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Math Help - differential equation

  1. #1
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    differential equation

    Use reduction of order to determine 2 linearly independent solutions of the differential equation

    (1 - x^2)y'' + 6xy' - 6y = 0 (-1< x < 1


    I have done:

    Assume y1 = x
    Then y2 = y1 v(x)

    using formula y2 = y1 int [(1/y1^2)^e -int [p dx]] dx

    y2 = 1/5x^6 - x^4 + 3x^2 + 1


    Can anyone please verify that I have done this correctly as I'm not too sure I've used the formula properly...

    Thanks in advance!
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  2. #2
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    Quote Originally Posted by hunkydory19 View Post
    Use reduction of order to determine 2 linearly independent solutions of the differential equation

    (1 - x^2)y'' + 6xy' - 6y = 0 (-1< x < 1


    I have done:

    Assume y1 = x
    Then y2 = y1 v(x)

    using formula y2 = y1 int [(1/y1^2)^e -int [p dx]] dx

    y2 = 1/5x^6 - x^4 + 3x^2 + 1


    Can anyone please verify that I have done this correctly as I'm not too sure I've used the formula properly...

    Thanks in advance!
    Have you substituted your solution for y2 into the DE? What do you find?
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  3. #3
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    I found that all of the terms canceled to leave 0, so does this definitely mean that it is a solution? I thought it might just be a coincidence!
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  4. #4
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    Quote Originally Posted by hunkydory19 View Post
    Use reduction of order to determine 2 linearly independent solutions of the differential equation

    (1 - x^2)y'' + 6xy' - 6y = 0 (-1< x < 1
    If y_1 is a solution the other solution is y_2 = y_1\int \frac{W}{y_1} dx. Where W is the Wronskian.
    What you do is write,
    y'' + \frac{6x}{1-x^2} y' - 6\frac{1}{1-x^2}y=0
    Then by Abel's theorem the Wronskain is (up to a constant) W = \exp \left( - \int \frac{6x}{1-x^2} dx \right) .
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