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Homework Help: 2nd order ode w/complex conjugate roots

  1. Jan 13, 2009 #1
    x''+x'+2x=0 x(0)=2 x'(0)=0

    I've taken the characteristic equation and reduced the roots to

    1/2 +- Sqrt(7/4)i of the form

    a +- bi (i = sqrt(-1)

    Then i put the homogeneous solution into the form of e[tex]^{}at[/tex]*(B1cos(bt)+B2sin(bt))

    for B1 i used the first i.c. and found that B1=2

    for B2 i used the i.c. for x' and found B2 to be 1/(sqrt7/4)

    now i need to get it into the form of Ae[tex]^{}at[/tex]sin(bt+phi)

    where A = sqrt(B1^2+B2^2) and phi is acos(B1/A)

    the problem is that A is not the hypotenuse of of a right triangle. I didn't know if i screwed up my b values along the way or if i should go to the law of cosines, but i saw no mention of such a problem so I figured i may have made an error beforehand. any help is appreciated.
  2. jcsd
  3. Jan 13, 2009 #2
    Actually I dont understand what your problem is... :D

    What do you mean by "A is not the hypotenuse of a right triangle"??, What right triangle?
  4. Jan 13, 2009 #3
    I mean that in this case law of cosines would be required to find A

    so i can't reduce the homogeneous solution by the method i was given (A = sqrt(B1^2+B2^2) , so i'm wondering if i made a mistake beforehand.

  5. Jan 13, 2009 #4
    I think you made a tiny arithmetic error, you have x'' + x' + 2x = 0, let's say the char. equation is a^2 + a + 2 = 0, quadratic formula gives us roots

    [tex] \frac{-1 \pm \sqrt{-7}}{2} = -\frac{1}{2} \pm i\sqrt{\frac{7}{4}} [/tex]
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