Question

y'-2y/x=1+1/x

Всё, что дано

Answer 1

$y'-\frac{2y}{x}=1+\frac{1}{x}\\\\y=uv\; ,\; \; y'=u'v+uv'\\\\u'v+uv'+\frac{2uv}{x}=\frac{x+1}{x}\\\\u'v+u(v'+\frac{2v}{x})=\frac{x+1}{x}\\\\a)\; \; \frac{dv}{dx}=-\frac{2v}{x}\; \; ,\; \; \int \frac{dv}{v}=-2\int\frac{dx}{x}\; ,\; \; ln|v|=-2ln|x|\\\\v=x^{-2}\; ,\; \; v=\frac{1}{x^2}\\\\b)\; \; u'\cdot \frac{1}{x^2}=\frac{1}{x}\; \; ,\; \; \frac{du}{dx}=\frac{x^2}{x}\; ,\; \; \int du=\int x\, dx\\\\u=\frac{x^2}{2}+C\\\\c)\; \; y=\frac{1}{x^2}\cdot (\frac{x^2}{2}+C)\; \; ,\; \; y=\frac{1}{2}+\frac{C}{x^2}$