Reread Example 3 and then discuss, with reference to Theorem 1.2.1, the existence and uniqueness of a solution of the initial-value problem consisting of xy' - 4y = x⁶e^x and the given initial condition.

(a) y(0) = 0

(b) y(0) = y₀, y₀ > 0

(c) y(x0) = y₀, x₀ . 0, y₀ > 0

Theorem 1.2.1

Let R be a rectangular region in the xy plane de­ned by a ≤ x ≤ b, c ≤ y ≤ d that contains the point (x₀, y₀) in its interior. If f (x, y) and ∂f /∂y are continuous on R, then there exists some interval I₀: (x₀ - h, x₀ + h), h > 0, contained in [a, b], and a unique function y(x), defi­ned on I₀, that is a solution of the initial value problem (2).