Question

Based on the graph of the general solution to the differential equation dy over dx equals 2 times x minus 2 times y comma which of the following statements is true?

The slopes along the y-axis are horizontal.
The slopes along the x-axis are horizontal.
The slopes are all positive in Quadrant 4.
The slopes are all positive in Quadrant 1.

Answer 1

The slopes along the y-axis are horizontal.

Explanation:

The statement "The slopes along the y-axis are horizontal" is true based on the graph of the general solution to the differential equation dy/dx = 2x - 2y. This is because when x = 0, the slope of the solution curve is equal to -2y, which means the slope is horizontal or zero when y = 0.

The statement "The slopes along the x-axis are horizontal" is false because the slope of the solution curve at the point (0,0) is -2(0) = 0, which means the slope is horizontal at this point only.

The statement "The slopes are all positive in Quadrant 4" is false because the slope is negative in Quadrant 4.

The statement "The slopes are all positive in Quadrant 1" is false because the slope is negative in Quadrant 1.

Therefore, the only true statement is "The slopes along the y-axis are horizontal."