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Question about solving ODE with Complex eigenvalue

  1. Sep 11, 2012 #1
    For example,
    ODE: y'' + y = 0
    solve this problem using MAPLE
    f(x) = _C1*sin(x)+_C2*cos(x)

    My question is Eigenvalue for D^2+1=0 is +i, -i
    so general solution is f(x) = C1*exp(i*x)+C2*exp(-i*x)
    according to Euler's formula f(x) = C1( cos(x)+i*sin(x) ) + C2*( cos(x)-i*sin(x) )
    it is different from the general solution generated by MAPLE
    why?

    Thanks!
     
  2. jcsd
  3. Sep 11, 2012 #2

    AlephZero

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    Science Advisor
    Homework Helper

    Your solution is the general solution assuming f(x) is complex, and your constants C1 and C2 are also complex. You can rearrange it as
    f(x) = (C1 + C2) cos(x) + i (C1 - C2) sin(x)
    or
    f(x) = A1 cos(x) + A2 sin(x)
    wherne A1 and A2 are complex constants.
    If course if you want to restrict f(x) to be a real function, A1 and A2 must be real. That condition is the equivalent to C1 and C2 being complex conjugates, so that C1 + C2 is real and C1 - C2 is imaginary.
     
  4. Sep 11, 2012 #3
    AlephZero, thank you! Really helpful!

     
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