The solution of differential equation (y x-1) y d x=x d y is

Step-by-step solution

Given differential equation is (y x-1) y d x=x d y d y d x = (y x-1) y x = y^2 x x - y x d y d x + y x = y^2 x x 1 y^2 d y d x + y⁻¹ x = x x ....(i) Put y⁻¹=v -y⁻² d y d x = d v d x y⁻² d y d x =- d v d x From Eq. (i), we have - d v d x + v x = x x d v d x - v x =- x x (ii) This is linear differential equation. Here, IF =e^ (- 1 x ) d x =e^ - x = 1 x So, solution is v IF = IF Q d x+C aligned & v 1 x &= 1 x (- x x ) d x+C & v 1 x aligned =- x x^2 d x+C=- [ x ( -1 x )+ 1 x 1 x d x ]+C 1 x y = x x + 1 x +C [ v= 1 y ]