Suppose f is a non-negative continuous function whose support [a, b] is a subset of (0, 1). Show that there is a unique solution to the ordinary differential equation F(x) = 2F(x)f (x), F(1) = 1, F...

Suppose f is a non-negative continuous function whose support [a, b] is a subset of (0, 1). Show that there is a unique solution to the ordinary differential equation F(x) = 2F(x)f (x), F(1) = 1, F (1) = 0, that F is everywhere positive, and F is bounded on [0,∞).