9. differential equations — ગણિત ધોરણ 12 સાયન્સ — Question

$\left(x^2-x y\right) d y=\left(x y+y^2\right) d x$

$\Rightarrow \frac{d y}{d x}=\frac{x y+y^2}{x^2-x y}$

which is a homogeneous differential eqn

Puty $= vx$

$\frac{d y}{d x}=v+x \frac{d v}{d x}$

So, eqn $(1)$ becomes

$v + x \frac{ dv }{ dx }=\frac{ v + v ^2}{1- v }$

$\Rightarrow x \frac{ dv }{ dx }=\frac{2 v ^2}{1- v }$

$\Rightarrow \frac{1- v }{2 v ^2} dv =\frac{ dx }{ x }$

Integrating both sides, we get

$\Rightarrow \int \frac{1- v }{ v ^2} dv =2 \int \frac{ dx }{ x }$

$\Rightarrow \frac{-1}{ v }-\log v =2 \log x +\log C$

$\Rightarrow \frac{- x }{ y }-\log y +\log x =2 \log x +\log C$

$\Rightarrow \frac{- x }{ y }=\log xy C$

$\Rightarrow ce ^{-x y}=x y \quad$ where $c=\frac{1}{C}$