Differential equation 2*x^2*y*y'+y^2=2

Given the equation:
$$2 x^{2} y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} + y^{2}{\left(x \right)} = 2$$
This differential equation has the form:

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

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

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

Divide both parts of the equation by g_2(y)
$$- \frac{2 - y^{2}{\left(x \right)}}{2 y{\left(x \right)}}$$
we get
$$\frac{2 y{\left(x \right)} \frac{d}{d x} y{\left(x \right)}}{y^{2}{\left(x \right)} - 2} = - \frac{1}{x^{2}}$$
We separated the variables x and y.

Now, multiply the both equation sides by dx,
then the equation will be the
$$\frac{2 dx y{\left(x \right)} \frac{d}{d x} y{\left(x \right)}}{y^{2}{\left(x \right)} - 2} = - \frac{dx}{x^{2}}$$
or
$$\frac{2 dy y{\left(x \right)}}{y^{2}{\left(x \right)} - 2} = - \frac{dx}{x^{2}}$$

Take the integrals from the both equation sides:
- the integral of the left side by y,
- the integral of the right side by x.
$$\int \frac{2 y}{y^{2} - 2}\, dy = \int \left(- \frac{1}{x^{2}}\right)\, dx$$
Detailed solution of the integral with y
Detailed solution of the integral with x
Take this integrals
$$\log{\left(y^{2} - 2 \right)} = Const + \frac{1}{x}$$
Detailed solution of the equation
We get the simple equation with the unknown variable y.
(Const - it is a constant)

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