KC Sinha Mathematics Solution Class 12 Chapter 12 द्वितीय कोटि का अवकलज (Second Order Derivative) Exercise 12.1 (Q13-Q15)

Exercise 12.1

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Question 13

यदि(If) $y=\sin ^{-1} x$ , दिखाएँ कि (show that) $\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}=0$
Sol :
$y=\sin ^{-1} x$

Differentiating w.r.t x

$\frac{d y}{d x}=\frac{1}{\sqrt{1-x^{2}}}$

Again , Differentiating w.r.t x

$\frac{d^{2} y}{d x^{2}}=\frac{0 \cdot \sqrt{1-x^{2}}-1 \cdot \frac{1}{2 \sqrt{1-x^{2}}} \times(-2x)}{(\sqrt{1-x^{2}})^{2}}$

$\frac{d^{2} y}{d x^{2}}=\frac{\frac{x}{\sqrt{1-x^{2}}}}{1-x^{2}}$

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

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

$\left(1-x^{2}\right) \frac{d^{2} y}{dx^{2}}-x \frac{d y}{d x}=0$

Question 14

यदि(If) $y=\left(\tan ^{-1} x\right)^{2}$ , दिखाएँ कि (show that) $\left(x^{2}+1\right)^{2} y_{2}+2 x\left(x^{2}+1\right) y_{1}-2=0$
Sol :
$y=\left(\tan ^{-1} x\right)^{2}$

Differentiating w.r.t x

$y_{1}=2\left(\tan ^{-1} x\right) \times \frac{1}{1+x^{2}}$

$y_{1}=\frac{2 \tan ^{-1} x}{1+x^{2}}$

$2 \tan ^{-1} x=\left(1+x^{2}\right) y_{1}$

Again , Differentiating w.r.t x

$y_{2}=\frac{2 \times \frac{1}{1+x^{2}} \times\left(1-x^{2}\right)-2 \tan ^{-1} x \times 2 x}{\left(1+x^{2}\right)^{2}}$

$y_{2}=\frac{2-4 x \tan ^{-1} x}{\left(1+x^{2}\right)^{2}}$

$\left(1+x^{2}\right)^{2} \cdot y_{2}=2-2 x \cdot 2 \tan ^{-1} x$

$\left(x^{2}+1\right)^{2} y_{2}=2-2 x\left(1+x^{2}\right) y_{1}$

$\left(x^{2}+1\right)^{2} y_{2}+2 x\left(x^{2}+1\right) y_{1}-2=0$

Question 15

यदि (If) y=3cos(log x)+4sin(log x) दिखाएँ कि (show that) $x^{2} y_{2}+x y_{1}+y=0$
Sol :
y=3cos(log x)+4sin(log x)

Differentiating w.r.t x

$y_{1}=-3 \cdot \sin (\log x) \times \frac{1}{x}+4 \cos (\log x) \times \frac{1}{x}$

$y_{1}=\frac{1}{x}[-3 \sin (\log x)+4 \cos (\log x)]$

Again , Differentiating w.r.t x

$y_{2}=\frac{-1}{x^{2}}\left[-3 \sin\left(\log x\right)+4 \cos \left(\log x\right)\right]+\frac{1}{x}\left[\frac{-3cos(log x)}{x}+\frac{4\{-sin(\log x)\}}{x}\right]$

$y_{2}=-\frac{1}{x^{2}}\left[-3 \sin \left(\log {x}\right)+4 \cos (\log x)\right]+\frac{1}{x^{2}}\left[-3 \cos \left(\log x\right)-4sin(\log x)\right]$

$x^{2} y_{2}=-\left[-3 \sin \left(\log x\right)+4 \cos (\log x)\right]-[3cos(\log x)+4\sin (\log x)]$

$x^{2} y_{2}=-x y_{1}-y$

$x^{2} y_{2}+x y_{1}+y_{2}=0$