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Particular Solutions of Undetermined Coefficients

  1. Oct 17, 2011 #1
    1. The problem statement, all variables and given/known data

    Find a particular solution y_c to the complex differential equation
    y'' + 2 y' + 3 y = -1 exp(2 i t)

    y_c=

    Using the solution y_c, construct a particular solution y_{1p} to the following differential equation
    y'' + 2 y' + 3 y = -1 sin(2 t)
    y_{1p} =

    Again using the solution y_c, construct a particular solution y_{2p} to the following differential equation
    y'' + 2 y' + 3 y = -1 cos(2 t)
    y_{2p} =

    2. Relevant equations

    y = Ae^(2it)

    3. The attempt at a solution

    I let y'' + 2y' + 3y = 0 and then solve getting
    e^(-t)(cos(sqrt2t) + sin(sqrt2t)) + g(h)

    then solve for the g(h) by first finding A

    y = Ae^(2it), y' = A2ie^(2it), y'' = -A4e^(2it)

    substitute and solve for A getting -1/(4i - 1)


    → y(t) = e^(-t)(cos(sqrt2t) + sin(sqrt2t)) -1/(4i-1)e^(2it)
    the Constants of integration can be anything for this problem so i just let them equal 1


    but this is still not the right answer

    please help
     
  2. jcsd
  3. Oct 18, 2011 #2

    Mark44

    Staff: Mentor

    Is the right answer yp = (1/17)(1 + 4i)e2it? If so, all I did was to convert -1/(4i - 1) by multiplying by the complex conjugate of the denominator over itself.

    BTW, when you write cos(sqrt2t) + sin(sqrt2t), I can't tell if you mean
    [itex]\sqrt{2t}[/itex] or [itex]\sqrt{2}t[/itex].

    If you don't use LaTeX, write it like this: sqrt(2)t or t*sqrt(2).
     
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