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Math Help - help with variation of parameters question

  1. #1
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    help with variation of parameters question

    Question: solve: 2x^2y'' + 5xy' + y = x^2 - x

    My attempt so far: y'' + (5y'/2x) + (y/x^2) = 1/2 - 1/(2x)

    yh: λ^2 + (5/2)λ +1 = 0
    (2λ+1)(λ+2) = 0
    λ = -1/2, -2

    yh= c1e^(-1/2x) + c2e^(-2x)

    Try yp = Ae^(-1/2x) + Bxe^(-1/2x) + C, but e^(-1/2x) is part of yh.

    yp=
    Axe^(-1/2x) + Bx^2e^(-1/2x) + C


    is this correct so far?
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  2. #2
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    Answer - No! The method you're trying is the method of undetermined coefficients. In this method we "guess" the form of y_p. In your case since the RHS is quadratic try a general quadratic.
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  3. #3
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    Quote Originally Posted by linalg123 View Post
    Question: solve: 2x^2y'' + 5xy' + y = x^2 - x

    My attempt so far: y'' + (5y'/2x) + (y/x^2) = 1/2 - 1/(2x)

    yh: λ^2 + (5/2)λ +1 = 0
    (2λ+1)(λ+2) = 0
    λ = -1/2, -2

    yh= c1e^(-1/2x) + c2e^(-2x)

    Try yp = Ae^(-1/2x) + Bxe^(-1/2x) + C, but e^(-1/2x) is part of yh.

    yp=
    Axe^(-1/2x) + Bx^2e^(-1/2x) + C


    is this correct so far?
    Not to mention that the homogeneous solution method is incorrect. You can only use that for constant coefficients. Use the substitution x = e^t to put it into constant coefficient format.

    -Dan
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    i got the homogeneous solution is c1x^(-1/2) + c2x^(-1)

    for the particular solution i tried
    V1'x^(-1/2) + v2'x^(-1) = 0
    -1/2v1'x^(-3/2) - v2'x^(-2) = x^2 - x

    but i got the wrong answer.. is this the right approach?
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  5. #5
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    Did you do what topsquark suggested? You can NOT find the solution to the homogenous solution the way you did, because it only works for constant coefficients.

    You need to first make the substitution x = e^t --> t = ln(x).

    So dy/dx = (dy/dt)(dt/dx) = (dy/dt)(1/x) = e^(-t)(dy/dt).

    Get a similar expression for the second derivative, substitute and go from there.
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    so y'' = [(e^-t)(dy/dt) + (e^-t)(d^2y/dt^2)](e^-t)
    y''= (e^-2t)(dy/dt) + (e^-2t)(d^2y/dt^2)

    y' = e^(-t)(dy/dt)

    y= ?

    2e^(2t)[(e^-2t)(dy/dt) + (e^-2t)(d^2y/dt^2)] + 5e^t[e^(-t)(dy/dt)] + y = e^2t - e^t

    2dy/dt + 2(d^2y/dt^2) + 5(dy/dt) + y = e^2t - e^t

    how do i get rid of y?
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  7. #7
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by linalg123 View Post
    so y'' = [(e^-t)(dy/dt) + (e^-t)(d^2y/dt^2)](e^-t)
    y''= (e^-2t)(dy/dt) + (e^-2t)(d^2y/dt^2)
    Close. y'' = -e^{-2t}(dy/dt) + e^{-2t}(d^2y/dt^2)

    I eventually get
    2(d^2y/dt^2) + 3(dy/dt) + y = e^{2t} - e^t

    Now you can solve the homogeneous equation. I get
    yh(t) = Ae^{-t/2} + Be^{-t}

    so your particular solution is going to be of the form
    yp(t) = Ce^{2t} + De^t

    -Dan
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