Question

x dy/dx = y - x

Answer 1

Solution.

$$x\frac{dy}{dx}=y-x.$$

This is linear inhomogeneous differential equation of the 1st order.

$$xy'-y=-x,\newline y'-\frac{1}{x}y=-1.$$

The solution of the equation is written in the form

$$y=y_c+y_p,$$

where $y_c$ is the solution of homogeneous equation, and $y_p$ is a particular solution of the inhomogeneous equation.

$y'-\frac{1}{x}y=0,$

$\frac{dy}{y}=\frac{dx}{x},$

$\int\frac{dy}{y}=\int\frac{dx}{x},$

$\ln{|y|}=\ln{|x|}+C_1,$

$|y|=|x|e^{C_1},$

from here $y_c=Cx,$ where $C$ is some constant, $C\neq 0.$

$y_p=u(x)x,$

$u'x+u-\frac{1}{x}ux=-1,$

$u'x=-1,$

$\frac{du}{dx}=-\frac{1}{x},$

$u=-\ln|x|,$ from here

$y_p=-\ln{|x|}x.$

So, $y=Cx-x\ln{|x|}.$

Answer. $y=Cx-x\ln{|x|}.$