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Does the initial value problem y' =3*y^2/3 , y(0)=10^-7

have an unique solution in a neighborhood of x=0?.

Okay, I applied the Existence and Uniqueness theorem and took the partial derivative of y' with respect to y and got 2*y^-1/3. Now since my starting point is (0,10^-7), my y' is continuous so it passes the test. And my i.v.p. does in fact have a unique solution at that starting point.

Okay, now don't you find it weird that my functions aren't depending on x?

And the solutions manual goes on to say both functions y' and partial of y' w.r.t. y are defined on this rectangle R={(x,y): -1<x<1, (1/2)10^-7 <y<(2)10^-7} What?!? Where did those numbers come from? Why can't x be equal to or greater than 1? My y' and partial of y' w.r.t. y don't even depend on x so why is x bounded?

Any help will be greatly appreciated, as always...Karen