Step-by-Step Solution:

We need to identify the correct differential equation from the slope field shown in the image.

The slope field is determined by the differential equation $\frac{dy}{dx} = f(x, y)$, where $f(x, y)$ represents the slope at any given point $(x, y)$.

Observing the Slope Field:

1. Horizontal slopes (dy/dx = 0): The slope lines appear horizontal along the $x$-axis ($y = 0$).

   • This suggests that $f(x, y) = 0$ when $y = 0$.

2. Slope behavior with $x$ and $y$:

   • Above the $x$-axis ($y > 0$), the slope increases (positive).
   • Below the $x$-axis ($y < 0$), the slope decreases (negative).

3. Symmetry: The slope pattern seems symmetric in both quadrants above and below the $x$-axis, indicating that the equation might include terms like $y^2$, which is symmetric about the $x$-axis.

Testing Each Option:

We evaluate the given options against the slope field:

(A) $\frac{dy}{dx} = 1 + y^2$:

• At $y = 0$, $\frac{dy}{dx} = 1$. This implies a uniform slope of 1 along the $x$-axis.
• The slope in the field at $y = 0$ appears constant but NOT 1 everywhere. Eliminate (A).

(B) $\frac{dy}{dx} = x - y$:

• At $y = 0$, $\frac{dy}{dx} = x$, meaning the slope varies linearly with $x$.
• At $x = 0$, $\frac{dy}{dx} = -y$, which means the slope becomes negative below the $x$-axis and positive above.
• This fits the slope behavior in the field. Possible answer.

(C) $\frac{dy}{dx} = 1 + x^2$:

• At $y = 0$, $\frac{dy}{dx} = 1 + x^2$, which gives a parabola-like slope pattern along the $x$-axis.
• This does NOT match the observed field, where slopes along $y = 0$ are not parabolic. Eliminate (C).

(D) $\frac{dy}{dx} = 1 - y^2 + x^2$:

• At $y = 0$, $\frac{dy}{dx} = 1 + x^2$, which again suggests a parabolic slope pattern.
• This does NOT match the slope field. Eliminate (D).

Conclusion:

The correct answer is (B) $\frac{dy}{dx} = x - y$, as it aligns with the slope patterns observed in the field.

Let me know if you need further explanation!

Here are 5 related questions you can try:

1. How do slope fields visually represent solutions to differential equations?
2. How can you sketch a slope field given a differential equation?
3. What does symmetry in a slope field tell us about the differential equation?
4. How does the behavior of $\frac{dy}{dx}$ at $y = 0$ help narrow down the options?
5. How can you verify the correctness of a solution to a slope field problem?

Tip: Always look for key features like horizontal slopes, symmetry, and patterns along axes when matching slope fields to equations.