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Math Help - proving differential

  1. #1
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    proving differential

    given that x^2 y=3cos 2x show that x^2 d^2y/dx^2 + 4x dy/dx +(4^2+2)y= 0

    i have

    x^2y + x^2 dy/dx = -2x(3sinx) + 3cos 2x did i differenciate the RHS correctly.. that is  3cos 2x

    differentiating it again i have x^2y + x^2 dy/dx+x^2 d^2y/dx^2 = -2x(3sinx) + x^2 y

    now rewriting x^2 d^2y/dx^2+2x^2 dy/dx+2x(3sinx)=0

    i cant get that to look like x^2 d^2y/dx^2 + 4x dy/dx +(4^2+2)y= 0

    did i do something wrong in the process.
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  2. #2
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    y=\frac{3cos(2x)}{x^2}

    \frac{dy}{dx}=\frac{-6cos(2x)}{x^3}-\frac{6sin(2x)}{x^2}

    \frac{d^2y}{dx^2}=[\frac{18}{x^4}-\frac{12}{x^2}]*cos(2x)+\frac{24sin(2x)}{x^3}

    x^2*([\frac{18}{x^4}-\frac{12}{x^2}]*cos(2x)+\frac{24sin(2x)}{x^3}) +4x(\frac{-6cos(2x)}{x^3}-\frac{6sin(2x)}{x^2}) +18(\frac{3cos(2x)}{x^2})

    I am not going to simplify because I am lazy. You can check to see if it works and your math is the same.
    *I made a correction I used x^3 by accident at first.
    Last edited by dwsmith; April 9th 2010 at 09:49 AM. Reason: Wrong power used
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  3. #3
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    am trying to follow what your doing but am a bit confused can you explain your steps and bit and show me what your getting at.. did you make y the subject for your first line... in that case shouldnt it be x^2 for the denominator
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  4. #4
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    I used x cubed by accident at first but I edited my original post to fix it.

    I just solved for y and took the first and second derivative.
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  5. #5
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    i have been trying to simplify the equation but it has been giving me some problems. i would like to see how you would do it if its not too much of a problem.
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  6. #6
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    Quote Originally Posted by sigma1 View Post
    given that x^2 y=3cos 2x show that x^2 d^2y/dx^2 + 4x dy/dx +(4^2+2)y= 0

    i have

    x^2y + x^2 dy/dx = -2x(3sinx) + 3cos 2x did i differenciate the RHS correctly.. that is  3cos 2x

    differentiating it again i have x^2y + x^2 dy/dx+x^2 d^2y/dx^2 = -2x(3sinx) + x^2 y

    now rewriting x^2 d^2y/dx^2+2x^2 dy/dx+2x(3sinx)=0

    i cant get that to look like x^2 d^2y/dx^2 + 4x dy/dx +(4^2+2)y= 0

    did i do something wrong in the process.
    differentiating once:

    2xy+x^2y'=-6 \sin(2x)

    Differentiating again:

    2y+2xy'+2xy+x^2y''=-12 \cos(2x)

    Substituting on the right:

    2y+4xy+x^2y''=-4x^2y

    now simplify

    CB
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