Separation of variables is a method of solving ordinary and partial differential equations.
For an ordinary differential equation
(1)
where
(2)
If the integrals can be done in closed form and the resulting equation can be solved for exponential growth and decay (Stewart 2001).
For a partial differential equation in a function of
the form
(3)
breaking the resulting equation into a set of independent ordinary differential equations, solving these for
This technique works because if the product of functions of independent variables is a constant, each function must separately be a constant. Success requires choice
of an appropriate coordinate system and may not be attainable at all depending on
the equation. Separation of variables was first used by L'Hospital in 1750. It is
especially useful in solving equations arising in mathematical physics, such as Laplace's equation , the Helmholtz
differential equation , and the Schrödinger equation.
See also Helmholtz Differential Equation ,
Laplace's Equation ,
Partial
Differential Equation ,
Stäckel Determinant Explore this
topic in the MathWorld classroom
Portions of this entry contributed by John
Renze
Explore with Wolfram|Alpha
References Arfken, G. "Separation of Variables" and "Separation of Variables--Ordinary Differential Equations." §2.6 and §8.3 in Mathematical
Methods for Physicists, 3rd ed. Bateman, H. Partial
Differential Equations of Mathematical Physics. Brown,
J. W. and Churchill, R. V. Fourier
Series and Boundary Value Problems, 5th ed. Byerly,
W. E. An
Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal
Harmonics, with Applications to Problems in Mathematical Physics. Courant, R. and Hilbert, D. Methods
of Mathematical Physics, Vol. 1. Courant,
R. and Hilbert, D. Methods
of Mathematical Physics, Vol. 2. Eisenhart,
L. P. "Separable Systems in Euclidean 3-Space." Physical Review 45 ,
427-428, 1934. Eisenhart, L. P. "Separable Systems of Stäckel."
Ann. Math. 35 , 284-305, 1934. Eisenhart, L. P. "Potentials
for Which Schroedinger Equations Are Separable." Phys. Rev. 74 ,
87-89, 1948. Frank, P. and Mises, R. von. Die
Differential- und Integralgleichungen der Mechanik und Physik, 8th ed., erster mathematischer
Teil. Frank, P. and Mises,
R. von. Die
Differential- und Integralgleichungen der Mechanik und Physik, 8th ed., zweiter physikalischer
Teil. Hildebrand, F. B.
Advanced
Calculus for Engineers. Jeffreys,
S. H. and Jeffreys, B. S. Methods
of Mathematical Physics, 3rd ed. Kellogg, O. D. Foundations
of Potential Theory. Lense, J. Reihenentwicklungen
in der mathematischen Physik. Maxwell,
J. C. A
Treatise on Electricity and Magnetism, Vol. 1, unabridged 3rd ed. Maxwell, J. C. A
Treatise on Electricity and Magnetism, Vol. 2, unabridged 3rd ed. Miller, W. Jr. Symmetry
and Separation of Variables. Moon,
P. and Spencer, D. E. "Separability Conditions for the Laplace and Helmholtz
Equations." J. Franklin Inst. 253 , 585-600, 1952. Moon,
P. and Spencer, D. E. "Theorems on Separability in Riemannian n -spaceProc. Amer. Math. Soc. 3 ,
635-642, 1952. Moon, P. and Spencer, D. E. "Recent Investigations
of the Separation of Laplace's Equation." Proc. Amer. Math. Soc. 4 ,
302-307, 1953. Moon, P. and Spencer, D. E. "Separability in
a Class of Coordinate Systems." J. Franklin Inst. 254 , 227-242,
1952. Moon, P. and Spencer, D. E. Field
Theory for Engineers. Moon,
P. and Spencer, D. E. "Eleven Coordinate Systems." §1 in Field
Theory Handbook, Including Coordinate Systems, Differential Equations, and Their
Solutions, 2nd ed. Morse,
P. M. and Feshbach, H. "Separable Coordinates" and "Table of
Separable Coordinates in Three Dimensions." §5.1 in Methods
of Theoretical Physics, Part I. Murnaghan, F. D. Introduction
to Applied Mathematics. Smythe, W. R.
Static
and Dynamic Electricity, 3rd ed., rev. pr. Sommerfeld,
A. Partial
Differential Equations in Physics. Stewart,
J. Calculus:
Concepts and Contexts, 2nd ed. Weber, E. Electromagnetic
Fields: Theory and Applications. Webster,
A. G. Partial
Differential Equations of Mathematical Physics, 2nd corr. ed. Referenced on Wolfram|Alpha Separation of Variables
Cite this as:
Renze, John and Weisstein, Eric W. "Separation of Variables." From MathWorld https://mathworld.wolfram.com/SeparationofVariables.html
Subject classifications
is
nonzero in a neighborhood of the initial value, the solution is given implicitly
by
(which are two pretty big "if"s), then a complete solution to the problem
has been obtained. The most important equation for which this technique applies is
,
the equation for exponential growth and decay (Stewart 2001).
and variables 
,
,
..., separation of variables can be applied by making a substitution of
the form
, 
, ..., and then plugging them back into the original equation.
