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Non-linear ODE with IV problem

  1. Feb 28, 2012 #1
    1. The problem statement, all variables and given/known data

    I just need to know how to start this. I've never seen a piece wise ODE before and I don't really know where to start to be honest. I know it's non-linear and it's of order one.

    dy/dx = (1/3)y - 3, y > 0 and dy/dx = -(1/3)y - 3 ≤ 0. y(0) = 1 with y(x) [itex]\in[/itex] C0 [0,[itex]\infty[/itex])

    I'm trying to find the non-trivial solution.

    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 28, 2012 #2

    Dick

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    Just solve the two parts separately.
     
  4. Feb 28, 2012 #3
    Thanks very much. That should be ok. :)
     
  5. Feb 28, 2012 #4
    So, will I have two solutions in the end?
     
  6. Feb 28, 2012 #5

    Dick

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    You'll have a single solution defined in two pieces. Are you sure that the pieces are y>0 and y<=0? Not x?
     
    Last edited: Feb 28, 2012
  7. Feb 28, 2012 #6

    HallsofIvy

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    No. You will have a single solution- a single function given by one formula for x< 0 and second formula for x> 0- a "piecewise function".

    I clearly type too slowly!
     
    Last edited: Feb 28, 2012
  8. Feb 28, 2012 #7
    Thanks guys.
     
  9. Apr 10, 2012 #8
    It's definitely a linear ODE right?
     
  10. Apr 10, 2012 #9

    Ray Vickson

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    It is important to know if you mean that
    [tex] y' = \frac{y}{3} - 3, \; y > 0, \text{ and } y' = -\frac{y}{3} - 3, \;
    y \leq 0, [/tex]
    or
    [tex] y' = \frac{y}{3} - 3, \; x > 0, \text{ and } y' = -\frac{y}{3} -3, \; x \leq 0.[/tex]
    Both systems have [itex]C^0[/itex] solutions; one of them has [itex]C^1[/itex] solutions, but the other does not (except for one very particular choice of initial conditions).

    RGV
     
  11. Apr 10, 2012 #10
    The first one you wrote. Cheers.
     
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