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 ⊛)