The differential equation dy dx equals the quotient of x and y squared.
will have a slope field with negative slopes in quadrant I
will have a slope field with positive slopes in all quadrants
will produce a slope field with columns of parallel tangents
Well, let's see if we can solve this equation with a little bit of humor!
So, you've got dy/dx = x/y^2... that's quite a mouthful! Let's break it down.
First, we need to find the slope at each point in the xy-plane. The slope is given by the ratio of x over y squared. That means the slope will change depending on the values of x and y.
Now, in quadrant I, both x and y are positive. And since the slope is given by the ratio x/y^2, if y is positive, then y^2 is positive too. That means the slope will be positive in quadrant I. So, the first option about negative slopes in quadrant I is not correct.
As for the second option, positive slopes in all quadrants, well, that's not correct either. Since the slope depends on the values of x and y, it won't be positive in all quadrants. So, we can cross that one out.
Now, let's talk about the third option, columns of parallel tangents. This is actually possible with this equation! Since the slope depends on the ratio x/y^2, if you have multiple points with the same x-coordinate but different y-coordinates, the slopes will be the same. This can create columns of parallel tangents in the slope field.
So, to sum it up, the correct answer is that the equation will produce a slope field with columns of parallel tangents. Hope that brought a smile to your face!
To determine the slope field of the given differential equation, we need to rewrite it in the form of "dy/dx = f(x, y)". In this case, the equation is dy/dx = x/y^2.
Now, let's analyze the equation. Since we have x in the numerator, the slope dy/dx will be positive when x > 0 and negative when x < 0, regardless of the value of y. Therefore, the statement "will have a slope field with negative slopes in quadrant I" is incorrect.
Since the equation does not depend on y, the value of dy/dx will be the same regardless of y. Therefore, the slope field will have a constant value of dy/dx for every point in the plane. This implies that the statement "will produce a slope field with columns of parallel tangents" is incorrect.
The correct statement is that the slope field will have positive slopes in quadrant I (when x > 0) and negative slopes in quadrant II (when x < 0). In quadrants III and IV, the slopes will be positive again since the negative value of x squared will become positive. Therefore, the statement "will have a slope field with positive slopes in all quadrants" is correct.
To determine the slope field of a differential equation, we can rewrite it in the form dy/dx = f(x, y), where f(x, y) represents the function on the right side of the equation.
In this case, the given differential equation is dy/dx = x / y^2.
To create a slope field, we need to evaluate the function f(x, y) at different points on the xy-plane and plot the corresponding slopes (represented by short line segments) at those points.
Let's analyze the given equation dy/dx = x / y^2 to understand the behavior of the slopes:
1. If x > 0 and y > 0: In this case, both the numerator and the denominator of f(x, y) are positive. Therefore, the slope dy/dx is positive for any point in the first quadrant (where both x and y are positive).
2. If x < 0 and y > 0: The numerator x is negative, but the denominator y^2 is positive. Thus, the slope dy/dx is negative for any point in the second quadrant (where x is negative and y is positive).
3. If x < 0 and y < 0: Both the numerator and the denominator are negative, so their quotient is positive. Consequently, the slope dy/dx is positive for any point in the third quadrant (where both x and y are negative).
4. If x > 0 and y < 0: Here, the numerator x is positive, but the denominator y^2 is negative, resulting in a negative slope dy/dx for any point in the fourth quadrant (where x is positive and y is negative).
Based on this analysis, we can deduce that the slope field of the given differential equation will have positive slopes in all four quadrants (options 2), but it will not have negative slopes in quadrant I (option 1).
Regarding the option 3, a "column of parallel tangents" refers to a slope field where the slopes are the same for each value of x. However, since the slope dy/dx = x / y^2 varies with both x and y in this equation, it does not produce a slope field with columns of parallel tangents.
Therefore, the correct answer is option 2: The given differential equation will have a slope field with positive slopes in all quadrants.
why all the words?
dy/dx = x/y^2
clearly
in QI,IV all the slopes are positive
in QII,III all the slopes are negative
at the x-axis, all the tangents will be vertical, but anywhere else they will not be. See
http://www.wolframalpha.com/input/?i=slope+field&a=*C.slope+field-_*Calculator.dflt-&f2={x%2C1%2Fy2}&f=VectorPlot.vectorfunction\u005f{x%2C1%2Fy2}&f3=x&f=VectorPlot.vectorplotvariable1\u005fx&f4=-10&f=VectorPlot.vectorplotlowerrange1\u005f-10&f5=10&f=VectorPlot.vectorplotupperrange1\u005f10&f6=y&f=VectorPlot.vectorplotvariable2\u005fy&f7=0.5&f=VectorPlot.vectorplotlowerrange2_0.5&f8=1&f=VectorPlot.vectorplotupperrange2\u005f1