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Provide solution for RD Sharma maths class 12 chapter Higher Order Derivatives exercise 11.1 question 22

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You must know about the derivative of logarithm function and cos x and sin x


y=3cos(log\: x)+4sin(log\: x)


Let\: \: y=3cos(log\: x)+4sin(log\: x)

        \begin{aligned} &\text { Differentiating both sides w.r.t } x\\ &\frac{d y}{d x}=-3 \sin (\log x) \frac{d(\log x)}{d x}+4 \cos (\log x) \frac{d(\log x)}{d x} \end{aligned}

        \begin{aligned} &\text { By using product rule of derivation }\\ &\frac{d y}{d x}=\frac{-3 \sin (\log x)}{x}+\frac{4 \cos (\log x)}{x}\\ &x \frac{d y}{d x}=-3 \sin (\log x)+4 \cos (\log x) \end{aligned}

        \begin{aligned} &\text { Again differentiating both sides w.r.t } x \text { , }\\ &x \frac{d}{d x}\left(\frac{d y}{d x}\right)+\frac{d y}{d x} \frac{d}{d x}(x)=\frac{d}{d x}[-3 \sin (\log x)+4 \cos (\log x)] \end{aligned}

        \begin{aligned} &\text { By using product rule of derivation, }\\ &x \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}(1)=-3 \cos (\log x) \frac{d}{d x}(\log x)-4 \sin (\log x) \frac{d}{d x}(\log x)\\ &x \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}(1)=\frac{-3 \cos (\log x)}{x}-\frac{4 \sin (\log x)}{x} \end{aligned}

        \begin{aligned} &x \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}=\frac{-[3 \cos (\log x)+4 \sin (\log x)]}{x} \\ &x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}=-[3 \cos (\log x)+4 \sin (\log x)] \\ &x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}=-y \end{aligned}

        \begin{aligned} &\therefore x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+y=0\\ &\text { or }\\ &x^{2} y_{2}+x y_{1}+y=0 \end{aligned}

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Gurleen Kaur

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