Question

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

Answer 1

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

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