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1

Example

$x\cdot y'-y-y\cdot\ln\left(\frac{y}{x}\right)=x^3y\cdot\ln\left(\frac{y}{x}\right)$
2

Grouping terms

$-y\cdot x^3\ln\left(\frac{y}{x}\right)-y\ln\left(\frac{y}{x}\right)-y+y'x=0$
3

Subtract $y'x$ from both sides of the equation

$-y\cdot x^3\ln\left(\frac{y}{x}\right)-y\ln\left(\frac{y}{x}\right)-y=-1\cdot y'x$
4

The logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator

$-y\cdot x^3\left(\ln\left(y\right)-\ln\left(x\right)\right)-y\ln\left(\frac{y}{x}\right)-y=-1\cdot y'x$
5

Multiply $\left(-\ln\left(y\right)+\ln\left(x\right)\right)$ by $y$

$x^3\left(y\ln\left(x\right)-y\ln\left(y\right)\right)-y\ln\left(\frac{y}{x}\right)-y=-1\cdot y'x$
6

Multiplying polynomials $x^3$ and $-y\ln\left(y\right)+y\ln\left(x\right)$

$-y\ln\left(\frac{y}{x}\right)-y+x^3y\ln\left(x\right)-x^3y\ln\left(y\right)=-1\cdot y'x$
7

Using the power rule of logarithms

$-y\ln\left(\frac{y}{x}\right)-y+\ln\left(x^{x^3y}\right)-x^3y\ln\left(y\right)=-1\cdot y'x$
8

Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$

$-y\ln\left(\frac{y}{x}\right)-y+x^3y\ln\left(x\right)-x^3y\ln\left(y\right)=-1\cdot y'x$

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