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Math Help - second order differential equation

  1. #1
    Junior Member
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    Exclamation second order differential equation

    show that y1(x) = e^(2+i)x and y2(x) = e^(2-i)x, i=sqrt(-1) are two linearly independent functions

    hence obtain a second order linear differential equation with constant coefficients each that y1(x) and y2(x) are its two fundamental solutions.

    my attempt :

    for the first part, I use the definition of wroskian = y1y2'-y2y1' and show it not equal to zero... ok

    the second part, I don't know how to do it...

    how to get the second order differential equation????

    is that setting : (r+2+i)(r+2-i) to get the auxillary equation??? is that possible??? can someone show me to solve this problem???
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  2. #2
    MHF Contributor chisigma's Avatar
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    Suppose that s_{1} = 2 + i and s_{2} = 2 - i are solution of a second order equation is s. In that case the equation is...

    (s-s_{1})\cdot (s-s_{2})= s^{2} - 4\cdot s + 5 = 0 (1)

    ... so that the second order ODE is...

     y^{''} - 4\cdot y^{'} + 5 \cdot y =0 (2)

    Kind regards

    \chi \sigma
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