Calculus of a Single Variable, Chapter 6, 6.3, Section 6.3, Problem 8

An ordinary differential equation (ODE) has differential equation for a function with single variable. A first order ODE follows y' =f(x,y).
The y' can be denoted as (dy)/(dx) to be able to express in a variable separable differential equation: N(y)dy= M(x)dx .
To be able to follow this, we let y'=(dy)/(dx) on the given first order ODE: xy'=y :
xy' = y
x(dy)/(dx) = y
Cross-multiply to rearrange it into:
(dy)/y= (dx)/x
Applying direct integration on both sides:
int (dy)/y= int (dx)/x
Apply basic integration formula for logarithm: int (du)/u = ln|u|+C .
ln|y|= ln|x|+C
y = e^(ln|x| + C)
= Ce^ln|x| since e^C is a constant
y = Cx