The differential equation dy/dx= -x/y
I. will have a slope field with negative slopes in all quadrants
II. will have a slope field with positive slopes in all quadrants
III. will produce a slope field with rows of parallel tangents
Choices:
A. I only
B. II only
C. III only
D. None of these
Please help!! thank you in advance and I greatly appreciate it. :)
dy/dx= -x/y
I. is clearly false, since in QII and QIV -x/y > 0
II. is clearly false, since in QI and QIII -x/y < 0
III. is clearly false, since for a given constant value of y, dy/dx depends on x
Take a visit to https://www.desmos.com/calculator/p7vd3cdmei and type in your function for dy/dx.
Well, let me lace up my intellectual clown shoes and try to help you out!
When we have a differential equation of the form dy/dx = -x/y, we can determine some information about its slope field.
To get an idea of the slope, let's rewrite the equation as ydy = -xdx. Integrating both sides, we get (1/2)y^2 = (-1/2)x^2 + C, where C is an arbitrary constant.
Now let's analyze the options:
I. The slope field having negative slopes in all quadrants: This is not the case, as the slope depends on both x and y. For example, when y > 0 and x > 0, ydy/dx = -x/y is negative, but when y < 0 and x > 0, it becomes positive.
II. The slope field having positive slopes in all quadrants: Similarly, this is not true. The signs of the slopes vary depending on the values of x and y.
III. The slope field producing rows of parallel tangents: This is the most plausible option. Given the equation (1/2)y^2 = (-1/2)x^2 + C, we can see that the curves formed by different values of C (constant of integration) produce rows of parallel tangents.
Therefore, the correct answer is C. III only.
I hope my clownish logic was able to assist you!
To determine the characteristics of the slope field generated by the differential equation dy/dx = -x/y, we need to analyze the equation.
First, let's rewrite the equation as y dy = -x dx. Integrating both sides, we get (1/2)y^2 = (-1/2)x^2 + C, where C is a constant of integration.
From this equation, we can conclude the following:
1. The slope field will have negative slopes in all quadrants:
- Solving for y^2, we get y^2 = -x^2 + 2C. Since the right-hand side is always negative, y^2 is negative or zero. Therefore, y can be negative or zero.
- This means that the slope dy/dx = -x/y will always be negative, as the numerator (x) is negative while the denominator (y) can be negative or zero. Therefore, statement I is true.
2. The slope field will not have positive slopes in all quadrants:
- From the equation (1/2)y^2 = (-1/2)x^2 + C, as discussed earlier, y can be negative or zero. This means that the denominator (y) can never be positive. Therefore, statement II is false.
3. The slope field will not produce rows of parallel tangents:
- To have rows of parallel tangents, the slope dy/dx should be constant along any horizontal line. However, in this case, the slope dy/dx = -x/y varies with both x and y. Thus, statement III is false.
Based on the analysis, the correct answer is D. None of these.
To determine the characteristics of the slope field for the given differential equation dy/dx = -x/y, we can analyze the equation and use the properties of slope fields.
First, let's consider the equation itself. The slope of the field at any point is given by -x/y. This means that the slope is negative when x and y have opposite signs and positive when they have the same signs.
Now, let's evaluate each of the choices:
I. The choice states that the slope field will have negative slopes in all quadrants. For this to be true, the slopes should be negative regardless of the values of x and y. However, since the slope is determined by -x/y, it is clear that the slopes will not always be negative. Therefore, choice I is incorrect.
II. The choice states that the slope field will have positive slopes in all quadrants. Similarly to the previous choice, the slopes will not always be positive since they depend on the values of x and y. Hence, choice II is also incorrect.
III. The choice states that the slope field will produce rows of parallel tangents. This means that if we draw lines tangent to the slope field at various points, the tangents will be parallel and form rows. To determine if this is true, we need to analyze the given differential equation further.
If we rewrite the equation as xy' + y = 0, we can recognize that it is a separable equation. To solve separable equations, we can rewrite it as dy/y = -xdx and integrate both sides. Integrating the left side yields ln|y|, and integrating the right side gives -0.5x^2.
So we have ln|y| = -0.5x^2 + C, where C is the constant of integration.
By exponentiating both sides, we get |y| = e^(-0.5x^2+C). Now, note that e^C is just another constant, so we can rewrite the equation as |y| = Ke^(-0.5x^2), where K is a positive constant.
From this equation, we can see that for any value of x, the value of y will be either K times e^(-0.5x^2) or -K times e^(-0.5x^2). When we draw the slope field based on these values of y, we will indeed see rows of parallel tangents.
Therefore, choice III is correct.
In summary, the correct answer is C. III only.