Explore BrainMass

Explore BrainMass

    Existence and Uniqueness of Solution to an ODE

    Not what you're looking for? Search our solutions OR ask your own Custom question.

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

    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}
    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

    NB: â- means vector value.

    © BrainMass Inc. brainmass.com March 4, 2021, 11:31 pm ad1c9bdddf


    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.