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Differential Equations with Discontinuous Forcing Functions

  1. Nov 23, 2014 #1
    1. The problem statement, all variables and given/known data
    Solve the given initial-value problem.
    [itex] y'' = 1 - u(t-1) [/itex]
    [itex] y(0) = 0 [/itex]
    [itex] y'(0) = 0 [/itex]
    2. Relevant equations


    3. The attempt at a solution
    First I took the Laplace transform of both sides:
    [itex] \mathcal{L}(y'') = \mathcal{L}(1 - u(t-1)) [/itex]
    [itex] s^{2}Y(s) - sy(0) - y'(0) = \mathcal{L}(1) - \mathcal{L}(u(t-1)) [/itex]
    [itex] s^{2}Y(s) = \frac{1-e^{s}}{s} [/itex]
    [itex] s^{2}Y(s) = (1-e^{s})\frac{1}{s} [/itex]
    [itex] Y(s) = (1-e^{s})\frac{1}{s^{3}} [/itex]
    At this point I am sort of stuck, the solution given in the back of the book is : [itex] \frac{1}{2}t^{2} - \frac{1}{2}u(t-1)(t-1)^{2} [/itex]
    I'm having a hard time seeing how my work is going to end up as the solution given, so I am thinking maybe I didn't do something right here..
     
  2. jcsd
  3. Nov 23, 2014 #2
    I think I may have figured out what I was doing wrong, I forgot to factor my answer...I'll post a solution momentarily...
     
  4. Nov 23, 2014 #3
    Ok, so did figure out what I was doing wrong...I'm sorry if I've wasted anyone's time
     
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