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  4. If cos -1((y/2))= log e((x/5))5,|y|< 2, then :

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Q. If $\cos ^{-1}\left(\frac{y}{2}\right)=\log _{e}\left(\frac{x}{5}\right)^{5},|y|< 2$, then :

JEE MainJEE Main 2022Continuity and Differentiability

Solution:

$\cos ^{-1}\left(\frac{y}{2}\right)=\log _{e}\left(\frac{x}{5}\right)^{5}$
$\cos ^{-1}\left(\frac{y}{2}\right)=5 \log _{e}\left(\frac{x}{5}\right)$
$\frac{-1}{\sqrt{1-\frac{y^{2}}{4}}} \cdot \frac{y^{\prime}}{2}=5 \cdot \frac{1}{\frac{x}{5}} \times \frac{1}{5}$
$\Rightarrow \frac{-y^{\prime}}{\sqrt{4-y^{2}}}=\frac{5}{x}$
$-x y^{\prime}=5 \sqrt{4-y^{2}}$
$-x y^{\prime \prime}-y^{\prime}=5 \cdot \frac{1}{2 \sqrt{4-y^{2}}}\left(-2 y y^{\prime}\right)$
$\Rightarrow x y^{\prime \prime}+y^{\prime}=\frac{5 y^{\prime} \cdot y}{\sqrt{4-y^{2}}}$
$x y^{\prime \prime}+y^{\prime}=5 \cdot\left(\frac{-5}{x}\right) y$
$x^{2} y^{\prime \prime}+x y^{\prime}=-25 y$