Question Number 102298 by Ar Brandon last updated on 08/Jul/20

sinx∙(dy/dx)−ycosx=y^3 sin^2 xcosx

Answered by john santu last updated on 08/Jul/20

(dy/dx)−ycot x=sin xcos x .y^3   set v = y^(−2)  ⇒(dv/dx) = −2y^(−3)  (dy/dx)  (dy/dx) = −(1/2)y^3  (dv/dx)  ⇔ −(1/2)y^3  (dv/dx)−ycot x=sin xcos x y^3   (dv/dx) +2cot x.v = sin 2x  IF u(x)=e^(∫2cot x dx)  = e^(2ln(sin x)) =sin ^2 x  v = ((∫2sin ^3 x d(sin x) +C)/(sin ^2 x))  ((sin ^2 x)/y^2 ) = (1/2)sin ^4 x + C   (1/y^2 ) = (1/2)sin ^2 x + C. csc^2 x   (JS ⊛)