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Math Help - second order ODE

  1. #1
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    second order ODE

    Hello,

    I have this problem:

    solve this ODE

    u'' + u = 0

    so the Characteristic Equation is going to be:
     r^2 + 1 = 0

    i'm going to get a complex root correct? I'm really unsure on how to approach complex roots.
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  2. #2
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    Quote Originally Posted by dlbsd View Post
    Hello,

    I have this problem:

    solve this ODE

    u'' + u = 0

    so the Characteristic Equation is going to be:
     r^2 + 1 = 0

    i'm going to get a complex root correct? I'm really unsure on how to approach complex roots.
    If the roots of the auxillary equation are r = a \pm i b then the homogenous solution to the differential equation is u = e^{at} (A \cos (bt) + B \sin (bt)).
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  3. #3
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    Since, u''=-u

    A function differentiated twice gives the negative of the same function, so it can be either A*cosine(bx) or B*sine(bx). By principle of superposition of the results (which is saying if two solutions are there bundle them up together)... u(x)=A cos (bx) + B sin (bx)
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