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Homework Help: Method of Undetermined Coefficients

  1. Feb 27, 2017 #1
    A question on my homework:

    Solve the DE using undetermined coefficients.

    y''' - 5y'' + 6y' = 8+2sinx

    Ok, supposedly easy. I use a homogenous equation to solve for yc.

    y''' -5y'' +6y' = 0
    m=0, -1, 6
    yc = c1 + c2e-x+c3e6x

    Then use a trial solution to find yp, but this is where I get confused...

    g(x) = 8+2sinx
    yp = A+Bcosx+Csinx


    yp' = -Bsinx+Ccosx
    yp'' = -Bcosx-Csinx
    yp''' = Bsinx-Ccosx

    Bsinx-Ccosx+5Bcosx+5Csinx-6Bsinx+6Ccosx = 8+2sinx

    (-5B+5C) = 2
    (5C+5B) = 0

    B= -1/5

    Here where I run into a problem -- because there is a constant in g(x), but only yp's derivatives are being added together, there is no way to make is equal 8+2sinx, or at least not as far as I can see. Would I just make A=8?
  2. jcsd
  3. Feb 27, 2017 #2
    The problem comes from the fact that any constant will satisfy the homogenous form of the differential equation (you can see this since you found a constant ##c_1## term in your general solution to the homogenous equation).

    The trick is the same as finding second solutions to a homogenous equation with double roots in the auxillary polynomial, which is to add an extra factor of ##x##. So try using ##Ax+B\cos x+C\sin x## for your trial solution instead. I expect you'd find that ##A## does indeed equal 8.
  4. Feb 28, 2017 #3

    Ray Vickson

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    An additive constant on the right is always easy to eliminate: just use ##z = y' - 8/6##, which obeys the DE
    $$z'' -5 z' + 6z = 2 \sin x$$
    Last edited: Feb 28, 2017
  5. Feb 28, 2017 #4


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    You can solve the equation for y'=z without any difficulty, and get y(x) by integrating.
  6. Feb 28, 2017 #5
    Thanks everyone! I don't think I've learned the second method in my classes yet, but the additive constant thing seemed apparent when I looked at the problem again. :)
  7. Mar 2, 2017 #6


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    It helps to solve the characteristic equation correctly: [itex]m^3 - 5m^2 + 6m = m(m^2 - 5m + 6) = m(m-2)(m-3)[/itex].

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