Which are true about the differential equation dy/dx = 2x(4-y)
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
I is true sometimes -- e.g., at (2,5)
II is true sometimes -- e.g., at (1,1),(-1,5),(-1,1),(1,-5)
III is clearly false. In a column at x=h, the tangents will get more negative as y increases.
I think the question is poorly worded.
To understand the characteristics of the given differential equation dy/dx = 2x(4-y), let's break it down.
First, it is useful to rewrite the differential equation in a different form. We can rearrange it as dy/(4-y) = 2xdx and integrate both sides.
∫dy/(4-y) = ∫2xdx
Integrating the left-hand side, we get -ln|4-y| = x^2 + C1, where C1 is the constant of integration.
Taking the exponential of both sides and simplifying, we get |4-y| = e^(-x^2-C1). Since the exponential function e^(-x^2-C1) is always positive, we can remove the absolute value sign.
4 - y = e^(-x^2-C1)
Rearranging this equation, we have:
y = 4 - e^(-x^2-C1)
Now that we have the solution to the differential equation, we can analyze its properties:
I. "Will have a slope field with negative slopes in quadrant I"
To determine the slope at any point on the graph, we need to calculate dy/dx. In the given equation, dy/dx = 2x(4-y). When x > 0, the term 2x is positive. Additionally, since the term (4-y) is always positive or zero (as y ranges from -∞ to 4), the product 2x(4-y) will be positive or zero. Therefore, the slope dy/dx will be positive or non-negative for x > 0. Hence, statement I is false.
II. "Will have a slope field with positive slopes in all quadrants"
Following the previous explanation, we see that the slope dy/dx is positive or non-negative for x > 0. Now, when x < 0, the term 2x is negative. Again, since (4-y) is positive or zero, their product 2x(4-y) will be negative or zero. Thus, dy/dx will be negative or non-positive for x < 0. Therefore, the statement II is false.
III. "Will produce a slope field with columns of parallel tangents"
To determine whether the slope fields have columns of parallel tangents, we need to examine if dy/dx has a constant value for different values of x. From the given differential equation, dy/dx = 2x(4-y). We notice that dy/dx depends on both x and y, so its value will vary as we move along the graph. Hence, the statement III is false.
In conclusion, none of the statements are true about the given differential equation dy/dx = 2x(4-y).