Question: Variation of Parameters. The classic derivation of the general solution formula for a homogeneous system of differential equations is based on a method called variation of parameters. Consider the system
x'(t) = A(t)x(t)+b(t)
Let X(t) be a fundamental matrix of solutions for the corresponding homogeneous equation. It is known that any solution to the homogeneous equation can be written as X(t)y, where y is an n-vector of (fixed) parameters. It is conceivable then that It might be helpful to express the general solution to the nonhomogeneous equation m the form x(1) = X(t)y(t), where now y(t) is an n-vector of varying parameters.
(a) Using the suggested form as a trial solution, find a system of differential equations that yet) must satisfy.
(b) Solve the system in (a) by integration and thereby obtain the general solution to the original system.
(c) Convert the result of (b) to one using the state-transition matrix, and verify that it is the same as the result given in the text.