Final answer:
To sketch the slope field for the differential equation y'(x) = 2x(y - 1), calculate and plot line segments at grid points within the window of -2 < x < 2, -2 < y < 2, with slopes derived from the equation at each point.
Step-by-step explanation:
The slope field for the differential equation y'(x) = 2x(y − 1) represents the direction of the solution curves at any given point in the plane. To sketch a slope field, you essentially plot tiny line segments with slopes corresponding to the value of the differential equation at grid points within the given window of −2 < x < 2, −2 < y < 2.
To create a slope field, first create a grid over the specified x and y ranges. At each grid point (x, y), calculate the slope given by the differential equation. For example, at the point (1, 1), the slope is 2 × 1 × (1 − 1) = 0, indicating that there would be a horizontal line segment at that grid point. Continue this process at multiple points to get a comprehensive visual approximation of the direction field.
It is important to have a systematic approach when plotting the slope field, often starting from the center and working outwards in a grid pattern. The more points you calculate, the more accurate your slope field will represent the solution curves of the differential equation.