Interval of Definition. By looking at an initial value problem dy/dx = f (x,y) with...

Interval of Definition. By looking at an initial value problem dy/dx = f (x,y) with y (x₀) = y₀, it is not always possible to determine the domain of the solution y (x) or the interval over which the function y (x) satisfies the differential equation.

(a) Solve the equation dy/dx = xy³.

(b) Give explicitly the solutions to the initial value problem with y (0) = 2

(c) Determine the domains of the solutions in part (b).

(d) As found in part (c), the domains of the solutions depend on the initial conditions. For the initial value problem dy/dx = xy³ with y (0) = a, a > 0, show that as a approaches zero from the right the domain approaches the whole real line (-∞,∞ ) and as a approaches + ∞ the domain shrinks to a single point.

(e) Sketch the solutions to the initial value problem