7.3. EULER’S METHOD
409
7.3 Euler’s method
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What is Euler’s method and how can we use it to approximate the solution to an
initial value problem?
• How accurate is Euler’s method?
Introduction
In Section 7.2, we saw how a slope field can be used to sketch solutions to a differential
equation. In particular, the slope field is a plot of a large collection of tangent lines to a
large number of solutions of the differential equation, and we sketch a single solution by
simply following these tangent lines. With a little more thought, we may use this same idea
to numerically approximate the solutions of a differential equation.
Preview Activity 7.3. Consider the initial value problem
dy
dt
=
1
2
(y + 1), y(0) = 0.
(a) Use the differential equation to find the slope of the tangent line to the solution
y(t) at t = 0. Then use the given initial value to find the equation of the tangent
line at t = 0.
(b) Sketch the tangent line on the axes below on the interval 0 ≤ t ≤ 2 and use it to
approximate y(2), the value of the solution at t = 2.
1 2 3 4 5 6 7
1
2
3
4
5
6
7
t
y