1. The problem asks to find the differential equation that generates the given direction field. 2. From the description, the vector field has a vertical component generally pointing along the y-axis and a horizontal component that changes direction at $x=0$. 3. This suggests the slope field is defined by $\frac{dy}{dx} = \frac{\text{vertical component}}{\text{horizontal component}}$. 4. Since the horizontal component changes sign at $x=0$ and the vertical component is mostly vertical, a simple model is $\frac{dy}{dx} = -\frac{x}{y}$. 5. This differential equation means the slope at any point $(x,y)$ is $-x/y$, which matches the symmetry and direction changes described. 6. To verify, rearrange: $y\,dy = -x\,dx$. 7. Integrate both sides: $\int y\,dy = -\int x\,dx$ gives $\frac{y^2}{2} = -\frac{x^2}{2} + C$. 8. Multiply both sides by 2: $y^2 + x^2 = 2C$, which represents circles centered at the origin. 9. The direction field of $\frac{dy}{dx} = -\frac{x}{y}$ corresponds to circles, consistent with the vector field's symmetry and direction changes. Final answer: The differential equation is $$\frac{dy}{dx} = -\frac{x}{y}$$.