Ma 2a
Fall 2010
(analytical track)
PROBLEM SET 1
Due on Monday, October 4
[All references are to J. C. Robinson]
I.
1D autonomous equations (formula solution)
1. [Problem 8.7] Show that for
k
6
= 0 the solution of the IVP
˙
x
=
kx

x
2
,
x
(0) =
x
0
,
is
x
(
t
) =
ke
kt
x
0
x
0
(
e
kt

1) +
k
.
Using this explicit solution describe the behavior of
x
(
t
) as
t
→ ∞
for
k <
0 and
k >
0.
2. [Problem 8.11] Assuming that
f
(
x
) is continuously differentiable, show that if
the solution of
˙
x
=
f
(
x
)
,
x
(0) =
x
0
blows up to
x
= +
∞
in finite time, then
Z
∞
x
0
dx
f
(
x
)
<
∞
.
II.
1D autonomous equations (qualitative approach)
34.
[Problem 7.1] For each of the following equations draw the phase diagram,
labeling the stationary points as stable or unstable:
(iii) ˙
x
= (1 +
x
)(2

x
) sin
x
,
(v) ˙
x
=
x
2

x
4
.
5. [Problem 7.9] Shaw that for autonomous scalar equations, if a stationary point
is attracting, then it must also be stable.
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 Spring '10
 323
 Critical Point, Stationary point, 1D autonomous equations, autonomous scalar equations

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