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To match the differential equations with their respective slope fields in Figure 17 and Figure 18, we need to analyze the behavior of the slopes (direction fields) shown in the figures and compare them to the form of the given differential equations.

Let's examine each equation and the slope fields:

### 1. **Equation A: \( y' = 2x + y \)**
   - **Form**: The slopes depend on both \( x \) and \( y \).
   - **Behavior**: The slopes should increase as both \( x \) and \( y \) increase. At \( x = 0 \), the slopes are controlled solely by \( y \), so the slope field should exhibit vertical changes based on the \( y \)-value. Away from \( x = 0 \), the field should change with both \( x \) and \( y \).
   
### 2. **Equation B: \( y' = 3 - y \)**
   - **Form**: The slopes depend only on \( y \), not \( x \).
   - **Behavior**: The slope field should be horizontal (no variation) in the \( x \)-direction. As \( y \) increases, the slope decreases since the equation is negatively dependent on \( y \). When \( y = 3 \), the slope becomes zero, indicating horizontal slopes.

### 3. **Equation C: \( y' = \sin(x) \sin(y) \)**
   - **Form**: The slopes are the product of \( \sin(x) \) and \( \sin(y) \), making this equation periodic in both \( x \) and \( y \).
   - **Behavior**: This should produce a more complex pattern with periodic changes in the direction field, showing wave-like structures.

### 4. **Equation D: \( y' = x(3 - y) \)**
   - **Form**: The slopes depend on both \( x \) and \( y \), but they vanish when \( y = 3 \), as the term \( (3 - y) \) becomes zero.
   - **Behavior**: At \( y = 3 \), the slopes should be horizontal, but they change with both \( x \) and \( y \) elsewhere. Near \( x = 0 \), the slope should depend only on \( y \).

### **Figure 17 Analysis**
   - This figure shows slopes that are constant along vertical lines, meaning the slope depends on \( y \) only. As \( y \) increases, the slope changes significantly, indicating a linear relationship with \( y \).
   - **Conclusion**: This is consistent with **Equation B: \( y' = 3 - y \)**, where the slope only changes with \( y \).

### **Figure 18 Analysis**
   - The slopes are zero when \( y = 3 \) and change with both \( x \) and \( y \). The slopes are positive for large positive \( x \) and negative for large negative \( x \), indicating dependence on both \( x \) and \( y \).
   - **Conclusion**: This matches **Equation D: \( y' = x(3 - y) \)**, where the slopes vanish at \( y = 3 \) and depend on \( x \) elsewhere.

### **Final Match**:
- **Figure 17**: \( y' = 3 - y \) (Equation B)
- **Figure 18**: \( y' = x(3 - y) \) (Equation D)

Would you like further clarification or details on any specific aspect?

Here are some related questions you may find helpful:
1. How do we analyze a slope field based on a differential equation?
2. What does it mean when a slope field shows zero slopes at certain values of \( x \) or \( y \)?
3. Can you explain how to interpret nonlinear slope fields such as \( y' = \sin(x) \sin(y) \)?
4. Why does the equation \( y' = 3 - y \) produce constant slopes along vertical lines?
5. How can you verify a differential equation using its slope field?

**Tip**: When matching differential equations with slope fields, focus on key points like where the slopes are zero and how they change with \( x \) and \( y \).

Match the differential equation with its slope field; justify your answers. Options: A) y' = 2x + y, B) y' = 3 - y, C) y' = sin(x)sin(y), D) y' = x(3 - y). Figures 17 and 18 show the slope fields.

Learn how to match differential equations with their corresponding slope fields, including detailed analysis of the equations y' = 2x + y, y' = 3 - y, y' = sin(x)sin(y), and y' = x(3 - y). This problem covers key concepts in slope fields and their visual representations, ideal for undergraduate students in a Differential Equations course.

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