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  • Provide solution for RD Sharma maths class 12 chapter Higher Order Derivatives exercise 11.1 question 34

Provide solution for RD Sharma maths class 12 chapter Higher Order Derivatives exercise 11.1 question 34

Answers (1)

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Answer:

Proved

Hint:

You must know the derivative of exponential and cos inverse function

Given:

y=e^{a \cos ^{-1} x} , \text { prove }\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}-a^{2} y=0

Solution:

y=e^{a \cos ^{-1} x}

        \begin{aligned} &\text { Taking logarithm both sides }\\ &\log y=a \cos ^{-1} x \log c\\ &\log y=a \cos ^{-1} x\\ &\frac{1}{y} \frac{d y}{d x}=a \times\left(\frac{-1}{\sqrt{1-x^{2}}}\right)\\ &\frac{d y}{d x}=\frac{a y}{\sqrt{1-x^{2}}} \end{aligned}

        \begin{aligned} &\text { Squaring both sides, }\\ &\left(\frac{d y}{d x}\right)^{2}=\frac{a^{2} y^{2}}{\left(1-x^{2}\right)}\\ &\left(1-x^{2}\right)\left(\frac{d y}{d x}\right)^{2}=a^{2} y^{2} \end{aligned}

        \begin{aligned} &\text { Again differentiate, }\\ &\left(\frac{d y}{d x}\right)^{2}(-2 x)+\left(1-x^{2}\right) \times 2 \frac{d y}{d x} \frac{d^{2} y}{d x^{2}}=a^{2} \cdot 2 y \cdot \frac{d y}{d x}\\ &-x \frac{d y}{d x}+\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}=a^{2} y\\ &\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}-a^{2} y=0 \end{aligned}

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