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    Systems of Ordinary Differential Equations

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    Solve the matrix differential equation X^'=AX where X= [x_1,〖 x〗_2 ]^T=[■([email protected]_2 )] and A=[■(3&[email protected]&-1)].
    Find the eigenvalue(s) of A by solving |λ-A|=0
    Solve the linear equation (λ-A)u=0 to get the eigenvector(s) u= 〖[u_1,u_2]〗^2
    Find the fundamental matrix Φ(t)
    What is the Wronskian for Φ?
    Use the result from a to c to express the general solution

    © BrainMass Inc. brainmass.com December 24, 2021, 10:06 pm ad1c9bdddf
    https://brainmass.com/math/ordinary-differential-equations/systems-ordinary-differential-equations-444172

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    Solve the matrix differential equation X^'=AX where X= [x_1,〖 x〗_2 ]^T=[■([email protected]_2 )] and A=[■(3&[email protected]&-1)].
    Find the eigenvalue(s) of A by solving |λ-A|=0
    We have

    whence the eigenvalues are and .
    Solve the linear equation (λ-A)u=0 to get the eigenvector(s) u= 〖[u_1,u_2]〗^2
    The eigenvector may be solved from the equation

    Thus we have , whence

    is an eigenvector of A with corresponding eigenvalue 4.
    The eigenvector may be solved from the equation

    Thus we have , whence

    is an eigenvector of A with corresponding eigenvalue .

    Find the fundamental matrix Φ(t)
    The fundamental matrix is a matrix with columns . Thus we have
    .
    What is the Wronskian for Φ?
    The Wronskian is given by

    Use the result from a to c to express the general solution
    The general solution to the system is given by

    where and are arbitrary constants.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 10:06 pm ad1c9bdddf>
    https://brainmass.com/math/ordinary-differential-equations/systems-ordinary-differential-equations-444172

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