(a) Verify that y1 = x3 and are linearly independent solutions of the differential e...

(a) Verify that y₁ = x³ and are linearly independent solutions of the differential equation on the interval

(b) 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) Find a solution of the differential equation satisfying

(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

(reference theorem 4.1.2)