Hey can someone help me with this? the back of the book simply says, "This is true due to the uniqueness theorem."
Suppose that x is a solution to the initial value problem:
Show that 0<x(t)<1 for all t for which x is defined.
I set x'=f(t,x). I took the partial derivative of f(t,x) with respect to x. Unless I am mistaken, I believe both f(t,x) and its partial derivative are continuous over a small enough rectangle R containing the point (1/2,0). Since it is given that x is a solution, it means that x is unique.
Why does 0<x(t)<1 have to be true?