James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 9.2 Direction Fields and Euler’s Method 597
Euler
Leonhard Euler (1707–1783) was the
leading mathematician of the mid-18th
century and the most prolific mathematician
of all time. He was born in
Switzerland but spent most of his career
at the academies of science supported
by Catherine the Great in St. Petersburg
and Frederick the Great in Berlin. The
collected works of Euler (pronounced
Oiler) fill about 100 large volumes. As
the French physicist Arago said, “Euler
calculated without apparent effort, as
men breathe or as eagles sustain themselves
in the air.” Euler’s calculations and
writings were not diminished by raising
13 children or being totally blind for
the last 17 years of his life. In fact, when
blind, he dictated his discoveries to his
helpers from his prodigious memory
and imagination. His treatises on calculus
and most other mathematical
subjects became the standard for mathematics
instruction and the equation
e i 1 1 − 0 that he discovered brings
together the five most famous numbers
in all of mathematics.
Example 4 In Example 2 we discussed a simple electric circuit with resistance
12 V, inductance 4 H, and a battery with voltage 60 V. If the switch is closed when
t − 0, we modeled the current I at time t by the initial-value problem
dI
dt
− 15 2 3I Is0d − 0
Estimate the current in the circuit half a second after the switch is closed.
SOLUtion We use Euler’s method with Fst, Id − 15 2 3I, t 0 − 0, I 0 − 0, and step size
h − 0.1 second:
So the current after 0.5 s is
I 1 − 0 1 0.1s15 2 3 ? 0d − 1.5
I 2 − 1.5 1 0.1s15 2 3 ? 1.5d − 2.55
I 3 − 2.55 1 0.1s15 2 3 ? 2.55d − 3.285
I 4 − 3.285 1 0.1s15 2 3 ? 3.285d − 3.7995
I 5 − 3.7995 1 0.1s15 2 3 ? 3.7995d − 4.15965
Is0.5d < 4.16 A
n
1. A direction field for the differential equation y9 − x cos y is
shown.
(a) Sketch the graphs of the solutions that satisfy the given
initial conditions.
(i) ys0d − 0 (ii) ys0d − 0.5
(iii) ys0d − 1 (iv) ys0d − 1.6
(b) Find all the equilibrium solutions.
y
2.0
2. A direction field for the differential equation y9 − tan( 1 2 y)
is shown.
(a) Sketch the graphs of the solutions that satisfy the given
initial conditions.
(i) ys0d − 1 (ii) ys0d − 0.2
(iii) ys0d − 2 (iv) ys1d − 3
(b) Find all the equilibrium solutions.
y
4
1.5
3
1.0
2
0.5
1
_2 _1 0 1 2 x
_2 _1 0 1 2 x
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