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    Ordinary differential equation

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    Solve the ode y'' - y' +4y = 0 as a system of first order odes.

    © BrainMass Inc. brainmass.com December 24, 2021, 7:44 pm ad1c9bdddf
    https://brainmass.com/math/ordinary-differential-equations/ordinary-differential-equation-example-problem-215011

    SOLUTION This solution is FREE courtesy of BrainMass!

    set x_1 = y and x_2 = y' so that

    x_1 ' = y' and

    x_2 ' = y '' = y ' - 4y = x_2 - 4x_1

    hence the given equation is equivalent to the system

    X ' =

    (0 1)
    (-4 1) X

    where X = [x_1 x_2] is a column vector.

    The eigenvalues of the matrix A on the right side are

    (1/2)(1 +/- i.sqrt(15))

    and an eigenvector corresponding to the " + " eigenvalue is

    [ 1 - i.sqrt(15), 8]

    so the solution we get from this eigenvalue/eigenvector pair is

    X(t) = e^( (1/2 + i.(sqrt(15)/2)) t } [1 - i.sqrt(15) 8]

    = e^{(1/2)t} (cos ((sqrt(15)/2) t) + i. sin ((sqrt(15)/2) t)) [ 1 - i.sqrt(15) 8]

    which can be written in the form U(t) + i. V(t) where

    U(t) = [e^{(1/2) t} (cos (sqrt(15)/2) t) + sqrt(15). sin (sqrt(15)/2) t); 8e^{(1/2) t} cos( sqrt(15)/2) t)]

    V(t) = [e^{(1/2)t} (sin (sqrt(15)/2) t) -- sqrt(15). cos (sqrt(15)/2) t); 8e^{(1/2)t} sin (sqrt(15)/2) t)]

    Finally, since U and V form a linearly independent solution set, the general solution to the system X ' = AX is

    X(t) = c_1 U(t) + c_2 V(t).

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

    © BrainMass Inc. brainmass.com December 24, 2021, 7:44 pm ad1c9bdddf>
    https://brainmass.com/math/ordinary-differential-equations/ordinary-differential-equation-example-problem-215011

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