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 edit

Name	Order	Equation
Abel's differential equation of the first kind	1	${\frac {dy}{dx}}=f_{o}(x)+f_{1}(x)y+f_{2}(x)y^{2}+f_{3}(x)y^{3}$
Abel's differential equation of the second kind	1	$(g_{o}(x)+g_{1}(x)y){\frac {dy}{dx}}=f_{o}(x)+f_{1}(x)y+f_{2}(x)y^{2}+f_{3}(x)y^{3}$
Bernoulli equation	1	${\frac {dy}{dx}}+P(x)y=Q(x)y^{n}$
Chrystal's equation	1	$\left({\frac {dy}{dx}}\right)^{2}+Ax{\frac {dy}{dx}}+By+Cx^{2}=0$
Clairaut's equation	1	$y=x{\frac {dy}{dx}}+f\left({\frac {dy}{dx}}\right)$
D'Alembert's equation	1	$y=xf\left({\frac {dy}{dx}}\right)+g\left({\frac {dy}{dx}}\right)$
Darboux equation	1	${\frac {dy}{dx}}={\frac {P(x,y)+yR(x,y)}{Q(x,y)+xR(x,y)}}$
Euler's differential equation	1	${\frac {dy}{dx}}+{\frac {\sqrt {a_{0}+a_{1}y+a_{2}y^{2}+a_{3}y^{3}+a_{4}y^{4}}}{\sqrt {a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}}}}=0$
Ivey's equation	2	${\frac {d^{2}y}{dx^{2}}}-{\frac {1}{y}}\left({\frac {dy}{dx}}\right)^{2}+{\frac {2}{x}}{\frac {dy}{dx}}+ky^{2}=0$
Jacobi's differential equation	1	${\frac {dy}{dx}}={\frac {Axy+By^{2}+ax+by+c}{Ax^{2}+Bxy+\alpha x+\beta y+\gamma }}$
Painlevé I transcendent	2	${\frac {d^{2}y}{dt^{2}}}=6y^{2}+t$
Painlevé II transcendent	2	${\frac {d^{2}y}{dt^{2}}}=2y^{3}+ty+\alpha$
Painlevé III transcendent	2	$ty{\frac {d^{2}y}{dt^{2}}}=t\left({\frac {dy}{dt}}\right)^{2}-y{\frac {dy}{dt}}+\delta t+\beta y+\alpha y^{3}+\gamma ty^{4}$
Painlevé IV transcendent	2	$y{\frac {d^{2}y}{dt^{2}}}={\tfrac {1}{2}}\left({\frac {dy}{dt}}\right)^{2}+\beta +2(t^{2}-\alpha )y^{2}+4ty^{3}+{\tfrac {3}{2}}y^{4}$
Painlevé V transcendent	2	${\frac {d^{2}y}{dt^{2}}}=\left({\frac {1}{2y}}+{\frac {1}{y-1}}\right)\left({\frac {dy}{dt}}\right)^{2}-{\frac {1}{t}}{\frac {dy}{dt}}+{\frac {(y-1)^{2}}{t^{2}}}\left(\alpha y+{\frac {\beta }{y}}\right)+\gamma {\frac {y}{t}}+\delta {\frac {y(y+1)}{y-1}}$
Painlevé VI transcendent	2	${\frac {d^{2}y}{dt^{2}}}={\frac {1}{2}}\left({\frac {1}{y}}+{\frac {1}{y-1}}+{\frac {1}{y-t}}\right)\left({\frac {dy}{dt}}\right)^{2}-\left({\frac {1}{t}}+{\frac {1}{t-1}}+{\frac {1}{y-t}}\right){\frac {dy}{dt}}+{\frac {y(y-1)(y-t)}{t^{2}(t-1)^{2}}}\left(\alpha +\beta {\frac {t}{y^{2}}}+\gamma {\frac {t-1}{(y-1)^{2}}}+\delta {\frac {t(t-1)}{(y-t)^{2}}}\right)$
Riccati equation	1	${\frac {dy}{dx}}+Q(x)y+R(x)y^{2}=P(x)$

Physics edit

Name	Order	Equation	Applications	Reference
Bellman's equation or Emden-Fowler's equation	2	${\frac {d}{dt}}\left(t^{\rho }{\frac {du}{dt}}\right)=t^{\sigma }u^{\rho }$ (Emden-Fowler) which reduces to ${\frac {d^{2}u}{dx^{2}}}=\phi ^{2}u^{p}$ if $\sigma +\rho =0$ (Bellman)	Diffusion in a slab	^[1]
Besant-Rayleigh-Plesset equation	2	$R{\frac {d^{2}R}{dt^{2}}}+{\frac {3}{2}}\left({\frac {dR}{dt}}\right)^{2}+{\frac {4\nu _{L}}{R}}{\frac {dR}{dt}}+{\frac {2\sigma }{\rho _{L}R}}+{\frac {\Delta P(t)}{\rho }}=0$	Spherical bubble in fluid dynamics	^[2]
Blasius equation	3	${\frac {d^{3}y}{dx^{3}}}+y{\frac {d^{2}y}{dx^{2}}}=0$	Blasius boundary layer	^[3]
Chandrasekhar's white dwarf equation	2	${\frac {1}{x^{2}}}{\frac {d}{dx}}\left(x^{2}{\frac {dy}{dx}}\right)+(y^{2}-c)^{3/2}=0$	Gravitational potential of white dwarf in astrophysics	^[4]
De Boer-Ludford equation	2	${\frac {d^{2}y}{dx^{2}}}-xy=y\|y\|^{\alpha },\ \alpha >0$	Plasma physics	^[5]
Emden–Chandrasekhar equation	2	${\frac {1}{\xi ^{2}}}{\frac {d}{d\xi }}\left(\xi ^{2}{\frac {d\psi }{d\xi }}\right)=e^{-\psi }$	Astrophysics	^[4]
Falkner–Skan equation	3	${\frac {d^{3}y}{dx^{3}}}+y{\frac {d^{2}y}{dx^{2}}}+\beta \left[1-\left({\frac {dy}{dx}}\right)^{2}\right]=0$	Falkner–Skan boundary layer	^[6]
Kidder equation	2	${\sqrt {1-\alpha y}}{\frac {d^{2}y}{dx^{2}}}+2x{\frac {dy}{dx}}=0,\ 0\leq \alpha \leq 1$	Flow through porous medium	^[7]
Krogdahl equation	2	${\frac {d^{2}Q}{d\tau ^{2}}}=-Q+{\frac {2}{3}}\lambda Q^{2}+{\frac {14}{27}}\lambda ^{2}Q^{3}+\mu (1-Q^{2}){\frac {dQ}{d\tau }}+{\frac {2}{3}}\lambda (1-\lambda Q)\left({\frac {dQ}{d\tau }}\right)^{2}+\cdots$	Stellar pulsation in astrophysics	^[8]
Lane–Emden equation	2	${\frac {1}{\xi ^{2}}}{\frac {d}{d\xi }}\left({\xi ^{2}{\frac {d\theta }{d\xi }}}\right)+\theta ^{n}=0$	Astrophysics	^[9]
Langmuir-Blodgett equation	2	${\sqrt {y}}{\frac {d^{2}y}{dx^{2}}}=e^{x}$
Langmuir-Boguslavski equation	2	${\frac {d}{dx}}\left(x^{n}{\frac {dy}{dx}}\right)={\frac {1}{\sqrt {y}}}$
Liñán's equation	2	${\frac {d^{2}y}{d\zeta ^{2}}}=(y^{2}-\zeta ^{2})e^{-\delta ^{1/3}(y+\gamma \zeta )}$	Combustion	^[10]
Rayleigh equation	2	${\frac {d^{2}y}{dx^{2}}}+k{\frac {dy}{dx}}+m\left({\frac {dy}{dx}}\right)^{3}+n^{2}y=0$	Hydrodynamic stability
Stuart–Landau equation	1	${\frac {dA}{dt}}=\sigma A-{\frac {l}{2}}A\|A\|^{2}$	Hydrodynamic stability	^[11]
Taylor–Maccoll equation	2	$(c^{2}-f'^{2})f''+c^{2}\cot \theta f'+(2c^{2}-f'^{2})f=0,\quad c=c(f^{2}+f'^{2})$ where $f'={\frac {df}{d\theta }}$	Flow behind a conical shock wave	^[12]
Thomas–Fermi equation	2	${\frac {d^{2}y}{dx^{2}}}={\frac {1}{\sqrt {x}}}y^{3/2}$	Quantum mechanics^[13]	^[14]

Engineering edit

Name	Order	Equation	Applications
Langmuir equation	2	$3y{\frac {d^{2}y}{dx^{2}}}+\left({\frac {dy}{dx}}\right)^{2}+4y{\frac {dy}{dx}}+y^{2}=1$	Environmental Engineering
Van der Pol equation	2	${d^{2}x \over dt^{2}}-\mu (1-x^{2}){dx \over dt}+x=0$	Oscillators
Duffing equation	2	${\frac {d^{2}x}{dt^{2}}}+\mu {\frac {dx}{dt}}+\alpha x+\beta x^{3}=\gamma \cos \omega t$	Oscillators

References edit

^ 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.
^ 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: 145–162. doi:10.1017/S0022112001006693. ISSN 0022-1120.
^ Boyd, John P. (2008-01). "The Blasius Function: Computations Before Computers, the Value of Tricks, Undergraduate Projects, and Open Research Problems". SIAM Review. 50 (4): 791–804. doi:10.1137/070681594. ISSN 0036-1445. {{cite journal}}: Check date values in: |date= (help)
^ ^a ^b 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.
^ Hastings, S. P.; McLeod, J. B. (1980). [http://link.springer.com/10.1007/BF00283254 "A boundary value problem associated with the second painlev� transcendent and the Korteweg-de Vries equation"]. Archive for Rational Mechanics and Analysis. 73 (1): 31–51. doi:10.1007/BF00283254. ISSN 0003-9527. {{cite journal}}: replacement character in |title= at position 60 (help)
^ Stewartson, K. (1954-07). "Further solutions of the Falkner-Skan equation". Mathematical Proceedings of the Cambridge Philosophical Society. 50 (3): 454–465. doi:10.1017/S030500410002956X. ISSN 0305-0041. {{cite journal}}: Check date values in: |date= (help)
^ Iacono, Roberto; Boyd, John P. (2015-07). "The Kidder Equation:". Studies in Applied Mathematics. 135 (1): 63–85. doi:10.1111/sapm.12073. ISSN 0022-2526. {{cite journal}}: Check date values in: |date= (help)
^ Krogdahl, Wasley S. (1955-07). "Stellar Pulsation as a Limit-Cycle Phenomenon". The Astrophysical Journal. 122: 43. doi:10.1086/146052. ISSN 0004-637X. {{cite journal}}: Check date values in: |date= (help)
^ 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. doi:10.2475/ajs.s2-50.148.57. ISSN 0002-9599.
^ Liñán, Amable (1974-07). "The asymptotic structure of counterflow diffusion flames for large activation energies". Acta Astronautica. 1 (7–8): 1007–1039. doi:10.1016/0094-5765(74)90066-6. {{cite journal}}: Check date values in: |date= (help)
^ "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
^ "The air pressure on a cone moving at high speeds.—I". Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. 139 (838): 278–297. 1933-02. doi:10.1098/rspa.1933.0017. ISSN 0950-1207. {{cite journal}}: Check date values in: |date= (help)
^ Davis, Harold Thayer. Introduction to nonlinear differential and integral equations. Courier Corporation, 1962.
^ 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. doi:10.1103/PhysRev.75.1561. ISSN 0031-899X.

[1] 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.

[2] 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: 145–162. doi:10.1017/S0022112001006693. ISSN 0022-1120.

[3] Boyd, John P. (2008-01). "The Blasius Function: Computations Before Computers, the Value of Tricks, Undergraduate Projects, and Open Research Problems". SIAM Review. 50 (4): 791–804. doi:10.1137/070681594. ISSN 0036-1445. {{cite journal}}: Check date values in: |date= (help)

[:0-4] 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.

[5] Hastings, S. P.; McLeod, J. B. (1980). [http://link.springer.com/10.1007/BF00283254 "A boundary value problem associated with the second painlev� transcendent and the Korteweg-de Vries equation"]. Archive for Rational Mechanics and Analysis. 73 (1): 31–51. doi:10.1007/BF00283254. ISSN 0003-9527. {{cite journal}}: replacement character in |title= at position 60 (help)

[6] Stewartson, K. (1954-07). "Further solutions of the Falkner-Skan equation". Mathematical Proceedings of the Cambridge Philosophical Society. 50 (3): 454–465. doi:10.1017/S030500410002956X. ISSN 0305-0041. {{cite journal}}: Check date values in: |date= (help)

[7] Iacono, Roberto; Boyd, John P. (2015-07). "The Kidder Equation:". Studies in Applied Mathematics. 135 (1): 63–85. doi:10.1111/sapm.12073. ISSN 0022-2526. {{cite journal}}: Check date values in: |date= (help)

[8] Krogdahl, Wasley S. (1955-07). "Stellar Pulsation as a Limit-Cycle Phenomenon". The Astrophysical Journal. 122: 43. doi:10.1086/146052. ISSN 0004-637X. {{cite journal}}: Check date values in: |date= (help)

[9] 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. doi:10.2475/ajs.s2-50.148.57. ISSN 0002-9599.

[10] Liñán, Amable (1974-07). "The asymptotic structure of counterflow diffusion flames for large activation energies". Acta Astronautica. 1 (7–8): 1007–1039. doi:10.1016/0094-5765(74)90066-6. {{cite journal}}: Check date values in: |date= (help)

[11] "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

[12] "The air pressure on a cone moving at high speeds.—I". Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. 139 (838): 278–297. 1933-02. doi:10.1098/rspa.1933.0017. ISSN 0950-1207. {{cite journal}}: Check date values in: |date= (help)

[13] Davis, Harold Thayer. Introduction to nonlinear differential and integral equations. Courier Corporation, 1962.

[14] 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. doi:10.1103/PhysRev.75.1561. ISSN 0031-899X.

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List of nonlinear ordinary differential equations

Contents

Mathematics edit

Physics edit

Engineering edit

See also edit

References edit