y 
′
 +cos( 
2
x+y
​
 )=cos( 
2
x−y
​
 )

The differential equation given is:

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

This is a non-linear first-order differential equation. To solve it, we can try the substitution method.

### Step 1: Substitution

Let \( z = 2x - y \). Then, the derivative of \( z \) with respect to \( x \) is:

\[
z' = 2 - y'
\]

Now, rewrite the original equation using \( z \):

\[
y' = 2 - z'
\]

Substituting \( y' = 2 - z' \) into the original equation:

\[
2 - z' + \cos(2x + y) = \cos(z)
\]

Since \( 2x + y = z + 2y \), we can rewrite \( \cos(2x + y) \) as:

\[
\cos(2x + y) = \cos(z + 2y)
\]

Thus, the equation becomes:

\[
2 - z' + \cos(z + 2y) = \cos(z)
\]

This equation is still quite complicated. Another approach is to look for possible simplifications or integrating factors. However, given the nonlinearity and the trigonometric terms, it suggests that finding an explicit solution might not be straightforward and could require numerical methods or further transformation.

### Step 2: Check for Possible Solutions or Simplifications

Let's explore potential solutions by trying a simple assumption \( y = y(x) \) that might simplify the equation.

#### Case: Assume \( y(x) = 0 \)
Substitute \( y(x) = 0 \) into the equation:

\[
0 + \cos(2x) = \cos(2x)
\]

This holds true, so \( y(x) = 0 \) is indeed a particular solution.

### Step 3: General Solution
Given the complexity of the non-linear trigonometric terms, a general closed-form solution may not be easily obtainable. However, \( y(x) = 0 \) is a particular solution. To find more solutions, you might need to use numerical methods or more advanced techniques.

Would you like more details or have any questions?

### Related Questions
1. How can we solve non-linear differential equations?
2. What is the substitution method in differential equations?
3. How can numerical methods be applied to solve differential equations?
4. What are some common types of non-linear differential equations?
5. How does symmetry in a differential equation help in finding solutions?

### Tip:
For non-linear differential equations, sometimes simple solutions (like \( y = 0 \)) can satisfy the equation. Always check if any assumptions or simplifications can lead to particular solutions before trying more complex methods.

Explore how to solve the non-linear differential equation y' + cos(2x + y) = cos(2x - y) step-by-step. This problem involves differential equations, trigonometric functions, and the substitution method for non-linear equations. Suitable for advanced learners.

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