The general solution of the differential equation (x y+y^2 ) d x- (x^2-2 x y ) d y=0 is

Step-by-step solution

array r (x y+y^2 ) d x- (x^2-2 x y ) d y=0 d y d x = (x y+y^2 ) (x^2-2 x y ) d y d x = y x + ( y x )^2 1-2 ( y x ) array Let y =v x d y d x =v+ x d v d x aligned & + x d v d x = v v 12 v x d v d x = v+v^2-v+2 v^2 1-2 v & x d v d x = 3 v^2 1-2 v ( 1 3 v^2 - 2 3 v ) d v= d x x aligned Integrating we get, -1 3 v - 2 3 v= x+ c₁ aligned & -x y -2 y x =3 x+ c₂ & -x y = c₂ x+2 y -x y = c₂ x y^2 & e^ - x y =c x y^2 c x y^2 e^ x y =1 aligned