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Math Help - Ordinary differential equations....

  1. #1
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    Ordinary differential equations....

    If { f1(x), f2(x) } is one set of two linearly independent solution on a<=x<=b and { g1(x), g2(x) } be another set of two linearly independent solution, then show that there exists a constant c not equal to 0 such that
    W [ g1(x), g2(x) ] = c W [ f1, f2] (x)
    for all a<=x<=b.

    where,
    W[] is wronskian, which is defined as

    W [ f1(x), f2(x)]= f1 (x) * f2 ' (x) - f1' (x) * f2' (x)


    (determinant involving f1, f2 and f1' , f2' )

    f1 ' (x) is derivative of f1(x) i.e., d( f1(x))/dx
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  2. #2
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    There is Liouville's theorem, which states that the Wronskian W of two independent solutions of the ODE y''+ay'+by=0 satisfies W'(t)={\rm e}^{-\int_{t_0}^{t} a(u)du}W(t_0). Apply this for W_1=W(f_1,f_2) and W_2=W(g_1,g_2), divide by parts, and integrate.
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  3. #3
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    Quote Originally Posted by niranjan View Post
    If { f1(x), f2(x) } is one set of two linearly independent solution on a<=x<=b and { g1(x), g2(x) } be another set of two linearly independent solution, then show that there exists a constant c not equal to 0 such that
    W [ g1(x), g2(x) ] = c W [ f1, f2] (x)
    for all a<=x<=b.

    where,
    W[] is wronskian, which is defined as

    W [ f1(x), f2(x)]= f1 (x) * f2 ' (x) - f1' (x) * f2' (x)


    (determinant involving f1, f2 and f1' , f2' )

    f1 ' (x) is derivative of f1(x) i.e., d( f1(x))/dx
    Are we to assume that this is a second order, linear, homogeneous differential equation? If not, it is not true. If yes, it would have been helpful to say so.
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  4. #4
    Super Member Rebesques's Avatar
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    Are we to assume that this is a second order, linear, homogeneous differential equation?

    Υeap, that's the only (college/uni course maybe?) setting
    where the question makes sense.
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