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Math Help - {x^2y''-20y=0, y(1)=1, y'(1)=gamma}

  1. #1
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    {x^2y''-20y=0, y(1)=1, y'(1)=gamma}

    Find (gamma) so that the solution of the initial value problem
    has finite limit as x → 0.

    I'm not sure I fully understand the problem. I can solve the homogeneous equation using the indicial equation, letting y = x^r:

    r^2 - r - 20 = 0
    r= -4 , 5

    y=c1 * x^-4 + c2 * x^5

    for y(1)=1, 1 = c1 + c2 => c1 = 1- c2

    for y'(1) = gamma, gamma = -4c1 + 5c2 => gamma = -4 + 4c2 + 5c2

    c1 = (-gamma/9 + 5/9)

    c2 = (gamma + 4)/9

    thus y= (-gamma/9 + 5/9)x^-4 + ((gamma + 4)/9)x^5

    but that doesn't seem to address the bit about the finite limit of the gamma function.... and my prof wants the answer in "gamma =" .......
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    Re: {x^2y''-20y=0, y(1)=1, y'(1)=gamma}

    Quote Originally Posted by shane18 View Post
    Find (gamma) so that the solution of the initial value problem
    has finite limit as x → 0.

    I'm not sure I fully understand the problem. I can solve the homogeneous equation using the indicial equation, letting y = x^r:

    r^2 - r - 20 = 0
    r= -4 , 5

    y=c1 * x^-4 + c2 * x^5

    for y(1)=1, 1 = c1 + c2 => c1 = 1- c2

    for y'(1) = gamma, gamma = -4c1 + 5c2 => gamma = -4 + 4c2 + 5c2

    c1 = (-gamma/9 + 5/9)

    c2 = (gamma + 4)/9

    thus y= (-gamma/9 + 5/9)x^-4 + ((gamma + 4)/9)x^5

    but that doesn't seem to address the bit about the finite limit of the gamma function.... and my prof wants the answer in "gamma =" .......
    Check your solution for x = 0. What term do you need to get rid of?

    -Dan
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    Re: {x^2y''-20y=0, y(1)=1, y'(1)=gamma}

    for x= 0, the term with x^ -4 would be undefined.

    Am I to disregard the r= -4 solution to the homogeneous equation, then?

    Also, do I define gamma in terms of x and y?
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    Forum Admin topsquark's Avatar
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    Re: {x^2y''-20y=0, y(1)=1, y'(1)=gamma}

    Quote Originally Posted by shane18 View Post
    for x= 0, the term with x^ -4 would be undefined.

    Am I to disregard the r= -4 solution to the homogeneous equation, then?

    Also, do I define gamma in terms of x and y?
    If the x^{-4} term is a problem then you need to find a gamma such that the term isn't there any more. What value of gamma would do that for you? (Hint: What is the coefficient of the x^{-4} term? Can you set that to zero?)

    -Dan
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