Question

Math

Posted 3 months ago

```
Which slope field is generated by the differential equation
$\frac{d y}{d x}=\frac{x-2}{y} \text { ? }$
Choose 1 answer:
(A)
(B)
(C)
(D)
(E)
```

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Answer from Sia

Posted 3 months ago

Solution by Steps

step 2

To determine the correct slope field, we need to analyze the behavior of the slopes based on the equation $\frac{d y}{d x} = \frac{x-2}{y}$

step 3

For $x > 2$, $\frac{x-2}{y}$ is positive if $y > 0$ and negative if $y < 0$

step 4

For $x < 2$, $\frac{x-2}{y}$ is negative if $y > 0$ and positive if $y < 0$

step 5

At $x = 2$, $\frac{d y}{d x} = 0$ for all $y$

step 6

Based on these observations, the slopes are positive in the first and third quadrants and negative in the second and fourth quadrants

step 7

The slopes appear to be steeper as the absolute value of $x$ increases

A

Key Concept

Slope Field Analysis

Explanation

The slope field of a differential equation $\frac{d y}{d x} = \frac{x-2}{y}$ shows the behavior of the slopes based on the values of $x$ and $y$. By analyzing the signs and steepness of the slopes in different quadrants, we can match the correct graph.

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