The Chemistry Maths Book, Second Edition

(Grace) #1

Solutions to exercises 643


Section 12.8


















34.y 1 = 1 ae


2 x

1 + 1 (b 1 − 1 x)e


−x

35.y 1 = 1 (a 1 + 1 bx 1 + 1 x


2

2 2)e


4 x





37.y 1 = 1 (a 1 − 13 x 2 4) 1 cos 12 x 1 + 1 b 1 sin 12 x







  1. (ii)I


h

(t) 1 = 1 ae


−(α+β)t

1 + 1 be


−(α−β)t

whereα 1 = 1 R 22 L,β


2

1 = 1 α


2

1 − 1 (1 2 LC)


Chapter 13


Section 13.2














= 1 ae


3 x

1 + 1 be


− 3 x

ifa


0

1 = 1 a 1 + 1 b, a


1

1 = 1 3(a 1 − 1 b)










Section 13.3


6.r


1

1 = 1 r


2

1 = 1 − 1 7.r 1 = 1 ±n


8.r


1

1 = 1 r


2

1 = 10 9.r


1

1 = 1 − 2 ,r


2

1 = 1 − 3



  1. (i) (ii)y 1 = 1 (a 1 + 1 b 1 ln 1 x)x


r

yax bx


rr

=+


12

++


!







!










⋅⋅


!










ax x x x


1

47 10

2


4


25


7


258


10






ya=+x x x


!







!










⋅⋅


!









0 


36 9

1


1


3


14


6


147


9






yaxa x


xxx


=+ −−






−−−






10

2

468

1


357










!






!






!














a xx x


1

35

3


3


1


3


3


3


5


() () ()






ya


xx


=+


!






!














0

24

1


3


2


3


4


() ()






ya x


m

m

=


=


0

0


ya x m ae


m

m

x

==


=


0

3

0

0

3

!



−+


1


39


(cos sin )53 3xx


yae be


x


e


xx x

=+ −−+


32 −− 3

1


42


1


3


yae be x x


xx

=+ − +


32 −

1


39


(cos sin )53 3


yae be e


xx x

=+ +


32 3−−

1


3


yabxe x x x


x

=+() (− + + +)


423

1


32


19 12 8


yae be


x
xx

=+ −−


32 −

1


42


y


x


p

=− −


1


42


11.y 1 = 1 ax


122

1 + 1 bx 12.y 1 = 1 (a 1 + 1 b 1 ln 1 x)x


13.y 1 = 1 ax


− 122

1 cos 1 x 14.y


1

1 = 1 ae


x





Section 13.4







  1. (i)P


1

1

1 = 1 sin 1 θ


(ii)


P


4

3

1 = 11051 sin


3

1 θ 1 cos 1 θ,P


4

4

1 = 11051 sin


4

1 θ


Section 13.5



  1. (i)H


5

1 = 132 x


5

1 − 1160 x


3

1 + 1120 x


(iii)H


6

1 = 164 x


6

1 − 1480 x


4

1 + 1720 x


2

1 − 1120


Section 13.6



  1. (i)


24.L


4

1 = 1 x


4

1 − 116 x


3

1 + 172 x


2

1 − 196 x 1 + 124


Section 13.7







  1. (i)


(ii)


Chapter 14


Section 14.2


2.c 2 D 1 = 1 0: V 1 = 1 A 1 + 1 Bx


c 2 D 1 = 1 λ


2

1 > 1 0: V 1 = 1 Ae


λx

1 + 1 Be


−λx

c 2 D 1 = 1 −λ


2

1 < 1 0: V 1 = 1 A 1 cos 1 λx 1 + 1 B 1 sin 1 λx


J


x
x

x


x


x



=−






















52

2

23


1


3


π


cos sin


J


x
x

x


x


x


52

2

23


1


3


=−



















π


sin cos




















6

1


35 2!!


x





J


xxx


2

22

2


1


2


1


13 2


1


24 2


=
























!!! !!













4


−−










nn n


x


()( )


(!)


12


3


2

3




ya


n


x


nn


=− x











0

22

2

1


1


1


(!) 2


()


(!)


P


4

222

15


2


=−sin ( cosθθ 71 )


P


4

13

5


2


=−sin ( cosθθθ 73 cos )


Px x x x


6

642

1


16


() (=−+−231 315 105 5)


ya


x


x


b


x


x


=+


cos sin 22

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