Understanding the Differential Equation
This differential equation can be identified as a first-order equation. It takes the form:

Equation Structure
- The equation given is \( (x + \tan y) dy = \sin^2 y \, dx \).
- It can be rearranged to express a relationship between \( dy \) and \( dx \).

Rearranging the Equation
- Divide both sides by \( dx \):
\[
\frac{dy}{dx} = \frac{\sin^2 y}{x + \tan y}
\]

Type of Differential Equation
- This is a first-order, separable differential equation, allowing for separation of variables.

Separation of Variables
- To solve it, we can separate the variables \( y \) and \( x \):
\[
\frac{dy}{\sin^2 y} = \frac{dx}{x + \tan y}
\]

Integrating Both Sides
- Integrate both sides:
- The left side with respect to \( y \)
- The right side with respect to \( x \)

General Solution
- After integration, you will obtain an implicit solution involving \( y \) and \( x \).
- The implicit solution may need further simplification or rearrangement to find \( y \) explicitly in terms of \( x \).

Concluding Remarks
- This equation illustrates the behavior of solutions in the \( xy \)-plane, providing insights into the relationship between the variables.
- Further analysis may involve examining specific initial conditions for particular solutions.