12. match the given differential equation with the slope field shown. put the letter in the little box on…

Explanation:

Step1: Analyze slope - sign properties

For $\frac{dy}{dx}=\sin x$, the slope depends only on $x$. The values of $\sin x$ range from - 1 to 1. When $x = 0,\frac{dy}{dx}=0$; when $x=\frac{\pi}{2},\frac{dy}{dx} = 1$; when $x=\frac{3\pi}{2},\frac{dy}{dx}=-1$. For $\frac{dy}{dx}=\frac{-x}{y}$, at the origin $(0,0)$ the slope is undefined. In the first - quadrant ($x>0,y>0$), the slope is negative; in the second - quadrant ($x < 0,y>0$), the slope is positive; in the third - quadrant ($x<0,y < 0$), the slope is negative; in the fourth - quadrant ($x>0,y < 0$), the slope is positive. For $\frac{dy}{dx}=x^{2}$, since $x^{2}\geq0$ for all real $x$, and $\frac{dy}{dx}=0$ when $x = 0$. The slope is symmetric about the $y$ - axis. For $\frac{dy}{dx}=y^{2}$, since $y^{2}\geq0$ for all real $y$, and $\frac{dy}{dx}=0$ when $y = 0$. The slope does not depend on $x$ directly. For $\frac{dy}{dx}=\frac{-y}{x}$, at the origin $(0,0)$ the slope is undefined. In the first - quadrant ($x>0,y>0$), the slope is negative; in the second - quadrant ($x < 0,y>0$), the slope is positive; in the third - quadrant ($x<0,y < 0$), the slope is negative; in the fourth - quadrant ($x>0,y < 0$), the slope is positive. But it has a different behavior near the axes compared to $\frac{dy}{dx}=\frac{-x}{y}$. For $\frac{dy}{dx}=\frac{-4}{y}$, at $y = 0$ the slope is undefined. When $y>0$, the slope is negative; when $y < 0$, the slope is positive.

Step2: Match with slope - fields

Without seeing the actual slope - fields, we can make general observations. If a slope - field has horizontal tangents at $x = 0$ and the slopes vary periodically with $x$, it is likely $\frac{dy}{dx}=\sin x$. If a slope - field has a symmetric pattern about the $y$ - axis and non - negative slopes, it is likely $\frac{dy}{dx}=x^{2}$. If a slope - field has horizontal tangents along the $x$ - axis ($y = 0$) and non - negative slopes, it is likely $\frac{dy}{dx}=y^{2}$.

Since the actual slope - fields are not labeled for reference in the text part, we assume we would analyze the following characteristics:

• For $\frac{dy}{dx}=\sin x$: Look for a slope - field where the slopes repeat periodically in the $x$ - direction and are zero at $x = n\pi,n\in\mathbb{Z}$.
• For $\frac{dy}{dx}=\frac{-x}{y}$: Look for a slope - field with four distinct quadrant - based slope signs and an undefined slope at the origin.
• For $\frac{dy}{dx}=x^{2}$: Look for a slope - field that is symmetric about the $y$ - axis and has non - negative slopes, with zero slope at $x = 0$.
• For $\frac{dy}{dx}=y^{2}$: Look for a slope - field with zero slope along the $x$ - axis and non - negative slopes.
• For $\frac{dy}{dx}=\frac{-y}{x}$: Look for a slope - field with four distinct quadrant - based slope signs and an undefined slope at the origin, with a different behavior near the axes compared to $\frac{dy}{dx}=\frac{-x}{y}$.
• For $\frac{dy}{dx}=\frac{-4}{y}$: Look for a slope - field with an undefined slope at $y = 0$, negative slopes for $y>0$ and positive slopes for $y < 0$.

To actually match them, we would need to analyze the specific features of each slope - field such as the location of horizontal and vertical tangents, the sign of the slopes in different quadrants, and any symmetry properties.

Since the slope - fields are not named or described in detail in the text, we cannot give a definite answer. But the general method to solve this problem is as above.

If we assume we have slope - fields named as slope - field 1, slope - field 2, etc. and we analyze them one by one:

1. For $\frac{dy}{dx}=\sin x$: Check for periodic behavior of slopes with respect to $x$ and zero - slope at $x=n\pi$.
2. For $\frac{dy}{dx}=\frac{-x}{y}$: Check for quadrant - based sign changes and an undefined slope at the origin.
3. For $\frac{dy}{dx}=x^{2}$: Check for $y$ - axis symmetry and non - negative slopes with zero slope at $x = 0$.
4. For $\frac{dy}{dx}=y^{2}$: Check for zero slope along the $x$ - axis and non - negative slopes.
5. For $\frac{dy}{dx}=\frac{-y}{x}$: Check for quadrant - based sign changes and an undefined slope at the origin with a different behavior near the axes compared to $\frac{dy}{dx}=\frac{-x}{y}$.
6. For $\frac{dy}{dx}=\frac{-4}{y}$: Check for an undefined slope at $y = 0$, negative slopes for $y>0$ and positive slopes for $y < 0$.

Since the actual matching cannot be done without the labeled slope - fields, we cannot provide a final answer in the form of "a corresponds to slope - field X" etc. But the above steps show how to approach the problem.

If we had to make a general statement about the process:

1. First, analyze the sign and value of $\frac{dy}{dx}$ for different values of $x$ and $y$ for each differential equation.
2. Then, compare these characteristics with the features of the given slope - fields such as the direction of line segments (positive/negative slope), location of horizontal/vertical tangents, and symmetry properties.