(d) Euler’s method state that, given
y
(
t
0
) =
y
0
,
t
0
and
h
, then
y
(
t
1
) =
y
(
t
0
) +
hf
(
t
0
, y
0
). In this
case, we have that
f
: Y =
bracketleftbigg
y
u
bracketrightbigg
,
y
(
t
0
) : Y(0) =
bracketleftbigg
2
1
bracketrightbigg
= Y
0
,
t
0
:
t
0
= 0,
h
= 0
.
5. Now since
Y(0) =
bracketleftbigg
2
1
bracketrightbigg
, then
Y
1
=
Y
0
+
h
Y
0
=
bracketleftbigg
2
1
bracketrightbigg
+ 0
.
5
bracketleftbigg
2
1
bracketrightbigg
=
bracketleftbigg
3
2
3
bracketrightbigg
, so
y
(0
.
5) = 3 and
u
(0
.
5) =
3
2
.
(e) With the step size
h
= 0
.
5, Euler’s method takes the form
Y
k
+1
=
Y
k
+ 0
.
5Y
k
=
3
2
Y
k
.
In this case, the amplication factor is
3
2
, which is greater than one so Euler’s method will be
unstable with
h
= 0
.
5.
(f) For backward Euler,
Y
k
+1
=
Y
k
+ 0
.
5Y
k
+1
=

2Y
k
=
(

2)
k
+1
Y
0
.
Since

(

2)

>
1, then backward Euler will be unstable for this problem with
h
= 0
.
5.
5.
Purpose:
Identifying properties of methods.
From [Hea02, pp.417–418,#9.9]
.
Exercise
:
For each
property listed below, state which of the following two ODE methods has or have the given property:
y
k
+1
=
y
k
+
h
2
(3
f
(
t
k
, y
k
)

f
(
t
k
−
1
, y
k
−
1
))
(1)
y
k
+1
=
y
k
+
h
2
(
f
(
t
k
, y
k
)

f
(
t
k
+1
, y
k
+1
))
(2)
Properties:
(a) Secondorder accurate
(b) Singlestep method
(c) Implicit method
(d) Unconditionally stable
(e) Good for solving stiff ODEs
6.
Purpose
: Determining the accuracy of a method through Taylor expansions.
Note to Instructor:
You may need to help the students by giving them a bit more background on how to Taylor expand
about points other than
t
n
+1
.
For example,
y
(
t
n
−
1
) =
y
(
t
n
)

hy
′
(
t
n
) +
h
2
2
y
′′
(
t
n
)

h
3
3!
h
′′′
(
t
n
) +
O
(
h
4
)
From [Hea02, pp.418,#9.12]
.
22