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Homework Help: Nonhomogeneous Equations; Method of Undetermined Coefficient

  1. Feb 14, 2010 #1
    I'm stuck on just one problem.

    1. The problem statement, all variables and given/known data

    2y'' + 3y' + y = t2 + 3sin(t)

    2. Relevant equations

    It says in the lesson that if you have a polynomial, guess a solution is

    "Yi(t)= Ts(A0tn + A1tn-1 + .......... + An)

    where s is the smallest nonnegative integer (s=0,1, or 2) that will ensure that no terms in Yi(t) is a solution of the corresponding homogeneous equation."

    And I don't really understand that jargon. Maybe someone could dumb it down for me.

    3. The attempt at a solution

    Since the solution is the sum of two different functions, t2 and 3sin(t), I can solve the differential equation for each individually, and them add them together at the end since the sum of the solutions to a differential equation is also a solution.

    I got the solution to
    2y'' + 3y' + y = 3sin(t).

    It is Y1(t) = (-3/10)sin(t) -(9/10)cos(t)

    But Y2(t) is giving me trouble.

    I guessed Y2(t) = A0t2 + A1t + A2 but ended up with a system equations that still had t in it; thus I couldn't solve it.

    Set me on the right track.
     
  2. jcsd
  3. Feb 14, 2010 #2

    rock.freak667

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    Homework Helper

    Just take your PI as

    y=A+Bt+Ct2+Dsint+Ecost

    then sub that into the different equation and equate coefficients.
     
  4. Feb 14, 2010 #3
    I already tried that, bro.

    Does

    ring a bell?
     
  5. Feb 14, 2010 #4

    rock.freak667

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    Homework Helper

    Sorry, I was used to doing the two together and not separately. Well just substitute y=A0+A1t+A2t2 into the equation. the coefficients of t and the constant on the right side would just be zero.


    Though when doing the undetermined coefficient method, I find it best to solve for the homogeneous solution first.
     
  6. Feb 14, 2010 #5

    HallsofIvy

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    Science Advisor

    Then show us exactly what you did. Putting that into the left side will give a quadratic and you can set coefficients of corresponding terms equal. (If there is no "corresponding" term on one side, its coefficient is 0.)

     
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