QJEEM22Ma2a7f: JEE Main 2022 Mathematics - Class 12 JEEM22Ma2a7f Solution: If , then :

{\cos ^{ - 1}}\left( {{y \over 2}} \right) = {\log _e}{\left( {{x \over 5}} \right)^5}\,\,\,\,\,\,\,\,\,|y| < 2

Differentiating on both side

$- {1 \over {\sqrt {1 - {{\left( {{y \over 2}} \right)}^2}} }} \times {{y'} \over 2} = {5 \over {{x \over 5}}} \times {1 \over 5}$

${{ - xy'} \over 2} = 5\sqrt {1 - {{\left( {{y \over 2}} \right)}^2}}$

Square on both side

${{{x^2}y{'^2}} \over 4} = 25\left( {{{4 - {y^2}} \over 4}} \right)$

Diff on both side

$2xy{'^2} + 2y'y''{x^2} = - 25 \times 2yy'$

$xy' + y''{x^2} + 25y = 0$