QJEEM22Mac13e: JEE Main 2022 Mathematics - Class 12 JEEM22Mac13e Solution: Let be the solution of the differential equation . If for some…

${{dy} \over {dx}} = {y \over x}{{(4{y^2} + 2{x^2})} \over {(3{y^2} + {x^2})}}$

Put $y = vx$

$\Rightarrow {{dy} \over {dx}} = v + x{{dv} \over {dx}}$

$\Rightarrow v + x{{dv} \over {dx}} = {{v(4{v^2} + 2)} \over {(3{v^2} + 1)}}$

$\Rightarrow x{{dv} \over {dx}} = v\left( {{{(4{v^2} + 2 - 3{v^2} - 1)} \over {3{v^2} + 1}}} \right)$

$\Rightarrow \int {(3{v^2} + 1){{dv} \over {{v^3} + v}} = \int {{{dx} \over x}} }$

$\Rightarrow \ln |{v^3} + v| = \ln x + c$

$\Rightarrow \ln \left| {{{\left( {{y \over x}} \right)}^3} + \left( {{y \over x}} \right)} \right| = \ln x + C$

$\downarrow \,y(1) = 1$

$\Rightarrow C = \ln 2$

$\therefore$ for $y(2)$

$\ln \left( {{{{y^3}} \over 8} + {y \over 2}} \right) = 2\ln 2 \Rightarrow {{{y^3}} \over 8} + {y \over 2} = 4$

$\Rightarrow [y(2)] = 2$

$\Rightarrow n = 3$