Question

Решить уравнение: x(y"+1)=-y'

Answer 1

$\displaystyle x(y''+1)=-y'\\z=y';z'=y''\\x(z'+1)=-z\\xz'+z=-x\\xz'+z=0\\\frac{dz}{z}=-\frac{dx}{x}\\ln|z|=-ln|x|+ln|C|\\z=\frac{C(x)}{x};z'=\frac{C'(x)}{x}-\frac{C(x)}{x^2}\\C'(x)-\frac{C(x)}{x}+\frac{C(x)}{x}=-x\\C'(x)=-x\to C(x)=-\frac{x^2}{2}+C_1\\z=-\frac{x}{2}+\frac{C_1}{x}=y'\\y=\int(-\frac{x}{2}+\frac{C_1}{x})dx=-\frac{x^2}{4}+C_1ln|x|+C_2$