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The differential equation provided is:

\[
y' = \cos 2x + y^2 \cos 2x
\]

This can be simplified as:

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

This is a first-order differential equation in terms of \( y \) and \( x \), and it could be solved using methods such as separation of variables if possible or integrating factor methods, depending on the form upon further analysis.

Would you like a step-by-step solution to solve this differential equation?

### Related Questions
1. How can we separate variables in this equation to solve for \( y \)?
2. What methods are there to solve first-order differential equations with trigonometric functions?
3. Can we apply an integrating factor in this scenario?
4. How does the cosine function affect the solution process of a differential equation?
5. What are the initial conditions required to determine a unique solution to this differential equation?

### Tip
When dealing with trigonometric functions in differential equations, consider if transformations (like using trigonometric identities) can simplify the terms.

Solve the first-order differential equation y' = cos(2x) + y^2 cos(2x) using methods like separation of variables or integrating factor. Understand trigonometric functions within differential equations and explore potential solutions.

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