integroitumistekijän

Integroitumistekijän, commonly referred to as the integrating factor, is a standard technique in the theory of linear first-order differential equations. It involves multiplying the equation by a function μ(x) chosen so that the left-hand side becomes the derivative of μ(x) y.

For a differential equation of the form y'(x) + p(x) y(x) = q(x), the integrating factor μ(x) is determined by μ'(x) = p(x) μ(x). On intervals where p is continuous, μ(x) = exp(∫ p(x) dx). Multiplying the equation by μ gives (μ y)' = μ q. Integrating both sides yields μ(x) y(x) = ∫ μ(x) q(x) dx + C, and thus y(x) = [∫ μ(x) q(x) dx + C] / μ(x).

This method requires p and q to be continuous on the chosen interval to guarantee a well-defined μ and a unique solution for a given initial condition. The integrating factor can also be viewed as a tool to convert a non-exact differential equation into an exact one, enabling straightforward integration.

Extensions and related concepts include applying the idea to linear systems, where a matrix integrating factor e^{∫ A(x) dx} can solve systems of the form y' = A(x) y + b(x). The integrating factor is widely used in physics, engineering, and applied mathematics to solve problems in circuit analysis, heat and diffusion equations, and population dynamics, among others.

Historically, the integrating factor emerged with the development of linear differential equation theory in the 18th and 19th centuries as a practical method for obtaining explicit solutions.