Diffrencial equation( Non-Exact)

Question

(1+ cos(x))y' = (sin²(x)+sin²(x) cos(x)- ysin(x))

Answer 1

y' - sinx/(1 + cosx)y = sin²x; Integrated factor M(x) = e^{∫-sinx/(1 + cosx)dx} = 1 + cosx;

y'(1 + cosx) - ysinx = sin²x(1 + cosx); [y(1 + cosx)]' = sin²x(1 + cosx); y(1 + cosx) = ∫sin²x(1 + cosx)dx;

y(1 + cosx) = ∫[(1 - cos2x)/2 + sin²xcosx)dx; y(1 + cosx) = x/2 - sin2x/4 + sin³x/3 + C;

y = (x/2 - sin2x/4 + sin³x/3 + C)/(1 + cosx);