Question

Дифференциальное уравненик x(y−1)dy/dx =y^2,x=e,y=1

Answer 1

Ответ:

$x(y-1)\frac{dy}{dx} =y^{2} , where: x = e, y = 1\\\frac{y-1}{y^{2} } dy=\frac{dx}{x} \\\int\frac{y-1}{y^{2} }dy=\int\frac{dx}{x}\\\int(\frac{1}{y}-\frac{1}{y^{2}})dy = \int\frac{dx}{x}\\\int\frac{1}{y}dy - \int\frac{1}{y^{2}}dy = \int\frac{dx}{x}\\ln|y|-(-\frac{1}{y})=ln|x|+C\\C = ln(1)-ln(e)+\frac{1}{1} =0-1+1=0\\ln|y|-ln|x|=-\frac{1}{y}\\ln\frac{y}{x} = -\frac{1}{y}\\\frac{y}{x}=e^{-\frac{1}{y}}\\\frac{y}{x}=\frac{1}{e^{\frac{1}{y}}}\\ye^{\frac{1}{y}}=x$

Пошаговое объяснение: