Which of the following best represents the direction field for the differential equation dy/dx = -x/(4y)? A - HSC - SSCE Mathematics Extension 1 - Question 7 - 2020 - Paper 1

To find the correct direction field, we need to analyze the differential equation:

dy/dx = -x/(4y).

This equation suggests that the slope of the direction field at any point (x, y) in the plane is dependent on both the x and y coordinates. The negative sign indicates that as x increases, dy/dx becomes more negative, suggesting a downward trend in the direction field in the right half of the coordinate system.

Next, we identify the behavior in the different quadrants:

• For x > 0 and y > 0, dy/dx is negative, so the arrows point downward.
• For x < 0 and y > 0, dy/dx is also negative; arrows point downwards.
• For x > 0 and y < 0, dy/dx is positive, so arrows point upwards.
• For x < 0 and y < 0, dy/dx is positive, indicating that arrows point upwards.

By visualizing these behaviors across the quadrants, we can compare them to the given options A, B, C, and D.