Knowlpedia

Definition

Integrating factor for a scalar linear ODE

Multiplication by exp of an integral turns y prime plus a(t)y into one derivative.

For continuous a,fa,f, the y′+a(t)y=f(t)y'+a(t)y=f(t) has integrating factor

μ(t)=exp⁡(∫t0ta(s) ds).\mu(t)=\exp\left(\int_{t_0}^ta(s)\,ds\right).

The product rule gives (μy)′=μf(\mu y)'=\mu f, so

y(t)=μ(t)−1(y(t0)+∫t0tμ(s)f(s) ds).y(t)=\mu(t)^{-1}\left(y(t_0)+\int_{t_0}^t\mu(s)f(s)\,ds\right).
Regularity and matrices

For integrable a,fa,f, the same formula gives an absolutely continuous solution and the equation holds almost everywhere. For matrix coefficients at different times, multiplication need not commute; replacing a scalar integral exponential by a matrix exponential requires a commutation condition. The general matrix construction uses a .