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  • Provide Solution for RD Sharma Class 12 Chapter 11 Higher Order Derivatives Exercise Multiple Choice Question Question 12

Provide Solution for RD Sharma Class 12 Chapter 11 Higher Order Derivatives Exercise Multiple Choice Question Question 12

Answers (1)

Answer:   Option (c)

Hint:

We must know about the derivative of  \cos x and logarithm.

Given:

                y=a \cos \left(\log _{e} x\right)+b \sin \left(\log _{e} x\right)

Explanation:

                y=a \cos \left(\log _{e} x\right)+b \sin \left(\log _{e} x\right)

Differentiate both side with respect to  x

                \frac{d y}{d x}=\frac{a[-\sin (\log x)]}{x}-\frac{b[\cos (\log x)]}{x} \\

                \frac{d y}{d x}=\frac{-[a \sin (\log x)+b \cos (\log x)]}{x} \

                \begin{aligned} \ &x \frac{d y}{d x}=-[a \sin (\log x)+b \cos (\log x)] \end{aligned}

Again differentiating with respect to  x ,

                \begin{aligned} &x \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}=-\left[\frac{a \cos (\log x)}{x}-\frac{b \sin (\log x)}{x}\right] \\ \end{aligned}

                \begin{aligned} &x \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}=-\left[\frac{a \cos (\log x)}{x}-\frac{b \sin (\log x)}{x}\right] \\ \end{aligned}

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

                x \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}+\frac{y}{x}=0

                \begin{aligned} \\ &x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+y=0 \end{aligned}    or

                x^{2}y_{2}+xy_{1}+y=0

Hence option the value of   x^{2}y_{2}+xy_{1}   is (-y)

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