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:
Applying direct integration on both sides:
`int (dy)/y= int (dx)/x`
Apply basic integration formula for logarithm: `int (du)/u = ln|u|+C` .
`y = e^(ln|x| + C)`
` = Ce^ln|x|` since `e^C` is a constant
`y = Cx`