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(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 (‑∞, ∞).
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