Differential equation ydy=-xdx

Given the equation:
$$y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} = - x$$
This differential equation has the form:
$$f_1(x)*g_1(y)*y' = f_2(x)*g_2(y)$$
where
$$f_{1}{\left(x \right)} = 1$$
$$g_{1}{\left(y \right)} = 1$$
$$f_{2}{\left(x \right)} = - x$$
$$g_{2}{\left(y \right)} = \frac{1}{y{\left(x \right)}}$$
We give the equation to the form:
$$\frac{g_1(y)}{g_2(y)}*y'= \frac{f_2(x)}{f_1(x)}$$
Divide both parts of the equation by $g_{2}{\left(y{\left(x \right)} \right)}$
$$\frac{1}{y{\left(x \right)}}$$
we get
$$y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} = - x$$
We separated the variables x and y.

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

Take the integrals from the both equation sides:
- the integral of the left side by y,
- the integral of the right side by x.
$$\int y\, dy = \int \left(- x\right)\, dx$$
Detailed solution of the integral with y
Detailed solution of the integral with x
Take this integrals
$$\frac{y^{2}}{2} = - \frac{x^{2}}{2} + Const$$
Detailed solution of the equation
We get the simple equation with the unknown variable y.
(Const - it is a constant)

Solution is:
$$y_{1} = y{\left(x \right)} = - \sqrt{- x^{2} + C_{1}}$$
$$y_{2} = y{\left(x \right)} = \sqrt{- x^{2} + C_{1}}$$