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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

 (1)

where and are given continuous functions, can be made integrable by letting be a function such that

 (2)

and

 (3)

Then would be the integrating factor such that multiplying by gives the expression

 (4) (5)

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

 (6)

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