Explore BrainMass

Explore BrainMass

    Advanced Calculus: The Existence Theorem for Nonlinear Differential Equations

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

    Please see the attached file for the fully formatted problems.

    Let g(x,y) be Lipschitz continuous. Let ? (x) = y , and for n > 0 define ? (x) = y +
    Prove that ? (x)  ?(x) on [x - , x + ], for some > 0, where ?(x) solves the ODE ?'(x) = g(x, ?(x)), and ?(x ) = y

    © BrainMass Inc. brainmass.com March 4, 2021, 5:56 pm ad1c9bdddf


    Solution Preview

    Please see the attached file for the complete solution.
    Thanks for using BrainMass.

    is Lipschitz continuous, then it satisfies the Lipschitz condition. There exists , such that is continuous, so we can suppose is continous in a rectangular region , . Since is a closed region, can reach its maximum value in this ...

    Solution Summary

    The existence theorem for nonlinear differential equations is discussed.