Correct : a

I have the full question. This is a Cauchy-Euler (equidimensional) equation. Let me solve it.
Given: x²y″ = 6y, i.e., x²y″ − 6y = 0
This is a Cauchy-Euler equation of the form x²y″ + pxy′ + qy = 0, with p = 0, q = −6.
Assume solution y = x^m:
y′ = mx^m−1, y″ = m(m−1)x^m−2
Substituting:
x² × m(m−1)x^m−2 − 6x^m = 0
m(m−1)x^m − 6x^m = 0
m(m−1) − 6 = 0
m² − m − 6 = 0
(m−3)(m+2) = 0
m = 3 or m = −2
General solution:
y(x) = ax³ + bx⁻² = ax³ + b/x²
Your page''s answer of A is correct ✓ 1. EXPLANATION: The equation x²y″ = 6y, rewritten as x²y″ − 6y = 0, is a Cauchy-Euler (equidimensional) equation. For such equations, assume a trial solution y = x^m.
Substituting y = x^m, y′ = mx^m−1, y″ = m(m−1)x^m−2 into the equation:
x² × m(m−1)x^m−2 − 6x^m = 0
m(m−1) − 6 = 0
m² − m − 6 = 0
(m−3)(m+2) = 0
This gives two roots: m = 3 and m = −2. Since the roots are real and distinct, the general solution is a linear combination of x³ and x⁻²:
y(x) = ax³ + b/x²
Correct answer: A) y(x) = ax³ + b/x² ✓