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Variations of parameters

  1. Jul 30, 2008 #1
    1. The problem statement, all variables and given/known data
    [tex] 2y\prime\prime +2y\prime + y = 4 \sqrt{x}[/tex]


    3. The attempt at a solution

    charecteristic equation: [tex] x^2+x+\frac{1}{2}[/tex]
    roots: [tex]\frac{1}{2}\pm\frac{1}{2}i[/tex]

    homogenous solution: [tex] a \sin{\frac{1}{2}x} + b \cos{\frac{1}{2}x}[/tex]

    Wronskian: [tex] \left(\begin{array}{cc}\sin{\frac{1}{2}x}&\cos{\frac{1}{2}x}\\\frac{1}{2}\cos{\frac{1}{2}x}&-\frac{1}{2}\sin{\frac{1}{2}x}\end{array}\right) = -\frac{1}{2}[/tex]

    It would be nice to know if up to here, everyone else gets the same answer. After this I get into non elementary functions which is no where near the level of difficulty included within the course I'm taking. I haven't attempted to solve what comes next, frankly because I have no clue where to even begin. I think I probably made a mistake above, if someone can point me in the right direction that would be awesome. The method used is variation of parameters.

    [tex]y_{1}=\sin{\frac{1}{2}x} [/tex]

    [tex]y_{2}=\cos{\frac{1}{2}x}[/tex]

    [tex] u_{1} = \int{4\sqrt{x}\cos{\frac{1}{2}x}dx} [/tex]

    [tex]u_{2} = \int{-4\sqrt{x}\sin{\frac{1}{2}x}dx}[/tex]

    ick
     
    Last edited: Jul 30, 2008
  2. jcsd
  3. Jul 30, 2008 #2
    With [itex]x^2+x+\frac{1}{2}[/itex] as your characteristic equation, I think your roots should be [itex]-\frac{1}{2}\pm\frac{1}{2}i[/itex]. And then with the complex roots, the homogeneous solution should be in the form of

    [tex]y=c_1e^{\lambda t}\cos(\mu t)+c_2e^{\lambda t}\sin(\mu t)[/tex]

    where the roots come from the form of [itex]\lambda \pm i\mu[/itex]. Your form was missing the exponential term. Hope that helps a bit.
     
  4. Jul 30, 2008 #3

    Defennder

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    Homework Helper

    Yeah, jeffreydk is right. Your roots are incorrect and you neglected the exponential.
     
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