1) Which are true about the differential equation dy/dx = (x)/(x-4)
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
see "Bryan's" post
To determine which statements are true about the differential equation dy/dx = (x)/(x-4), we can analyze its properties.
First, let's calculate the slopes of the differential equation. By rewriting the equation as (x-4)(dy/dx) = x, we can separate variables and integrate both sides:
∫1 dy = ∫(x/(x-4)) dx
=> y = ∫(x/(x-4)) dx
Now, let's find the antiderivative:
Since x/(x-4) can be expressed as (x-4+4)/(x-4) = 1 + (4/(x-4)), we can rewrite the integral as:
y = ∫(1 + 4/(x-4)) dx
The integral of 1 is x, and the integral of 4/(x-4) can be found using a substitution method, taking u = x-4:
∫(4/(x-4)) dx = 4 ∫(1/u) du = 4 ln(|u|) = 4 ln(|x-4|)
Therefore, the solution to the differential equation is:
y = x + 4 ln(|x-4|) + C
Now let's analyze the statements:
I. The slope field with negative slopes in quadrant I: This statement is not true. By graphing the equation f(x) = (x)/(x-4), we can see that the slope is positive in quadrant I for all values of x > 4.
II. The slope field with positive slopes in all quadrants: This statement is not true either. As previously mentioned, the slope is positive only for x > 4.
III. The slope field with columns of parallel tangents: This statement is true. By observing the graph, we can see that for x = 4, which is a vertical asymptote, the slope approaches positive or negative infinity depending on which side of the asymptote we are on. This results in the formation of columns of parallel tangents.
Therefore, the correct statement is III - the differential equation will produce a slope field with columns of parallel tangents.