a diff eq question:
http://img98.imageshack.us/img98/8682/diffequv9.jpg
the top part is the problem, the 2nd [seemingly superimposed] part is a graph accompanying the solution in the back of the book.
for clarification, the tick marked on the x axis of the top graph [part of the question] is x=c.
that graph they have there in the question looks to me like a combo of y=x^3 on x<0, y=0 on x:[0,c], y=(x-c)^3 on x>c
also, perhaps by "left half" they mean (x-c)^3 with x<c ?
theorem 1: existence and uniqueness of solutions
"suppose that the function f(x,y) and its partial derivative with respect to y are continuous on some rectangle R in xy-plane that contains the point (a,b) in its interior. then, for some open interval I containing the point 'a', the initial value problem
dy/dx=f(x,y), y(a)=b
has one and only one solution that is defined on the interval I"
now, i understand up to "Can you also use the 'left half'" --
it's clear enough that the piecewise function satisfies the diff eq (the derivs of the pieces = the diff eq y'=3y^(2/3) with the corresponding pieces subbed in for y).
however, the rest is unclear to me.
the answer in the back of the book states: The initial value problem y'=3y^(2/3), y(a)=b always has infinitely many solutions defined for all x. However, if b != 0 then it has a unique solution near x=a. (see the bottom part of the image for the accompanying graph)