Differential equation ylnydx+xdy=0

Given the equation:
$$x \frac{d}{d x} y{\left(x \right)} + y{\left(x \right)} \log{\left(y{\left(x \right)} \right)} = 0$$
This differential equation has the form:

f1(x)*g1(y)*y' = f2(x)*g2(y),

where
$$\operatorname{f_{1}}{\left(x \right)} = 1$$
$$\operatorname{g_{1}}{\left(y \right)} = 1$$
$$\operatorname{f_{2}}{\left(x \right)} = - \frac{1}{x}$$
$$\operatorname{g_{2}}{\left(y \right)} = y{\left(x \right)} \log{\left(y{\left(x \right)} \right)}$$
We give the equation to the form:

g1(y)/g2(y)*y'= f2(x)/f1(x).

Divide both parts of the equation by g2(y)
$$y{\left(x \right)} \log{\left(y{\left(x \right)} \right)}$$
we get
$$\frac{\frac{d}{d x} y{\left(x \right)}}{y{\left(x \right)} \log{\left(y{\left(x \right)} \right)}} = - \frac{1}{x}$$
We separated the variables x and y.

Now, multiply the both equation sides by dx,
then the equation will be the
$$\frac{dx \frac{d}{d x} y{\left(x \right)}}{y{\left(x \right)} \log{\left(y{\left(x \right)} \right)}} = - \frac{dx}{x}$$
or
$$\frac{dy}{y{\left(x \right)} \log{\left(y{\left(x \right)} \right)}} = - \frac{dx}{x}$$

Take the integrals from the both equation sides:
- the integral of the left side by y,
- the integral of the right side by x.
$$\int \frac{1}{y \log{\left(y \right)}}\, dy = \int \left(- \frac{1}{x}\right)\, dx$$
Detailed solution of the integral with y
Detailed solution of the integral with x
Take this integrals
$$\log{\left(\log{\left(y \right)} \right)} = Const - \log{\left(x \right)}$$
Detailed solution of the equation
We get the simple equation with the unknown variable y.
(Const - it is a constant)

Solution is:
$$\operatorname{y_{1}} = y{\left(x \right)} = e^{\frac{C_{1}}{x}}$$