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ODE A2 - 1

ODE Chapter 2 AAssignment

2-1 For each of the following initial value problems, (i) state the largest interval in whichthere will be a unique solution, (ii) solve it, if you can.

10

2

=

=+

)(

sin)(tan)(

ywhere

xyxya

30

1

=

=+

)(

)(

ywhere

xx

yyf

10

2 2

=

=+

)(

cossin)(sin)(cos)(

ywhere

xxyxyxb

10

2

=

=+

)(

sin)(cos)(

ywhere

xyeyxg x

21

2

)1(

12)(

=

+=+

ywhere

xxyyxc

10 =

=+

)(

sin)(sin)(

ywhere

xyxyxh

30

41

=

=

)(

)(

ywhere

x

yyd

42

2

eeywhere

xxyyxxi

=

=+

)(

ln)ln()(

30

141

=

=

)(

),()()(

ywhere

xyyxe

2-2 For each of the following initial value problems determine whether or not the

Fundamental Existence and Uniqueness Theorem (for 1storder ordinary differential

equations) guarantees that the problem has a unique solution in a neighbourhood of

the point indicated.

0151

== )(,)( ywhereyxyi 10

2

== )(,)( ywherex

y

yiii

0)1(,)( 51 == ywherexyyii 1)0(,)(2

== ywherey

xyiii

2-3 Does the initial value problem ( ) 02 == ywherexyy ,sin have a unique solution

in any neighbourhood of2=x ? If so

(a) find the unique solution, and

(b) show how the existence follows from the Fundamental Existence and

Uniqueness Theorem.If not

(a) write down at least two solutions, and

(b) show why the Fundamental Existence and Uniqueness Theorem fails to apply

to this initial value problem.

2-4 Show how the Existence and Uniqueness Theorem guarantees the existence of a

unique solution of ),( yxxy += where y(2) = 5 near x= 2.

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ODE A2 - 2

2-5 (a) State the largest open interval in which you can be sure that the initial value

problem 10312

1==

+ )(,

)(ywherey

xy

will have a unique solution and,

(b) Solve the initial value problem.(c) What is the largest x-interval in which you have found a continuous solution

2-6 (a) Find all constant solutions and the general solution of

1

2 2

+=

x

y

dx

dy )(

(b) Solve1

2 2

+=

x

y

dx

dy )(, where y(0) = 0.

2-7 Verify that the solution of the initial value problem ( )311 +++= )ln()( xxxy iscontinuous on ( ) ,1 .

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ODE A2 - 3

Answers to ODE Chapter 2 AAssignment

2-1 (a) xxy coscos,, 3222

2+=

(b) xxy coscos,, 3222

2

+=

(c) ( )2

1

12

1

340

2

2

++=x

xxy,,

(d) ( ) )()ln()(,, xxxy += 131141 (e) same as (d)

(f) ( ) ( )3111 +++= )ln()(,, xxxy

(g) lyanalyticalevaluatedbecannotdxx

eFI

x

=

cos..,,

2

22

(h) no unique solution at y(0) = 1

(i) ( )x

xxy

ln,,

421

22

=

2-2 (i) Yes, (ii) No, (iii) No, (iv) Yes.

2-3 No.

2-5 (a)

( )1,

(b)x

xy

+=1

312 )(

(c) ( )1,

2-6 (a) 2=ySolutionntConsta :

Cxy

SolutionGeneral +=+

)ln(: 12

1

(b)2

11

2

1=

+

)ln( xy