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Find solution of initial value problem - 1st order non-linear ODE

  1. Jun 23, 2011 #1
    Hey,
    we have to solve the following problem for our ODE class.

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

    Find the solution of the initial value problem
    dx/dt = (x^2 + t*x - t^2)/t^2

    with t≠0 , x(t_0) = x_0

    Describe the (maximal) domain of definition of the solution.

    3. The attempt at a solution
    Well, I know that this is a 1st order nonlinear ODE. Unfortunately I got no clue how to deal them.
    I tried this:
    dx/dt = (x^2 + t*x - t^2)/t^2
    = x^2/t^2 + x/t -1

    Now substitute: u = x/t -> x=ut , x'=u't+u
    Therefore we get:
    u't+u = u^2+u-1
    t* du/dt +u = u^2+u-1 //-u
    t* du/dt = u^2 -1

    0= t*u' -u^2 +1
    which is my dead end.

    Is the idea ok? What could I do?

    Kind regards,
    mihyaeru

    PS: How can i insert a fraction?
     
  2. jcsd
  3. Jun 23, 2011 #2
    I just noticed that we might be able to solve this via applying the Riccati equation????
     
  4. Jun 23, 2011 #3

    ehild

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    This is a separable differential equation, you must be able to solve it
    :smile: !

    ehild
     
  5. Jun 23, 2011 #4
    As far as I know it is a separation of variables case, iff there would be no t in front of the du/dt. Or no -1.
    You think of the solution
    du / (u^2 -1) = dt / t ,
    don't you?

    But anyway thanks for your answer =)
     
  6. Jun 23, 2011 #5

    LCKurtz

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    That doesn't matter. Just keep the t with the dt.
    That's exactly what he is thinking of. Use partial fractions on the left side and integrate both sides.
     
  7. Jun 24, 2011 #6

    ehild

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    I do. And you should be able to integrate both sides.

    ehild
     
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