The differential equation dy/dx = x/y^2
Possible answers:
I. will have a slope field with negative slopes in quadrant I
II. will have a slope field with positive slopes in all quadrants
III.will produce a slope field with columns of parallel tangents
IV. None of these
x/y^2 has positive values in QI
x/y^2 has negative values in QII,QIII
For any constant x, the slopes get less steep farther from the x-axis
so, what do you think?
My answer is III. Is that correct?
I think I like D
If you look at the slopes in any column (constant x) they are not parallel
desmos.com can produce slope fields
thank you Oobleck. I opened a new tab with Desmos.com but I don't know how to use it. Can you tell me how can I do for dy/dx= x/y^2 in desmos.com pleaseeee
just do
google "desmos slope fields"
and it will provide many hits. The top one in my list was
https://www.desmos.com/calculator/p7vd3cdmei
and then I just changed the formula
generally, googling "desmos xxx" will provide good help
thank you so muchhhh
To determine which statement(s) about the given differential equation is true, we can look at the properties and characteristics of the equation.
The given differential equation is dy/dx = x/y^2.
To understand the slope field, we can rewrite the differential equation as xdy - y^2dx = 0.
This equation is not solvable explicitly for y(x). So instead, let's analyze the slope field by considering some key properties of the differential equation.
The slope of the solution curve at any point (x, y) is given by dy/dx = x/y^2.
Now, let's look at each statement and analyze them one by one:
I. The statement "will have a slope field with negative slopes in quadrant I"
To check if this statement is true, we can substitute some positive and negative values of x and y into the given differential equation.
For example, when x = 1 and y = 1, dy/dx = 1/1^2 = 1, which is positive. If we choose other points in quadrant I, we will consistently get positive slope values. Thus, statement I is false.
II. The statement "will have a slope field with positive slopes in all quadrants"
Following the same method as above, we can choose points in different quadrants to check the slope values.
For example, if we choose x = -1 and y = -1 in the differential equation, we get dy/dx = (-1)/(-1)^2 = -1, which is negative. Therefore, statement II is false.
III. The statement "will produce a slope field with columns of parallel tangents"
To verify this statement, we need to analyze the behavior of the slope values as we move along the x-axis.
By substituting a fixed value of x and various values of y into the differential equation, we can confirm if the slopes remain the same along columns or change.
For instance, if we substitute x = 1 and y = 1, we get dy/dx = 1/1^2 = 1. Similarly, if we choose other points on the line x = 1, we will consistently get a slope of 1. Therefore, statement III is true.
IV. None of these
Since statement III is true, we can conclude that the correct answer is IV. None of these.
To summarize:
I. False
II. False
III. True
IV. None of these