If y=a xⁿ+1}+b x^{-n}, then x² {d² y}{d x²}=

Correct Answer is : n(n+1)y{n}({n}+1) yn(n+1)y

Correct Answer is : n(n+1)y{n}({n}+1) yn(n+1)y

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MHTCET 10 May 2024 Shift 1 Paper

Question : 138 of 150

Marks: +1, -0

y=a x^{n+1}+b x^{-n}

x^{2} \frac{d^{2} y}{d x^{2}}=

Solution:

Concept:

Differentiate the given function twice and express the second derivative in terms of

y

Explanation:

Given

y = a x^{n+1} + b x^{-n}

Differentiating with respect to

x

\frac{dy}{dx} = a(n+1)x^{n} + b(-n)x^{-n-1}

= a(n+1)x^{n} - b n x^{-n-1}

Differentiating again with respect to

x

\frac{d^{2}y}{dx^{2}} = a(n+1)n x^{n-1} - b n(-n-1)x^{-n-2}

= a n(n+1)x^{n-1} + b n(n+1)x^{-n-2}

Factor out

\frac{n(n+1)}{x^{2}}

\frac{d^{2}y}{dx^{2}} = \frac{n(n+1)}{x^{2}}\left[a x^{n+1} + b x^{-n}\right]

Since

y = a x^{n+1} + b x^{-n}

, we get:

\frac{d^{2}y}{dx^{2}} = \frac{n(n+1)}{x^{2}} y

Multiplying both sides by

x^{2}

x^{2}\frac{d^{2}y}{dx^{2}} = n(n+1)y

Answer:

x^{2}\frac{d^{2}y}{dx^{2}} = n(n+1)y

Hence, the correct option is A.

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