(a) Verify that y 1 x 3 and y 2 x 3 are linearly independent solutions of the differential equation x 2 y ' ' 4xy ' 6y 0 on the interval (, ) (b) For the functions y 1 and y 2 in part (a), show that W(y 1 , y 2 ) 0 for every real number x Does this result violate Theorem 4 1 3 Explain (c) Verify th...

a From the graphs of y 1 x 3 and y 2 x 3 we see that the functions are linearly independent since th ...

(a) Verify that y 1 = x 3 and y 2 = |x| 3 are linearly independent...

Question:

(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 (‑∞, ∞).