Final answer:
The differential equation is y' = -y^3, and its slope field is sketched by drawing segments with slopes determined by y's value at various points. For initial conditions y(0) = 1, y(0) = 0, and y(0) = -1, the corresponding solution curves are sketched according to the slope field, illustrating different behaviors.
Step-by-step explanation:
The differential equation in question is y' = -y^3, which suggests that the slope at any given point on the graph depends on the cube of the y-coordinate at that point. A slope field is a visual representation showing the slopes (direction field) of solutions to the differential equation at various points.
To sketch the slope field for this differential equation, we plot small line segments that represent the slope of the solution curves at various (x, y) points in the plane. Because the right side of the equation involves only y, the slope at any point depends solely on the y-coordinate. Consequently, the slope field will have horizontal rows of line segments with the same slope.
For the initial conditions given (y(0) = 1, y(0) = 0, and y(0) = -1), we draw lines that start from these points and follow the pattern of slopes indicated by the slope field:
- For y(0) = 1, we have y' = -1, so the solution curve will decrease rapidly initially because the slope is steep and negative.
- For y(0) = 0, the slope is y' = 0, so the solution curve will be a horizontal line.
- For y(0) = -1, the slope is y' = -(-1)^3 = 1, so the solution curve will increase rapidly.
These solution curves will help you visualize the behavior of solutions under different initial conditions. When sketching by hand, accuracy is not essential, so the sketches should only convey a rough idea of the direction and curvature of the solution curves in relation to the slope field.