(a) Verify that y₁ =­ x³ and y₂ =­ |x|³ are linearly independent solutions of the differential equation x²y'' ‑ 4xy' + 6y =­ 0 on the interval (‑∞, ∞).

(b) For the functions y₁ and y₂ in part (a), show that W(y₁, y₂) = 0 for every real number x. Does this result violate Theorem 4.1.3? Explain.

(c) Verify that Y₁ =­ x³ and Y₂ =­ x² are also linearly independent solutions of the differential equation in part (a) on the interval (‑∞, ∞).

(d) Besides the functions y₁, y₂, Y₁, and Y₂ in parts (a) and (c), fi­nd a solution of the differential equation that satis­fies y(0) = 0, y'(0) = 0.

(e) By the superposition principle, Theorem 4.1.2, both linear combinations y =­ c₁y₁ + c₂y₂ and Y =­ c₁Y₁ + c₂Y₂ are solutions of the differential equation. Discuss whether one, both, or neither of the linear combinations is a general solution of the differential equation on the interval (‑∞, ∞).