This chapter introduces the concepts of direction fields, trajectories, and solution curves with illustrative examples, such as the motion of a yacht adrift on a lake and the motion of a vertically moving object with a velocity-dependent drag force acting on it. The chapter begins with the scenario of a yacht on a lake drifting with the currents, where the motion of the yacht is governed by a simple ordinary differential equation. A direction field for the differential equation is constructed by covering a map of the lake with a rectangular array of regularly spaced points and then assigning a direction arrow to each of them to indicate the direction in which the yacht would move were it located at one of these points. Examples, such as the one described here, are used to demonstrate that the direction field for a differential equation is a useful tool for gaining some insight into the qualitative behavior of the solutions of the equation, even if closed-form solutions of the equation cannot be found. Besides direction fields, even more useful information about solution curves (graphs of solutions) can be obtained by using the curve-sketching techniques of calculus. One of the examples presented in this chapter is to determine all of the important features of the graph of the solution of the innocuous-looking initial value problem \(\displaystyle y' = x^2+y^2-1, \quad y(0) = 0. \qquad \hbox{(1)} \) One might be inclined to think that (1) has a relatively simple solution. But as it turns out, its solution cannot be expressed in terms of the familiar functions of calculus. Even so, all of the important features of its graph can be gleaned from the sign changes of \(y'\) and \(y''\) , such as intervals of increase and decrease, concavity, end behavior, and whether there are any local extrema and points of inflection. The difference between solution curves and trajectories is explained and illustrated with their graphs by imagining that the position \((x,y)\) at time t of a small piece of driftwood floating in a body of water can be modeled by the system of differential equations \(\displaystyle \frac {dx}{dt} = y, \quad \frac {dy}{dt} = -{\omega }^2x, \qquad \hbox{(2)} \) where \(\omega \) denotes some positive constant. Graphs of some of the trajectories (ellipses) of (2) in the phase plane (xy-plane) are compared to their counterparts (elliptical helices) in txy-space.

Direction Fields, Trajectories, Solution Curves

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发表日期：

2026年

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Leigh C. Becker

摘要

This chapter introduces the concepts of direction fields, trajectories, and solution curves with illustrative examples, such as the motion of a yacht adrift on a lake and the motion of a vertically moving object with a velocity-dependent drag force acting on it. The chapter begins with the scenario of a yacht on a lake drifting with the currents, where the motion of the yacht is governed by a simple ordinary differential equation. A direction field for the differential equation is constructed by covering a map of the lake with a rectangular array of regularly spaced points and then assigning a direction arrow to each of them to indicate the direction in which the yacht would move were it located at one of these points. Examples, such as the one described here, are used to demonstrate that the direction field for a differential equation is a useful tool for gaining some insight into the qualitative behavior of the solutions of the equation, even if closed-form solutions of the equation cannot be found. Besides direction fields, even more useful information about solution curves (graphs of solutions) can be obtained by using the curve-sketching techniques of calculus. One of the examples presented in this chapter is to determine all of the important features of the graph of the solution of the innocuous-looking initial value problem \(\displaystyle y' = x^2+y^2-1, \quad y(0) = 0. \qquad \hbox{(1)} \) One might be inclined to think that (1) has a relatively simple solution. But as it turns out, its solution cannot be expressed in terms of the familiar functions of calculus. Even so, all of the important features of its graph can be gleaned from the sign changes of \(y'\) and \(y''\) , such as intervals of increase and decrease, concavity, end behavior, and whether there are any local extrema and points of inflection. The difference between solution curves and trajectories is explained and illustrated with their graphs by imagining that the position \((x,y)\) at time t of a small piece of driftwood floating in a body of water can be modeled by the system of differential equations \(\displaystyle \frac {dx}{dt} = y, \quad \frac {dy}{dt} = -{\omega }^2x, \qquad \hbox{(2)} \) where \(\omega \) denotes some positive constant. Graphs of some of the trajectories (ellipses) of (2) in the phase plane (xy-plane) are compared to their counterparts (elliptical helices) in txy-space.