(a) Are solutions to this ODE stable?
(b) Is Euler’s method stable for this ODE using this stepsize?
(c) Compute the numerical value for the approximate solution at
t
= 0
.
5 given by Euler’s method.
(d) Is the backward Euler (BE) method stable for this ODE using this stepsize?
(e) Compute the numerical value for the approximate solution at
t
= 0
.
5 given by the backward
Euler method.
Solution:
(a) The solution to this ODE is
y
(
t
) =
e
−
5
t
and is stable since
λ
=

5
<
0 for the generic ODE
y
′
=
λy
as discussed in class.
(b) Verify that Euler’s method is not stable for this ODE using a stepsize of
h
= 0
.
5 by computing
the amplification factor which is greater than 1,

1 +
λh

=

1 + (

5)(0
.
5)

= 1
.
5
>
1
(c) The numerical value for the approximation solution at
t
= 0
.
5 is,
y
1
=
y
0
+
hf
(
t
0
, y
0
) = 1 + 0
.
5(

5(1)) =

1
.
5
(d) The Backward Euler method is stable for this ODE using a
h
= 0
.
5 stepsize, and can be verified
by computing the amplification factor,
vextendsingle
vextendsingle
vextendsingle
vextendsingle
1
1

λh
vextendsingle
vextendsingle
vextendsingle
vextendsingle
= 0
.
286
<
1
In fact, the BE method is stable for this ODE for any step size.
(e) The numerical value for the approximate solution using this method at
t
= 0
.
5 is,
y
1
=
y
0
+
h
(

5
y
1
)
⇒
y
1
=
y
0
1 + 5
h
= 0
.
2857
20
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3.
Purpose:
Comparing results for Forward Euler versus Backward Euler.
From [Hea02, p.417,#9.5]
.
Exercise
:
With the initial value of
y
0
= 1 at
t
0
= 0 and a time step of
h
= 1
,
compute the
approximate solution value
y
1
at time
t
1
= 1 for the ODE
y
′
=

y
using each of the following two
numerical methods. (Your answers should be numbers, not formulas.)
(a) Euler’s method
(b) Backward Euler method
Solution:
(a) Euler’s method is defined as
y
n
+1
=
y
n
+ Δ
tf
(
y
n
).
Thus,
y
1
=
y
0
+
h
(

y
) = 1 + 1(

0) = 1
(b) The Backward Euler method is defined as
y
n
+1
=
y
n
+ Δ
tf
(
y
n
+1
).
Thus,
y
1
=
y
0
+
h
(

y
1
)
⇒
y
1
(1 +
h
) =
y
0
⇒
y
1
=
1
1+1
=
1
2
4.
Purpose:
Converting a second order ODE to a first order system, determining stability of solutions
and stability of Euler’s method and Backward Euler method on this system.
Note to Instructor:
If you have chosen not to present the slides on stability of an ODE solution, you
may wish to skip part (c) of this exercise.
From [Hea02, p.417,#9.7]
.
Exercise
:
Consider the IVP
y
′′
=
y
for
t
≥
0 with initial values
y
(0) = 1 and
y
′
(0) = 2
.
(a) Express this secondorder ODE as an equivalent system of two firstorder ODEs.
(b) What are the corresponding intial conditions for the system of ODEs in part (a)?
(c) Are solutions of this system stable?
(d) Perform one step of Euler’s method for this ODE system using a step size of
h
= 0
.
5
.
(e) Is Euler’s method stable for this problem using this step size?
(f) Is the backward Euler method stable for this problem using this step size?
Solution:
(a) Recall the given equation
y
′′
=
y
. Let
u
=
y
′
. Differentiating the latter equation, we get
u
′
=
y
′′
.
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 Spring '13
 MRR
 Math, Logic, Numerical Analysis, Equations, phase portrait, Euler, Numerical ordinary differential equations, Jacobian

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