1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
    m3-5m2+6m=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

    Right...?

    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
    C=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

    User Avatar
    Science Advisor
    Homework Helper

    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

    ehild

    User Avatar
    Homework Helper
    Gold Member

    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

    pasmith

    User Avatar
    Homework Helper

    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].

     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Method of Undetermined Coefficients
Loading...