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    Differential equation solved with variables separable method

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    Let x: [0, infinity) -> R and y: [0, infinity) -> R be solutions to the system of differential equations:

    x' = - x

    y' = - sin y

    With initial condition:

    x(0) = y(0) = alpha, where alpha belongs to [0, pi)

    (a) Show that |x(t)| =< alpha for all t >= 0

    (b) Show that | y(t) - x(t) | =< alpha( 1 - e^-t) for all t >= 0. ( e here is exponential function)

    Please justify every step and claim, and if you use any theorems refer to them.

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    Solution Summary

    A given differential equation with initial conditions is solved using variables separable method.