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Math Help - A domain restriction on the solution of a differential equation

  1. #1
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    A domain restriction on the solution of a differential equation

    I'm an AP Calc teacher. A problem I gave my students states:

    Find the general solution to the exact differential equation.

    y'=5 sec^2 x - 1.5 \sqrt x

    y(0)=7

    So taking the anti derivative,

    y=5 tan x - x^{3/2} + 7

    That's fine. The solution also notes:

    (0 < x < \pi/2)

    Can anyone explain why this domain restriction is in place? I understand that there is a discontinuity at pi/2, but why does it choose this specific interval? Is it because the initial value is given at 0?

    Thanks for any help.
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  2. #2
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    Re: A domain restriction on the solution of a differential equation

    at pi/2 you would be dividing by 0 in the differential equation.

    I'd think, since an initial condition is given at 0, that the actual domain is [0, pi/2)

    You could choose any domain that includes zero and doesn't include (k+1/2)pi, k an integer, and those are
    (-pi/2, 0], and [0, pi/2), however you also have a square root of x in your diff eq. That rules out the first interval if you are working on the real numbers.

    So that leaves you only with [0,pi/2)
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