Exponential Solution of a Differential Equation
for a Linear Operator

Samuel Alcántara Montes
Universidad Autónoma Metropolitana
México, DF
hsbh@hp9000a1.uam.mx The differential equation dY /dt = A(t) Y(t) where A(t) is a linear operator and Y(t) is another operator that satisfies the differential equation and the initial condition Y(0) = I, where I denotes the identity operator was investigated by W. Magnus [1] who proposed a technique to solve the differential equation using the polarization derivative the multiple commutator operator and the Bernoulli numbers. In this work an alternative method is given which solves the same differential equation using a very simple identity and allows to bring down the exponent in the exponential solution . Some applications are given.

References:

1

W. Magnus, (1954): Comm. Pure and App. Math. Vol VII, 649-673.