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Ordinary Differential Equation

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An ordinary differential equation (frequently called an "ODE," "diff eq," or "diffy Q") is an equality involving a function and its derivatives. An ODE of order n is an equation of the form

F(x,y,y^',...,y^((n)))=0,
(1)

where y is a function of x, y^'=dy/dx is the first derivative with respect to x, and y^((n))=d^ny/dx^n is the nth derivative with respect to x.

Nonhomogeneous ordinary differential equations can be solved if the general solution to the homogenous version is known, in which case the undetermined coefficients method or variation of parameters can be used to find the particular solution.

Many ordinary differential equations can be solved exactly in the Wolfram Language using DSolve[eqn, y, x], and numerically using NDSolve[eqn, y, {x, xmin, xmax}].

An ODE of order n is said to be linear if it is of the form

a_n(x)y^((n))+a_(n-1)(x)y^((n-1))+...+a_1(x)y^'+a_0(x)y=Q(x).
(2)

A linear ODE where Q(x)=0 is said to be homogeneous. Confusingly, an ODE of the form

y^'=f(y/x)
(3)

is also sometimes called "homogeneous."

In general, an nth-order ODE has n linearly independent solutions. Furthermore, any linear combination of linearly independent functions solutions is also a solution.

Simple theories exist for first-order (integrating factor) and second-order (Sturm-Liouville theory) ordinary differential equations, and arbitrary ODEs with linear constant coefficients can be solved when they are of certain factorable forms. Integral transforms such as the Laplace transform can also be used to solve classes of linear ODEs. Morse and Feshbach (1953, pp. 667-674) give canonical forms and solutions for second-order ordinary differential equations.

While there are many general techniques for analytically solving classes of ODEs, the only practical solution technique for complicated equations is to use numerical methods (Milne 1970, Jeffreys and Jeffreys 1988). The most popular of these is the Runge-Kutta method, but many others have been developed, including the collocation method and Galerkin method. A vast amount of research and huge numbers of publications have been devoted to the numerical solution of differential equations, both ordinary and partial (PDEs) as a result of their importance in fields as diverse as physics, engineering, economics, and electronics.

The solutions to an ODE satisfy existence and uniqueness properties. These can be formally established by Picard's existence theorem for certain classes of ODEs. Let a system of first-order ODE be given by

(dx_i)/(dt)=f_i(x_1,...,x_n,t),
(4)

for i=1, ..., n and let the functions f_i(x_1,...,x_n,t), where i=1, ..., n, all be defined in a domain D of the (n+1)-dimensional space of the variables x_1, ..., x_n, t. Let these functions be continuous in D and have continuous first partial derivatives partialf_i/partialx_j for i=1, ..., n and j=1, ..., n in D. Let (x_1^0,...,x_n^0) be in D. Then there exists a solution of (4) given by

x_1=x_1(t),...,x_n=x_n(t)
(5)

for t_0-delta<t<t_0+delta (where delta>0) satisfying the initial conditions

x_1(t_0)=x_1^0,...,x_n(t_0)=x_n^0.
(6)

Furthermore, the solution is unique, so that if

x_1=x_1^*(t),...,x_n=x_n^*(t)
(7)

is a second solution of (◇) for t_0-delta<t<t_0+delta satisfying (◇), then x_i(t)=x_i^*(t) for t_0-delta<t<t_0+delta. Because every nth-order ODE can be expressed as a system of n first-order ODEs, this theorem also applies to the single nth-order ODE.

An exact first-order ordinary differential equation is one of the form

p(x,y)dx+q(x,y)dy=0,
(8)

where

(partialp)/(partialy)=(partialq)/(partialx).
(9)

An equation of the form (◇) with

(partialp)/(partialy)!=(partialq)/(partialx)
(10)

is said to be nonexact. If

((partialp)/(partialy)-(partialq)/(partialx))/q=f(x)
(11)

in (◇), it has an x-dependent integrating factor. If

((partialq)/(partialx)-(partialp)/(partialy))/(xp-yq)=f(xy)
(12)

in (◇), it has an xy-dependent integrating factor. If

((partialq)/(partialx)-(partialp)/(partialy))/p=f(y)
(13)

in (◇), it has a y-dependent integrating factor.

Other special first-order types include cross multiple equations

yf(xy)dx+xg(xy)dy=0,
(14)

homogeneous equations

y^'=f(y/x),
(15)

linear equations

y^'+p(x)y=q(x),
(16)

and separable equations

y^'=X(x)Y(y).
(17)

Special classes of second-order ordinary differential equations include

y^('')=f(y,y^')
(18)

(x missing) and

y^('')=f(x,y^')
(19)

(y missing). A second-order linear homogeneous ODE

y^('')+P(x)y^'+Q(x)y=0
(20)

for which

(Q^'(x)+2P(x)Q(x))/(2[Q(x)]^(3/2))=[constant]
(21)

can be transformed to one with constant coefficients.

The undamped equation of simple harmonic motion is

y^('')+omega_0^2y=0,
(22)

which becomes

y^('')+betay^'+omega_0^2y=0
(23)

when damped, and

y^('')+betay^'+omega_0^2y=Acos(omegat+delta)
(24)

when both forced and damped.

Systems with constant coefficients are of the form

(dx)/(dt)=Ax(t)+p(t).
(25)

The following are examples of important ordinary differential equations which commonly arise in problems of mathematical physics.

Abel's differential equation

y^'=f_0(x)+f_1(x)y+f_2(x)y^2+f_3(x)y^3+...
(26)
[g_0(x)+g_1(x)y]y^'=f_0(x)+f_1(x)y+f_2(x)y^2+f_3(x)y^3.
(27)

Airy differential equation

y^('')-xy=0.
(28)

Anger differential equation

y^('')+(y^')/x+(1-(nu^2)/(x^2))y=(x-nu)/(pix^2)sin(nupi).
(29)

Baer differential equations

(x-a_1)(x-a_2)y^('')+1/2[2x-(a_1+a_2)]y^'-(p^2x+q^2)y=0,
(30)
(x-a_1)(x-a_2)y^('')+1/2[2x-(a_1+a_2)]y^'-(k^2x^2-p^2x+q^2)y=0.
(31)

Bernoulli differential equation

y^'+p(x)y=q(x)y^n.
(32)

Bessel differential equation

x^2y^('')+xy^'+(lambda^2x^2-n^2)y=0.
(33)

Binomial differential equation

(y^')^m=f(x,y).
(34)

Bôcher equation

y^('')+1/2[(m_1)/(x-a_1)+...+(m_(n-1))/(x-a_(n-1))]y^' +1/4[(A_0+A_1x+...+A_lx^l)/((x-a_1)^(m_1)(x-a_2)^(m_2)...(x-a_(n-1))^(m_(n-1)))]y=0.
(35)

Briot-Bouquet equation

x^my^'=f(x,y).
(36)

Chebyshev differential equation

(1-x^2)y^('')-xy^'+alpha^2y=0.
(37)

Clairaut's differential equation

y=xy^'+f(y^').
(38)

Confluent hypergeometric differential equation

xy^('')+(c-x)y^'-ay=0.
(39)

d'Alembert's equation

y=xf(y^')+g(y^').
(40)

Duffing differential equation

y^('')+omega_0^2y+betay^3=0.
(41)

Eckart differential equation

y^('')+[(alphaeta)/(1+eta)+(betaeta)/((1+eta)^2)+gamma]y=0,
(42)

where eta=e^(deltax).

Emden-Fowler differential equation

(x^py^')^'+/-x^sigmay^n=0.
(43)

Euler differential equation

x^2y^('')+axy^'+by=S(x).
(44)

Halm's differential equation

(1+x^2)^2+y^('')+lambday=0.
(45)

Hermite differential equation

y^('')-2xy^'+lambday=0.
(46)

Heun's differential equation

w^('')+(gamma/x+delta/(x-1)+epsilon/(x-a))w^'+(alphabetax-q)/(x(x-1)(x-a))w=0,
(47)

where w^'=dw/dx.

Hill's differential equation

y^('')+[theta_0+2sum_(n=1)^inftytheta_ncos(2nz)]y=0.
(48)

Hypergeometric differential equation

x(x-1)y^('')+[(1+alpha+beta)x-gamma]y^'+alphabetay=0.
(49)

Jacobi differential equation

(1-x^2)y^('')+[beta-alpha-(alpha+beta+2)x]y^'+n(n+alpha+beta+1)y=0.
(50)

Laguerre differential equation

xy^('')+(1-x)y^'+lambday=0.
(51)

Lamé's differential equation

(x^2-b^2)(x^2-c^2)z^('')+x(x^2-b^2+x^2-c^2)z^'-[m(m+1)x^2-(b^2+c^2)p]z=0,
(52)

where z^'=dz/dx.

Lane-Emden differential equation

1/(xi^2)d/(dxi)(xi^2(dtheta)/(dxi))+theta^n=0.
(53)

Legendre differential equation

(1-x^2)y^('')-2xy^'+alpha(alpha+1)y=0.
(54)

Linear constant coefficients

a_0y^((n))+...+a_(n-1)y^'+a_ny=p(x).
(55)

Lommel differential equation

x^2y^('')+xy^'-(x^2+nu^2)y=kx^(mu+1).
(56)

Löwner's differential equation

y^'=-y(1+kappa(x)y)/(1-kappa(x)y).
(57)

Malmstén's differential equation

y^('')+r/zy^'=(Az^m+s/(z^2))y.
(58)

Mathieu differential equation

V^('')+[a-2qcos(2v)]V=0,
(59)

where V^'=dV/dv.

Modified Bessel differential equation

x^2y^('')+xy^'-(x^2+n^2)y=0.
(60)

Modified spherical Bessel differential equation

r^2R^('')+2rR^'-[k^2r^2+n(n+1)]R=0,
(61)

where R^'=dR/dr

Rayleigh differential equation

y^('')-mu(1-1/3y^('2))y^'+y=0.
(62)

Riccati differential equation

w^'=q_0(x)+q_1(x)w+q_2(x)w^2.
(63)

Riemann P-Differential Equation

u^('')+[(1-alpha-alpha^')/(z-a)+(1-beta-beta^')/(z-b)+(1-gamma-gamma^')/(z-c)]u^' +[(alphaalpha^'(a-b)(a-c))/(z-a)+(betabeta^'(b-c)(b-a))/(z-b)+(gammagamma^'(c-a)(c-b))/(z-c)]u/((z-a)(z-b)(z-c))=0,
(64)

where u^'=du/dz.

Sharpe's differential equation

zy^('')+y^'+(z+A)y=0.
(65)

Spherical Bessel differential equation

r^2R^('')+2rR^'+[k^2r^2-n(n+1)]R=0,
(66)

where R^'=dR/dr.

Struve differential equation

z^2y^('')+zy^'+(z^2-nu^2)y=(4(1/2z)^(nu+1))/(sqrt(pi)Gamma(nu+1/2)).
(67)

Sturm-Liouville equation

d/(dx)[p(x)y^']+[lambdaw(x)-q(x)]y=0.
(68)

Gegenbauer differential equation

(1-x^2)y^('')-(2alpha+1)xy^'+n(n+2alpha)y=0.
(69)

van der Pol equation

y^('')-mu(1-y^2)y^'+y=0.
(70)

Weber differential equation

y^('')+(n+1/2-1/4z^2)y=0,
(71)

where y^'=dy/dz.

Whittaker differential equation

u^('')+u^'+(k/z+(1/4-m^2)/(z^2))u=0,
(72)

where u^'=du/dz.

See also

Adams' Method, First-Order Ordinary Differential Equation, Green's Function, Isocline, Laplace Transform, Leading Order Analysis, Majorant, Partial Differential Equation, Relaxation Methods, Runge-Kutta Method, Second-Order Ordinary Differential Equation, Simple Harmonic Motion, Undetermined Coefficients Method, Variation of Parameters Explore this topic in the MathWorld classroom

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References

Boyce, W. E. and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, 5th ed. New York: Wiley, 1992.Braun, M. Differential Equations and Their Applications, 4th ed. New York: Springer-Verlag, 1993.Carroll, J. "A Composite Integration Scheme for the Numerical Solution of Systems of Ordinary Differential Equations." J. Comput. Appl. Math. 25, 1-13, 1989.Coddington, E. A. An Introduction to Ordinary Differential Equations. New York: Dover, 1989.Forsyth, A. R. Theory of Differential Equations, 6 vols. New York: Dover, 1959.Forsyth, A. R. A Treatise on Differential Equations. New York: Dover, 1997.Fulford, G.; Forrester, P.; and Jones, A. Modelling with Differential and Difference Equations. New York: Cambridge University Press, 1997.Guterman, M. M. and Nitecki, Z. H. Differential Equations: A First Course, 3rd ed. Philadelphia, PA: Saunders, 1992.Hull, T. E.; Enright, W. H.; Fellen, B. M.; and Sedgwick, A. E. "Comparing Numerical Methods for Ordinary Differential Equations." SIAM J. Numer. Anal. 9, 603-637, 1972.Hull, T. E.; Enright, W. H.; Fellen, B. M.; and Sedgwick, A. E. "Erratum to 'Comparing Numerical Methods for Ordinary Differential Equations.' " SIAM J. Numer. Anal. 11, 681, 1974.Ince, E. L. Ordinary Differential Equations. New York: Dover, 1956.Jeffreys, H. and Jeffreys, B. S. "Numerical Solution of Differential Equations." Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 290-301, 1988.Kamke, E. Differentialgleichungen: Lösungsmethoden und Lösungen, Bd. 1: Gewöhnliche Differentialgleichungen, 9. Aufl. Stuttgart, Germany: Teubner, 1983.Milne, W. E. Numerical Solution of Differential Equations. New York: Dover, 1970.Morse, P. M. and Feshbach, H. "Ordinary Differential Equations." Ch. 5 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 492-675, 1953.Moulton, F. R. Differential Equations. New York: Dover, 1958.Polyanin, A. D. and Zaitsev, V. F. Handbook of Exact Solutions for Ordinary Differential Equations. Boca Raton, FL: CRC Press, 1995.Postel, F. and Zimmermann, P. "A Review of the ODE Solvers of Axiom, Derive, Macsyma, Maple, Mathematica, MuPad, and Reduce." Submitted to The 5th Rhine Workshop on Computer Algebra. July 26, 1996. http://www.loria.fr/~zimmerma/ComputerAlgebra/ode_comp.ps.gz.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Integration of Ordinary Differential Equations." Ch. 16 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 701-744, 1992.Simmons, G. F. Differential Equations, with Applications and Historical Notes, 2nd ed. New York: McGraw-Hill, 1991.Weisstein, E. W. "Books about Ordinary Differential Equations." http://www.ericweisstein.com/encyclopedias/books/OrdinaryDifferentialEquations.html.Zaitsev, V. F. and Polyanin, A. D. Spravochnik po obyknovennym differentsial'nym uravneniyam. Moscow: Fizmatlit, 2001.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, 1997.

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Ordinary Differential Equation

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