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 Mathematical Models

1.1.1 Verify by substitution that the given function is a solution of thegiven differential equation. Throughout these problems, primes denote derivatives with respect to x.

y’=3x2; yrrx3+7

(hec5 O

2

CCI.

0U C

CC.)

0

—

I.

-

I(J

—

II+

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_

—

Ic

t

I.

I

1)

I.

-

-

00

-

.I

Ii-

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-r

1.1.151bubstitute y = en into the given differential equation to determine“—---——-----áll values of the constant r for which y = enx is a solution of the

equation

I! Iy +y —‘2y=O

I! L(

rry —

e r/1/ 10 M Uf h o

1r- z(*+L)(r-))

So!

4

1.1.20 First verify that y(r) satisfies the given differential equation. Thendetermine a value of the constant C so that y(x) satisfies the giveninitial condition.

yl =—

y(x) = Ce + x — 1, y(O) = 10

IK—iD y-(

I

x 1) -Ce-t / y

- c1

5)

( / L

N/I” (o) C e

Jñl}{/ vq/

5

1.1.45 Suppose a population P of rodents satisfies the differential equationdP/dt = kP2. Initially, there are P(0) = 2 rodents, and their numberis increasing at the rate of dP/cIt = 1 rodent per month when thereare P = 10 rodents. How long will it take for this population to growto a hundred rodents? To a thousand? What’s happening here?

ii2

kHc

=2 /) -

C—k+j

L

/-?kf

lcdIZ(

/

(io) 10

P(q /00

I-) (10W

ICC)-

6

I/ /-4 -Mq

*0

Section 1.2 - Integrals as General and ParticularSolutions

,1-2Find a function y = f(x) satisfying the given differential equationand the prescribed initial condition.

=2x+1; y(O)=3.

y()= J(ti)( yty-C

7

1.2.6 Find a function y = f(x) satisfyingand the prescribed initial condition.

dx

the given differential equation

y(—4) = 0.

y( Jxx

\/ ()/

3LJ))

2+ ci

1,x vL

q[/

y(x)=3

i93

8

1.2.11 Find the position function x(t) of a moving particle with the givenacceleration a(t), initial position x0 = x(0), and initial velocity vo =

v(0).

a(t) = 50,

= 10,

= 20.

v(•) $o+ V0

O (o) 50 / O

(f) 5 t * Ic ±

(CL)1O() +

9

1.2.15 Find the position function x(t) of a moving particle with the givenacceleration a(t), initial position = x(O), and initial velocity vo =

V (0).

a(t) = 4(t + 3)2,

V0 = —1,

= 1.

(p5))

v()4 (o)’+(

V(o)-i

—57v()

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 aspeed of 60m/s. How tall is the building?

Vj (I7L). 1c/

55

xc

L) () ± o

lLvn t 5f

11

1.2.35 A stone is dropped from rest at an initial height Ii above the surfaceof 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 spacecraft propelled by the solar wind. Its aluminized sail provides it witha constant acceleration of O.OOlg = O.0098m/s2.Suppose this spacecraft starts from rest at time t = 0 and simultaneously fires a projectile (straight ahead in the same direction) that travels at one-tenth ofthe speed c = 3 x 108m/s of light. How long will it take the spacecraftto catch up with the projectile, and how far will it have traveled bythen?

L (7 v+ v

.—

13

(A k4- q c ye/J)

(A CL( /1-y )

1.3.1 and 1.3.6 See Below

FIGURE 1.3.16.

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.

= —, —

dy4. ----=x—v

d

3

7

‘

I III II \\S..

I I I I II I/%I\/ S.

II/I/II% S. S’.———.5

I I ‘.‘_—.___ I,•/•/ I//\I\s .—.——— f//S

II\/\\\ \:—-.- f///

‘i’’•’• ,

3

2

C

—3

Il//I III I /1/ ,‘//

-3 JllJJ—3 —2 —1 0 2

FIGURE 1.3.15.

dv2. —— =.‘ +y

dx

—--——f/I’’

— ..

—2 —i

FIGURE 1.3.18.

5. - = y — -v + I(lv

13)

/5=

LXIS

ofue

(14)

f the

(15)

Irigin

m

ider an Fig.

(16)

(17)

the lefthus they many

iranteeS

throughsolutionnstanCe,ihole x

—‘‘‘I/I,,

-/ / / I / / I I I/1//ill/i‘‘‘‘II?,’Il/Il/lii / Pt , ;i/1/Ill/Il

) 1 2

‘I,,,

f/I) / I / / I

I’ll/Ill,‘I//I/Ill

•f , / I I I If/I / Il/I I—

, I ‘ /i’

/ /

3

7 —II

I’

I I I /

‘IIll

Ill’/ I P1

‘I,,,/ Il / /

—I

—7

/ I I I S. S.

Il//S.I S. I S.

III‘‘I’’

II II Il//// I I I S.

II Ill IS.

————‘I’ll

—2 —1 0 I 2

‘‘‘S.

—‘S. S. S. I I / II I I // S. /

_\ S. I•I I I•S. I I/ I

‘ I I / I / /1 I

dy3. —=v—snx

dx

0 I 2

3

2

(1

F11 ,,TT / / /

I////I I// c/f//f//f

///./!I’/Il/I,,!,, //f,,,,/

Il/uI/I.’ /

j__

II f/I/fII /1/f —

—l

-3 —2-I

FIGURE 1.3.19.

- y+ I

(1

______________

—3-

—I

‘I S. 5

‘II S. 1/ S. ‘/ \ I I

z:.5.:::::: ‘.‘

‘ S. I S. S. S.

IS.S.\ /1-S.\ Il I I I I

- I•I / I:III // II,I S., I S.

—7

—3—3 —2 —l 0 I 2 3

5— .— -. /‘i ?///)I

5—---—— f-S I/I /:i/Il I/I I/Il

1/ / I /1 I 1/ I I I —

—2 —1 0 I 2

F1GUE 1.3 17 FIGURE 1.3.20.

1.3.9 See Below

Tt —— ,i,,,,, —

,,,j 1/

• dl f/I ft / /

f//I/Ill/f/I/I,.,,,f/I//f//f

.—---——-—-———— fI,,,Il,,_

——___L

____________

—3 —2 —I 0 I 2 3

FIGURE 1.3.21.

S \-—-‘•— ‘S/SI S 1Sk\s “‘‘.5/S

—I ‘‘

Il/Si/S S 55/51/Sis’.,, SS’.i

, 5/// 5’.’,’, ‘.55/ SkIS—— S /k’’,

Ilk /555’” S’.\kk /1SI/k/S il/I

I -

—3 —2 —I 0 I 2 3

or does not guarantee e.ii,steiice ofa solution oJ’the given initiali’ahie proble,n. If cvi.sie,ic’e is guaranteed, determine whether77meoremn I does or does not guarantee uniqueness of that solution.

ii. - = 2i2y2; v(I) = —1

12. =1nv; v(1)=’ 1

13. ‘./V; y(O) = I

14. = .1V; v(O)=0

15. = /‘‘; ‘‘(2) = 2

16. _=,/Z; (2)=z1

21. v’ = x + y. v(O) = 0; y(—4) =?

22. i”=v—x, y(4)=O; v(—4)=?

3

7

3

I

dv8. —=.i-—y

d,i.

FIGURE 1.3.24.

In Problcmiii: II ihivugli 20, determine whether Theorem I does

3

7

5/l__,\l\””’

5II!/—S’.’.5’ 5\’.S— Ii

.1 I I / — S S S S S S — / I S.II 11”.’SS 5’’——’ I

I/_—”SS S-”——/ / III//Sf-’’-’’ S “— III‘I ‘.---—-i i , r(5/// / I I I

‘( ‘_L.

—I

—2

I/Il’—I I/It’,‘Il/I,,Ill//fS/It/fIII I

I t (•/ I Ihi/II I II

——‘III If//il / I/ / / / I I SI/Ill?f/I/I,/1/1/‘, I P si I/SIllS I ‘, I

—2 —l 0 I 2 3

FIGURE 1.3.22.

9.di.

3 TT rE’r;

ith—I

—2/5 I”

— 1 / I ,S J

17. =.i — 1; v(0) =

18. = ,i — I; v(I) = 0

19. = In( 1 - .2). i’(O) = 0

20. — =x — ,,2. (0)= 1di:

in Problems 21 amid 22, fin use the method of Eiamnple 2to 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 thedesired tahie of the solution v(x).3-2-IO 2 3

FIGURE 1.3.23.

1.3.11 Determine whether Theorem 1 does or does not guarantee existenceof a solution of the given initial value problem. If existence is guaranteed, determine whether Theorem 1 does or does not guaranteeuniqueness of that solution.

— 2xy2 y(l) = —1.

•(\

cre iio5

raai/

n.Y i€ (,Ift( (i, -JL

16

1.3.15 Determine whether Theorem 1 does or does not guarantee existenceof a solution of the given initial value problem. If existence is guaranteed, determine whether Theorem 1 does or does not guaranteeuniqueness of that solution.

y

- I To

/ Cz/l) y—

____

1-

5, K)5 n( )

I

17

1.3.21 First use the method of Example 2 from the textbook to construct aslope field for the given differential equation. Then sketch the solution curve corresponding to the given initial condition. Finally, usethis solution curve to estimate the desired value of the solution y(x).

y’=x+y, y(O)=O;

X\f_z -3 (—i

____ __ __

3

_____

7

—

_

3 1 J___

_ _

Q 13

.QzL zL zL

___

0

___

r

__

y(([); 3

)

--5_

( // /1/

/ /

\\\\

/

\ \\z18

1.3.29 Verify that if c is a constant, then the function defined piecewise by

1 0 x<c,(x—c)3 >c

satisfies the differential equation y’ 3y for all x. Can you alsouse the “left half” of the cubic y = (x — c)3 in piecing together asolution curve of the differential equation? Sketch a variety of suchsolution curves. Is there a point (a, b) of the xy-plane such that theinitial value problem y’ = 3y, y(a) = b has either no solution or aunique solution that is defined for all x? Reconcile your answer withTheorem 1.

yl-

.O

3(x-c)

O3yoI 1 Cc 6c.

cc a A I i-y 5° 4 1G

19

QI•]$-

—..--

cD

c-

“Jc_

i

0

D

c_i

cJ-1:-,

—H

--(1)

H

-

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C

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