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7.3. EULER’S METHOD
Activity 7.7.
Consider the differential equation
dy
dt = 6y − y 2 .
(a) Sketch the slope field for this differential equation on the axes provided at left
below.
2
4
6
8
2
4
6
8 y
t
0.4
0.8
1.2
2
4
6
8
t
y
(b) Identify any equilibrium solutions and determine whether they are stable or
unstable.
(c) What is the long-term behavior of the solution that satisfies the initial value
y(0) = 1?
(d) Using the initial value y(0) = 1, use Euler’s method with ∆t = 0.2 to approximate the solution at t i = 0.2, 0.4, 0.6, 0.8, and 1.0. Sketch the points (t i , y i ) on
the axes provided at right in (a). (Note the different horizontal scale on the two
sets of axes.)
t i
y i
dy/dt
∆y
0.0
1.0000
0.2
0.4
0.6
0.8
1.0
(e) What happens if we apply Euler’s method to approximate the solution with
y(0) = 6?
⊳
7.3. EULER’S METHOD
Activity 7.7.
Consider the differential equation
dy
dt = 6y − y 2 .
(a) Sketch the slope field for this differential equation on the axes provided at left
below.
2
4
6
8
2
4
6
8 y
t
0.4
0.8
1.2
2
4
6
8
t
y
(b) Identify any equilibrium solutions and determine whether they are stable or
unstable.
(c) What is the long-term behavior of the solution that satisfies the initial value
y(0) = 1?
(d) Using the initial value y(0) = 1, use Euler’s method with ∆t = 0.2 to approximate the solution at t i = 0.2, 0.4, 0.6, 0.8, and 1.0. Sketch the points (t i , y i ) on
the axes provided at right in (a). (Note the different horizontal scale on the two
sets of axes.)
t i
y i
dy/dt
∆y
0.0
1.0000
0.2
0.4
0.6
0.8
1.0
(e) What happens if we apply Euler’s method to approximate the solution with
y(0) = 6?
⊳