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Annihilator method of undetermined coefficients.

  1. Jul 27, 2012 #1
    1. The problem statement, all variables and given/known data

    I found everything except step #5. Please tell me if I am correct

    Find a particular solution to

    (D - 1)(D[itex]^{2}[/itex] + 4D - 12)y = cos(t)

    using the annihilator approach of the method of undetermined coefficients.

    2. Relevant equations

    1) Find annihilator
    2) Find A = fundamental set of corresponding homogeneous equation
    3) Find B= fundamental set of the annihilated equation.
    4) B-A = y[itex]_{p}[/itex]
    5) Plug in y[itex]_{p}[/itex] to find the coefficients.

    3. The attempt at a solution

    I am going to skip typing my work for Steps 1-4, because it would take an insane amount of time.

    1) D[itex]^{2}[/itex]+1 annihilates cos(t)

    2) Set A = [e[itex]^{t}[/itex], e[itex]^{-6t}[/itex], e[itex]^{2t}[/itex]]
    3) Set B = [e[itex]^{t}[/itex], e[itex]^{-6t}[/itex], e[itex]^{2t}[/itex], cos(t), sin(t)]
    4) B-A = [cos(t), sin(t)]

    So y[itex]_{p}[/itex] = c[itex]_{1}[/itex]cos(t) + c[itex]_{2}[/itex]sin(t)

    5) Expanded equation:
    (D[itex]^{3}[/itex]+3D[itex]^{2}[/itex]-16D+12)(c[itex]_{1}[/itex]cos(t) + c[itex]_{2}[/itex]sin(t)) = cos(t)

    After fully expanding using FOIL,

    9c[itex]_{1}[/itex] - 17 c[itex]_{2}[/itex] = 1
    17c[itex]_{1}[/itex] + 9c[itex]_{2}[/itex] = 0

    I used matrix transformations to find
    c[itex]_{1}[/itex] = [itex]9/370[/itex] and
    c[itex]_{2}[/itex] = [itex]-17/370[/itex]

    Am I correct? These solutions seem way too messy compared to what he has given us in the past. In class, he solved Step 5 without actually FOILing the equation, which I did not quite follow, but if I could figure it out it would be much easier than spending several minutes doing monotonous algebra. I know for a fact he will give us an equation like this (with cos(t) and sin(t)) on the final exam because he did the same for the normal test.
     
    Last edited: Jul 27, 2012
  2. jcsd
  3. Jul 27, 2012 #2

    LCKurtz

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    Homework Helper
    Gold Member

    The way to check if it is correct is to plug the answer back into the equation and see if it works. To save you some time, I let Maple do the grunt work and your answer for ##y_p## is correct. Good work.
     
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