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  • Please solve RD Sharma class 12 chapter Higher Order Derivatives exercise 11.1 question 41 maths textbook solution

Please solve RD Sharma class 12 chapter Higher Order Derivatives exercise 11.1 question 41 maths textbook solution

Answers (1)

best_answer

Answer:

Proved

Hint:

You must know the derivative of cot inverse x

Given:

y=\left(\cot ^{-1} x\right)^{2}, \text { prove } y_{2}\left(x^{2}+1\right)^{2}+2 x\left(x^{2}+1\right) y_{1}=2

Solution:

y=\left(\cot ^{-1} x\right)^{2}

        \begin{aligned} &\frac{d y}{d x}=2 \cot ^{-1} x\left(\frac{-1}{1+x^{2}}\right)\\ &\left(1+x^{2}\right) \frac{d y}{d x}=-2 \cot ^{-1} x\\ &\text { Again differentiating }\\ &\left(1+x^{2}\right)^{2} \frac{d^{2} y}{d x^{2}}+2 x \frac{d y}{d x}=-2\left(\frac{-1}{1+x^{2}}\right) \end{aligned}

        \begin{aligned} &\left(1+x^{2}\right)^{2} \frac{d^{2} y}{d x^{2}}+2 x \frac{d y}{d x}=\left(\frac{2}{1+x^{2}}\right) \\ &\left(1+x^{2}\right)^{2} \frac{d^{2} y}{d x^{2}}+2 x\left(1+x^{2}\right) \frac{d y}{d x}=2 \\ &y_{2}\left(x^{2}+1\right)^{2}+2 x\left(x^{2}+1\right) y_{1}=2 \end{aligned}

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