Question Number 103958 by bemath last updated on 18/Jul/20

what is integrating factor  of (xy^2 −y) dx − x dy = 0

Answered by bramlex last updated on 18/Jul/20

(xy^2 −y) dx = x dy   (dy/dx) = ((xy^2 −y)/x) = y^2 −(y/x)  (dy/dx)+(y/x) = y^2   set v = y^(−1)  ⇒(dv/dx) = −y^(−2)  (dy/dx)  (dy/dx) = −y^2  (dv/dx) . substitute to  original equation   −y^2  (dv/dx) + (y/x) = y^2   (dv/dx)−(v/x) = −1 . so integrating  factor is u(x) = e^(−∫ (dx/x))  = e^(ln ((1/x)))   u(x) = (1/x) . solution for v(x)  v(x) = ((∫ −1.((1/x))dx +C)/(1/x))  v(x) = x { ln ((1/x)) + C }   (1/y) = −x ln (x) + Cx ★