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Abel's Formula

  1. Mar 13, 2014 #1
    1. Use Abel's Formula to find the Wronskian of two solutions of the given differential equation
    without solving the equation.


    x2y" - x(x+2)y' + (t + 2)y = 0

    2.

    Abel's Formula

    W(y1, y2)(x) = ce-∫p(x)dx





    3.

    I put it in the form of

    y" + p(x)y' + q(x)y = 0

    to find my p(x) to use for Abel's formula


    p(x) = - (x+2 / x)


    this would give:

    W(y1, y2)(x) = ce-∫(-)(x+2 / x)dx


    I'm not sure if I'm going in the right direction. I need some help.
     
    Last edited by a moderator: Mar 13, 2014
  2. jcsd
  3. Mar 13, 2014 #2

    Mark44

    Staff: Mentor

    Your use of parentheses is commendable, although you have them in the wrong place here and below.
    You should have p(x) = -(x + 2)/x

    What you wrote is the same as ##-(x + \frac 2 x)##.
    This seems OK to me (aside from the parentheses thing). Are you having trouble with the integration?
     
  4. Mar 13, 2014 #3
    After integrating I get:

    ce(x) + 2ln(x)

    Is this the answer? What about the value for c?
     
  5. Mar 13, 2014 #4

    Mark44

    Staff: Mentor

    You should simplify that.
    I believe that c is W(y1(x), y2(x))(x0). IOW, the Wronskian of the two functions, evaluated at some initial x value.
     
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