Question

Найдите общие решения уравнений

Answer 1

$\dfrac{dy}{dx}=x(1-y)\\ \int \dfrac{dy}{y-1}=-\int xdx\\ ln|y-1|=-\dfrac{x^2}{2}+C_1\\ |y-1|=C_2e^{-\dfrac{x^2}{2}}\\ y=1+Ce^{-\dfrac{x^2}{2}}$

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$y'-\dfrac{2y}{x+1}=(x+1)^2\\ y'*\dfrac{1}{(x+1)^2}+y*(-\dfrac{2}{(x+1)^3})=1\\ \left[(\dfrac{1}{(x+1)^2})'=\dfrac{-2}{(x+1)^3}\right]\\ (y*\dfrac{1}{(x+1)^2})'_x=1\\ y*\dfrac{1}{(x+1)^2}=x+C\\ y=(x+C)(x+1)^2$