4.39. Integrate the Bernoulli equation

$$
\frac{d y}{d x}+\frac{y}{x}=\frac{1}{x^{2}} \cdot \frac{1}{y^{2}}
$$

using two methods: the Bernoulli method and the method of variation of arbitrary constants, after first reducing it to a linear equation.
The problem is solved. 1st method. First, we will use the Bernoulli method.

1st step. Represent the unknown function $y(x)$ as a product of two new unknown functions $u(x)$ and $v(x)$: $y(x)=u(x) \cdot v(x)$, where $\frac{d y}{d x}=v(x) \frac{d u}{d x}+u(x) \frac{d v}{d x}$.

2nd step. Substitute the expressions for $y(x)$ and its derivative into the original equation:

$$
v(x) \frac{d u}{d x}+u(x) \frac{d v}{d x}+\frac{u(x) v(x)}{x}=\frac{1}{x^{2}} \cdot \frac{1}{(u(x) v(x))^{2}}
$$

and factor out $u(x)$ in the last two terms of the left side:

$$
v(x) \frac{d u}{d x}+u(x)\left[\frac{d v}{d x}+\frac{v(x)}{x}\right]=\frac{1}{x^{2}} \cdot \frac{1}{(u(x) v(x))^{2}}
$$

3rd step. Set the expression in the square brackets to zero: $\frac{d v}{d x}+\frac{v(x)}{x}=0$, and solve the resulting equation by the method of separation of variables: $v(x)=\frac{1}{x}$.

4th step. Substitute the found function into the equation obtained in the 2nd step, to get an equation for the unknown function $u(x)$:

$$
\frac{1}{x} \frac{d u}{d x}=\frac{1}{x^{2}} \cdot \frac{1}{\left(u(x) \frac{1}{x}\right)^{2}}
$$

Solve it by the method of separation of variables: $u^{2} d u=x d x$;

$$
\int u^{2} d u=\int x d x ; \quad \frac{u^{3}}{3}=\frac{x^{2}}{2}+C ; u(x)=\sqrt[3]{\frac{3 x^{2}}{2}+3 C}
$$

Denote $3 C=C_{1}$.

5th step. Write down the answer:

$$
y(x)=u(x) v(x)=\frac{1}{x} \cdot \sqrt[3]{\frac{3 x^{2}}{2}+C_{1}}=\sqrt[3]{\frac{3}{2 x}+\frac{C_{1}}{x^{3}}}
$$

2nd method. Now let's solve this equation by reducing it to a linear equation. We have $\alpha=-2$. Introduce the substitution: $u(x)=(y(x))^{1-a}=(y(x))^{1-(-2)}=(y(x))^{3}$. We get a linear first-order ODE with a non-zero right-hand side:

$$
\frac{d u}{d x}+\frac{3}{x} \cdot u=\frac{3}{x^{2}}
$$

Apply the method of variation of arbitrary constants.

1st step. Solve the corresponding homogeneous equation $\frac{d u}{d x}+\frac{3}{x} u=0$ by the method of separation of variables:

$$
\frac{d u}{u}=-3 \cdot \frac{d x}{x} ; \int \frac{d u}{u}=-3 \int \frac{d x}{x} ; \ln |u|=-3 \ln |x|+\ln C ; u(x)=\frac{C}{x^{3}} .
$$

2nd step. Replace the arbitrary constant of integration $C$ with a function $C(x)$ to be determined:

$$
u(x)=\frac{C(x)}{x^{3}}
$$

3rd step. Substitute the new expression for $y(x)$ into the original equation and obtain an equation for $C(x)$:

$$
\frac{d C}{d x} \cdot \frac{1}{x^{3}}=\frac{3}{x^{2}}, \text{ hence } \frac{d C}{d x}=3 x
$$

4th step. Solve the resulting equation by the method of separation of variables: $d C=3 x d x ; C(x)=\frac{3 x^{2}}{2}+C_{1}$.

5th step. Write down the final answer, returning to the sought function: $y(x)=\sqrt[3]{\left(\frac{3 x^{2}}{2}+C_{1}\right) \frac{1}{x^{3}}}=\sqrt[3]{\frac{3}{2 x}+\frac{C_{1}}{x^{3}}}$.