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    Existence and Uniqueness of Solution to an ODE

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    See the attached file.
    Express the 2nd order ODE
    d_t^2 u=(d^2 u)/(dt)^2 =sin?(u)+cos?(Ï?t) Ï? Z/{0}
    u(0)=a
    d_t u(0)=b
    as a system of 1st order ODEs and verify that there exists a global solution by invoking the global existence and uniqueness Theorem.

    Useful information:
    Global existence and uniqueness Theorem:
    The ordinary differential equation
    d_t â-u=â-f(t,â-u (t))
    â-u (0)=â-u_0
    has a unique solution if â-fâ??C^0 (I)Ã-Lipschitz(L_â?? (R)), f is continuous with respect to 1st variable and Lipschitz with respect to 2nd variable.

    Lipschitz Continuity: A function g:Iâ?'R is Lipschitz continuous if â??Î?>0 such that
    â?-g(â-x)-g(â-y)â?-â?¤Î?â?-â-x-â-yâ?-â??â-x,â-yâ??I.

    NB: â- means vector value.

    © BrainMass Inc. brainmass.com April 3, 2020, 10:16 pm ad1c9bdddf
    https://brainmass.com/math/ordinary-differential-equations/existence-uniqueness-solution-ode-429916

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    Solution Preview

    To express the 2nd order ODE as a system of 1st order ODEs, we let v = u_t. Then we have

    u_t = v
    v_t = sin u + cos(omega t)

    with the initial conditions

    u(0) = a
    v(0) = b

    These ODEs may be written in vector form as

    (1) d_t w = ...

    Solution Summary

    The solution demonstrates the existence and uniqueness of a given second order ordinary differential equations with given initial conditions.

    $2.19

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