Question

Решить дефиринцал
(x^2-2xy)y'=xy-y^2

Answer 1

$(x^2-2xy)y'=xy-y^2\\y'= frac{xy-y^2}{x^2-2xy} \\t=frac{y}{x}; ,; y=tx; ,; y'=t'x+t\\t'x+t= frac{tx^2-t^2x^2}{x^2-2tx^2} \\ t'x+t=frac{t-t^2}{1-2t} \\t'x= frac{t-t^2}{1-2t}-t\\t'x= frac{t-t^2-t+2t^2}{1-2t}\\ frac{dt}{dx} = frac{t^2}{x(1-2t)} \\int frac{(1-2t), dt}{t^2} =int frac{dx}{x}\\int (t^{-2}-frac{2}{t})dt=int frac{dx}{x}\\frac{t^{-1}}{-1}-2cdot ln|t|=ln|x|+C\\ -frac{x}{y}-2cdot lnleft | frac{y}{x} right |=ln|x|+C$