Question

Дифференциальные уравнения xy'+y=y^2

Answer 1

$xy'+y=y^2\\xy'=y^2-y\\y'=frac{y^2-y}{x}\\frac{dy}{dx}=frac{y^2-y}{x}\\int frac{dy}{y^2-y}=int frac{dx}{x}\\int frac{dy}{(y-frac{1}{2})^2-frac{1}{4}}=int frac{dx}{x}\\frac{1}{2cdot frac{1}{2}}cdot lnleft |frac{y-frac{1}{2}-frac{1}{2}}{y-frac{1}{2}+frac{1}{2}}right |=ln|x|+lnC\\lnleft |frac{y-1}{y}right |=ln|x|+lnC\\frac{y-1}{y}=cx\\1-frac{1}{y}=cx\\frac{1}{y}=1-cx\\y=frac{1}{1-cx}$