The next seven problems illustrate the fact that, if the hypotheses of Theorem 1 are not satisfied, then the initial value problem y' = f (x,y), y(a) = b may have either no solutions, finitely many solutions, or infinitely many solutions.

(a) Verify that if c is a constant, then the function defined piecewise by

satisfies the differential equation y' = 2√y for all x (including the point x = c). Construct a figure illustrating the fact that the initial value problem y' = 2 √y, y(0) = 0 has infinitely many different solutions.

(b) For what values of b does the initial value problem y' = 2√y, y(0) = b have

(i) No solution,

(ii) A unique solution that is defined for all x?

Transcribed Image Text:

y(x) = (x-c)² for x ≤c, for x> c