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7.2. QUALITATIVE BEHAVIOR OF SOLUTIONS TO DES
7.2 Qualitative behavior of solutions to DEs
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What is a slope field?
• How can we use a slope field to obtain qualitative information about the solutions
of a differential equation?
• What are stable and unstable equilibrium solutions of an autonomous differential
equation?
Introduction
In earlier work, we have used the tangent line to the graph of a function f at a point a to
approximate the values of f near a. The usefulness of this approximation is that we need
to know very little about the function; armed with only the value f (a) and the derivative
f ′ (a), we may find the equation of the tangent line and the approximation
f (x) ≈ f (a) + f
′ (a)(x − a).
Remember that a first-order differential equation gives us information about the
derivative of an unknown function. Since the derivative at a point tells us the slope of the
tangent line at this point, a differential equation gives us crucial information about the
tangent lines to the graph of a solution. We will use this information about the tangent
lines to create a slope field for the differential equation, which enables us to sketch solutions
to initial value problems. Our aim will be to understand the solutions qualitatively. That
is, we would like to understand the basic nature of solutions, such as their long-range
behavior, without precisely determining the value of a solution at a particular point.
Preview Activity 7.2. Let’s consider the initial value problem
dy
dt
= t − 2, y(0) = 1.
(a) Use the differential equation to find the slope of the tangent line to the solution
y(t) at t = 0. Then use the initial value to find the equation of the tangent line
at t = 0. Sketch this tangent line over the interval −0.25 ≤ t ≤ 0.25 on the axes
provided.
7.2. QUALITATIVE BEHAVIOR OF SOLUTIONS TO DES
7.2 Qualitative behavior of solutions to DEs
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What is a slope field?
• How can we use a slope field to obtain qualitative information about the solutions
of a differential equation?
• What are stable and unstable equilibrium solutions of an autonomous differential
equation?
Introduction
In earlier work, we have used the tangent line to the graph of a function f at a point a to
approximate the values of f near a. The usefulness of this approximation is that we need
to know very little about the function; armed with only the value f (a) and the derivative
f ′ (a), we may find the equation of the tangent line and the approximation
f (x) ≈ f (a) + f
′ (a)(x − a).
Remember that a first-order differential equation gives us information about the
derivative of an unknown function. Since the derivative at a point tells us the slope of the
tangent line at this point, a differential equation gives us crucial information about the
tangent lines to the graph of a solution. We will use this information about the tangent
lines to create a slope field for the differential equation, which enables us to sketch solutions
to initial value problems. Our aim will be to understand the solutions qualitatively. That
is, we would like to understand the basic nature of solutions, such as their long-range
behavior, without precisely determining the value of a solution at a particular point.
Preview Activity 7.2. Let’s consider the initial value problem
dy
dt
= t − 2, y(0) = 1.
(a) Use the differential equation to find the slope of the tangent line to the solution
y(t) at t = 0. Then use the initial value to find the equation of the tangent line
at t = 0. Sketch this tangent line over the interval −0.25 ≤ t ≤ 0.25 on the axes
provided.