\tau {\dot {w}}&=v+a-bw\end{aligned}}}$	Action potentials in neurons, oscillators	[57]
Hodgkin-Huxley equations	1	${\displaystyle {\begin{aligned}I&=C_{m}{\frac {{\mathrm {d} }V_{m}}{{\mathrm {d} }t}}+{\bar {g}}_{\text{K}}n^{4}(V_{m}-V_{K})+{\bar {g}}_{\text{Na}}m^{3}h(V_{m}-V_{Na})+{\bar {g}}_{l}(V_{m}-V_{l})\\{\frac {dn}{dt}}&=\alpha _{n}(V_{m})(1-n)-\beta _{n}(V_{m})n\\{\frac {dm}{dt}}&=\alpha _{m}(V_{m})(1-m)-\beta _{m}(V_{m})m\\{\frac {dh}{dt}}&=\alpha _{h}(V_{m})(1-h)-\beta _{h}(V_{m})h\end{aligned}}}$	Action potentials in neurons	[58]
Kuramoto model	1	${\displaystyle {\frac {d\theta _{i}}{dt}}=\omega _{i}+{\frac {1}{N}}\sum _{j=1}^{N}K_{ij}\sin(\theta _{j}-\theta _{i}),\qquad i=1\ldots N}$	Synchronization, coupled oscillators	[59]
Lotka–Volterra equations	1	${\displaystyle {\begin{aligned}{\frac {dx}{dt}}&=\alpha x-\beta xy\\{\frac {dy}{dt}}&=\delta xy-\gamma y\end{aligned}}}$	Population dynamics	[60]
Price equation	1	${\displaystyle {d \over {dt}}\mathbb {E} (x)=\underbrace {{\text{Cov}}(x,f)} _{\text{Selection effect}}+\underbrace {\mathbb {E} ({\dot {x}})} _{\text{Dynamic effect}}}$	Evolution and change in allele frequency over time	[61]
SIR model	1	${\displaystyle {\begin{aligned}{\frac {dS}{dt}}&=-\beta SI\\{\frac {dI}{dt}}&=\beta SI-\gamma I\\{\frac {dR}{dt}}&=\gamma I\end{aligned}}}$	Epidemiology	[62]

Economics and finance

Name	Order	Equation	Applications	Reference
Bass diffusion model	1	${\displaystyle {\frac {dF}{dt}}=(1-F)(p+qF)}$	A Riccati equation used in marketing to describe product adoption	[63]
Ramsey–Cass–Koopmans model	1	${\displaystyle {\begin{aligned}{\dot {k}}&=f(k)-(n+\delta )k-c\\{\dot {c}}&=\sigma (c)\left[f_{k}(k)-\delta -\rho \right]\cdot c\end{aligned}}}$	Neoclassical economics model of economic growth	[64] [65]
Solow–Swan model	1	${\displaystyle {\dot {k}}(t)=sk(t)^{\alpha }-(n+g+\delta )k(t)}$	Model of long run economic growth	[66]

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

1. Panayotounakos, Dimitrios E.; Zarmpoutis, Theodoros I. (2011-10-26). "Construction of Exact Parametric or Closed Form Solutions of Some Unsolvable Classes of Nonlinear ODEs (Abel's Nonlinear ODEs of the First Kind and Relative Degenerate Equations)". International Journal of Mathematics and Mathematical Sciences. 2011: e387429. doi: 10.1155/2011/387429 . ISSN   0161-1712.
2. Weisstein, Eric W. "Bernoulli Differential Equation". mathworld.wolfram.com. Retrieved 2024-06-02.
3. Hille, Einar (1894). Lectures on ordinary differential equations. Addison-Wesley Publishing Company. p. 675. ISBN   978-0201530834.
4. Weisstein, Eric W. "Briot-Bouquet Equation". mathworld.wolfram.com. Retrieved 2024-06-03.
5. Zwillinger, Daniel (1998). Handbook of differential equations (3rd ed.). San Diego, CA: Academic Press. p. 257. ISBN   978-0-12-784396-4.
6. Jones, G.Stephen (June 1962). "On the nonlinear differential-difference equation f′(x) = −αf(x − 1) {1 + f(x)}". Journal of Mathematical Analysis and Applications. 4 (3): 440–469. doi:10.1016/0022-247X(62)90041-0.
7. Marshall, Susan H.; Smith, Donald R. (2013). "Feedback, Control, and the Distribution of Prime Numbers". Mathematics Magazine. 86 (3): 189–203. doi:10.4169/math.mag.86.3.189. ISSN   0025-570X. JSTOR   10.4169/math.mag.86.3.189.
8. Chrystal (1897). "XXIV.—On the p-discriminant of a Differential Equation of the First Order, and on Certain Points in the General Theory of Envelopes connected therewith". Transactions of the Royal Society of Edinburgh. 38 (4): 803–824. doi:10.1017/s0080456800033494. ISSN   0080-4568.
9. "Clairaut equation - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2024-06-02.
10. Weisstein, Eric W. "d'Alembert's Equation". mathworld.wolfram.com. Retrieved 2024-06-02.
11. "Darboux equation - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2024-06-02.
12. Zwillinger, Daniel (1998). Handbook of differential equations (3rd ed.). San Diego, CA: Academic Press. p. 180. ISBN   978-0-12-784396-4.
13. Weisstein, Eric W. "Euler Differential Equation". mathworld.wolfram.com. Retrieved 2024-06-02.
14. "Jacobi equation - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2024-06-03.
15. Löwner, Karl (March 1923). "Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I". Mathematische Annalen (in German). 89 (1–2): 103–121. doi:10.1007/BF01448091. ISSN   0025-5831.
16. Weisstein, Eric W. "Logistic Equation". mathworld.wolfram.com. Retrieved 2024-06-03.
17. Lorenz, Edward N. (1 March 1963). "Deterministic Nonperiodic Flow". Journal of the Atmospheric Sciences. 20 (2): 130–141. Bibcode:1963JAtS...20..130L. doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2. ISSN   0022-4928.
18. Donaldson, S. K. (September 1984). "Nahm's equations and the classification of monopoles". Communications in Mathematical Physics. 96 (3): 387–407. Bibcode:1984CMaPh..96..387D. doi:10.1007/BF01214583. ISSN   0010-3616.
19. "Painlevé-type equations - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2024-06-03.
20. Rabinovich, M. I.; Fabrikant, A. L. (1979-08-01). "Stochastic wave self-modulation in nonequilibrium media". Zhurnal Eksperimentalnoi I Teoreticheskoi Fiziki. 77: 617–629. Bibcode:1979ZhETF..77..617R. ISSN   0044-4510.
21. "Riccati equation - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2024-06-03.
22. Peitgen, Heinz-Otto; Jürgens, Hartmut; Saupe, Dietmar (2004), "12.3 The Rössler Attractor", Chaos and Fractals: New Frontiers of Science, Springer, pp. 636–646
23. Mehta, B. N; Aris, R (1971-12-01). "A note on a form of the Emden-Fowler equation". Journal of Mathematical Analysis and Applications. 36 (3): 611–621. doi:10.1016/0022-247X(71)90043-6. ISSN   0022-247X.
24. Lin, Hao; Storey, Brian D.; Szeri, Andrew J. (2002-02-10). "Inertially driven inhomogeneities in violently collapsing bubbles: the validity of the Rayleigh–Plesset equation". Journal of Fluid Mechanics. 452 (1): 145–162. Bibcode:2002JFM...452..145L. doi:10.1017/S0022112001006693. ISSN   0022-1120.
25. Boyd, John P. (January 2008). "The Blasius Function: Computations Before Computers, the Value of Tricks, Undergraduate Projects, and Open Research Problems". SIAM Review. 50 (4): 791–804. Bibcode:2008SIAMR..50..791B. doi:10.1137/070681594. ISSN   0036-1445.
26. Chandrasekhar, Subrahmanyan (1970). An introduction to the study of stellar structure. Dover books on astronomy (Unabr. and corr. republ. of orig. publ. 1939 by Univ. of Chicago Pr ed.). New York: Dover Publ. ISBN   978-0-486-60413-8.
27. Hastings, S. P.; McLeod, J. B. (1980). "A boundary value problem associated with the second painlevé transcendent and the Korteweg-de Vries equation". Archive for Rational Mechanics and Analysis. 73 (1): 31–51. Bibcode:1980ArRMA..73...31H. doi:10.1007/BF00283254. ISSN   0003-9527.
28. Morris, Richard Michael; Leach, Peter Gavin Lawrence (2015). "The Ermakov-Pinney Equation: its varied origins and the effects of the introduction of symmetry-breaking functions". arXiv: 1510.08992 [math.CA].
29. Hawkins, Rachael M.; Lidsey, James E. (2002-07-30). "Ermakov-Pinney equation in scalar field cosmologies". Physical Review D. 66 (2): 023523. arXiv: astro-ph/0112139 . Bibcode:2002PhRvD..66b3523H. doi:10.1103/PhysRevD.66.023523.
30. Stewartson, K. (July 1954). "Further solutions of the Falkner-Skan equation". Mathematical Proceedings of the Cambridge Philosophical Society . 50 (3): 454–465. Bibcode:1954PCPS...50..454S. doi:10.1017/S030500410002956X. ISSN   0305-0041.
31. Nemiroff, Robert J.; Patla, Bijunath (2008-03-01). "Adventures in Friedmann cosmology: A detailed expansion of the cosmological Friedmann equations". American Journal of Physics. 76 (3): 265–276. arXiv: astro-ph/0703739 . Bibcode:2008AmJPh..76..265N. doi:10.1119/1.2830536. ISSN   0002-9505.
32. "Heisenberg representation - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2024-06-03.
33. Orverem, J. M.; Haruna, Y. (2023-07-08). "The Generalized Sundman Transformation and Differential Forms for Linearizing the Variable Frequency Oscillator Equation and the Modified Ivey's Equation". Fudma Journal of Sciences. 7 (3): 167–170. doi: 10.33003/fjs-2023-0703-1859 . ISSN   2616-1370.
34. Iacono, Roberto; Boyd, John P. (July 2015). "The Kidder Equation". Studies in Applied Mathematics. 135 (1): 63–85. doi:10.1111/sapm.12073. hdl: 2027.42/111960 . ISSN   0022-2526.
35. Krogdahl, Wasley S. (July 1955). "Stellar Pulsation as a Limit-Cycle Phenomenon". The Astrophysical Journal. 122: 43. Bibcode:1955ApJ...122...43K. doi:10.1086/146052. ISSN   0004-637X.
36. Rosenblat, S.; Shepherd, J. (July 1975). "On the Asymptotic Solution of the Lagerstrom Model Equation". SIAM Journal on Applied Mathematics. 29 (1): 110–120. doi:10.1137/0129012. ISSN   0036-1399.
37. Lane, H. J. (1870-07-01). "On the theoretical temperature of the Sun, under the hypothesis of a gaseous mass maintaining its volume by its internal heat, and depending on the laws of gases as known to terrestrial experiment". American Journal of Science. s2-50 (148): 57–74. Bibcode:1870AmJS...50...57L. doi:10.2475/ajs.s2-50.148.57. ISSN   0002-9599.
38. Liñán, Amable (July 1974). "The asymptotic structure of counterflow diffusion flames for large activation energies". Acta Astronautica. 1 (7–8): 1007–1039. Bibcode:1974AcAau...1.1007L. doi:10.1016/0094-5765(74)90066-6.
39. Lima, F. M. S.; Arun, P. (2006-10-01). "An accurate formula for the period of a simple pendulum oscillating beyond the small angle regime". American Journal of Physics. 74 (10): 892–895. arXiv: physics/0510206 . Bibcode:2006AmJPh..74..892L. doi:10.1119/1.2215616. ISSN   0002-9505.
40. Chambré, P. L. (1952-11-01). "On the Solution of the Poisson-Boltzmann Equation with Application to the Theory of Thermal Explosions". The Journal of Chemical Physics. 20 (11): 1795–1797. Bibcode:1952JChPh..20.1795C. doi:10.1063/1.1700291. ISSN   0021-9606.
41. "On the Problem of Turbulence", Collected Papers of L.D. Landau, Elsevier, pp. 387–391, 1965, doi:10.1016/b978-0-08-010586-4.50057-2, ISBN   978-0-08-010586-4 , retrieved 2024-06-02
42. Taylor, G. I.; MacColl, J. W. (February 1933). "The air pressure on a cone moving at high speeds". Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. 139 (838): 278–297. Bibcode:1933RSPSA.139..278T. doi: 10.1098/rspa.1933.0017 . ISSN   0950-1207.
43. Davis, Harold Thayer. Introduction to nonlinear differential and integral equations. Courier Corporation, 1962.
44. Feynman, R. P.; Metropolis, N.; Teller, E. (1949-05-15). "Equations of State of Elements Based on the Generalized Fermi-Thomas Theory". Physical Review. 75 (10): 1561–1573. Bibcode:1949PhRv...75.1561F. doi:10.1103/PhysRev.75.1561. ISSN   0031-899X.
45. Teschl, Gerald (2001), "Almost everything you always wanted to know about the Toda equation", Jahresbericht der Deutschen Mathematiker-Vereinigung, 103 (4): 149–162, MR   1879178
46. Trachenko, K.; Zaccone, A. (2021-06-14). "Slow stretched-exponential and fast compressed-exponential relaxation from local event dynamics". Journal of Physics: Condensed Matter. 33: 315101. arXiv: 2010.10440 . doi:10.1088/1361-648X/ac04cd. ISSN   0953-8984.
47. Ginzburg, V. V.; Gendelman, O. V.; Zaccone, A. (2024-02-23). "Unifying Physical Framework for Stretched-Exponential, Compressed-Exponential, and Logarithmic Relaxation Phenomena in Glassy Polymers". Macromolecules. 57 (5): 2520–2529. arXiv: 2311.09321 . doi:10.1021/acs.macromol.3c02480. ISSN   0024-9297.
48. Weisstein, Eric W. "Duffing Differential Equation". mathworld.wolfram.com. Retrieved 2024-06-02.
49. Hagedorn, Peter; Stadler, Wolfram (1988). Non-linear oscillations. The Oxford engineering science series. New York: Oxford University Press. p. 163. ISBN   978-0-19-856142-2.
50. Villari, Gabriele (April 1987). "On the qualitative behaviour of solutions of Liénard equation". Journal of Differential Equations. 67 (2): 269–277. Bibcode:1987JDE....67..269V. doi:10.1016/0022-0396(87)90150-1.
51. "Rayleigh equation - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2024-06-03.
52. Weisstein, Eric W. "van der Pol Equation". mathworld.wolfram.com. Retrieved 2024-06-02.
53. Prigogine, I.; Lefever, R. (1968-02-15). "Symmetry Breaking Instabilities in Dissipative Systems. II". The Journal of Chemical Physics. 48 (4): 1695–1700. Bibcode:1968JChPh..48.1695P. doi:10.1063/1.1668896. ISSN   0021-9606.
54. Tyson†, John J. (1979). "Oscillations, bistability, and echo waves in models of the Belousov-Zhabotinskii reaction". Annals of the New York Academy of Sciences. 316 (1): 279–295. Bibcode:1979NYASA.316..279T. doi:10.1111/j.1749-6632.1979.tb29475.x. ISSN   0077-8923.
55. Cushing, J. M.; Hudson, Jarred T. (March 2012). "Evolutionary dynamics and strong Allee effects". Journal of Biological Dynamics. 6 (2): 941–958. Bibcode:2012JBioD...6..941C. doi: 10.1080/17513758.2012.697196 . ISSN   1751-3758. PMID   22881366.
56. Arditi, Roger; Ginzburg, Lev R. (9 August 1989). "Coupling in predator-prey dynamics: Ratio-Dependence". Journal of Theoretical Biology. 139 (3): 311–326. Bibcode:1989JThBi.139..311A. doi:10.1016/S0022-5193(89)80211-5.
57. Sherwood, William Erik (2014), "FitzHugh–Nagumo Model", in Jaeger, Dieter; Jung, Ranu (eds.), Encyclopedia of Computational Neuroscience, New York, NY: Springer New York, pp. 1–11, doi:10.1007/978-1-4614-7320-6_147-1, ISBN   978-1-4614-7320-6 , retrieved 2024-06-02
58. Hodgkin, A. L.; Huxley, A. F. (1952-08-28). "A quantitative description of membrane current and its application to conduction and excitation in nerve". The Journal of Physiology. 117 (4): 500–544. doi:10.1113/jphysiol.1952.sp004764. ISSN   0022-3751. PMC   1392413 . PMID   12991237.
59. Strogatz, Steven H. (1 September 2000). "From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators". Physica D: Nonlinear Phenomena. 143 (1–4): 1–20. Bibcode:2000PhyD..143....1S. doi: 10.1016/S0167-2789(00)00094-4 .
60. Weisstein, Eric W. "Lotka-Volterra Equations". mathworld.wolfram.com. Retrieved 2024-06-02.
61. Frank, Steven A. (7 August 1995). "George Price's contributions to evolutionary genetics". Journal of Theoretical Biology. 175 (3): 373–388. Bibcode:1995JThBi.175..373F. doi:10.1006/jtbi.1995.0148. PMID   7475081.
62. Kermack, W. O.; McKendrick, A. G. (August 1927). "A contribution to the mathematical theory of epidemics". Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. 115 (772): 700–721. Bibcode:1927RSPSA.115..700K. doi:10.1098/rspa.1927.0118. ISSN   0950-1207.
63. Bass, Frank M. (1 January 1969). "A New Product Growth for Model Consumer Durables". Management Science. 15 (5): 215–227. doi:10.1287/mnsc.15.5.215. ISSN   0025-1909.
64. Roe, Terry L.; Smith, Rodney B. W.; Saracoǧlu, D. Șirin (2010). Multisector growth models: theory and application. New York: Springer. p. 56. ISBN   978-0-387-77357-5. OCLC   288985452.
65. Blanchard, Olivier; Fischer, Stanley (1989). Lectures on macroeconomics. Cambridge, Mass: MIT Press. p. 40. ISBN   978-0-262-02283-5.
66. Agénor, Pierre-Richard (2004). The economics of adjustment and growth (2nd ed.). Cambridge, Mass: Harvard University Press. pp. 439–460. ISBN   978-0-674-01578-4.