The general solution of the differential equation d y d x + y^2+y+1 x^2+x+1 =0 is

Step-by-step solution

aligned & d y d x + y^2+y+1 x^2+x+1 =0 & d y d x =- ( y^2+y+1 x^2+x-1 ) & d y y^2+y+1 =- d x x^2+x+1 & d y y^2+y+ 1 4 + 3 4 =- d x x^2+x+ 1 4 + 3 4 & d y (y+ 1 2 )^2+ ( 3 2 )^2 =- d x (x+ 1 2 )^2+ ( 3 2 )^2 & 1 ( 3 2 )^2 ⁻¹ [ y+ 1 2 ( 3 2 ) ]= -1 ( 3 2 )⁻¹ ⁻¹ [ x+ 1 2 ( 3 2 ) ]+C₁ aligned aligned & 2 3 ⁻¹ ( 2 y +1 3 )= -2 3 ⁻¹ ( 2 x +1 3 )+ C ₁ & 2 3 ⁻¹ [ (2 y +1) 3 + ( 2 x +1 3 ) 1- ( 2 y +1 3 ) ( 2 x +1 3 ) ]= C ₁ & 2 3 ⁻¹ [ 2(x+y+1) 3 3-(4 x y+2 x+2 y+1) 3 ]=C₁ & 2 3 ⁻¹ [ 2(x+y+1) 3 3 2(1-2 x y-x-y) ]=C₁ & align