Hello,

The given equation is,

dy / dx = 2x - y + 1 / x + y + 2

The above eqn can also be written as,

dy / dx = 2x + 1 - y / x + y + 2

Cross Multiplying,

xdy + ( y + 2 )dy = ( 2x + 1 )dx - ydx

So, ( xdy + ydx ) + ( y + 2 )dy - ( 2x + 1 )dx = 0

So, d( x.y ) + ( y + 2 )dy - ( 2x + 1 )dx = 0

Integrating on both sides,

∫ d ( x.y ) + ∫ ( y + 2 )dy - ∫ ( 2x + 1 )dx = 0

So, xy + ( y^2 / 2 ) + 2y - ( 2 x^2 / 2 ) - x + c = 0

So, -x^2 - x + 2y + xy + ( y^2 / 2 ) + c = 0

So, x^2 + x - 2y - xy - ( y^2 / 2 ) + c = 0

Hence, the solution of given differential equation is x^2 + x - 2y - xy - ( y^2 / 2 ) +c = 0

Best Wishes.