Question

Найти общее решение уравнения

Answer 1

$(1 + x^2)y' - 2xy = (1 + x^2)^2\\y' - \frac{2xy}{(1 + x^2)} = 1 + x^2\\y' + y(1 + x^2)\frac{d}{dx}(\frac{1}{1+x^2}) = 1 + x^2\\\frac{dy}{dx} \cdot \frac{1}{1+x^2} + y\frac{d}{dx}(\frac{1}{1+x^2}) = 1\\\fbox{d$(fg)$ = $f$d$g$ + $g$d$f$}\\\frac{d}{dx} (\frac{y}{1 + x^2}) = 1\\\int d(\frac{y}{1 + x^2}) = \int dx\\\frac{y}{(1 + x^2)} = x + c_1\\y = c_1(1 + x^2) + x + x^3$