Question

cos xdy + ysin xdx = dx
Помогите решить

Answer 1

Перепишем:
$Cos xdy + ysin xdx = dx|*frac{1}{dxcosx}\y'+ytgx=frac{1}{cosx}\y=uv;y'=u'v+v'u\u'v+v'u+uvtgx=frac{1}{cosx}\u'v+u(v'+vtgx)=frac{1}{cosx}\begin{cases}v'+vtgx=0\u'v=frac{1}{cosx}end{cases}\frac{dv}{dx}+vtgx=0|*frac{dx}{v}\frac{dv}{v}=-tgxdx\intfrac{dv}{v}=-int tgxdx\ln|v|=ln|cosx|\v=cosx\frac{ducosx}{dx}=frac{1}{cosx}|*frac{dx}{cosx}\du=frac{dx}{cos^2x}\int du=intfrac{dx}{cos^2x}\u=tgx+C\y=cosx(tgx+C)=sinx+Ccosx$