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Homework Help: Wronskians equations help

  1. Mar 19, 2010 #1
    Im doing my Differential Equation's homework, and I've come across some really hard problems, when it got really complicated I tuoght i was wrong (you know book's problems tend to workout nicely) . . .anyways I got REALLY stuck on this one if anyone can guide me thank you
    1. The problem statement, all variables and given/known data

    y''+3y'+2y = 1/(1+ex)

    2. Relevant equations

    Use variation of parameters (the chapter is about wronskians)

    3. The attempt at a solution

    first we find the complementary equation using the auxiliary equation (m2+3m+2) and find that
    yc=c1e-2x + c2e-x

    therefore y_1=e^-2x and y_2=e^-x

    then we use the Wronskian of
    | y_1 y_2 |
    | y'_1 y'_2 | = -e^-3x - ( -2e^-3x) therefore W= e^-3x

    W_1= using f(x) = 1/(1+e^x)

    | 0 y_2 |
    | f(x) y'_2 | = 1 / (e^x+e^2x) = u'

    to find yp=u1y1 + u2y2

    the problem is finding u, since the u' = W_1/W . . .which would yield

    e^3x / (e^x+e^2x)

    I would have to integrate that, I cant find anyway to do it, I tried integration by parts and partial fraction decomposition.
    I was thinking of looking it up in an integration table. But since its a problem from the book (and the book doesnt have an integration table) I dont think the book is THAT demanding as in to go look it up in another book.
    Im thinking maybe im doing something wrong. . .

    Thank you!
  2. jcsd
  3. Mar 19, 2010 #2
    Re: Wronskians

    e^3x / (e^x+e^2x) is basically e^2x/(1+e^x), right?

    did you try integration by parts ?
    or first do u=e^x
    and see if it helps
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