Question

Найдите частное решение уравнения xdy-ydx=ydy удовлетворяющие условию y(-1)=1

Answer 1

$xdy - ydx = ydy\\\\ -ydx = ydy - xdy\\\\ -ydx = (y - x)dy\\\\ -y\frac{dx}{dy} = y - x\\\\ \frac{dx}{dy} = -1 + \frac{x}{y}\\\\ x' - x\frac{1}{y} = -1\\\\ x = uv\\\\$

$u'v + v'u - uv\frac{1}{y} = -1\\\\ u'v +u(v' - v\frac{1}{y}) = -1\\\\ v' - v\frac{1}{y} = 0, \ v' = v\frac{1}{y}, \ \frac{dv}{v} = \frac{dy}{y}, \ \int \frac{dv}{v} = \int \frac{dy}{y}\\\\ \ln(v) = \ln(y), \ v = y\\\\ u'y = -1, \ du = -\frac{dy}{y}, \ \int du = -\int \frac{dy}{y}\\\\ u = -\ln(y) + C\\\\\ x = uv = (-\ln(y) + C)y\\\\ -1 = (-0 + C)*1, \ C = -1\\\\ \boxed{x = (-\ln(y) - 1)y}$