Math 2280 Assignment 1
-
Dylan Zwick
Spring 2013
Section 1.1 1, 12, 15, 20, 45
-
Section 1.2 1, 6, 11, 15, 27, 35, 43
-
Section 1.3 1, 6, 9, 11, 15, 21, 29
-
1
Section 1.1 Differential Equations and Mathe
matical Models
-
1.1.1 Verify by substitution that the given function is a solution of the
given differential equation. Throughout these problems, primes de
note derivatives with respect to x.
;
2
y’=3x
+7
3
yrrx
(hec5
2
O
C
I.
C
0
C
U
I.
I.
I.
—
.
C
C.)
—
II
+
-
-
—
0
I
I
-
I
0
1)
Q
-r
0
-
Ii
-
Ic
I
(I
)
t
(J
ç-J
en into the given differential equation to determine
“—---——-----áll values of the constant r for which y = enx is a solution of the
1.1.15 1
bubstitute y
=
equation
I
I!
y +y —‘2y=O
I!
L(
rry
er
—
10
/1/
1
r
z
(*+L)(r-))
So!
4
M Uf
ho
1.1.20 First verify that y(r) satisfies the given differential equation. Then
determine a value of the constant C so that y(x) satisfies the given
initial condition.
yl =
Ce + x
y(x) =
—
—
1,
y(O)
=
10
I
K—iD
x 1)
y-(
-Ce-t / y
I
N/I”
(o)
Ce
(
5)
-
/L
Jñl}{/
5
vq/
c1
1.1.45 Suppose a population P of rodents satisfies the differential equation
. Initially, there are P(0) = 2 rodents, and their number
2
dP/dt = kP
is increasing at the rate of dP/cIt = 1 rodent per month when there
are P = 10 rodents. How long will it take for this population to grow
to a hundred rodents? To a thousand? What’s happening here?
ii2
kHc
=2
/)
j
-
C—k+
L
/-?kf
/
IZ(
-
lcd
ICC)
(io) 10
P(q /00
6
/
0W
1
I-) (
I
*0
/-4
-Mq
Section 1.2
Solutions
-
Integrals as General and Particular
,1-2Find a function y = f(x) satisfying the given differential equation
and the prescribed initial condition.
=2x+1;
y()= J(ti)(
7
y(O)=3.
yty-C
1.2.6 Find a function y = f(x) satisfying the given differential equation
and the prescribed initial condition.
y(—4)
dx
y(
=
0.
Jxx
2+
1,x
L
\/
/
()
3
LJ))
y(x)=
i9
3
3
8
ci
v
q [/
1.2.11 Find the position function x(t) of a moving particle with the given
0 = x(0), and initial velocity vo =
acceleration a(t), initial position x
v(0).
a(t)
v(•)
$o+
=
50,
=
10,
=
20.
0
V
O (o)
(f)
50
5 t * Ic
(CL)
±
1O() +
9
/O
1.2.15 Find the position function x(t) of a moving particle with the given
= x(O), and initial velocity vo =
acceleration a(t), initial position
V (0).
a(t)
=
0
V
4(t +
=
=
3)2,
—1,
1.
(p5))
v()4 (o)’+(
V(o)
-i
—57
v()
x()
€+
5)
-J?(o)+C
1.2.27 A ball is thrown straight downward from the top of a tall building.
The initial speed of the ball is lOm/s. It strikes the ground with a
speed of 60m/s. How tall is the building?
Vj
(I7L).
1c/
55
xc
L)
lLvn t
()
±
5f
11
o
1.2.35 A stone is dropped from rest at an initial height Ii above the surface
of the earth. Show that the speed with which it strikes the ground is
v()
V
( F)
12
1.2.43 Arthur Clark’s The Windfrom the Sun (1963) describes Diana, a space
craft propelled by the solar wind. Its aluminized sail provides it with
a constant acceleration of O.OOlg = O.0098m/s
. Suppose this space
2
craft starts from rest at time t = 0 and simultaneously fires a projec
tile (straight ahead in the same direction) that travels at one-tenth of
the speed c = 3 x 10
m/s of light. How long will it take the spacecraft
8
to catch up with the projectile, and how far will it have traveled by
then?
L
(7
v+
v
.—
(A k4- q c
13
(A
CL(
y
/1-
)
ye/J)
1.3.1 and 1.3.6 See Below
In Problems I through 10, we have provided the slope field of
the indicated differential equation, together with one oi I11OFL’
solution curves. Sketch likely solution cune through the ad
ditional points marked in each slope field.
13)
dy
4. ----=x—v
d
3
I
=
LXIS
—,
III II
I I I I I
I
S.
I/%I\/
S. S’.———.5
II/I/II%
I,
I I
•/•/ I//\I\s
f//S
II\/\\\ \:—-.f///
2
/5=
\\S..
‘.‘_—.___
—
.—.———
‘i’’•’•
C
3
of
ue
,
—
‘‘‘I/I,,
I I
/1//ill/i
‘‘‘‘II?,’
Il/Il/li
i / Pt , ;i
/1/Ill/Il
-/ / / I / / I
7
—--——f/I’’
—
‘
..
—3
(14)
2
1
)
—i
—2
FIGURE 1.3.18.
f the
(15)
rigin
I
III
Il//I
,‘//
/1/
-3 JllJJ
0
—3 —2 —1
2
5.
FIGURE 1.3.15.
I
2.
dv
dx
(lv
y
=
-v + I
—
3
+y
=.‘
——
-
—
7
‘I
Ill
Ill’
/ I P1
‘I,,,
/ Il / /
II
I’
‘I,,,
/ I I I S. S.
Il//S.
I S. I S.
III
‘‘I’’
II II Il///
/ I I I S.
Ill IS.
II
—I
—7
—2
(16)
—1
—l
————‘I’ll
-3
/
_\
S.
‘
I
I
2
I
0
—2-I
FIGURE 1.3.19.
2
I
0
I
I /
/
I I
S. S. I I
I I
/ S.
I•I I I•S.
/ I
I / I / /1
—‘S.
dy
dx
y+ I
-
—=v—snx
3
,,TT
1
F
1
2
(1
y many
/ /
I////I I//
/
c/f//f//f
///./!I’/
Il/I,,!,,
Il/uI/I.’
/
//f,,,,/
j__
.::::::
5
z:.
(1
5—
—7
—3
—3
‘
S.
I S.
IS.S.\
S.\ Il
‘I
S.
I S.
-
5
1/ S.
—2
‘/
\
—l
F1GUE 1.3 17
I•I
/1I
I
I
/ I:
0
I
2
1/
—33
—2
/
i/Il
I
—1
FIGURE 1.3.20.
?///)I
/
I/I
I/Il
:
f-S
S. S.
I
I/I
/‘i
-.
5—---——
III // II,
I
S.
S.,
I
‘I
I I
.
—
‘.‘
—I
through
solution
nstanCe,
ihole x
‘‘‘S.
—
FIGURE 1.3.16.
3.
iranteeS
f/I/f
/1/f
II
II
I
ider a
n Fig.
the left
hus the
I I I /
I’ll/Ill,
‘I//I/Ill
•f , / I I
I I
f/I / Il/I I
—
/ /
‘ /
i’
,
m
(17)
I
/ I / /
f/I)
/1
0
I
1/
I
I
I
I
2
—
1.3.9 See Below
3
Tt ——
•
7
,i,
3
—
,,,,
1/
,,,j
dl f/I
ft / /
I
fI,,,Il,,_
.—---——-—-————
\-—-‘•— ‘S/S
S
“‘‘.5/S
S 1Sk\s
I
‘‘
—I
——
Il/Si/S
——
I
___L
—I
—2
2
I
0
/l__,\l\””’
—“—
5
5
II!/—S’.’.5’
S S S
.1 I I /
2
I
3
ii.
S S S
—
/
5’’——’
S-”——/
I//Sf-’’-’’
S
‘I
(5///
/
r
—I
—2
I
——‘III I
f//il / I
/ / / / I I S
I/Ill?
f/I/I,
/1/1/
‘,
S/It/f
I
III
(•/ I I
t
I
P
—1
v(1)=’ 1
12.
=1nv;
13.
‘./V;
y(O)
.1V;
v(O)=0
14.
s
=
I
=
i I/SIll
I
S I ‘,
hi/I
II
I I
—l
=
‘_L.
‘(
I/Il’—
I I/It’,
‘Il/I,,
Ill//f
v(I)
;
y
2
2i
=
I
II
III
‘.---—-i
i ,
I I I
/
I/_—”SS
-
Ii
I S.
5\’.S—
II 11”.’SS
—2
0
In Probl cmiii: II ihivugli 20, determine whether Theorem I does
or does not guarantee e.ii,steiice ofa solution oJ’the given initial
i’ahie proble,n. If cvi.sie,ic’e is guaranteed, determine whether
77meoremn I does or does not guarantee uniqueness of that so
lution.
dv
8. —=.i-—y
d,i.
7
/1
il/I
FIGURE 1.3.24.
FIGURE 1.3.21.
3
SkIS
S’.\kk
-
—I
—2
—3
3
SS’.i
‘.55/
5///
S
Ilk /555’”
SI/k/S
,
—3
55/51/S
S
is’.,,
5’.’,’,
/k’’,
I
0
3
2
15.
=
/‘‘;
‘‘(2)
=
2
FIGURE 1.3.22.
16. _=,/Z;
9.
di.
3
TT
17.
=.i
—
1;
v(0) =
18.
=
—
I;
v(I) = 0
= In( 1
19.
—I
/5 I”
—2
—
1 /
I ,S
2
J
3
20.
—
di:
=x
-
—
.2).
,,2.
i’(O) = 0
(0)= 1
in Problems 21 amid 22, fin use the method of Eiamnple 2
to construct a slope field for the given differential equation.
Then sketch the solution curve corresponding to the given mi—
tial condition. Fincillv use this solution curie to estimate the
desired tahie of the solution v(x).
21. v’ x + y.
22. i”=v—x,
=
FIGURE 1.3.23.
,i
rE’r;
ith
3-2-IO
(2)=z1
v(O) 0;
)=O;
4
y(
=
y(— =?
)
4
)=?
4
v(—
1.3.11 Determine whether Theorem 1 does or does not guarantee existence
of a solution of the given initial value problem. If existence is guar
anteed, determine whether Theorem 1 does or does not guarantee
uniqueness of that solution.
—
2
2xy
y(l)
=
—1.
•(\
cre
iio5
raai/
n.Y
Ift(
(i,
i€
-JL
(,
16
1.3.15 Determine whether Theorem 1 does or does not guarantee existence
of a solution of the given initial value problem. If existence is guar
anteed, determine whether Theorem 1 does or does not guarantee
uniqueness of that solution.
y
-
To
I
Cz/l)
/
y
1—
5,
I
17
K)5
n(
)
1.3.21 First use the method of Example 2 from the textbook to construct a
slope field for the given differential equation. Then sketch the solu
tion curve corresponding to the given initial condition. Finally, use
this solution curve to estimate the desired value of the solution y(x).
y(O)=O;
y’=x+y,
X\f_z -3
3
7
(—i
3
1
13
—
Q
.Q
0
zL zL zL
J___
r
--5_
(
/
y(([); 3
/ /
1/
/ /
)
/
\\\\
\
\\z
18
1.3.29 Verify that if c is a constant, then the function defined piecewise by
1
0
3
(x—c)
x<c,
>c
satisfies the differential equation y’
3y for all x. Can you also
3 in piecing together a
c)
use the “left half” of the cubic y = (x
solution curve of the differential equation? Sketch a variety of such
solution curves. Is there a point (a, b) of the xy-plane such that the
initial value problem y’ = 3y, y(a) = b has either no solution or a
unique solution that is defined for all x? Reconcile your answer with
Theorem 1.
—
.O
yl-
3(x-c)
O
3y
oI
1
Cc
cc
y
5°
6c.
A I
a
4 1G
19
i-
C
cD
“J
c_i
0
c_i
cJ
—H--
QI
-1:-,
•]$—..--
c-
D
(1)
-
•c-)
c_i
!
H
-
-j
‘
U
-___s
rI