Separation of variables is a method of solving ordinary and partial differential equations.
For an ordinary differential equation
(1)
where is
nonzero in a neighborhood of the initial value, the solution is given implicitly
by
(2)
If the integrals can be done in closed form and the resulting equation can be solved for
(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).
For a partial differential equation in a function
and variables ,
,
..., separation of variables can be applied by making a substitution of
the form
(3)
breaking the resulting equation into a set of independent ordinary differential equations, solving these for , , ..., and then plugging them back into the original equation.
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
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1955. Referenced on Wolfram|Alpha Separation of Variables
Cite this as:
Renze, John and Weisstein, Eric W. "Separation of Variables." From MathWorld --A
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