if
Question:

If $y^{2}+\log _{e}\left(\cos ^{2} x\right)=y, x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, then :

  1. (1) $y^{\prime \prime}(0)=0$

  2. (2) $\left|y^{\prime}(0)\right|+\left|y^{\prime \prime}(0)\right|=1$

  3. (3) $\left|y^{\prime \prime}(0)\right|=2$

  4. (4) $\left|y^{\prime}(0)\right|+\left|y^{\prime \prime}(0)\right|=3$


Correct Option: , 3

Solution:

$y^{2}+2 \log _{e}(\cos x)=y$……..(1)

$\Rightarrow 2 y y^{\prime}-2 \tan x=y^{\prime}$…..(2)

From(1), $y(0)=0$ or 1

$\therefore y^{\prime}(0)=0$

Again differentiating (2) we get,

$2\left(y^{\prime}\right)^{2}+2 y y^{\prime \prime}-2 \sec ^{2} x=y^{\prime \prime}$

Put $x=0, y(0)=0,1$ and $y^{\prime}(0)=0$,

we get, $\left|y^{\prime \prime}(0)\right|=2$

 

Administrator

Leave a comment

Please enter comment.
Please enter your name.