A slope field is a visual map of a first order differential equation, showing the direction a solution curve should travel at many points in the plane. Instead of solving the equation exactly, you can see the overall behavior of many possible solutions at once. This matters because many real systems, such as cooling objects, populations, motion with drag, and chemical reactions, are described by rates of change.
Slope fields turn an equation like dy/dx = f(x, y) into a picture you can interpret quickly.
Key Facts
- A slope field for dy/dx = f(x, y) places a short segment with slope f(x, y) at each point (x, y).
- Solution curves follow the local direction of the slope segments and should be tangent to them.
- For dy/dx = x + y, the slope at (2, 1) is 3, so the segment there rises 3 units for every 1 unit right.
- For dy/dx = y, slopes depend only on y, so all points on the same horizontal line have equal slope.
- For dy/dx = x, slopes depend only on x, so all points on the same vertical line have equal slope.
- An equilibrium solution occurs when dy/dx = 0 for all points on a curve or line, often giving a horizontal solution such as y = c.
Vocabulary
- Slope field
- A slope field is a grid of small line segments that shows the slope of solutions to a differential equation at many points.
- Differential equation
- A differential equation is an equation that relates a function to one or more of its derivatives.
- Solution curve
- A solution curve is a graph of a function that satisfies the differential equation and follows the slope field.
- Initial condition
- An initial condition gives a specific point, such as y(0) = 2, that selects one solution curve from a family of solutions.
- Equilibrium solution
- An equilibrium solution is a constant solution where the rate of change is zero and the graph is horizontal.
Common Mistakes to Avoid
- Drawing segments with the wrong slope sign is incorrect because positive slopes rise left to right and negative slopes fall left to right.
- Connecting slope marks point to point like a dot graph is incorrect because a solution curve should be smooth and tangent to nearby slope segments, not forced through every segment.
- Ignoring the initial condition is incorrect because the initial point determines which one of many possible solution curves should be drawn.
- Assuming every slope field depends on both x and y is incorrect because equations like dy/dx = y depend only on height, while dy/dx = x depends only on horizontal position.
Practice Questions
- 1 For the differential equation dy/dx = x - y, find the slope at (0, 2), (3, 1), and (-1, -1).
- 2 For dy/dx = y(4 - y), identify the equilibrium solutions and determine whether the slope is positive or negative at y = 1, y = 5, and y = -2.
- 3 A slope field has equal slopes along every horizontal row, and the slopes are zero along y = 0. Explain why dy/dx = y could match the field better than dy/dx = x.