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Nonhomogeneous 2nd order dif question?

  1. Jul 16, 2009 #1
    y''-2y''-3y=3e^2t find the general solution




    I have tried Ate^t, Ate^2, Ate^3
    none have worked they all leave extra variables that dont match up.
    is there another combination I could try?
     
  2. jcsd
  3. Jul 16, 2009 #2

    Mark44

    Staff: Mentor

    You need to do this in two parts:
    Find the solutions to the homogeneous equation y'' - 2y' -3y = 0.
    Find a particular solution to the nonhomogeneous equation y'' - 2y' -3y = 3e2t.

    Your general solution will be all solutions to the homogeneous equation plus the particular solution.

    For your homogeneous equation, a basis for your solution set is {e3t, e-t}.

    For a particular solution, you would ordinarily try a solution of the form yp = Ae3t, but that won't work in your nonhomogeneous equation, since this is a multiple of one of the solutions of the homogeneous equation. Instead, try yp = Ate3t, and solve for the value of A that works.
     
  4. Jul 16, 2009 #3
    opps wrote the wrong right side down.
    its y''-2y'-3y=-3te^-t

    i got the e^3t and e^-t already, the right side is still tricky
    I keep getting 6At(e^-t)-12A(t^2)(e^-t)=-3t(e^-t)... and that was using A(t^3)(e^-t)
    or
    2A(e^-t)-6At(e^-t)=-3t(e^-t) with using A(t^2)(e^-t)
    or
    -2A(e^t)=-3t(e^-t) using At(e^-t)

    I cant see what Im doing wrong? is there an error I missed or just a method I havent used yet?
     
  5. Jul 16, 2009 #4

    Mark44

    Staff: Mentor

    That makes us even. I misread the function on the right side of your original DE. I thought you had it as 3e3t, but what you had originally was 3e2t, and you have changed that now to -3te-t.

    What I said about the solution to the homogeneous equation is still valid. For your particular solution, try yp = At2e-t and solve for the constant A. In other words, with this function, calculate yp'' - 2yp' - 3yp = -3te-t. Group all of your terms by their power of t: t0, t, and t2. The coefficient of the t0 terms has to be zero, as does the coefficient of the t2 terms. The coefficient of the t term has to be -3.

    Your general solution will be y(t) = c1e3t + c2e-t + Ate-t (with a specific value in place of A).
     
  6. Jul 16, 2009 #5
    thats my problem everything that should cancel isnt canceling.

    2A(e^-t)-6At(e^-t)=-3t(e^-t) with using A(t^2)(e^-t)
    is what I am left with. What do I do with the 2A?
     
  7. Jul 16, 2009 #6

    Mark44

    Staff: Mentor

    How about this: yp = Ate-t + Bt2e-t? I confess I'm a little rusty on this.
     
  8. Jul 16, 2009 #7
    hey dont worry about it, thanks for the help. Ill try that I think that might work
     
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