ydx + (xy + 2x -ye^y)dy = 0, state the valid interval

The given differential equation is:

\[
y dx + (xy + 2x - ye^y) dy = 0
\]

To find the valid interval, we first need to solve the differential equation, identifying any potential restrictions on the solution.

### Step 1: Rearranging the equation
We rewrite the equation in a more convenient form:

\[
y \, dx = -(xy + 2x - ye^y) \, dy
\]

Dividing through by \( y \) (assuming \( y \neq 0 \)):

\[
dx = -\left(x + \frac{2x}{y} - e^y\right) \, dy
\]

This is now a first-order linear differential equation in \( x \) and \( y \).

### Step 2: Solving the equation
To solve this, we treat it as a separable differential equation. First, let's rearrange terms to separate variables:

\[
\frac{dx}{x + \frac{2x}{y} - e^y} = -dy
\]

At this point, solving the equation for \( x \) and \( y \) would typically involve integration. However, rather than proceed with the solution, we focus on the interval over which the solution is valid.

### Step 3: Determining the valid interval
The restrictions on the variables are as follows:

1. **Non-zero \( y \)**: Since we divided by \( y \), \( y \neq 0 \) is a restriction.
2. **Well-defined functions**: The solution involves exponential and polynomial terms, which are defined for all real numbers. However, any division by \( y \) imposes further restrictions.

Thus, the valid interval for \( y \) is any interval that **excludes \( y = 0 \)**. A typical interval would be \( (-\infty, 0) \cup (0, \infty) \).

### Conclusion:
The valid interval for the solution excludes \( y = 0 \), so it could be:

\[
(-\infty, 0) \cup (0, \infty)
\]

Would you like me to go further into the solution steps or clarify anything?

Here are 5 related questions:
1. How would the solution change if \( y = 0 \) were included?
2. What technique is used to solve separable differential equations?
3. How do we handle non-linear terms in differential equations?
4. What is the significance of integrating factors in linear equations?
5. Can you derive the exact solution from this equation?

**Tip:** When solving differential equations, always check for singular points that might restrict the solution's domain.

ydx + (xy + 2x - ye^y)dy = 0, state the valid interval

Learn how to solve the differential equation ydx + (xy + 2x - ye^y)dy = 0 by identifying the valid interval for the solution. This problem covers key concepts such as separable equations and linear first-order differential equations, making it suitable for advanced high school or college-level students.

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