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**Breaking down Question 1:**

The problem asks us to verify if the expression \( y^2 - 4x^2 + e^{xy} = C \) is a solution to the differential equation:

\[
(8x - y e^{xy}) - (2y + x e^{xy}) y' = 0
\]

### Step-by-Step Breakdown

1. **Differentiate the given solution expression with respect to \( x \):**
   To verify if \( y^2 - 4x^2 + e^{xy} = C \) is indeed a solution, we start by differentiating both sides of the equation with respect to \( x \):
   \[
   \frac{d}{dx}(y^2 - 4x^2 + e^{xy}) = \frac{d}{dx}(C)
   \]
   Since \( C \) is constant, \( \frac{d}{dx}(C) = 0 \).

2. **Apply the Product Rule and Chain Rule:**
   Differentiate each term on the left-hand side:
   \[
   2y \cdot y' - 8x + (y + xy') e^{xy} = 0
   \]
   Rearrange to group terms involving \( y' \):
   \[
   (2y + x e^{xy}) y' + y e^{xy} - 8x = 0
   \]

3. **Isolate \( y' \):**
   Move terms to isolate \( y' \):
   \[
   (2y + x e^{xy}) y' = 8x - y e^{xy}
   \]
   So,
   \[
   y' = \frac{8x - y e^{xy}}{2y + x e^{xy}}
   \]

4. **Substitute \( y' \) back into the differential equation:**
   Substitute this \( y' \) expression into the original differential equation:
   \[
   (8x - y e^{xy}) - (2y + x e^{xy}) \left(\frac{8x - y e^{xy}}{2y + x e^{xy}}\right) = 0
   \]
   Simplifying this expression shows that it holds true, confirming that \( y^2 - 4x^2 + e^{xy} = C \) is indeed a solution of the differential equation.

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### Related Questions

1. How would you verify if a different function is a solution to the same differential equation?
2. What steps are involved in isolating terms when solving for \( y' \) in implicit differentiation?
3. How does the presence of exponential terms like \( e^{xy} \) impact the solution process for differential equations?
4. What are alternative methods for confirming solutions to differential equations?
5. How does solving differential equations with constants, like \( C \), differ from those without?

### Tip:
When differentiating complex expressions, clearly separate and simplify terms involving derivatives to avoid errors in calculations.

To verify whether y^2 − 4x^2 + e^xy = C is a solution of (8x − y e^xy) − (2y + x e^xy)y' = 0

Explore a step-by-step breakdown to verify if y^2 - 4x^2 + e^xy = C is a solution of (8x − y e^xy) − (2y + x e^xy)y' = 0 using differentiation and exponential functions. Suitable for undergraduate engineering students.

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