Question

For $\text{n} \in \mathbb{N}$ if $y = \text{a}x^{\text{n}+1} + \text{b}x^{-\text{n}}$, then $x^2 \frac{\text{d}^2 y}{\text{d}x^2} =$

• A. $\text{n}(\text{n} - 1)y$
• B. $(\text{n} - 1)y$
• C. $\text{n}(\text{n} + 1)y$
• D. $(\text{n} + 1)y$

Hint:

Look for factorization after differentiation.

Answer 1

The Correct Option is C

Step 1: Differentiate $y$. \[ y = ax^{n+1} + bx^{-n} \] \[ \frac{dy}{dx} = a(n+1)x^n - bn x^{-n-1} \]
Step 2: Second derivative. \[ \frac{d^2y}{dx^2} = a(n+1)n x^{n-1} + b n(n+1)x^{-n-2} \]
Step 3: Multiply by $x^2$. \[ x^2 \frac{d^2y}{dx^2} = a n(n+1)x^{n+1} + b n(n+1)x^{-n} \] \[ = n(n+1)(ax^{n+1} + bx^{-n}) \]
Step 4: Conclusion. \[ x^2 \frac{d^2y}{dx^2} = n(n+1)y \]