The general solution of the differential equation (3 x y+y^2 ) d x+ (x^2+x y ) d y=0 is

Step-by-step solution

aligned & (3 x y+y^2 ) d x+ (x^2+x y ) d y=0 & d y d x =- (3 x y+y^2 ) (x^2+x y ) aligned Put y=v x d y d x =v+x d v d x aligned & v+x d v d x =- (3 v x^2+v^2 x^2 ) (x^2+v x^2 ) =- 3 v+v^2 1+v & x d v d x =- 3 v+v^2 1+v -v=- (2 v^2+4 v ) 1+v aligned aligned & 1+v (2 v^2+4 v ) d v= -1 x d x & 1 2 v+1 v^2+2 v d v=- d x x 1 4 2(v+1) v^2+2 v d v=- x+ c₁ aligned aligned & 1 4 ( v ^2+2 v )+ x = c ₁ 1 4 ( y ^2 x ^2 + 2 y x )+ x = c ₁ & 1 4 ( y ^2+2 xy x ^2 )+ x = c ₁ ( y ^2+2 xy x ^2 )+ x ^4=4 c ₁ & [ ( y ^2+2 xy x ^2 ) (