Integration Factor

An integration factor is a mathematical technique used to solve first-order linear ordinary differential equations. By multiplying the entire equation by an appropriate integration factor, the equation can be transformed into an exact differential equation, which can then be solved using direct integration.

The general form of a first-order linear ordinary differential equation is:

y'(x) + P(x)y(x) = Q(x)

To find the integration factor, denoted as μ(x), we compute the exponential of the integral of the function P(x):

μ(x) = e^∫P(x)dx

By multiplying both sides of the differential equation by the integration factor, the left side becomes the exact derivative of the product of μ(x) and y(x), which can then be integrated directly:

μ(x)y'(x) + μ(x)P(x)y(x) = μ(x)Q(x)

(μ(x)y(x))’ = μ(x)Q(x)

∫(μ(x)y(x))’dx = ∫μ(x)Q(x)dx