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Integrating Factor

An integrating factor is a function by which an ordinary differential equation can be multiplied in order to make it integrable. For example, a linear first-order ordinary differential equation of type

(dy)/(dx)+p(x)y(x)=q(x),
(1)

where p and q are given continuous functions, can be made integrable by letting v(x) be a function such that

v(x)=intp(x)dx
(2)

and

(dv(x))/(dx)=p(x).
(3)

Then e^(v(x)) would be the integrating factor such that multiplying by y(x) gives the expression

d/(dx)[e^(v(x))y(x)]=e^(v(x))[(dy(x))/(dx)+p(x)y(x)]
(4)
=e^(v(x))q(x)
(5)

using the product rule. Integrating both sides with respect to x then gives the solution

y(x)=e^(-v(x))inte^(v(x))q(x)dx.
(6)

See also

First-Order Ordinary Differential Equation, Ordinary Differential Equation

This entry contributed by Joakim Munkhammar

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References

Adams, R. A. Calculus: A Complete Course, 4th ed. Reading, MA: Addison Wesley, 1999.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 526-529, 1953.

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Integrating Factor

Cite this as:

Munkhammar, Joakim. "Integrating Factor." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/IntegratingFactor.html

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