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Homework Help: Exact solutions and the convergence of eulers method

  1. Jan 27, 2013 #1
    im having trouble with this question - http://i.imgur.com/Ars4J1b.png - more specifically with part a, as i have a good idea how to go about b. given the initial value problem

    y' = 1-t+y , y(t0)=y0

    show that the exact solution is


    we've only spoken of approximations in class and i've just been kind of guessing as to how i should go about it so far. Ive tried to look up the term exact solutions but havent found anything of much use.

    2. Relevant equations
    integrating factor: if dy/dt+ay=g(t), then μ(t) is such that dμ(t)/dt=aμ(t). Multiply both sides of the equation dy/dt+ay=g(t) by μ(t) to obtain μ(t)dy/dt+ayμ(t)=μ(t)g(t).

    3. The attempt at a solution
    rearranging the given formula, i was able to get


    where a = -1. thus, dμ(t)/dt=-μ(t) making μ(t)=e-t. multiplying bothsides give


    the left side can be obtained by the power-rule if the initial function was d(ye-t)/dt so we replace the leftside with this


    integrating bothsides and then solving for y gives


    however, i dont think this is the correct method. What should i be doing instead of this?
  2. jcsd
  3. Jan 27, 2013 #2


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    Your method is fine. You just have a mistake calculating your last line. Re-check that -2-t.
  4. Jan 27, 2013 #3
    Thanks! Im not sure how i was integrating that wrong, but a quick check on a calculator quickly resolved the issue. I appreciate it!
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