INTRODUCTION TO DIFFERENTIAL EQUATIONS, TEST 3VersionA, Sections 5-8, Spring 2009

Section _ Name $o\V\-'iJ"\.j

Instructions. You are allowed to use one 8 1/2 x 11 inch sheet of paper of notes. No books orelectronic equipment (including calculators, PDAs, computers, cell phones) are allowed. Do notcollaborate in any way. In order to receive credit, your answers must be clear, legible, and coherent.In case of an error in a test question, simply write in the correct answer. Questions 1~ 12 are 5points each; each box in question 13 is 2 points.

~l. If we write the Laplace transform of y as Y = £[y], then the Laplace transform of theinitial value problem y" - 2y' = 1, yeO) = 5, y'(O) = 1 isA) 82y - 28Y = 1, yeO) = 5, 8Y(0) = 1 B) 82y - 28Y =~, yeO) = 5, 8Y(0) = 1C) 82y - 28Y =~, yeO) = 5, Y'(O) = 1 D) 82y - 28Y - 58 -t~ = ~

SL ~ - s ~(a) -.;:/ (0) - 2 [ .s '( - ( 0)J:; .1~ ~ ~.$

.s \ .£;E ~ 2. In the process of solving the initial value problem y' + 3y = 3, yeO) = 0, y'(O) = 0 bythe Laplace transform method, we obtain the expression for the Laplace transform £[y] =

333A) - B) 8+3 C) -3 D) 38(8+3) ~ ( 3)8 8+ 88+

~'(+3i-:: ~o y=

F 3. In solving a differential equation by the Laplace transform method, we find that Y =68

£[y] = ~4' We conclude that y =8 +

A) sin2t B) 3sin2t D) 6 sin 2t D) cos2t E) 3cos2t F) 6cos2t

For y" + 4y = get), yeO) = 0, y'(O) = 0 , which of the answers below best matches the followingdescriptions?

L4. transfer function

~5. impulse response

1A) 82 + 4

C) Ltg]82 +4 D) ~ sin2t 1 t

E) :2 io sin(2r) get - r) dr

1

E) {t - 3 t > 3 F) { 0 t > 3o t<3 t-3 t<3

u, Ii) t (&- 3)

f[e~Hf/t)J:= F(S-3)

+' tt) ~ t F :: ~"l.

G 9. In solving an initial value problem by the Laplace transform method, we find that

£[y] = 2( 21 1) = 12- ~1' Which of the solutions y below are correct?s s + s s +

A) only a) is correct B) only b) is correct C) only c) is correctD) only a) and b) are correct E) only a) and c) are correct F) only b) and c) are correctG) all the choices below are correct H) none of the choices below are correct

;t[,tJ =- ~L

J [s,""-t .. ::: 1..<J ,'2.+\

AlO. With the notation x = (y, y', y")T, we can write the third-order differential equationy'" + y" + 2y' - 3y = 0 as the first-order system

£ -&-=--7. The inverse Laplace transform of ~ is

sA) te-3t B) te3t C) { t t > 3 D) { 0 t > 3o t<3 t t<3

-35e F/s) I ~ (f')= f ::1~ .-:')9:::

A8. The inverse Laplace transform of l/(s - 3)2 isA)te3t B)t-3 C) (t-3)e3t D){t-3 t>3o t < 3E) cannot be found from the table

b) lot rsin(t - r)dr c) t - sint

A) x'~ G 1 ~)x B) of ~ G -2 -1) C 2 -3)0 0 o x C) x' = 0 1 o x-2 -1 1 0 o 0 1

C 0 ~)x (-3 0 0) F)x'~G 0

~)xD) x' = 0 2 E) x' = 0 2 0 x 1o 0 -3 o 0 1 -2 -1

XI-=- J I - )("2-x, -)(v~'J I '></ -:.. Xs

X3 -:'011 XI_JII- (3 ~ I i' 3 XI - '2.)('2. - )( <3:3 - - d- 0 -t) -:::

~11. One of the eigenvalues of the real-valued matrix A is 5i, and the corresponding eigenvec-tor is (1,2 + 3i)T. The general solution to x' = Ax is

A ( cos 5t ) ( sin 5t ) B ( cos 5t ) ( sin 5t )) C1 cos 3t - sin 3t + C2 cos 3t + sin 3t ) Cl 2 cos 5t + 3 sin 5t + C2 3 cos 5t + 2 sin 5t

(COS 5t ) (0) ( 0) ( sin 5t )

C) Cl 2cos5t + C2 3sin5t D) C1 cos2t + C2 sin3t

~12. The matrix A has eigenvalues -1,3 and corresponding eigenvectors (1,0)T and (1,1)T.The solution to the initial value problem x' = Ax, x(O) = (5,4)T isA) Xl = e~t + 4e3t, X2 = 4e3t B) Xl = 4e~t + e3t, X2 = 4e3t C) Xl = 5e~t, X2 = 4e3tD) Xl = 5e3t, X2 = 4e~t

,..,.. (I \ t ( I \ ~{>dtF G, 0 ) ~ r- c:..... )) '"6

(1/)= -;;(o~~ c, (~) ~ c,(\) =)

13. (2 + 2 = 4 points each) For the following, classify the behavior [near (O,O)J()'ccording to thetype of phase plane diagram [ node (N), spiral (Sp), saddle (Sa), center (C) 1 and according towhether it is stable (St) or unstable (U).

diagram stability systema

b-t:x=y

C. iJ= -2xb

S'1 Ux = X - 2yiJ= -3y

c N LA x =X -2yiJ= 3y

dN St x = -x-2y

iJ= -3ye

Sf> U x=2yiJ= -x+y

fSet U x =x+2y

iJ=xg

4l:.t:s.. L\~,k-

h

~hWN S:-ti (c0Sf ~

j

~C S

(0 ,\ O--J < t- (:.q.. \ \ ::-2.. D ) -2 -r J

(I -'2\ -r=-\-"),o -3) J v

(I - L\o 3) -r ~ )j 3

(-~ ~~ ') ~ ~ - ~- 3

(~ "l. \ o=d (..~ (-r I \ _

\ I) -\ I-r) -(:;; 0 ~JA (I~r_~)

3

0(" '1.._ <" .4- \

~- .... I!:..II-I..('r - ___2

~ >r = -+1 !. / I~ g'2....