Logarithmic Derivative - Integrating Factors

Integrating Factors

The logarithmic derivative idea is closely connected to the integrating factor method for first-order differential equations. In operator terms, write

D = d/dx

and let M denote the operator of multiplication by some given function G(x). Then

M−1DM

can be written (by the product rule) as

D + M*

where M* now denotes the multiplication operator by the logarithmic derivative

G′/G.

In practice we are given an operator such as

D + F = L

and wish to solve equations

L(h) = f

for the function h, given f. This then reduces to solving

G′/G = F

which has as solution

exp(∫F)

with any indefinite integral of F.