The particular solution of differential equation ( x + y ) dy +( x - y ) d x =0 at x = y =1 is

Step-by-step solution

(x+y) d y+(x-y) d x=0 We have d y d x =- (x-y) x+y Put y=v x d y d x =v+x d v d x v+x d v d x = -(x-v x) x+v x = -x(1-v) x(1+v) = -(1-v) 1+v x d v d x = -1+v 1+v -v= -1+v-v-v^2 1+v x d v d x = - (1+v^2 ) 1+v (1+v) 1+v^2 d v=- d x x d v 1+v^2 + 1 2 2 v 1+v^2 d v=- d x x aligned & ⁻¹( v )+ 1 2 |1+ v ^2 |=- x + c ₁ & ⁻¹ ( y x )+ 1 2 |1+ y ^2 x ^2 |=- x + c ₁ aligned 2 ⁻¹ ( y x )+ | x^2+y^2 x^2 |+2 x= c [ c=2 c₁ ] 2 ⁻¹ ( y x )+ | x^2+y^2 x^2 x^2 |= c