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Differential Equations: Largest rectangle on ty-plane

  1. Feb 11, 2013 #1
    1. The problem statement, all variables and given/known data
    Find the largest open rectangle in the ty-plane that contains the initial value point and satisfies the following theorem:

    Let R be the open rectangle defined by a<t<b, α<y<β. Let f(t,y) be a function of two variables defined on R where f(t,y) and the partial derivative ∂f/∂y are continuous on R. Suppose (t0, y0) is a point in R. Then there is an open interval t-interval (c,d) contained in (a, b) and containing t0, in which there exists a unique solution of the initial value problem

    y'f(t,y), y(t0) = y0


    (t-a)(t-3)y'= 1+ sec(y) and y(π)=1

    2. Relevant equations



    3. The attempt at a solution

    So, when I take the partial derivative I get

    ∂f(t,y)/∂y = [sec(y)tan(y)](t-1)(t-3)/(t-1)(t-3)2

    From this I can see that:

    t≠1
    t≠3

    But y is available for all values.

    Hence the largest rectangle goes from 1 to 3 on the t axis and from -∞ to +∞ on the y axis. But this doesn't contain the initial value point. Am i missing something?
     
  2. jcsd
  3. Feb 12, 2013 #2

    mfb

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    In that case, you need a different rectangle.
     
  4. Feb 12, 2013 #3

    haruspex

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    I assume that should read y'=f(t,y)
    Later you have t-1, not t-a. Which is it?
    Why not just sec(y)tan(y)/((t-1)(t-3))?
    Think outside the box :wink:
     
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