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Thread: Missing Solutions of DE

  1. #1
    Dec 2009

    Missing Solutions of DE

    The question is about the initial value problem

    $\displaystyle \frac{dy}{dt} = \frac{t \ cos \ 2t}{y}$

    The question askes to find a one-parameter family of solutions to the DE, and then it asks: "Are there any solutions to the differential equation that are missing from the set of solutions that you found? Explain."


    I've found the general solution by separation of variables:

    $\displaystyle \int \ y \ dy = \int \ t \ cos 2t \ dt$

    $\displaystyle \frac{y^2}{2} = \frac{1}{4} (2t \ sin (2t)+ cos(2t)) + c$
    $\displaystyle \therefore y = \pm \sqrt{t \ sin (2t)+ \frac{1}{2} \ cos (2t) + k}$

    So, how do I know whether there are any missing solutions or not? Any explanation would be greatly appreciated.
    Last edited by demode; Mar 15th 2012 at 12:40 PM.
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