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Homework Help: 2nd order ODE

  1. Sep 14, 2007 #1
    1. The problem statement, all variables and given/known data

    y''-2ay'+a^2y=e^ax

    Find a general solution

    2. The attempt at a solution

    I've found the general solution of the homogeneous eq: Ce^ax+Dxe^ax

    Next, I must find a particular solution on the form Be^ax (*), right?

    The derivative of (*) is Bae^ax and the 2nd derivative is B(a^2)e^ax

    so that

    y''-2ay'+a^2y=0

    e^ax can never be 0, so I must have made a mistake...
     
  2. jcsd
  3. Sep 14, 2007 #2

    EnumaElish

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    How do you go from the general sol. Ce^ax+Dxe^ax to the particular sol. Be^ax? E.g., why not Bxe^ax?
     
  4. Sep 14, 2007 #3
    Bacause r(x)=e^ax, and my textbook tells me that the particular solution is then on the form Be^ax
     
  5. Sep 14, 2007 #4
    how can the particular soln be of the form Be^ax, when Ce^ax satisfied the homogeneous soln?

    also, the particular soln cant be of the form Bxe^ax since Dxe^ax also satisfied the homogeneous soln

    therefore, the particular soln must be of the form...?
     
  6. Sep 14, 2007 #5
    Ah, use of the modification rule twice?

    B(x^2)e^ax?
     
  7. Sep 14, 2007 #6

    EnumaElish

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    I was thinking that Be^ax = Ce^ax + Dxe^ax when B = C and D = 0.
     
  8. Sep 14, 2007 #7
    My book operates with the so-called modification rule. I got the corect solution this time.
     
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