QJEEM22Ma2277: JEE Main 2022 Mathematics - Class 12 JEEM22Ma2277 Solution: Let be the solution curve of the differential equation ,…

{{dy} \over {dx}} + {1 \over {{x^2} - 1}}y = \sqrt {{{x - 1} \over {x + 1}}} ,\,x > 1

Integrating factor I.F. $= {e^{\int {{1 \over {{x^2} - 1}}dx} }} = {e^{{1 \over 2}\ln \left| {{{x - 1} \over {x + 1}}} \right|}}$

$= \sqrt {{{x - 1} \over {x + 1}}}$

Solution of differential equation

$y\sqrt {{{x - 1} \over {x + 1}}} = \int {{{x - 1} \over {x + 1}}dx = \int {\left( {1 - {2 \over {x + 1}}} \right)dx} }$

$y\sqrt {{{x - 1} \over {x + 1}}} = x - 2\ln |x + 1| + C$

Curve passes through $\left( {2,\sqrt {{1 \over 3}} } \right)$

${1 \over {\sqrt 3 }} \times {1 \over {\sqrt 3 }} = 2 - 2\ln 3 + C$

$C = 2\ln 3 - {5 \over 3}$

$y(8) \times {{\sqrt 7 } \over 3} = 8 - 2\ln 9 + 2\ln 3 - {5 \over 3}$

$\sqrt 7 \,.\,y(8) = 19 - 6\ln 3$