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  • Please Solve RD Sharma Class 12 Chapter 11 Higher Order Derivatives Exercise Multiple Choice Questions Maths Textbook Solution Question 15

Please Solve RD Sharma Class 12 Chapter 11 Higher Order Derivatives Exercise Multiple Choice Questions Maths Textbook Solution Question 15

Answers (1)

Answer:

                (c)

Hint:

We must have known about the derivative of inverse trigonometric functions like  \sin^{-1}x .

Given:

                y=\sin \left(m \sin ^{-1} x\right)

Explanation:

                y=\sin \left(m \sin ^{-1} x\right)

Differentiating both sides with respect x,

                \frac{d y}{d x}=\cos \left(m \sin ^{-1} x\right) \times \frac{m}{\sqrt{1-x^{2}}}

                \begin{aligned} &\\ &\frac{d y}{d x}=\frac{m \cos \left(m \sin ^{-1} x\right)}{\sqrt{1-x^{2}}} \end{aligned}

Again differentiate with respect to x ,

                 \frac{d^{2} y}{d x^{2}}=m\left[\left(\sqrt{1-x^{2}}\right)\left(-\sin \left(m\left(\sin ^{-1} x\right)\right) \times \frac{m}{\sqrt{1-x^{2}}}\right)-\cos \left(m\left(\sin ^{-1} x\right)\right) \times \frac{1}{2 \sqrt{1-x^{2}}} \times(0-2 x)\right] \\

                \frac{d^{2} y}{d x^{2}}=m\left[\frac{\left(-m \sin \left(m\left(\sin ^{-1} x\right)\right)\right)+\cos \left(m\left(\sin ^{-1} x\right)\right) \times \frac{x}{\sqrt{1-x^{2}}}}{1-x^{2}}\right] 

                \begin{aligned} & \\ &\frac{d^{2} y}{d x^{2}}=m\left[\frac{\left(-m \sin \left(m\left(\sin ^{-1} x\right)\right)\right)}{1-x^{2}}+\frac{x \cos \left(m\left(\sin ^{-1} x\right)\right)}{\left(1-x^{2}\right) \sqrt{1-x^{2}}}\right] \end{aligned}

                \frac{d^{2} y}{d x^{2}}=m\left[\frac{\left(-m \sin \left(m\left(\sin ^{-1} x\right)\right)\right)}{1-x^{2}}+\frac{x \frac{d y}{d x}}{\left(1-x^{2}\right)}\right]

                \left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}=m x \frac{d y}{d x}-m^{2} \sin \left(m\left(\sin ^{-1} x\right)\right)

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

                \left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-m x \frac{d y}{d x}+m^{2} y=0     or

                \begin{aligned} & \\ &\left(1-x^{2}\right) y^{2}-m x y^{1}=-m^{2} y \end{aligned}

Posted by

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