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Oddly Formatted Second Order ODE

  1. Oct 20, 2014 #1
    1. The problem statement, all variables and given/known data
    u'' + w20*u = cos(wt)

    w refers to omega.

    2. Relevant equations


    3. The attempt at a solution

    I'm not sure where to begin on this. For starters, it's a multiple choice problem, and all the answers are given in terms of y, so I'm not sure if u is supposed to replace y' or something.

    Second of all, even if I start solving it, trying to use r2 + 1*w20 = 0, and then getting that r = w0, where do I go from there? I have a general solution of y = c1ew0t + c2e-w0t

    So then I say that the particular solution has to have the form A*cos(wt) + B*sin(wt), and the second derivative is -A*cos(wt) - B*sin(wt)

    So then A*cos(wt) - B*sin(wt) + w02*A*cos(wt) + B*sin(wt) = cos(wt)

    So then A*cos(wt) + w02*A*cos(wt) = 1*cos(wt)

    So then that gets me that A + w02*A = 1, and A = 1/(w02)

    The sine terms cancel (and there's no sine in the solution anyway)

    So then I have y = c1ew0t + c2e-w0t + (cos(wt))/(w02).

    The closest choice I see is c1*cos(w0t) + c2*sin(w0t) + (cos(wt))/w02, and that choice is wrong.
     
  2. jcsd
  3. Oct 20, 2014 #2

    vela

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    ##r=\omega_0## isn't a solution. Plugging it into the characteristic equation would give you ##2\omega_0^2 = 0##, which isn't true in general.

    You didn't differentiate correctly, you dropped a sign, and you made an algebra errors distributing ##\omega_0^2##.

     
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