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Math Help - integral expression for bessel functions...

  1. #1
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    integral expression for bessel functions...

    Can someone give me a hand with this problem? I cant figure out how to integrate things once I have y y' and y'' plugged into the bessel function.

    problem in a .png linked below

    I only need help with the integration on part A. Thanks in advance!

    http://www.hotlinkfiles.com/files/23...Picture4_1.png


    -Canopy
    Last edited by canopy; March 17th 2009 at 01:39 PM.
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  2. #2
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    Quote Originally Posted by canopy View Post
    Can someone give me a hand with this problem? I cant figure out how to integrate things once I have y y' and y'' plugged into the bessel function.

    problem in a .png linked below

    I only need help with the integration on part A. Thanks in advance!

    http://www.hotlinkfiles.com/files/23...Picture4_1.png


    -Canopy
    y = \int_0^{\pi/2} \cos( x \sin \theta) d \theta
    y' = \int_0^{\pi/2} - \sin \theta \sin( x \sin \theta) d \theta
    y'' = \int_0^{\pi/2} - \sin^2 \theta \cos( x \sin \theta) d \theta

    so

    x^2y'' + x y' + x^2 y =  \int_0^{\pi/2} - x^2\sin^2 \theta \cos( x \sin \theta) - x \sin \theta \sin( x \sin \theta) + x^2 \cos( x \sin \theta) d \theta
     =\int_0^{\pi/2} x^2(1-\sin^2) \theta \cos( x \sin \theta) - x \sin \theta \sin( x \sin \theta) d \theta
     =\int_0^{\pi/2} x^2\cos^2 \theta \cos( x \sin \theta) - x \sin \theta \sin( x \sin \theta) d \theta
     = \int_0^{\pi/2} d\left( x \cos \theta \sin( x \sin \theta) \right) = \left. x \cos \theta \sin( x \sin \theta) \right|_0^{\pi/2} = 0
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