The Chemistry Maths Book, Second Edition

(Grace) #1

642 Solutions to exercises



  1. (i)


(ii)



  1. 6 x


2




    1. 5 e




− 2 x

1 − 11


Section 11.3


13.y


2

1 = 12 x


3

1 + 1 c 14.



  1. 16.y


3

1 = 13 e


x

1 + 1 c



  1. 18.y 1 = 1 cx


19.y 1 = 111 − 1 x 20.


21.y 1 = 1 x 1 − 11 22.y 1 = 1 −ln(2 1 − 1 e


x

)


23.y 1 = 1 x(ln 1 x 1 + 1 2) 24.



  1. 27.y 1 = 1 ae


x

1 − 12 x 1 − 15


28.(x 1 − 1 y)


2

1 − 14 y 1 = 1 c


Section 11.4












Section 11.5


33.y 1 = 121 + 1 ce


− 2 x





35.y 1 = 1 e


− 3 x

(x 1 + 1 c)






37.y 1 = 1 ce


− 12 x

1 − 14 38.y 1 = 1 cos 1 x 1 + 1 c 1 cos


2

1 x










Section 11.6




















−+

1


1


23

23

kk


e


kkt

()


()

[]C ( )


ak k


kkkk


e


kt

=


+−








12

2311

1


1


1

[]C


ak k


kk


ee


kk


ee


kt kt kt

=











−− − −

12

21 31

13 2

kkt

kk


3

32






y


x


na


cx


n

a

=


++






+


1

1


y


b


a


ce


ax n

n

=+


−+

+ 1

() 1

y


x


=+xxxx xc





−+




2


22


2

2

sin cos sin


yce


x

=−


2

2
1

4


xt


ka


kk


e


kkt

()


()

=














−+


1

11

1


11

11


1


11

xa


nkt


nn−−

−=−()
τ

1 n

n


k


=


ln


yx


x
44

4


2


=










ln


y


x


x


22

1


3


1


=−












yy


x


x


2

3

3


+= −


y


ce


x

=






1


1


yae


x

=


3

y


xc


=







1


2


2

2


4

2

e


−x

xt


t


m


() t


cos


=







12


4


2

π


π


m


dx


dt


t


2

2

=cos 2 π


Section 11.7







  1. (i)I(t) 1 = 1 Ae


−t 2 RC

(ii)


Chapter 12


Section 12.2


4.y 1 = 1 ae


− 2 x

1 + 1 be


2 x 23

5.y 1 = 1 (a 1 + 1 bx)e


3 x

6.y 1 = 1 a 1 cos 12 x 1 + 1 b 1 sin 12 x


Section 12.3


7.y 1 = 1 ae


3 x

1 + 1 be


− 2 x





9.y 1 = 1 (a 1 + 1 bx)e


4 x

10.y 1 = 1 (a 1 + 1 bx)e


− 3 x 22

11.y 1 = 1 e


− 2 x

(A 1 cos 1 x 1 + 1 B 1 sin 1 x)






Section 12.4



  1. 14.x 1 = 1 e


−3(t−1)

(t 1 − 1 1)



  1. 16.x 1 = 1 (cos 1 t 1 − 1 sin 1 t)e


t

17.y 1 = 1 e


π− 2 x

(cos 12 x 1 − 1 e


π 22

1 sin 12 x)


18.y 1 = 1 −sin 13 x 19.y 1 = 1 xe


4(1−x)

20.y 1 = 12 e


− 2 x

21.θ


n

(t) 1 = 1 Ae


int 2

τ

1 + 11 Be


−int 2

τ

,n 1 = 10 ,± 1 ,± 2 ,=


Section 12.5



  1. (i)


(ii)


23.x(t) 1 = 1 (U


0

2 ω)sin 1 ωt


Section 12.6



  1. (i)x 2 l 1 = 10 , 124 , 122 , 324 , 1


(ii)x 2 l 1 = 10 , 125 , 225 , 325 , 425 , 1



  1. (i) ifnodd,


ifneven
ψ

n

x


l


nx


l


()= sin


2 π


ψ


n

x


l


nx


l


()= cos


2 π


A=,= 24 δ π


Aab=+,= ba


22 − 1

δ tan ( )


xt=


1


3


sin 3


xe e


tt

=+



2


3


1


3


2

y e ae be


x

ix
ix

=+


−− 32

11 2

11 2

()


yae be e


xxx

=+



()


25 25 2

×+








cosωωtRC tsinω


It ce


EC


RC


tRC

()


()


=+






− 0

2

1


ω


ω


+−




RtL tsinωω ωcos


It


E


RL


Le


Rt L

()=









0 −

222

ω


ω

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