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`