Question

For \(n\in\mathbb{N}\), if \[ y=ax^{n+1}+bx^{-n}, \] then \[ x^2\frac{d^2y}{dx^2} \] is equal to:

• A. \((n-1)y\)
• B. \(n(n-1)y\)
• C. \(n(n+1)y\)
• D. \((n+1)y\)

Hint:

While differentiating power functions, carefully apply the power rule to both positive and negative exponents. Negative exponents are very common in CUET and JEE level differentiation problems.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Repeat of Question 7. Focuses on the derivation of a homogeneous differential relationship from a power function.
Step 2: Key Formula or Approach:
Apply power rule twice.
Step 3: Detailed Explanation:
\( y = ax^{n+1} + bx^{-n} \)
\( y' = a(n+1)x^n - bnx^{-n-1} \)
\( y'' = an(n+1)x^{n-1} + bn(n+1)x^{-n-2} \)
Multiplying by \( x^2 \):
\( x^2y'' = n(n+1) [ax^{n+1} + bx^{-n}] = n(n+1)y \)
Step 4: Final Answer:
The correct choice is Option (C).