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Differential equation proof

  1. Apr 17, 2016 #1
    1. The problem statement, all variables and given/known data
    Given a function g(t)=acosωt + bsinωt, where a and b are constants, show that g(t) is the real part of the complex function: keeiωt for some k and Φ
    Remark: the complex expression ke is called a phasor. If we know that g(t) has the form kcos(ωt+Φ) then we need know only the constants k and Φ-the amplitude and the phase- to know the function g. Hence we can use the phasor ke as a notation for the function g(t)=keeiωt

    2. Relevant equations
    Euler's formula eiωt= cosωt +isinωt

    3. The attempt at a solution
    Not really sure where to start here except for expanding using euler;s as the first step. any help would be greatly appreciated.
     
  2. jcsd
  3. Apr 17, 2016 #2

    haruspex

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    So do that, for both exponential terms.
     
  4. Apr 17, 2016 #3
    okay so then I get k(cosΦt +isinΦt)(cosωt+isinωt)
    but what should this be equal too?
     
  5. Apr 17, 2016 #4

    haruspex

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    How do you get φt terms?
     
  6. Apr 17, 2016 #5
    umm I'm not sure. I didn't know we would be using that symbol. What does φ represent?
     
  7. Apr 17, 2016 #6

    haruspex

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    It is a constant. You need to find the values of k and φ which make the real part match g().
     
  8. Apr 17, 2016 #7

    haruspex

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    I think perhaps you did not understand my question. In your post #3, you had terms like sin(φt). I don't understand how you got those. Please post your working.
     
  9. Apr 17, 2016 #8
    Sorry I'm kind of lost. Where did φ come from?
     
  10. Apr 17, 2016 #9
    In my third post I only haveΦ k ω and t
     
  11. Apr 17, 2016 #10
  12. Apr 17, 2016 #11

    haruspex

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    Looks like a font issue. Are you saying Φ and φ look very different to you? In the font that comes up on my iPad they are almost the same, so I may have used the wrong one.
     
  13. Apr 17, 2016 #12
    yes they look very different. I think thats where the confusion was.
     
  14. Apr 17, 2016 #13

    haruspex

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    Ok, so please reread my posts with that in mind.
     
  15. Apr 17, 2016 #14
    But in regards to your earlier quastion. I used euler's formula to obtain sin(Φt) eiΦt=cosΦt +isin(Φt)
    right?
     
  16. Apr 17, 2016 #15
    that sin (Φt) is not apart of that calculation
     
  17. Apr 17, 2016 #16

    haruspex

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    According to your initial post, it is e, not eiφt.
     
  18. Apr 17, 2016 #17
    oops, my mistake, than diregard the t's
     
  19. Apr 17, 2016 #18

    haruspex

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    Ok. So multiply out the corrected version of the expression in your post #3.
     
  20. Apr 17, 2016 #19
    k(cosΦ +isinΦ)(cosω+isinω)
     
  21. Apr 17, 2016 #20

    haruspex

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    No, the other exponential term does have a t factor: eiωt.
     
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