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диференціальні рівняння 1 порядку

The equation provided is a first-order differential equation:

\[
y y'(1 + x^2) = 1 + y^2
\]

Let's work through this differential equation.

### Step 1: Separate the Variables
We aim to express the equation in a form where we can integrate both sides, so we'll try to separate the variables \( y \) and \( x \).

First, divide both sides of the equation by \( y(1 + x^2) \):

\[
y' = \frac{1 + y^2}{y(1 + x^2)}
\]

Now we can rewrite the equation as:

\[
\frac{dy}{dx} = \frac{1}{y} + \frac{y}{1 + x^2}
\]

### Step 2: Integrate Both Sides
We'll integrate each term separately.

#### 1. First term: \(\int \frac{1}{y} \, dy\)
This is a standard integral:

\[
\int \frac{1}{y} \, dy = \ln |y|
\]

#### 2. Second term: \(\int \frac{y}{1 + x^2} \, dx\)
This integral requires recognizing that \( \frac{y}{1 + x^2} \) involves a substitution or recognizing the antiderivative form, but it may become complex.

At this point, the equation suggests a more advanced integration technique or possibly a substitution to solve it fully.

Would you like a more detailed step-by-step solution for the integration, or would you prefer a numerical approach to solving this equation?

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Here are some relative questions you might find interesting:
1. What is a method to solve more complex separable differential equations?
2. How do we recognize a first-order separable differential equation?
3. Can numerical methods be used to solve differential equations when analytical solutions are difficult?
4. How does the substitution method work in integrating complex terms?
5. What is the role of boundary conditions in solving differential equations?

**Tip:** Always check if variables can be separated in differential equations—it simplifies the integration process.

Learn how to solve the first-order differential equation yy'(1 + x^2) = 1 + y^2 by separating variables and integrating both sides. This problem involves advanced calculus concepts such as separable differential equations and integration techniques.

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