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    Initial-value problem

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    Using the method of undetermined coefficients, find the solution of the system:

    X'=AX + B

    that satisfies the initial condition:

    X(0)=( 0
    A and B are matrices defined in the attached Notepad file.
    Note: When solving the homogeneous soln, exhibit a fundamental matrix psi(t) and also the special fundamental matrix phi(t) satisfying phi(0)=I.

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    Solution Preview

    Solution:For this system X'=AX+B (1)
    We first need to know the given matrices A and B. If they are matrices with constant numbers, it is easy to solve them. For example, if we know A=P'CP where C=diag(a1,a2,...,an) and P'=P^(-1). P^(-1) is the inverse of P. Note that such P is called orthogonal matrix. Many matrices can easily find such matrix P such that A=P'CP.
    Then substituting A=P^(-1)CP , (1) becomes
    PX'=CPX+PB (2)
    where C=diag{a1,a2,...,an}.
    Let Y=PX and D=PB,then (2) is as follows
    Y'=CY+D (3)
    Obviously Y=-C^(-1)D is a special solution for (3). In order to solve (3), we only need to solve the linear diff. equations (4) as follows
    Y'=CY (4)
    In fact, IF Y=(y1,y2,...,yn), then (4) can rewrite as follows.

    Solution Summary

    This shows how to find the solution of a system using the method of undetermined coefficients.