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Thread: 1.3.13 Verify that the given function is a solution of the differential equation

  1. #1
    Super Member bigwave's Avatar
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    1.3.13 Verify that the given function is a solution of the differential equation

    $\textsf{Verify the following that the given function is a solution of the differential equation}$

    $y^{\prime\prime}+y=\sec t
    \quad 0<t<\pi/2
    \quad y=(\cos{t})\ln{t}+t\sin{t}$

    ok presume this is a plug in thing
    but the y'' is a little daunting
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  2. #2
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    Re: 1.3.13 Verify that the given function is a solution of the differential equation

    it doesn't seem to be a solution.

    Are there any typos?
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  3. #3
    Super Member bigwave's Avatar
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    Re: 1.3.13 Verify that the given function is a solution of the differential equation

    1.3.13 Verify that the given function is a solution of the differential equation-13.png
    its number 13
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    MHF Contributor MarkFL's Avatar
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    Re: 1.3.13 Verify that the given function is a solution of the differential equation

    The given solution is:

    $\displaystyle y=\cos(t)\ln(\cos(t))+t\sin(t)$

    You initially stated the argument for the log function was $\displaystyle t$ instead of $\displaystyle \cos(t)$.
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  5. #5
    Super Member bigwave's Avatar
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    Re: 1.3.13 Verify that the given function is a solution of the differential equation

    ok so if
    $$\displaystyle y=\cos(t)\ln(\cos(t))+t\sin(t)$$
    then
    $$\displaystyle y'= t \cos(t) - \sin(t) \log(\cos(t))$$
    and
    $$y''=(y')^2$$
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    MHF Contributor MarkFL's Avatar
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    Re: 1.3.13 Verify that the given function is a solution of the differential equation

    I get:

    $\displaystyle y=\cos(t)\ln(\cos(t))+t\sin(t)$

    $\displaystyle y'=-\sin(t)-\sin(t)\ln(\cos(t))+t\cos(t)+\sin(t)=-\sin(t)\ln(\cos(t))+t\cos(t)$

    $\displaystyle y''=\sin^2(t)\sec(t)-\cos(t)\ln(\cos(t))-t\sin(t)+\cos(t)=\cos(t)+\sin^2(t)\sec(t)-y=(\cos^2(t)+\sin^2(t))\sec(t)-y=\sec(t)-y$

    And thus:

    $\displaystyle y''+y=\sec(t)$
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  7. #7
    Super Member bigwave's Avatar
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    Re: 1.3.13 Verify that the given function is a solution of the differential equation

    whoa....

    not sure how you got $y''$ is that from $W\vert A$
    Last edited by bigwave; Dec 10th 2018 at 12:55 PM. Reason: spelling
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  8. #8
    MHF Contributor MarkFL's Avatar
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    Re: 1.3.13 Verify that the given function is a solution of the differential equation

    Quote Originally Posted by bigwave View Post
    whoa....

    not sure how you got $y''$ is that from $W\vert A$
    No, I just did it by hand with some mental simplifications.
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