Math 2280 Assignment 6
-
Dylan Zwick
Spring 2013
Section 3.7 1, 5, 10, 17, 19
-
Section 3.8 1, 3, 5, 8, 13
-
Section 4.1
-
1, 3, 13, 15, 22
1
Section 3.7 Electrical Circuits
-
3.7.1 This problem deals with the RL circuit pictured below. It is a se
ries circuit containing an inductor with an inductance of L henries,
a resistor with a resistance of R ohms, and a source of electromotive
force (emf), but no capacitor. In this case the equation governing our
system is the first-order equation
LI’ + RI
=
E(t).
L
Suppose that L = 5H, R = 25Q, and the source E of emf is a battery
supplying 100V to the circuit. Suppose also that the switch has been
in position 1 for a long time, so that a steady current of 4A is flowing
in the circuit. At time t = 0, the switch is thrown to position 2, so
thatl(0)=4andE=Ofort>0.Findl(t).
2
cj:
U
0)
I
0
0
0)
0
ti
cJ-\
U”
(o
cj
q)
(I
rh
H
UNH±
•11
I
C
(1
o
LI
c)
c•)
I•
LI
.
—
CI
II
S—
-4-i
0
,-.
LI
LI
-4-i
0
AI
o
cI-
•U
-4-.
C)
0
H
•.
0
cL)
0
NJ
Lr’
cj
(r
c)
I’
H
-,
(I
cD
(
tI
N.?
4
9
QJ
qj
0
HH
\J,i
This problem deals with an RC circuit pictured below, containing
a resistor (R ohms), a capacitor (C farads), a switch, a source of emf,
but no inductor. This system is governed by the linear first-order
differential equation
3.7.10
-
R+-Q=E(t).
for the charge
Q’(t).
Q
=
Q(t) on the capacitor at time t. Note that 1(t)
=
C
cos wt is applied to the RC
Suppose an emf of voltage E(t) =
circuit at time t = 0 (with the switch closed), and Q (0) = 0. Substitute
8 (t) = A cos wt + B sin wt in the differential equation to show that
Q
the steady periodic charge on the capacitor is
8 p (t)
Q
where 3
=
C
0
E
=
C
R
2
\/l+w
tan’ (wRC).
5
cos (wt
—
3)
-4
ci)
ci)
C
‘
L
c_i
\J
•‘
---.
±
-t
—_\
c—
-‘
0
.c
1—
(\J
4.
t3:
3
3
1
-1
LJJ
‘SI
0
C-,
Th
(
0
Ic
LUL!
J
‘I
r\
r
‘—S
c-
N
0
0
cm
4’
—
C’
-Th
cm
cm
-A-
1Th
U
Q
(-
-(--
-
(Th
-h
L
(
0
(1;
D
0
L
0
U
0
rJ
C—
‘4
0
(‘
—
I
+
C’
Lc
c
-4--
ON
—
cm
•‘-
II
cm
cm
cm
0
II
cm
cm
I
ci)
C’)
C
n
rJ)
(D
(J-J
3.7.17 For the RLC circuit pictured below find the current 1(t) using the
given values of R, L, C and V(t), and the given initial values.
L
Rzrrl6Q,L=2H,C=.02F;
E(t)
/6 • 50
=
100V; 1(0)
=
0, Q(O)
t
—1±
(it)
—C
e
L
((s)
-
I(o)
e
1
T) -(
5.
(00
=
=
I(*)=
=
?r*Z
_cI
L
-
Yi4(1()
e
—
()
()-
z c))
(c3c—75
C)
(0
(l=0
I()
-
e-15
7
7
CL
Y()
—75
Section 3.8 Endpoint Problems and Eigenvalues
-
3.8.1 For the eigenvalue problem
y”+)y=O; y’(O) =O,y(l)=O,
first determine whether ) = 0 is an eigenvalue; then find the positive
eigenvalues and associated eigenfunctions.
A
y
((a)
A
A
Ac
y ‘()
-
()
A A f ()
o)
y
(
1
i1(K)
+
s
8
(;
()
0
$0)
1
*1jVi
( A
To(,Or1
wôikc
)
kY
(iivl 5al Ii4)
eJ7eIVaftt(’.
01
/
‘
J
y()
(af
:2?
In-Fc&1
9
C
3.8.3 Same instructions as Problem 3.8.1, but for the eigenvalue problem:
y”+y
0;
=
y(—7r)
=
O,y(ir)
y&
C
0J()tfifi4()
-A
y
0.
=
c()
(d
1
Aoi()t
\/(T1
y(-
co5(
0
c
( ) 81 ( )
Z A co ( ) o
A
,z\V0) /H0
-
=
5
(
4
3ol
o()
FLI
A =0
;o
1
(
(())
i1
1Z4
)
H efl
t)
5i
r (J
07
C)
I
A=o
\/(
4-h,i
SJi()
[r f’
4he,i
Q)7vcu1
ft€
e
1
--
I’efl
10
y(
cs(’fLx)
14
(
L
/
ii
i]
3.8.5 Same instructions as Problem 3.8.1, but for the eigenvalue problem:
y”
y(() A
t
(G5(*1
o
\y =
L
ti
A
+
0;
y(—2)
0, y’(2)
=
=
0.
/( Acar(x)Bc i
-A X
)
ci(vr
-s
A
0
oc)
-
ii’i
\/((
0
nc
() d
(&f(7 ) o
n
7
e
(
‘4
CdS
‘3— 4
‘3
2
()
+cfl(7
4
1.3
L
e
z)=I
y1
JK i±i-tA-,5
E?
[
no
)\c
‘n
(3ti\
)\
1
•Lj
L
VUJc(I(Aej
k=
cf
/
If Ay
\1X’’
AP
ii
()
(i11Tr1
JJ
E ie n
y(
n
weil
(,)D
Ccñ
n
c
ü,i
j
((Lt)]T
3.8.8
-
Consider the eigenvalue problem
y”+Ay=
y(O)
0;
=0
y(l)= y’(l)(notatypo).;
all its eigenvalues are nonnegative.
(a) Show that ).
Yo() = x.
=
0 is an eigenvalue with associated eigenfunction
(b) Show that the remaining eigenfunctions are given by (x)
Th
y
x, where /3 is the nth positive root of the equation tan z =
3
sin /
z. Draw a sketch showing these roots. Deduce from this sketch
(2n + 1)ir/2 when n is large.
that
Bo
y(i)A
o
y( x.
5
\/
e
()
ct
/
e
1417
in ci /1( J /
A70
(05
.4
/ (a)
yI
y
‘(
I
()
t
()
0
()
()
y (i)
‘(1)
i()
(°5
ri(v)
( K)
c
12
Co()
(P o ,t }ke
1/vtI rc1k)
More room, if necessary, for Problem 3.8.8.
,-f
5’o
tOci
J
24
roc
it
-
ftfr1c
oE
i-s’
4A-e
,iL4
)
1
(A,
1c’
I
/45
ye
ocCc(fJ
13
c
-/-
//9rc’K/m/Jev
3.8.13
-
Consider the eigenvalue problem
y”+2y’+AyO;
(a) Show that A
y(O)=y(l)=O.
1 is not an eigenvalue.
(b) Show that there is no eigenvalue A such that A
<
1.
ir + 1, with
2
(c) Show that the nth positive eigenvalue is A = n
associated eigenfunction y(X) = e sin (n7rx).
1’7rI
roch
-
-f
y(H A e
A
O
y
50
(
I
y()
U
SO
ere
A
f
riv(qf
ke 4
/1 c
1+Zy Ao
e
A=c
y(
(i)
Ae÷Oe
€
fG/cIi 04
‘C7
/t(e
c1
-
)
—
N
j
-
I
(Ti
\j
-
+
—
—
;
c
\Th
°
—0
c
0
N
s—.
II
+
—p.
0
rD
Section 4.1 First-Order Systems and Applications
-
4.1.1
Transform the given differential equation into an equivalent system
of first-order differential equations.
-
x” + 3x’ + 7x
D
xl
50/
-)1€
5yfk
=
.
2
t
7
i
1
7
r
xre
:
j
16
4.1.3
Transform the given differential equation into an equivalent system
of first-order differential equations.
-
(4)
‘I
+ 6x
—
3x’ + x
=
cos 3t.
/
I
3
X
+ (0;
—
(+)
I
(
‘I
I
I
17
—
1
toyJ
4.1.13
Find the particular solution to the system of differential equations
below. Use a computer system or graphing calculator to construct a
direction field and typical solution curves for the given system.
-
x’=—2y, y’=2x; x(O)=1,y(O)=O.
y
y
y
y
t
y () A
y -?A//)2c(?)
.-
/
18
Af(7cs(?)
More room, if necessary, for Problem 4.1.13.
Pirec
o(
/
‘4
lvi
V
7’
Ndfe;
5hoI) be
40
CI(cIJ
A
19
1•
I
bei
4.1.15
y
Find the general solution to the system of differential equations
below. Use a computer system or graphing calculator to construct a
direction field and typical solution curves for the given system.
-
-
/
y’=-8x.
y
y ()
A
y ()
&ffl era)
x ()
y If)
yO
4
Y
/ ()
()
y ‘(i)
5/
-
5/
(z )
!:
si
() +
A
f
20
cc
()
(?)
More room, if necessary, for Problem 4.1.15.
Pirecio
21
Beginning with the general solution of the system from Prob
2+y
lem 13, calculate x
2 to show that the trajectories are circles.
4.1.22 (a)
(b)
-
-
Show similarly that the trajectories of the system from Problem
2 +y
.
2
15 are ellipses of the form 16x
2 = C
i -A j’ (Ze) 8 s
y() A coC7)fI?)
X(
X
(]
y ()
Ac
A cc
2
D
b)
A
A Ifl (?) G5
()
(0$
()
c[rc
ejq/ 5 /
y
A 611 “1 (Z5 )
L+
-
(A o(?#) mi
I
(
.
i:
•i4/?)
c(
(5(2t
4
22
C
T(?
ri1(7J
0
I-i
-t
\
C
C
LID
rr
riD
C