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2nd order differential equation: undetermined coefficients

  1. Dec 6, 2014 #1
    1. The problem statement, all variables and given/known data

    y''-y=t-4e^(-t)


    2. Relevant equations

    method of undetermined coefficients

    3. The attempt at a solution

    solving for characteristic equation first

    y''-y=0

    r^2-1=0

    c_1e^(-t)+c_2e^(t)

    RHS

    particular solution

    t-4e^(-t)

    y_p(t)= At+B+Ce^(-t)

    y_pt'(t)=A-Ce^(-t)

    y_p''(t)=Ce^(-t)

    plug into LHS

    Ce^(-t)-At-B-Ce^(-t)=t-4e^(-t)

    -A=t
    A=-1

    B=0

    C cancels on the left.... Im not sure how to any further?

    Thanks
     
  2. jcsd
  3. Dec 6, 2014 #2

    Mark44

    Staff: Mentor

    No, this won't work, since e-t is already a solution in the complementary equation. Do you know what to do when you run into this?
     
  4. Dec 6, 2014 #3
    You add a coefficient t

    so the guess becomes At+B+Cte^(-t) right? When I plug that back in, it still cancels out
     
  5. Dec 6, 2014 #4

    Mark44

    Staff: Mentor

    I'm not sure what you mean.
    If yp = At + B + Cte-t, then yp'' - yp has to equal t - 4e-t. Take yp and its second derivative, and substitute them into your nonhomogeneous equation to get A, B, and C.
     
  6. Dec 6, 2014 #5
    Got it... I was making a mistake with my derivative... I feel stupid.

    Thanks Mark44!
     
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