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Math Help - Not getting the right answer in Reduction of Order Problem

  1. #1
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    Not getting the right answer in Reduction of Order Problem

    x''=\frac{t+1}{t}x'-\frac{1}{t}x

    with x_1(t)=e^t

    The goal is to find the other solution.

    my attempt
    x_2=ux_1
    x_2'=e^{t}u'+e^{t}u
    x_2''=e^tu''+2e^tu'+e^tu

    Substituting into the original equation I get...
    u''=-\frac{t-1}{t}u'

    substituting with u'' with v'...
    v'=-\frac{t-1}{t}v
    I integrate to
    ln v=ln t-t+c1
    v=t\cdot e^{-t}=u'
    integrating again...
    u=(-t-1)e^{-t}

    x_2=(-t-1)e^{-t}\cdot e^t
    x_2=-t-1

    but the answers supposed to be t-1 where did I go wrong? thanks
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  2. #2
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    Why do you think you're wrong? If you substitute your answer into the original ODE you get

    0  = \dfrac{t+1}{t} (-1) - \dfrac{1}{t} \left(-t-1\right)

    which I believe identically satisfies your ODE!
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  3. #3
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    Because of this...

    http://www.mathhelpforum.com/math-he...wn-181831.html

    So there are more than two solutions?
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  4. #4
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    If you look, there's a slight typo (on the negative sign).
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  5. #5
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    Who has the typo? On the maple I only see that he moved the right part of the equation to the left side which seems like it's allowed...
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  6. #6
    Behold, the power of SARDINES!
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    Quote Originally Posted by Carbon View Post
    Who has the typo? On the maple I only see that he moved the right part of the equation to the left side which seems like it's allowed...
    In my finial line in the other thread I wrote

    e^{t}e^{-t}(t+1)=t-1

    It should be

    t+1
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