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Derivatives of trigonometric functions

  1. Apr 22, 2010 #1
    1. The problem statement, all variables and given/known data

    Find the constants A and B such that the function y = Asinx + Bcosx satisfies the differential equation y'' + y' - 2y = sinx

    2. Relevant equations

    None

    3. The attempt at a solution

    My attempt: y = Asin x + Bcosx

    y' = Acosx - Bsinx

    y'' = - Asin x - Bcosx

    y'' + y' - 2y = sin x
    - Asinx - Bcosx + Acosx - Bsinx - 2Asinx + 2Bcosx = sinx
    - 3Asinx + Bcosx + Acosx - Bsinx = sinx...

    I dont know if my working out is correct, but I couldnt figure out what to do from there.
     
  2. jcsd
  3. Apr 22, 2010 #2
    You're almost there.

    Now factorize the sines and the cosines: (-3A-B)sinx + (-3B+A)cosx=sinx.

    For this to be true we must have -3A-B=1 and -3B+A=0.

    I think you'll be able to take over from here, right?

    edit: Oops, thanks radou. Equations have been corrected
     
    Last edited: Apr 22, 2010
  4. Apr 22, 2010 #3

    radou

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    "+ 2Bcosx " sold be "- 2Bcosx".
     
  5. Apr 22, 2010 #4
    Howcome you spilt the equations into 2? and why does the one equal 1 and the other equal 0? Aren't they suppose to represent the rads of the sin and cos graph?
     
  6. Apr 22, 2010 #5
    You have (-3A-B)sinx + (-3B+A)cosx = 1sinx + 0cosx.

    This equation has to hold for all x. So the only way this will work is if you have -3A-B = 1 and -3B+A=0.
     
  7. Apr 23, 2010 #6
    Oh ok that sort of makes sense. What rule is this by the way? Cause I have never heard it when doing trig before

    Edit: Nevermind I understand now :) Thanks
     
    Last edited: Apr 23, 2010
  8. Apr 23, 2010 #7

    gabbagabbahey

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    Whenever you have two orthogonal functions [itex]f(x)[/itex] and [itex]g(x)[/itex], and an equation of the form [itex]\alpha f(x)+\beta g(x)=\gamma f(x)+\delta g(x)[/itex], the only way it can be satisfied for all [itex]x[/itex] is if [itex]\alpha=\gamma[/itex] and [itex]\beta=\delta[/itex].This can be proved using the definition of orthogonality. Sin(x) and Cos(x) are orthogonal functions.

    If you are unfamiliar with orthogonality of functions, you can also prove that [itex]\alpha\sin x+\beta\cos x=0[/itex] can only hold for all [itex]x[/itex] if [itex]\alpha=0[/itex] and [itex]\beta=0[/itex] by differentiating the equation, and solving the system of two equations (the original and the differentiated one), for your two unknowns.
     
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