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28 August, 02:48

Show that y=sin (t) is a solution to (dydt) 2=1-y2. Enter your answers below in terms of the independent variable t in the order in which the terms were given. Be sure you can justify your answer.

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  1. 28 August, 03:13
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    y = sin (t) is a solution to the differential equation

    (dy/dt) ² = 1 - y²

    Step-by-step explanation:

    Given (dy/dt) ² = 1 - y²

    Suppose y = sin (t) is a solution, then it satisfies the differential equation.

    That is

    [d (sin (t)) ]² = 1 - y²

    Let y = sin (t)

    dy/dt = d (sin (t)) = cos (t)

    (dy/dt) ² = cos²t

    But cos²t + sin²t = 1

    => 1 - sin²t = cos²t

    So

    (dy/dt) ² = 1 - sin²t

    Since sin²t = (sint) ² = y²,

    we have

    (dy/dt) ² = 1 - y²

    as required.
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