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  • Explain solution RD Sharma class 12 Chapter 11 Higher Order Derivatives Exercise Multiple Choice Question question 22

Explain solution RD Sharma class 12 Chapter 11 Higher Order Derivatives Exercise Multiple Choice Question question 22

Answers (1)

Answer:

                (b)

Hint:

We must have known about the derivative of  x

Given:

                y^{\frac{1}{n}}+y^{\frac{-1}{n}}=2 x  

Explanation:

                y^{\frac{1}{n}}+y^{\frac{-1}{n}}=2 x

Differentiate with respect to  x

{\left[\frac{1}{n} y^{\frac{1}{n}-1}-\frac{1}{n} y^{\frac{-1}{n}-1}\right] \frac{d y}{d x}=2} \\

\frac{1}{n} y^{-1}\left[y^{\frac{1}{n}}-y^{\frac{-1}{n}}\right] \frac{d y}{d x}=2 \\

\frac{1}{n y}\left[y^{\frac{1}{n}}-y^{\frac{-1}{n}}\right] \frac{d y}{d x}=2

\begin{aligned} \\ &{\left[y^{\frac{1}{n}}-y^{\frac{-1}{n}}\right] \frac{d y}{d x}=2 x y} \end{aligned}

Squaring both sides,

                \left[y^{\frac{2}{n}}-y^{\frac{-2}{n}}-2\right]\left(\frac{d y}{d x}\right)^{2}=4 x^{2} y^{2}                                                                                                  … (i)

Now

Given    y^{\frac{1}{n}}+y^{\frac{-1}{n}}=2 x

Squaring both sides,

                y^{\frac{2}{n}}+y^{\frac{-2}{n}}+2=4 x^{2}

                \begin{aligned} & \\ &y^{\frac{2}{n}}+y^{\frac{-2}{n}}=4 x^{2}-2 \end{aligned}

Putting this value in (i)

                4 x^{2} y^{2}=\left[4 x^{2}-2-2\right]\left(\frac{d y}{d x}\right)^{2} \\

                4 x^{2} y^{2}=\left[4 x^{2}-4\right]\left(\frac{d y}{d x}\right)^{2}

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

Differentiate with respect to  x ,

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

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

{\left[n^{2} y-1\right] y_{2}+x y_{1}=n^{2} y}

\begin{aligned} & \\ &{\left[x^{2}-1\right] y_{2}+x y_{1}=n^{2} y} \end{aligned} 

Posted by

infoexpert27

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