Updated 2013

1.3-8   dy/dx = x^2 - y
=====   Draw threaded curves on the given graphic through the blue 
        dots.
        
 The graphic is reproduced online. Print a copy and draw the
 threaded curves, according to these rules.

    1. Solution curves don't cross.
    2. Threaded curves pass direction field tangent segments at
       nearly the same slope.
    3. Threaded curves can exit the graphic top, bottom, left or
       right. A threaded curve y that passes through point (x0,y0)
       satisfies the initial condition y(x0)=y0.

 Reference: The graphic ready to print.
http://www.math.utah.edu/~gustafso/exercise1.3-8-EdwardsPenney.jpg

 Reference: Lecture slides on direction fields.
http://www.math.utah.edu/~gustafso/s2013/2250/Picard+DirectionFields.pdf

1.3-14  dy/dx = y^(1/3), y(0)=0
======  The problem header is a long paragraph summarizing the Peano 
        and Picard theorems. You are asked to test hypotheses for
        the two theorems, ask report which theorems apply to the
        given problem.

Reference: Lecture slides on Peano and Picard theorems.
http://www.math.utah.edu/~gustafso/s2013/2250/Picard+DirectionFields.pdf

    PEANO THEOREM (not in the textbook, but in 1.3-14 header)
    Let f be continuous on a box B containing the initial point
    (x0,y0). Then there is a solution of y'=f(x,y) satisfying
    y(x0)=y0 passing edge-to-edge through a smaller box inside B.

    PICARD THEOREM (Theorem 1, section 1.3)
    Let f and its partial f_y be continuous on a box B containing
    the initial point (x0,y0). Then there is a solution of y'=f(x,y)
    satisfying y(x0)=y0 passing edge-to-edge through a smaller box
    inside B. The solution is unique inside the smaller box.

 Email Question
 ==============
 On problem 1.3-14 I'm kind of confused on what its asking for when
 it says the existence or uniqueness is guaranteed in some
 neighborhood of x=0. Do I make a domain with the center being 0,0?
 Is this the same equation you gave in class when you did the
 example y'=y^(1/3) and y(0)=0?

 ANSWER: 
 Create a box B with center x=0, y=0 and verify that f(x,y)=y^(1/3)
 is continuous on the box. Then consider any box C with center x=0,
 y=0. Verify that the partial of f on y is discontinuous in C. These
 two checks verify that Peano's theorem applies but Picard-Lindelof
 fails to apply. Please refer to the Peano and Picard slides,
 instead of the textbook, which since the second edition has had
 some disastrous edits applied to Theorem 1, making problem 1.3-14
 unintelligible. The original Theorem 1, to which the problem
 refers, contained both statements: Peano and Picard-Lindelof.