We are tasked with analyzing the differential equation:

$\sin(y) \, dx + x \cos(y) \, dx = y \, dx + x \, dy$

Step 1: Rearrange the terms

First, collect all terms involving $dx$ and $dy$: $\sin(y) \, dx + x \cos(y) \, dx - y \, dx = x \, dy$

Factor $dx$ from the left-hand side: $\big(\sin(y) + x \cos(y) - y\big) \, dx = x \, dy$

Step 2: Write in standard form

Divide through by $x$ (assuming $x \neq 0$) to normalize the equation: $\frac{\sin(y) + x \cos(y) - y}{x} \, dx = dy$

This is the standard form of the equation.

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Step 3: Check for exactness or integrating factor

To simplify or solve, you might test:

1. Exactness: If $M(x, y) = \frac{\sin(y)}{x} + \cos(y) - \frac{y}{x}$ and $N(x, y) = 1$, then check if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$.
2. Potential Substitution: Look for substitutions like $z = x + y$ or other methods depending on the structure.

Would you like me to explore exactness, find an integrating factor, or suggest substitutions? Let me know! 😊

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Follow-up Questions:

1. Would you like detailed steps to test for exactness in this differential equation?
2. Should I attempt solving this equation using a substitution like $u = x + y$?
3. Would you prefer to express this equation geometrically or discuss its physical interpretation?
4. Should we explore numerical solutions or approximations?
5. Do you want to dive deeper into how integrating factors are used?

Tip: Always check for exactness first; it's the most straightforward path to solving differential equations!