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  • Need solution for RD Sharma maths class 12 chapter Higher Order Derivatives exercise 11.1 question 39

Need solution for RD Sharma maths class 12 chapter Higher Order Derivatives exercise 11.1 question 39

Answers (1)

Answer:

Proved

Hint:

You must know the derivative of sin and logarithm function

Given:

y=\sin (\log x) , \text { prove } x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+y=0

Solution:

y=\sin (\log x)

        \begin{aligned} &\frac{d y}{d x}=\cos (\log x) \times \frac{1}{x}\\ &=\frac{\cos (\log x)}{x}\\ &\text { Again differentiating }\\ &\frac{d^{2} y}{d x^{2}}=\frac{x\left[-\sin (\log x) \times \frac{1}{x}\right]-\cos (\log x)}{x^{2}}\\ &=\frac{-\cos (\log x)-\sin (\log x)}{x^{2}} \end{aligned}

        \begin{aligned} &\text { Now, }\\ &\text { LHS }=x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+y\\ &=\frac{x^{2}\{-\cos (\log x)-\sin (\log x)\}}{x^{2}}+\frac{x \cos (\log x)}{x}+\sin (\log x)\\ &=0=\mathrm{RHS} \end{aligned}

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