https://file.aiquickdraw.com/mathbot/file/1729327691819xdk0bq17.jpg

The equation provided in the image is:

\[
\text{(tan}^{-1} y - x) dy = (1 + y^2) dx
\]

Let's solve this differential equation.

### Step-by-step solution:

1. **Rearrange the equation:**

\[
(\tan^{-1} y - x) dy = (1 + y^2) dx
\]

This is a first-order differential equation. Let's attempt to separate the variables \(x\) and \(y\) for easier integration.

2. **Rewrite the equation in a separable form:**

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

Now, integrate both sides.

3. **Integration of the left-hand side:**

The integral of \(\frac{\tan^{-1} y - x}{1 + y^2} dy\) can be split:

\[
\int \frac{\tan^{-1} y}{1 + y^2} dy - \int x \, dy
\]

- The first integral \(\int \frac{\tan^{-1} y}{1 + y^2} dy\) looks complicated, but upon simplification, it can be solved using advanced integration techniques.
- The second integral \(- \int x \, dy = -x \cdot y\), assuming \(x\) is independent of \(y\).

4. **Integration of the right-hand side:**

The integral of \(dx\) is simply:

\[
\int dx = x
\]

5. **Combine the results:**

We now combine both sides and introduce the constant of integration, \(C\), to arrive at the general solution:

\[
\text{Solution: (Left side expression)} = x + C
\]

Where the left side involves solving the integration of \(\frac{\tan^{-1} y}{1 + y^2}\), and then we can equate it to \(x + C\).

Would you like me to go deeper into the integration process for the left-hand side, or explain any other step in detail?

---

### Further Questions:

1. What are first-order differential equations?
2. How do you approach solving separable differential equations?
3. What is the significance of the inverse trigonometric functions like \(\tan^{-1} y\) in calculus?
4. Can you give examples of applications where such differential equations appear?
5. How does the method of integration by parts help in more complex integrals?

**Tip:** When solving integrals involving inverse trigonometric functions, use substitutions to simplify the expression.

Solve the differential equation (tan⁻¹ y - x) dy = (1 + y²) dx.

Solve the first-order differential equation (tan⁻¹ y - x) dy = (1 + y²) dx using integration techniques. This solution covers separable differential equations and inverse trigonometric functions, suitable for undergraduate students.

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