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Finding Intervals of Solutions to ODE's

  1. Feb 7, 2013 #1
    1. The problem statement, all variables and given/known data

    Consider the IVP

    [itex]\frac{dy}{dt}[/itex] = t2 + y2, y(0)=(0)

    and let B be the rectangle [0,a] x [-b,b]

    a) the solution to this problem exists for

    0≤t≤min{a, [itex]\frac{b}{a2+b2}[/itex]

    b) that min{a,[itex]\frac{1}{2}[/itex]a} is largest when a=[itex]\frac{1}{\sqrt{2}}[/itex]
    c) Deduce an interval 0≤t≤α on which the solution to this problem exists and is unique.

    2. Relevant equations



    3. The attempt at a solution

    for a) f(t,y)= t2 + y2

    the local Lipschitz condition [itex]\frac{∂f}{∂y}[/itex] = 2y is continuous for all (t,y)

    so M=max(t,y)[itex]\inB[/itex]|f(t,y)| = a2+b2

    and from what we did i see the b/b2+a2

    it's really the other two questions that I'm really confused on. Part b) and c). Any help would be great!
     
  2. jcsd
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