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General solution of initial value problem -dont understand problem is asking me?

  1. Nov 27, 2009 #1
    General solution of initial value problem --dont understand problem is asking me??

    1. The problem statement, all variables and given/known data

    Find a value for y-sub-0 for which the solution of the initial value problem:
    y' - y = 1+ 3sin t y(0) - y-sub-0
    remains finite as t approaches infinity.

    (i called it "y-sub-0" , just because i cant do subscript)

    book answer says y-sub-0 = -5/2 but i dont understand how or why they have that.



    2. Relevant equations



    3. The attempt at a solution

    first of all, i got the general solution (but then i am stuck)

    y' - y = 1+ 3sin t

    get integrating factor...
    u(t) = e^t

    (e^t)y = integ: (1+ 3sin t)e^t

    expand RHS...

    (e^t)y = integ: e^t + integ (e^t)(3sin t)

    integrate RHS using integ by parts on the last term of RHS (this is where i suspect i am messing up??)
    (e^t)y = e^t + 3/4e^t(sin t + cos t) + e^t + c

    divide both sides by e^t to isolate y, and cancel where applicable

    y = e^t + 3/4(sin t - cos t) + 1 + c/e^t

    then, apply initial condition , y(0) = ysub0

    y = 1/4 + c

    BUT --this is where i am confused. because the question asks what value of y will remain finite as t approaches infinity? But they just said that t = 0 in the initial conditions ( y(0) = y-sub-0 )

    so i dont understand what they are asking me.. can you please rephrase what value i am looking for?

    book answer says y = -5/2, but i dont see how or why that is the answer...
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 27, 2009 #2

    Dick

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    Science Advisor
    Homework Helper

    Re: General solution of initial value problem --dont understand problem is asking me?

    For one thing, the integrating factor should be e^(-t), shouldn't it? As for the overall concept, if you find the correct solution, it should have a term like c*e^(t). e^(t) goes to infinity, so you want to arrange your initial conditions so that you can set c=0.
     
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