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ODE Chapter 3 EAssignment

Approximate Solution on Direction Fields

3-1 (ODE Nov 1999 Q6) The solution curve of the initial value problem

1)0(,2

1)sin( =+= yyxxy

dx

dy

goes through precisely one of the five pointsA, .,Eindicated below on the

direction field for the differential equation. Determine that point.

E

D

C

B

A

(E)

(D)

(C)

(B)

(A)

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3-2 (ODE Jun 2000 Q5) The solution curve of the initial value problem

1)0(,2

1)sin( =+= yyxxy

dx

dy

goes through precisely one of the five pointsA, .,Eindicated below on thedirection field for the differential equation. Determine that point.

E

D

CB

A

(E)

(D)

(C)(B)

(A)

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3-3 (ODE Oct 2002 Q6) The diagram below represents the direction field of

2

1(E)

1

2(D)

1

2(C)

2

1(B)

21(A)

+

+

+

+

x

y

x

y

x

y

x

y

xy

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Long Term Behavior of Differential Equations

3-4 (ODE Nov 1999 Q7) It is known that the behavior asxincreases fromx0= 0 of the

solutiony(x) of the differential equation ),( yxfy = , whose direction field is below,

satisfying 0)0( yy = , depends only on the value of 0y .

+>

==

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3-5 (ODE Jun 2000 Q6) It is known that the behavior asxincreases fromx0= 0 of the

solutiony(x) of the differential equation ),( yxfy = , whose direction field is below,

satisfying 0)0( yy = , depends only on the value of 0y .

==

xas1)(then,51If(E)1)(then,1If(D)

as0)(then,1If(C)

as)(then,5If(B)

5)(then,5If(A)

0

0

0

0

0

xyyxyy

xxyy

xxyy

xyy

3-6 (ODE Jun 2000 Q13) A population P satisfies the initial value problem

1000)0(),500(20 == PPPdt

dP

Then Pwill

decreasethenfirstatincrease(E)

increasethenfirstatdecrease(D)

towardstendandtallforincrease(C)

valuelimitingaapproachandtallfordecrease(B)

valuelimitingaapproachandtallforincrease(A)

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3-7 (ODE Undated Q7) The behavior of the solutiony(x) of the differential equation

)1)(2( yyy += satisfying 0)0( yy = depends only on the value of 0y . Which of

the following is NOT true

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3-10 (ODE Aug 2006 Q4) For the autonomous DE

=

2sin)1( y

yedx

dy y , indicate the

constant solutions and the regions of positive and negative slope on a direction field

to determine how many of the following statements are true

=

=

=

=

xasxfthenfif

xasxfthenfif

xallforxfthenfif

xasxfthenfif

)(0)0(

2)()0(

1)(1)0(

1)()0(

21

21

.4(E)

.3(D)

.2(C)

.1(B)

.0(A)

3-11 (ODE Mar 2007 Q4) For the autonomous DE

=

2sin)1( y

yedx

dy y , indicate

the constant solutions and the regions of positive and negative slope on a directionfield to determine how many of the following statements are true

=

=

=

=

xasxfthenfif

xasxfthenfif

xallforxfthenfif

xasxfthenfif

)(0)0(

2)()0(

1)(1)0(

1)()0(

21

21

.4(E)

.3(D)

.2(C)

.1(B)

.0(A)