Question
Let g be a function such that g(y) > 0 for all y. Which of the following could be a slope field for the differential equation \(\frac{dy}{dx}=(x^2−1)g(y)\) ?
Answer/Explanation
Question
Shown above is a slope field for which of the following differential equations?
(A)\(\frac{dy}{dx}=1+x\) (B)\(\frac{dy}{dx}=x^2\) (C)\(\frac{dy}{dx}=x+y\) (D)\(\frac{dy}{dx}=\frac{x}{y}\) (E)\(\frac{dy}{dx}=lny\)
Answer/Explanation
Ans:C
Question
The figure below shows a slope field for one of the differential equations given below. Identify the equation.
(A) \(\frac{\mathrm{d} y}{\mathrm{d} x}=y-x\)
(B) \(\frac{\mathrm{d} y}{\mathrm{d} x}=-xy\)
(C) \(\frac{\mathrm{d} y}{\mathrm{d} x}=2x\)
(D) \(\frac{\mathrm{d} y}{\mathrm{d} x}=\frac{x}{y}\)
Answer/Explanation
Ans:(B)
By observing different properties of the slope field, we can eliminate the answer choices. Notice that in the column x = 0, the slope is constantly 0. So when x = 0, the slope is independent of the changing y-value in that column. This eliminates the choices (A) \(\frac{\mathrm{d} y}{\mathrm{d} x}=y-x\) and (E)\(\frac{\mathrm{d} y}{\mathrm{d} x}=-2y\)
Question
The figure below shows a slope field for one of the differential equations given below. Identify the equation.
(A)\(\frac{\mathrm{d} y}{\mathrm{d} x}=x\)
(B) \(\frac{\mathrm{d} y}{\mathrm{d} x}=y-x\)
(C) \(\frac{\mathrm{d} y}{\mathrm{d} x}=-x\)
(D) \(\frac{\mathrm{d} y}{\mathrm{d} x}=2y\)
Answer/Explanation
Ans:(A)
If you look vertically at any column of tangents, you’ll notice that the tangents have the same slope. (Points in the same column have the same x-coordinate but different y-coordinates.) Therefore, the numerical value of \(\frac{\mathrm{d} y}{\mathrm{d} x}\) (which represents the slope of the tangent) depends only the x-coordinate of the point and is independent of the y-coordinate. Only choices (A) and (C) satisfy these conditions. Also notice that the tangents have positive slope when x ≥ 0 and negative slope when x ≤ 0.
