## Transcribed Text

1. Equilibrium points of ODE
Find the equilibrium points of the following system of ODEs.
(1) The function z : R → R
2
satisfies
z˙(t) =
1 2
1 2
z(t).
(2) The functions x, y : I ⊂ R → R satisfy
x˙(t) = x(t) − y
2
(t)
y˙(t) = x
3
(t) + y
2
(t)
(3) The functions x, y, z : I ⊂ R → R satisfy
x˙(t) = −y(t) − z(t)
y˙(t) = x + ay(t)
z˙(t) = b + z(t)(x(t) − c)
where a, b, c > 0 and c
2 − 4ab > 0.
This system is known as Roessler attractor in the literature. It is an example of chaotic dynamics.
(4) The function u : I ⊂ R → R
3
satisfies
u˙(t) = F(u(t)), where F : R
3 → R
3
is defined as F
x
y
z
=
x
2 + yz
y
2 + xz
z
2 + xy
This system is known as the Hamilton ODE associated to three-dimensional Ricci flow.
2. Equilibrium points of ODE II
Please rewrite the following ODEs of second order into a system of first order ODE. Then find the
equilibrium points of the system of first order ODE.
(1) The function x : I ⊂ R → R satisfies
x¨(t) + 2βx˙(t) + ω
2
0x(t) = 0.
for constants β > 0 and ω0 > 0.
(2) The function ϕ : I ⊂ R → R satisfies
ϕ¨(t) + 2βϕ˙(t) + ω
2
0
sin(ϕ(t)) = 0
for constants β > 0 and ω0 > 0.
1
2 MATH 2030 (FALL 2019) ORDINARY DIFFERENTIAL EQUATIONS EXERCISE SHEET NR. 10
3. Review: Linear equations with constant coefficents
Please solve the initial value problem
z˙(t) =
0 4
4 0
z(t)
z(0) = z0 ∈ R
2
.
for the unknown function z : R → R
2
. Moreover, please answer the following questions
(1) What are the equilibrium points of the above system?
(2) For which z0 ∈ R
2 do we have limt→∞ z(t) = 0 ∈ R
2
?
(3) Is the origin a stable equilibrium point?
4. Review: Integration
Please decide, which of the following improper integrals exist and compute their value.
(1) Consider for α > 0 the expression
Z ∞
1
1
x
α
dx.
(2) Consider for s > 0 the expression
Z ∞
0
exp(−ts) dt.
(3) Consider for s > 0 the expression
Z ∞
0
exp(−ts)t
2 dt.
(4) Consider for s > 0 the expression
Z ∞
0
exp(−ts) sin(t) dt.
Reminder:
Let a ∈ R and f a continous function on R. We define the improper integral R ∞
a
f(x) dx by
Z ∞
a
f(x) dx = lim
b→∞ Z b
a
f(x) dx,
if the limit on the right-hand side exists.

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction
of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice.
Unethical use is strictly forbidden.