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	${\frac {dy}{dx}}=f_{o}(x)+f_{1}(x)y+f_{2}(x)y^{2}+f_{3}(x)y^{3}$	Class of differential equation which may be solved implicitly	^[1]
Abel's differential equation of the second kind	1	$(g_{o}(x)+g_{1}(x)y){\frac {dy}{dx}}=f_{o}(x)+f_{1}(x)y+f_{2}(x)y^{2}+f_{3}(x)y^{3}$	Class of differential equation which may be solved implicitly	^[1]
Bernoulli equation	1	${\frac {dy}{dx}}+P(x)y=Q(x)y^{n}$	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	${\frac {dx}{dt}}=(a-x(t-1))x(t)$ or the related form $f'(x)={\frac {-f(x)f({\sqrt {x}})}{2x}}$	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	$\left({\frac {dy}{dx}}\right)^{2}+Ax{\frac {dy}{dx}}+By+Cx^{2}=0$	Generalization of Clairaut's equation with a singular solution	^[8]
Clairaut's equation	1	$y=x{\frac {dy}{dx}}+f\left({\frac {dy}{dx}}\right)$	Particular case of d'Alembert's equation which may be solved exactly	^[9]
d'Alembert's equation or Lagrange's equation	1	$y=xf\left({\frac {dy}{dx}}\right)+g\left({\frac {dy}{dx}}\right)$	May be solved exactly	^[10]
Darboux equation	1	${\frac {dy}{dx}}={\frac {P(x,y)+yR(x,y)}{Q(x,y)+xR(x,y)}}$	Can be reduced to a Bernoulli differential equation; a general case of the Jacobi equation	^[11]
Elliptic function	1	$y'={\sqrt {\left(1-y^{2}\right)\left(1-k^{2}y^{2}\right)}}$	Equation for which the elliptic functions are solutions	^[12]
Euler's differential equation	1	${\frac {dy}{dx}}+{\frac {\sqrt {a_{0}+a_{1}y+a_{2}y^{2}+a_{3}y^{3}+a_{4}y^{4}}}{\sqrt {a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}}}}=0$	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	^[13]
Jacobi equation	1	${\frac {dy}{dx}}={\frac {Axy+By^{2}+ax+by+c}{Ax^{2}+Bxy+\alpha x+\beta y+\gamma }}$	Special case of the Darboux equation, integrable in closed form	^[14]
Loewner differential equation	1	${\frac {dw}{dt}}=-wp_{t}(w)$	Important in complex analysis and geometric function theory	^[15]
Logistic differential equation (sometimes known as the Verhulst model)	2	${\frac {d}{dx}}f(x)=f(x){\big (}1-f(x){\big )}$	Special case of the Bernoulli differential equation; many applications including in population dynamics	^[16]
Lorenz attractor	1	${\begin{aligned}{\frac {\mathrm {d} x}{\mathrm {d} t}}&=\sigma (y-x)\\{\frac {\mathrm {d} y}{\mathrm {d} t}}&=x(\rho -z)-y\\{\frac {\mathrm {d} z}{\mathrm {d} t}}&=xy-\beta z\end{aligned}}$	Chaos theory, dynamical systems, meteorology	^[17]
Nahm equations	1	${\begin{aligned}{\frac {dT_{1}}{dz}}&=[T_{2},T_{3}]\\{\frac {dT_{2}}{dz}}&=[T_{3},T_{1}]\\{\frac {dT_{3}}{dz}}&=[T_{1},T_{2}]\end{aligned}}$	Differential geometry, gauge theory, mathematical physics, magnetic monopoles	^[18]
Painlevé I transcendent	2	${\frac {d^{2}y}{dt^{2}}}=6y^{2}+t$	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	${\frac {d^{2}y}{dt^{2}}}=2y^{3}+ty+\alpha$	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	^[19]
Painlevé III transcendent	2	$ty{\frac {d^{2}y}{dt^{2}}}=t\left({\frac {dy}{dt}}\right)^{2}-y{\frac {dy}{dt}}+\delta t+\beta y+\alpha y^{3}+\gamma ty^{4}$	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 IV transcendent	2	$y{\frac {d^{2}y}{dt^{2}}}={\tfrac {1}{2}}\left({\frac {dy}{dt}}\right)^{2}+\beta +2(t^{2}-\alpha )y^{2}+4ty^{3}+{\tfrac {3}{2}}y^{4}$	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 V transcendent	2	${\frac {d^{2}y}{dt^{2}}}=\left({\frac {1}{2y}}+{\frac {1}{y-1}}\right)\left({\frac {dy}{dt}}\right)^{2}-{\frac {1}{t}}{\frac {dy}{dt}}+{\frac {(y-1)^{2}}{t^{2}}}\left(\alpha y+{\frac {\beta }{y}}\right)+\gamma {\frac {y}{t}}+\delta {\frac {y(y+1)}{y-1}}$	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 VI transcendent	2	${\begin{aligned}{\frac {d^{2}y}{dt^{2}}}&={\frac {1}{2}}\left({\frac {1}{y}}+{\frac {1}{y-1}}+{\frac {1}{y-t}}\right)\left({\frac {dy}{dt}}\right)^{2}-\left({\frac {1}{t}}+{\frac {1}{t-1}}+{\frac {1}{y-t}}\right){\frac {dy}{dt}}\\&+{\frac {y(y-1)(y-t)}{t^{2}(t-1)^{2}}}\left(\alpha +\beta {\frac {t}{y^{2}}}+\gamma {\frac {t-1}{(y-1)^{2}}}+\delta {\frac {t(t-1)}{(y-t)^{2}}}\right)\\\end{aligned}}$	All of the other Painlevé transcendents are degenerations of the sixth	^[19]
Rabinovich–Fabrikant equations	1	${\begin{aligned}{\dot {x}}&=y(z-1+x^{2})+\gamma x\\{\dot {y}}&=x(3z+1-x^{2})+\gamma y\\{\dot {z}}&=-2z(\alpha +xy)\end{aligned}}$	Chaos theory, dynamical systems	^[20]
Riccati equation	1	${\frac {dy}{dx}}+Q(x)y+R(x)y^{2}=P(x)$	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	${\begin{aligned}{\frac {dx}{dt}}&=-y-z\\{\frac {dy}{dt}}&=x+ay\\{\frac {dz}{dt}}&=b+z(x-c)\end{aligned}}$	Chaos theory, dynamical systems	^[22]

Physics

Name	Order	Equation	Applications	Reference
Bellman's equation or Emden-Fowler's equation	2	${\frac {d}{dt}}\left(t^{\rho }{\frac {du}{dt}}\right)=t^{\sigma }u^{\rho }$ (Emden-Fowler) which reduces to ${\frac {d^{2}u}{dx^{2}}}=\phi ^{2}u^{p}$ if $\sigma +\rho =0$ (Bellman)	Diffusion in a slab	^[23]
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	^[24]
Blasius equation	3	${\frac {d^{3}y}{dx^{3}}}+y{\frac {d^{2}y}{dx^{2}}}=0$	Blasius boundary layer	^[25]
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	^[26]
De Boer-Ludford equation	2	${\frac {d^{2}y}{dx^{2}}}-xy=y\|y\|^{\alpha },\ \alpha >0$	Plasma physics	^[27]
Emden–Chandrasekhar equation	2	${\frac {1}{\xi ^{2}}}{\frac {d}{d\xi }}\left(\xi ^{2}{\frac {d\psi }{d\xi }}\right)=e^{-\psi }$	Astrophysics	^[26]
Ermakov-Pinney equation	2	${\ddot {x}}+\omega ^{2}x={\frac {h^{2}}{x^{3}}}$	Electromagnetism, oscillation, scalar field cosmologies	^[28]^[29]
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	^[30]
Friedmann equations	2	${\frac {\ddot {a}}{a}}=-{\frac {4\pi G}{3}}\left(\rho +{\frac {3p}{c^{2}}}\right)+{\frac {\Lambda c^{2}}{3}}$ and ${\frac {{\dot {a}}^{2}+kc^{2}}{a^{2}}}={\frac {8\pi G\rho +\Lambda c^{2}}{3}}$	Physical cosmology	^[31]
Heisenberg equation of motion	1	${\frac {d{\hat {A(t)}}}{dt}}={\frac {i}{\hbar }}[{\hat {\mathcal {H}}},{\hat {A}}]$	Quantum mechanics	^[32]
Ivey's equation	2	${\frac {d^{2}y}{dx^{2}}}-{\frac {1}{y}}\left({\frac {dy}{dx}}\right)^{2}+{\frac {2}{x}}{\frac {dy}{dx}}+ky^{2}=0$	Space charge theory	^[33]
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	^[34]
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	^[35]
Lagerstrom equation	2	$y''+{\frac {k}{r}}y'+\epsilon y'y=0$	One dimensional viscous flow at low Reynolds numbers	^[36]
Lane–Emden equation or polytropic differential equation	2	${\frac {1}{\xi ^{2}}}{\frac {d}{d\xi }}\left({\xi ^{2}{\frac {d\theta }{d\xi }}}\right)+\theta ^{n}=0$	Astrophysics	^[37]
Liñán's equation	2	${\frac {d^{2}y}{d\zeta ^{2}}}=(y^{2}-\zeta ^{2})e^{-\delta ^{1/3}(y+\gamma \zeta )}$	Combustion	^[38]
Pendulum equation	2	${\frac {d^{2}\theta }{dt^{2}}}+{\frac {g}{\ell }}\sin \theta =0$	Mechanics	^[39]
Poisson–Boltzmann equation (1d case)	2	${\frac {d^{2}\theta }{dz^{2}}}+{\frac {k}{z}}{\frac {d\theta }{dz}}=-\delta e^{\theta }$	Inflammability and the theory of thermal explosions	^[40]
Stuart–Landau equation	1	${\frac {dA}{dt}}=\sigma A-{\frac {l}{2}}A\|A\|^{2}$	Hydrodynamic stability	^[41]
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	^[42]
Thomas–Fermi equation	2	${\frac {d^{2}y}{dx^{2}}}={\frac {1}{\sqrt {x}}}y^{3/2}$	Quantum mechanics^[43]	^[44]
Toda lattice	1	${\begin{aligned}{\dot {a}}(n,t)&=a(n,t){\Big (}b(n+1,t)-b(n,t){\Big )}\\{\dot {b}}(n,t)&=2{\Big (}a(n,t)^{2}-a(n-1,t)^{2}{\Big )}\end{aligned}}$ where ${\begin{aligned}a(n,t)&={\frac {1}{2}}{\rm {e}}^{-(q(n+1,t)-q(n,t))/2}\\b(n,t)&=-{\frac {1}{2}}p(n,t)\end{aligned}}$	Model of one-dimensional crystal in solid-state physics, Langmuir oscillations in plasma, quantum cohomology; notable for being a completely integrable system	^[45]

Engineering

Name	Order	Equation	Applications	Reference
Duffing equation	2	${\frac {d^{2}x}{dt^{2}}}+\mu {\frac {dx}{dt}}+\alpha x+\beta x^{3}=\gamma \cos \omega t$	Oscillators, hysteresis, chaotic dynamical systems	^[46]
Lewis regulator	2	$y''+(1-\|y\|)y'+y=0$	Oscillators	^[47]
Liénard equation	2	${d^{2}x \over dt^{2}}+f(x){dx \over dt}+g(x)=0$ with $f$ odd and $g$ even	Oscillators, electrical engineering, dynamical systems	^[48]
Rayleigh equation	2	$y''+F(y')+y=0$	Oscillators (especially auto-oscillation), acoustics; the Van der Pol equation is a Rayleigh equation	^[49]
Van der Pol equation	2	${d^{2}x \over dt^{2}}-\mu (1-x^{2}){dx \over dt}+x=0$	Oscillators, electrical engineering, chaotic dynamical systems	^[50]

Chemistry

Name	Order	Equation	Applications	Reference
Brusselator	1	${\begin{aligned}{d \over dt}\left\{X\right\}&=\left\{A\right\}+\left\{X\right\}^{2}\left\{Y\right\}-\left\{B\right\}\left\{X\right\}-\left\{X\right\}\\{d \over dt}\left\{Y\right\}&=\left\{B\right\}\left\{X\right\}-\left\{X\right\}^{2}\left\{Y\right\}\end{aligned}}$	A type of autocatalytic reaction modelled at constant concentration	^[51]
Oregonator	1	${\begin{aligned}{\frac {d[X]}{dt}}&=k_{I}[A][Y]-k_{II}[X][Y]+k_{III}[A][X]-2k_{IV}[X]^{2}\\{\frac {d[Y]}{dt}}&=-k_{I}[A][Y]-k_{II}[X][Y]+{\frac {1}{2}}fk_{V}[B][Z]\\{\frac {d[Z]}{dt}}&=2k_{III}[A][X]-k_{V}[B][Z]\end{aligned}}$	A type of autocatalytic reaction modelled at constant concentration	^[52]

Biology and medicine

Name	Order	Equation	Applications	Reference
Allee effect	1	${\frac {dN}{dt}}=-rN\left(1-{\frac {N}{A}}\right)\left(1-{\frac {N}{K}}\right)$	Population biology	^[53]
Arditi–Ginzburg equations	1	${\begin{aligned}{\frac {dN}{dt}}&=f(N)\,N-g{\left(\!{\tfrac {N}{P}}\!\right)}P\\{\frac {dP}{dt}}&=e\,g{\left(\!{\tfrac {N}{P}}\!\right)}P-uP\end{aligned}}$	Population dynamics	^[54]
FitzHugh–Nagumo model or Bonhoeffer-van der Pol model	1	${\begin{aligned}{\dot {v}}&=v-{\frac {v^{3}}{3}}-w+RI_{\rm {ext}}\\\tau {\dot {w}}&=v+a-bw\end{aligned}}$	Action potentials in neurons, oscillators	^[55]
Hodgkin-Huxley equations	1	${\begin{aligned}I&=C_{m}{\frac {{\mathrm {d} }V_{m}}{{\mathrm {d} }t}}+{\bar {g}}_{\text{K}}n^{4}(V_{m}-V_{K})+{\bar {g}}_{\text{Na}}m^{3}h(V_{m}-V_{Na})+{\bar {g}}_{l}(V_{m}-V_{l})\\{\frac {dn}{dt}}&=\alpha _{n}(V_{m})(1-n)-\beta _{n}(V_{m})n\\{\frac {dm}{dt}}&=\alpha _{m}(V_{m})(1-m)-\beta _{m}(V_{m})m\\{\frac {dh}{dt}}&=\alpha _{h}(V_{m})(1-h)-\beta _{h}(V_{m})h\end{aligned}}$	Action potentials in neurons	^[56]
Kuramoto model	1	${\frac {d\theta _{i}}{dt}}=\omega _{i}+{\frac {1}{N}}\sum _{j=1}^{N}K_{ij}\sin(\theta _{j}-\theta _{i}),\qquad i=1\ldots N$	Synchronization, coupled oscillators	^[57]
Lotka–Volterra equations	1	${\begin{aligned}{\frac {dx}{dt}}&=\alpha x-\beta xy\\{\frac {dy}{dt}}&=\delta xy-\gamma y\end{aligned}}$	Population dynamics	^[58]
Price equation	1	${d \over {dt}}\mathbb {E} (x)=\underbrace {{\text{Cov}}(x,f)} _{\text{Selection effect}}+\underbrace {\mathbb {E} ({\dot {x}})} _{\text{Dynamic effect}}$	Evolution and change in allele frequency over time	^[59]
SIR model	1	${\begin{aligned}{\frac {dS}{dt}}&=-\beta SI\\{\frac {dI}{dt}}&=\beta SI-\gamma I\\{\frac {dR}{dt}}&=\gamma I\end{aligned}}$	Epidemiology	^[60]

Economics and finance

Name	Order	Equation	Applications	Reference
Bass diffusion model	1	${\frac {dF}{dt}}=(1-F)(p+qF)$	A Riccati equation used in marketing to describe product adoption	^[61]
Ramsey–Cass–Koopmans model	1	${\begin{aligned}{\dot {k}}&=f(k)-(n+\delta )k-c\\{\dot {c}}&=\sigma (c)\left[f_{k}(k)-\delta -\rho \right]\cdot c\end{aligned}}$	Neoclassical economics model of economic growth	^[62]^[63]
Solow–Swan model	1	${\dot {k}}(t)=sk(t)^{\alpha }-(n+g+\delta )k(t)$	Model of long run economic growth	^[64]

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^ 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.
^ 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
^ Weisstein, Eric W. "Duffing Differential Equation". mathworld.wolfram.com. Retrieved 2024-06-02.
^ 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.
^ 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.
^ "Rayleigh equation - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2024-06-03.
^ Weisstein, Eric W. "van der Pol Equation". mathworld.wolfram.com. Retrieved 2024-06-02.
^ 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.
^ 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.
^ 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.
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^ 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
^ 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.
^ 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.
^ Weisstein, Eric W. "Lotka-Volterra Equations". mathworld.wolfram.com. Retrieved 2024-06-02.
^ 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.
^ 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.
^ 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.
^ 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.
^ Blanchard, Olivier; Fischer, Stanley (1989). Lectures on macroeconomics. Cambridge, Mass: MIT Press. p. 40. ISBN 978-0-262-02283-5.
^ 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.

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