Differential Equations and Slope Fields

Match the differential equation to its slope field given the options: some equations contain only $x$, some contain only $y$, and some contain both $x$ and $y$.

In this problem, you are tasked with matching a differential equation to its corresponding slope field. Slope fields, also known as direction fields, provide a graphical representation of the solutions of a first-order differential equation without explicitly solving the equation. They plot small line segments at various points in the plane, with each segment's slope representing the solution's derivative at that point. This visual tool helps in understanding the behavior and direction of solutions over the plane.

When solving such a problem, it's important to understand that the segments in the slope field reflect the rate of change dictated by the differential equation at specific points. Equations that involve only $x$ will likely show variations in the slope that depend solely on horizontal changes, while equations involving only $y$ will depict a pattern that varies with vertical changes. Equations involving both $x$ and $y$ will showcase more complex interactions. Identifying how these variations correspond to the differential equation's format is crucial for matching to the correct slope field. By analyzing the resulting patterns and slopes, you can infer which differential equation corresponds to each visual representation.