The solution of the differential equation (3 x y+y² ) d x+ (x²+x y ) d y=0 is

Step-by-step solution

Homogeneous equation can be written in the form of Put aligned d y d x &=- 3 x y+y² x²+x y y=& v x and d y d x =v+x d v d x , we get aligned v+x d v d x =- 3 x² v+x² v² x²+x² v x d v d x = -2 v(v+2) v+1 array ll & 1 x d x=- (v+1) 2 v(v+2) d v & - 2 x =- [ 1 2(v+2) + 1 2 v ] d v array On integrating, we get aligned &-2 _ e x= 1 2 (v+2)+ 1 2 v- c & v(v+2) x⁴=c² & y x ( y x +2 ) x⁴=c² & ( v= y x ) aligned Hence, required solution is (y²+2 x y ) x²=c²