Integrating factor

An integrating factor is a function multiplied into a differential equation to make it easier to solve, often by turning it into an exact equation.

First-order linear differential equation: A differential equation of the form y′ + P(x)y = Q(x), with unknown function y and given functions P and Q. Its standard integrating factor turns the left side into the derivative of a product.
Ordinary differential equation: An equation involving an unknown function of one independent variable and one or more of its derivatives. Integrating factors are chiefly used to transform ordinary differential equations.
Linear differential equation: A differential equation in which the unknown function and its derivatives appear linearly. The familiar first-order integrating-factor formula is one instance of a broader linear-equation method.
Exact equation: An equation whose differential expression is the differential of a scalar function. An integrating factor is often sought precisely because the original equation is not exact.
Exact differential equation: An equation M(x,y) dx + N(x,y) dy = 0 whose left side is the differential of a function of x and y. An integrating factor can make a non-exact equation exact.
Differential form: A mathematical object that can be integrated along curves or regions and expressed using differentials such as dx and dy. Exactness and its transformation by a multiplier are expressed in differential-form notation.
Continuity equation: An equation expressing conservation of a quantity, such as mass or charge, through a local balance of fluxes and sources. In one dimension, suitable conservation equations can reduce to linear ordinary differential equations solved with a multiplier.
Homogeneous differential equation: A differential equation whose terms share a specified scaling property, or whose forcing term is zero in a linear equation. Some homogeneous first-order equations are solved by substitution rather than an integrating factor.
Product rule: The calculus rule (uv)′ = u′v + uv′ for differentiating a product of functions. The linear-equation method recognizes its left side as the derivative of μ(x)y.
Total derivative: The derivative of a composite function with respect to its independent variable, accounting for every dependency. The method turns an equation into a total derivative that can be integrated.

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Differential equationRelated: It converts a first-order linear equation into an integrable form.