Question

x+xy+y'(y+xy)=0 помогите решить общее уравнение

Answer 1

$displaystyle x+xy+y'(y+xy)=0\\x(y+1)+y'(x+1)y=0\\y'(x+1)y=-x(y+1)\\frac{dy}{dx}*frac{y}{y+1}=-frac{x}{x+1}\\\frac{ydy}{y+1}=-frac{xdx}{x+1}\\\ intlimits frac{ydy}{y+1}= intlimits -frac{xdx}{x+1}\\\ intlimits(1-frac{1}{y+1})dy=- intlimits(1-frac{1}{x+1})dx\\\ intlimits dy-intlimitsfrac{dy}{1+y}=-intlimits dx +intlimits frac{dx}{1+x}\\\intlimits dy-intlimitsfrac{d(y+1)}{y+1}=-intlimits dx +intlimits frac{d(x+1)}{x+1}$

$boxed{y-ln|y+1|=-x+ln|x+1|+C}$