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    Solve an IVP ODE using the Method of Variation of Parameters

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    Solve an IVP ODE using the method of variation of parameters

    Find the solution of the system X'
    using the method of variation of parameters

    2 0 0 cos(t)
    X' = -1 0 -1 X + sin(t)
    1 1 2 e^-t

    that satisfies the intial condition

    ( 0 )
    X(0) = 1
    -1

    © BrainMass Inc. brainmass.com October 9, 2019, 3:32 pm ad1c9bdddf
    https://brainmass.com/math/numerical-analysis/solve-ivp-ode-method-variation-parameters-6525

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    Solution.

    From what you provided, we can rewrite the differential equations as follows. Let X=(x(t),y(t),z(t))', or X=(x,y,z)' in short, then we have
    x'=2x+cost (1)
    y'=-x-z+sint (2)
    z'=x+y+2z+e(-t) (3)
    By (1), it is a first order ordinary linear differential equation. You can use a formula in your text book and get the following solution.
    x(t)=Ae^(2t)+1/5*sint-2/5*cost.
    Using the initial condition x(0)=0, we can determine the constant A. By a simple calculation we get
    A=2/5
    So ...

    Solution Summary

    An Initial Value Problem - Ordinary Differential Equation is solved using the method of variation of parameters.

    $2.19