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The given differential equation is:

\[
\left( -4e^x \sin(y) + 3y \right) dx + \left( 4x - 2e^x \cos(y) \right) dy = 0
\]

This is a first-order differential equation of the form:

\[
M(x, y) dx + N(x, y) dy = 0
\]

where:
- \( M(x, y) = -4e^x \sin(y) + 3y \)
- \( N(x, y) = 4x - 2e^x \cos(y) \)

### Step 1: Check if the equation is exact

To verify if the equation is exact, we use the "mixed partials" condition, which states that a differential equation is exact if:

\[
\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}
\]

1. First, calculate \( \frac{\partial M}{\partial y} \):

\[
\frac{\partial M}{\partial y} = \frac{\partial}{\partial y} \left( -4e^x \sin(y) + 3y \right)
= -4e^x \cos(y) + 3
\]

2. Now, calculate \( \frac{\partial N}{\partial x} \):

\[
\frac{\partial N}{\partial x} = \frac{\partial}{\partial x} \left( 4x - 2e^x \cos(y) \right)
= 4 - 2e^x \cos(y)
\]

Since \( \frac{\partial M}{\partial y} = -4e^x \cos(y) + 3 \) and \( \frac{\partial N}{\partial x} = 4 - 2e^x \cos(y) \), these are not equal, meaning the equation is **not exact**.

Would you like more details or have any specific questions about this? 

Here are five relative questions you can explore next:
1. How can we modify a non-exact equation to make it exact?
2. What are integrating factors, and how can they be applied to this equation?
3. What is the general solution for a non-exact differential equation?
4. How do you solve an exact differential equation once verified as exact?
5. What is the significance of the level curves of \( F(x, y) \)?

**Tip:** When dealing with exact differential equations, always start by checking the mixed partials condition to see if it's solvable using the exact method.

Use the 'mixed partials' check to see if the following differential equation is exact: (-4e^x sin(y) + 3y)dx + (4x - 2e^x cos(y))dy = 0

Learn how to verify if the given differential equation (-4e^x sin(y) + 3y)dx + (4x - 2e^x cos(y))dy = 0 is exact using the mixed partials condition. Detailed step-by-step solution provided, focusing on key concepts of exactness in differential equations and related partial derivatives.

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