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Thread: Show that Bessel function satisfies ODE

  1. #1
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    Show that Bessel function satisfies ODE

    Show that the function, [IMG]file:///C:/Users/Vince/AppData/Local/Temp/msohtmlclip1/01/clip_image002.png[/IMG], satisfies the equation [IMG]file:///C:/Users/Vince/AppData/Local/Temp/msohtmlclip1/01/clip_image004.png[/IMG].

    Attempt at the solution,
    [IMG]file:///C:/Users/Vince/AppData/Local/Temp/msohtmlclip1/01/clip_image006.png[/IMG]
    [IMG]file:///C:/Users/Vince/AppData/Local/Temp/msohtmlclip1/01/clip_image008.png[/IMG]

    Therefore, by substituting into ODE of interest,
    [IMG]file:///C:/Users/Vince/AppData/Local/Temp/msohtmlclip1/01/clip_image010.png[/IMG]
    [IMG]file:///C:/Users/Vince/AppData/Local/Temp/msohtmlclip1/01/clip_image012.png[/IMG]
    [IMG]file:///C:/Users/Vince/AppData/Local/Temp/msohtmlclip1/01/clip_image014.png[/IMG]

    But,
    [IMG]file:///C:/Users/Vince/AppData/Local/Temp/msohtmlclip1/01/clip_image016.png[/IMG]
    [IMG]file:///C:/Users/Vince/AppData/Local/Temp/msohtmlclip1/01/clip_image018.png[/IMG]
    [IMG]file:///C:/Users/Vince/AppData/Local/Temp/msohtmlclip1/01/clip_image020.png[/IMG]
    [IMG]file:///C:/Users/Vince/AppData/Local/Temp/msohtmlclip1/01/clip_image022.png[/IMG]

    Substituting back into ODE,
    [IMG]file:///C:/Users/Vince/AppData/Local/Temp/msohtmlclip1/01/clip_image024.png[/IMG]
    [IMG]file:///C:/Users/Vince/AppData/Local/Temp/msohtmlclip1/01/clip_image026.png[/IMG]
    [IMG]file:///C:/Users/Vince/AppData/Local/Temp/msohtmlclip1/01/clip_image028.png[/IMG]
    [IMG]file:///C:/Users/Vince/AppData/Local/Temp/msohtmlclip1/01/clip_image026.png[/IMG]
    [IMG]file:///C:/Users/Vince/AppData/Local/Temp/msohtmlclip1/01/clip_image030.png[/IMG]
    [IMG]file:///C:/Users/Vince/AppData/Local/Temp/msohtmlclip1/01/clip_image032.png[/IMG]
    [IMG]file:///C:/Users/Vince/AppData/Local/Temp/msohtmlclip1/01/clip_image034.png[/IMG]
    [IMG]file:///C:/Users/Vince/AppData/Local/Temp/msohtmlclip1/01/clip_image036.png[/IMG]
    [IMG]file:///C:/Users/Vince/AppData/Local/Temp/msohtmlclip1/01/clip_image038.png[/IMG]
    [IMG]file:///C:/Users/Vince/AppData/Local/Temp/msohtmlclip1/01/clip_image040.png[/IMG]

    At this point I am stuck. I tried playing around with the bottom limit of the summation on the LHS of the equation but Im not too sure what that means in terms of factorials. Can anyone help?
    P.S. I am new to this forum and am not sure if the Microsoft equation editor is popping up correctly, if it isn't showing then I will change it, let me know.
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  2. #2
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    Re: Show that Bessel function satisfies ODE

    All those image files are local on your machine.

    The forum server has no way to access them for display.
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