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Superposition and variation of parameters

  1. Nov 12, 2008 #1
    1. The problem statement, all variables and given/known data

    y''+2y'+y = 4t^2 - 3 + (e^-t)/t

    of course i evaluated the general soltuion to be c1e^-1t + c2te^-1t

    but now how do you do the right part? i tried y=At^2+Bt+c+1/(Dt+E)*e^-t as a solution but after differentiating it twice and putting it into the eqaution i got...

    (4e^t/Dt+E)-(4De^t/(Dt+E^2))+(2D^2e^t/(Dt+E)^3)+At^2+4At+Bt+2A+2B+C = 4t^2-3+(e^-t)/t

    and i don't know what to do with that. i found a=4 and b=0 and c=-11 but that's about all i did, i'm unsure how to find the rest of the letters to complete the probleme

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Nov 13, 2008 #2


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    Staff Emeritus
    Science Advisor

    Since (1/t) e-t is NOT one of the solutions one would expect to get as a solution to a homogeneous d.e. with constant coefficients, "undetermined coefficients" will not work. You titled this "superposition and variation of parameters". Have you tried variation of parameters to get a specific solution for the (1/t)e-t?
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