Differential Equations 2280
Midterm Exam 3
Exam Date: 22 April 2016 at 12:50pm
co
Instructions: This in-class exam is 50 minutes. No calculators, notes, tables or books. No answer
check is
expected. Details count 3/4, answers count 1/4.
Chapter 3
1. (Linear Constant Equations of Order
(
ii)
(a) [30%] Find by variation of parameters a particular solution
for the equation p”
steps in variation of parameters. Check the answer by quadrature.
‘
=
. Show all
2
x
(b) [40%] Find the Beats solution for the forced undampeci spring-ma.ss problem
x” + 256x
=
231 cos(5t),
x(0)
=
x’(O)
=
0.
It is known that this solution is the sum of two harmonic oscillations of different frequencies. To save
time, please dont convert to phase-amplitude form.
(c) [20%] Given rnx”(t) -F cx’(t) + kx(t) = 0, which represents a damped spring-mass system, assume
m = 9, c = 24, k = 16. Determine, if the equation is over-damped critically damped or under-damped.
To save time, do not solve for x(t).
,
(d) [10%] Determine the practical resonance frequency w for the RLC current equation
21” + 71’ + 501
f(
z.
-
,
=
500 sin(wt).
‘
1
C
3I
2
Use this page to start your solution.
0
L
j LJ d
ia
ijit
-
(5l
II
i-
I’
CD
cS7
‘
n
r--
3-’
‘1
D
4
-
‘—I
_ji
N
-
C
L*
—;-
-
‘•=
I
¶
çi
I
3-,
‘
çt/
cfl
C-)
--
—çc-
--
C)
-4-
I
CD
—
I
if
7
cV
ci-
-)
C,
(1
C)
—
€-
L
4
c_p
—
N
-+
,
3
i_
(-b’-.
-i
c57
cfl
I
if
)__
3
-
0
H
2280 Exam 3 S2016
Chapters 4 and 5
2. (Systems of Differential Equations)
k(a)
[30%j The 3 x 3 matrix
(4
A=I 1
0
1 1
4 1
0 4
I iN
has eigeiivalues A
=
3, 4, 5. One Euler solution vector is fleAt with A
=
3 and
Find two
0)
= Af(t).
—
=
more Euler solution vectors and then display the vector general solution (t) of
(b) f40%] The 3 x 3 triangular matrix
(3
A=I 0
0
1
0’\
3 1
0 4
)
represents a. linear cascade, such as found in brine tank models.
Part 1. Use the linear integrating factor method to find the vector general solution (t) of
t) = A(t).
Part 2. The eigenanalysis method fails for this example. Cite two different methods, besides the
Af(t). Don’t show
linear integrating factor method, which apply to solve the system t)
why
the method applies.
and
method
each
precisely
explain
solution details for these methods, but
(c) [30%] Time Cayley-Hamiltoim-Ziebur shortcut. applies especially to the system
x’ =x+4y,
which has complex eigcnvalues A
=
y’
=
—4x+y,
1 ± 4i.
Part 1. Show the details of the method, finally displaying formulas for x(t),y(t).
Part 2. Report a fundamental matrix ((t).
\O 0 o)J
75:
f-’
:
oo)cj
r
)
0
—bcc
c1—+
(0)
CO
C
U
CQ
Use this page to start your solution.
e:
/tIe4
(
c-3=I
0
4
-
-I-4
< D’
—
o—
—cOO
.—
cyc
4)
0)
U
--
IN
0
4)
IQ)
3—
—
—
-
-U
-U
c-)
cJ
-P
(34)
4)
c)
(_)
4)
5’
N
U
•1—
:1-
U
\-
::5—)
II
-
Th
Q)
-
C)
—
Q)
c.4
>
Il
‘I-
C)
4-)
-
___—,
—)
)
s
-)
C)
t
C-)
—
(3
C-)
ci)
4)
ci-)
(_)
4.1
rt’
-
:;:
9)
-
-
.
c
——
)D_
LiLt
L-
(Sn
-,
C)
CJ
(_)
d
-,
;
.--j--
-)
c)
U
-I
c:i)
-II
jjj
_7__
c)
(si
--
.U
-U
4
C-)
—
—
£
N
U
L)
-
-j-)
3-
c)
c%I
c)
0
—-
_\
!
)
1
L
4
I
\\
..)//(\
)
c.)\
-
0
-4)
-
1
çq
L,
cL.)
)
4.)
c_)
ti
3
C.
-
0
I
j/L
_)_
L._i £
:,.-
0
:-
2280 Exam 3 52016
1°
Chapter 6
ems)
3. (Linear and Nonlinear Dynamical Syst
A
0 is stable or unstable. Then classify the
ification into improper or proper
0 as a saddle, center, spiral or node. Sub-class
librium
[20%] Determine whether the unique equi
(a)
equilibrium point
node is not required.
=
d
(-1 1’\
l}
d—2
(b) [30%] Consider the nonimear dynamical system
xl
=
=
An equilibrium point is x
equilibrium point..
(c)
130%]
—8, y
2,
2y
-2V+3
2
x—
—
2x(z—2y).
rized system at this
—4. Compute the Jacobian matrix of the linea
Consider the soft nonlinear spring system
{
2+
‘
r,
the phase portrait classification saddle, cente
(1) Determine the stability at t = oo and
bian
Jaco
= Ad, where A is the
l system
spiral or node at = for the linear dynamica
matrix of this system at x = 2, y = 0.
2, y = 0 as a saddle, center, spiral or node
(2) Apply the Pasting Theorem to classify x
uss all details of the application of the theorem.
for the nonlinear dynamical system. Disc
Details count 75%.
above.
lusions of the Pasting Theorem used in part (c)
(d) [20%] State the hypotheses and the conc
Accuracy and completeness expected.
f_i-a
I
—I—
=
-—2
i
/
-Ih’? fi
js
-
(1
--
°LY
Use this page to start your solution.
\
)
d j
(-- (I
-2
32.
\
Un
it
DTh
0’
0
—
N
D
—
+
D
çc
(
—
-
a
0
I::
L>
=-.,
—
c’’=_’I
c
(_2
I
S
—
—
c-A
C
‘-p
.s.
CL..
E
_c
L)
0
C.)
“C
\J’
ç—
-
(7
4-
c)
1<
c-SI
4
0
L4
•-‘
3
C
C
-
t.
a
—
4-
‘-)
—
0
c
111
(cç)