7 use an appropriate substitution to change the differential

7- use an appropriate substitution to change the differential equation to either a separable or linear equation.  (show work and explanation)

Solution

x*dx - y*dx + x*dy = 0

x*dy = (y - x)*dx

dy/dx = (y - x)/x

dy/dx = y/x - 1

use y/x = u

y = u*x

dy/dx = u + x*du/dx

u + x*du/dx = u - 1

x*du/dx = -1

du/dx = -1/x

du = -dx/x

u = -ln x + c

y/x = -ln x + c

y = -x*ln x + c*x