Let's analyze the direction field and match it to the correct differential equation from the options. ### **Step-by-Step Solution** #### 1. **Identify a Key Point and Slope** Let’s use the point **(0, 4)** as shown in the plot. The arrow at this point is **pointing straight right**, which means the slope (dy/dx) at this point is **0**. #### 2. **Plug (x, y) = (0, 4) into Each Option:** Let’s test each equation to see which one gives dy/dx = 0 at (0, 4): --- ##### **Option A: \( y' = x(3-x) \)** - At (0,4): \( y' = 0 \times (3-0) = 0 \) - **Possible!** --- ##### **Option B: \( y' = \cos(y) \)** - At (0,4): \( y' = \cos(4) \neq 0 \) - **No, not correct.** --- ##### **Option C: \( y' = y(4-y) \)** - At (0,4): \( y' = 4(4-4) = 4 \times 0 = 0 \) - **Possible!** --- ##### **Option D: \( y' = e^{-x} \)** - At (0,4): \( y' = e^{0} = 1 \) - **No, not correct.** --- ##### **Option E: \( y' = y(3-y) \)** - At (0,4): \( y' = 4(3-4) = 4 \times (-1) = -4 \) - **No, not correct.** --- ##### **Option F: \( y' = \cos(4-y) \)** - At (0,4): \( y' = \cos(4-4) = \cos(0) = 1 \) - **No, not correct.** --- #### 3. **Check More Points to Distinguish Between the Remaining Options (A and C)** Let's check the point (0, 0): ##### **For Option A: \( y' = x(3-x) \)** - At (0,0): \( y' = 0 \) - The slope at all points along the y-axis is zero (horizontal arrows). ##### **For Option C: \( y' = y(4-y) \)** - At (0,0): \( y' = 0 \) - The slope at (0,0) is also zero. Let's check a point off the y-axis, say (2, 4): ##### **For Option A: \( y' = x(3-x) \)** - At (2, 4): \( y' = 2(3-2) = 2 \times 1 = 2 \) - Slope is positive (pointing upwards). ##### **For Option C: \( y' = y(4-y) \)** - At (2, 4): \( y' = 4(4-4) = 0 \) - Slope is zero (horizontal). From the plot, at (2,4), the arrow is **horizontal**, so the slope is zero, matching **Option C**. --- ### **Final Answer** The correct differential equation is: \[ \boxed{y' = y(4-y)} \] #### **Explanation:** - The direction field shows horizontal arrows (slope 0) at \( y = 0 \) and \( y = 4 \). - This matches \( y' = y(4-y) \), which is zero when \( y = 0 \) or \( y = 4 \). - The other equations do not give zero slope at these points.