Editing Logarithmic derivative (section)

==Integrating factors==
The logarithmic derivative idea is closely connected to the [[integrating factor]] method for [[first-order differential equation]]s. In [[Operator (mathematics)|operator]] terms, write <math display="block"> D = \frac{d}{dx} </math> and let ''M'' denote the operator of multiplication by some given function ''G''(''x''). Then <math display="block"> M^{-1} D M </math> can be written (by the [[product rule]]) as <math display="block">D + M^{*} </math> where <math> M^{*} </math> now denotes the multiplication operator by the logarithmic derivative <math display="block"> \frac{G'}{G}</math>

In practice we are given an operator such as
<math display="block"> D + F = L </math>
and wish to solve equations
<math display="block"> L(h) = f </math>
for the function ''h'', given ''f''. This then reduces to solving
<math display="block"> \frac{G'}{G} = F </math>
which has as solution
<math display="block"> \exp \textstyle ( \int F ) </math>
with any [[indefinite integral]] of ''F''.{{Citation needed|date=August 2021}}