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.
Read more about this topic: Logarithmic Derivative
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