Advanced Engineering Mathematics Solutions (6-1~6-7&Ch6review)
Transcript of Advanced Engineering Mathematics Solutions (6-1~6-7&Ch6review)
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Solution 6-1~6-7 and Ch6 review
Solution 6-1 No. 6
t et f t 4sinh {f(t)}
2
53
2
444sinh
t et et et et t et e
534538
215335
215131212
53
s s s s s s s s
s s
t et e
No. 11
bt
bt t t f
,0
0 ,
2
0002
020
s
e se
s
bse s
bsbeb
t s
st e s
st tebt
st dt tet f
21
2 s s
bse s
bsbe
No. 22
Since 0 101
2
1
21 t t t t t dt edt t e
Set st then sd
s dt t ,
s s sd
st t
st t
d eedt e
1011
01
01
Remark: 0
1 lim,t t
t f t f kt Met f cannot be met.
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No. 26
,25215
s
s s F find t f
1
s F 1
6.254.2
56.2
s s 1
4.251
s 1 t t
s ee55
51
4.26.2
No. 32
,1
b sa s s F find t f
If ab 1 ab
s F 1 1 b sa s
11 bt at ab ee 1
If ab 1 s F 1
at
a ste
21
No. 43
,10422
76
s s
s s F find t f
1 s F 1 8212
76
s
s 1
421
273
s
s 1
2221
2113
s
s
1
t et e t t
s s
s2sin2cos3
4
1
2221
241
2221
13
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Solution 6-2 No. 4
00' ,00 ,109"
y ye y y t
Set sY t y
,1
109" s
y y 1
109)0(')0()(2 s
sY y sy sY s
Insert 00',00 y y
1
1092
s sY s
232
31
23211
921
9211
921
11
92110
s s
s s s s
s s s
s s s s
sY
t y -1 t t t e sY 3sin313cos
No. 12
52' ,32 ,03'2"
y y y y y Set yt y y y y yt ~ ,0'~2',0~20,2t,2 t,2
Let sY t y , sY y ~~~
0~~3)0(~)~(~~2)0('~)0(~)~(~~ 2 sY y sY s y y s sY s Substitute 50'~ ,30~ y y into the above equation
0~~33)~(~~25~3)~(~~ 2 sY sY s s sY s 11~3)~(~3~2~ 2 s sY s s
3~21
1~27
3~22~11~3)~(
~ s s s s
s sY
)(~ y 1 32127)~(~ ee sY
Restore 2t
23212
27 t t eet y
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No. 25
422
1
s s s F
And 1 t s 2
2
4221 sin
Then 1
t t d t t s s 22
422
402
20
22
2
4221 cos11coscossin
No. 29
a s sas s
s F 2
123
1
And 1 at a s e 1
Then 1
aat et
a
aet aa s s d e
1
00
1
1
t
a
aet aa
ae
a s sd
00
11
21
2111
a
at eat
a
at ea
t
a sa
s
a sa
a sa
s
a sa
a s sas s 2
1
2
121
21
2
12
1
21
231
-122
123
1
a
at eat
aas s
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Solution 6-3 No. 4
t t t f 0 ,2cos t ut t tut tut tut ut ut t f 2cos2cos2cos2cos2cos
s s
s s se
s s
s se
s s et f
s s 144422 2222222
No. 8
21 2 t t t f
212 t ut ut t f
21 22 t ut t ut
2424211121 22 t ut t t ut t
2422422111211 22 t ut ut t ut t ut ut t ut
s
e se
se
se
se
se s s s s s st f
2
2
2
3
2
234222
No.16
4/2 23 seet f s s
s s s s s s s s se se
eeee 32
2132
21
22
32
3131 3212213212
21 t uet uet uet uet f t t t t
31 3232211212
21 t ueet uee t t t t
332sinh112sinh t ut t ut
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No. 28
R = 1 k , L = 1 H, 0v if t 0 and V t sin40 if t )(sin40)()(')( t tut vt Lit Ri
t ut t tut vii )sin(40)(sin40)('1000
0)0(:0100012
40 ii s sI s I s
se
se s s
s s s
e s I s
e s I s s s
10001
110
11040
1100040;
1401000
2
3
622
t uet t t i t 1000
1610
40 sin1000cos
t u
t et t t i 1000sin1000cos1610
40
0 t ii
t et t t iii 1000cossin1000
11040 6
No. 32
0,10,10 2 v F C R if 4t and V e t 1014 36 if 4t
)(t vt Ri C t Q
410144101410010 43126036 t ueet uet vd ii t t t
3
4126101410010 s
see s
s I s I
34125101410
3
4125101410 ;1 s
see
s s
s
see
s s I s I
s
s s s s
s see ee s I 41253
610
20310
41251014 10
46201043410125
t ueeet i t t
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4 ,361028205104364102012510
40 ,0 t t et eet et eet iii
t t ii
No. 35
oterwise ,0
3 , cos9900 F10C ,1 2
t V t t v H L
3cos9900cos99003cos9900 t tut tut ut ut t v 33cos9900cos9900 t ut t ut
t vC
t qt Li )(' t vd it i t 0100)('
d it i t
0100' t v
12
39900
1299001000
s
s se
s
s se s
s I i s sI
s I t i 00 i
12
39900
129900
1002
s
s se
s
s se s
s I s
100212
329900
100212
29900)( s s
se s
s s
se s s I
s
s s
s
s see 3
100210000
12100
100210000
12100
t i 1 s I 3310sin10003sin10010sin1000sin100 t ut t t ut t
t i 0 0t i 3 t ii t t t t t i 10sin1000sin10010sin1000sin100 3 t iii
310sin10003sin10010sin1000sin100 t t t t t i 010sin1000sin10010sin1000sin100 t t t t
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oterwise ,0
3 , cos9900 F10C ,1 2
t V t t v H L
1 )(' t vC
t qt Li t
;2
2t i L
dt t dq
dt t dv
C dt
t dq
dt
t id
Or 2 100
2
2
dt t dv
dt
t id t i
3cos9900 t ut ut t v
33sin9900sin99003sin9900 t ut t ut t ut ut dt
t dv
Transform both sides of (2)
12
39900
1299001000'02
s
se
s
se s I i si s I s
t v t 3 , 9900,0
vv
03,99003
vv
dt
t dv Laplace transformation
t sevv sedt
dt dv st e ;9900
3 ;3990033333
t sevv sedt
dt dv st e
se se s
se
s
se s I i si s I s 399009900
12
39900
1299001000'02
From (1) At t 0 0000 v L C
qdt
di
Since 00' 00
iq and 00 i
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se s
se s
s I s 3112
1990012
1199001002
se s s
se s s
s I 3112
11002
990012
111002
9900
s
s
s
s
s
s
s
see 3
12
2
10029900
12
2
10029900
s s s
s
s see s I 3
100210000
12100
100210000
12100
t i 1 s I
3310sin10003sin10010sin1000sin100 t ut t t ut t
t i 0 0t i 3 t ii t t t t t i 10sin1000sin10010sin1000sin100 3 t iii
310sin10003sin10010sin1000sin100 t t t t t i 010sin1000sin10010sin1000sin100 t t t t
No. 39
2 t ,0
20 , 1000 F5.0C ,1,2R
t V t v H L
t vC
t qt Ri
dt t di
L Or 1 022 t vt d i I
dt t di
21000 t ut ut v Laplace transform on both side of 1
s
se s s
s I s I i s sI
2110002
20 s I t i 00 i
s
se s s
s I s s 211000222
222
2
222
11 0 0 0
s s
se
s s
s I
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121
2
121
11000 s
se
s s I
t i 1
22sin2
1000sin1000
t ut t
et t
e s I
20 t i t t et i s i n1000
t ii 2s i n21 0 0 0s i n1 0 0 0 t t et t et i
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Solution 6-4
No. 3 0(0)y' ,2y(0) ,29" t y y
Make Laplace transformation on both sides of the equation
0(0)y' ,2y(0) ;90'0 22 s
e sY y sy sY s
are then inserted
se s sY s 22 29
92
2
922
s
se
s
s sY
t y 1
23cos313cos2223sin313cos2 t tut t ut t sY
92
2
922
s
se
s
s sY
t y 1 d t t sY t 0 23
1 3sin3cos2
2
0 t i t t y 3c o s2
2 t i
t t t t t y 3cos3cos23sin3cos231
231
No. 6
se sY y s sY y sy sY s 5040'02
3(0)y' ,0y(0) are then inserted
se sY s sY sY s 5432
se sY s s 3542
1221223
5425423
s
se
s s s
se
s s sY
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t y 1 11sinsin3 122 t ut et e sY t t
1221223
s
se
s sY
t y 1 d t et e sY t t t 0
2212 1sinsin3
10 t i t et y t s i n3 2
1t ii 1s i ns i n3 122 t et et y t t
No. 9
1000),2(212'2" 2 ) , y'( ) y( t eet u y y y t
s s
see s
ee sY y s sY y sy sY s 221
22
112 2020'0
1000 ) , y'( ) y(
s s
see
s
ee sY s sY sY s 22
1
22
1
12 221
s s see
s ee sY s s 22
1
22
112 122
222
22
2221
22
2221
1
2221
s s
see
s s s
see
s s s s s sY
222
2222
22253
51
151
22253
51
151
2221
s s
see s
s s
s
s s s
s
s s see
121
2222
121
53
51
151
121
53
51
151
121
1
s
see s
s
s
s s
s
s see
121
2222
121
521
51
151
121
521
51
151
121
1
s
see s
s
s
s s
s
s see
t y 1 sY
2sincossin22
51
52
51
51
t ueet et eet e t t t t t
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)sincos3()sincos3()sincos2(sincos2( 333101222
51 et t eeet t ee t t t t
)(3103
101)(2
52
51
52)2
1(3)21(2
)sincos3(sincos t t t t et t et t ee
)(3103)(2
52
101
101)2
1(3)21(2
sincos)( t t t t eet t eet y
No. 11 1000 ,2123" ) , y'( ) y( t t u y y y
s s
se e sY y s sY y sy sY s 22 2030'0
1(0)y' ,0y(0) are then inserted
s s
se e sY s sY sY s 22 231
s s se e sY s s 22 123
21232 s s s s
212
21211
s s
se s s s
se s s
sY
s
s s s
s s s s s ee2
21
11
221
112
1
21
11
t y 1 sY 21 22212
211
212 t ueet ueeee t t t t t t
No. 125(0)y' ,2y(0)),(100255'2" t t y y y
s s
e sY y s sY y sy sY s 1005020'02
252 5(0)y' ,2y(0)
s s
e sY s sY s sY s 100542522
252
s s
e s sY s s 10012522
252
522
100
522225
52212
s s
se
s s s s s
s sY
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2221100
252
522100
252
522100
52212
252
52212
s
se
s s s s
se
s
s
s s
se
s s
s
s
s
s s
s
t y 1
t ut et sY t
2sin5052
t tuet t 2sin5052
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Solution 6-5
No. 4
d t t d t t t t t 02
10 coscoscoscoscoscos
d t t t 02
1 cos2cos
t t t 02
121 cos2sin
0sincossin2
1
2
1
2
1 t t t t
t t t t sincossin
21
21
21
t t t
cossin121
No. 6
t ba baebt et babt t t babt at d eed eeee 000
ba
bt eat eba
t baebt e 1
No. 8
t d t yt y t 24 0
22
04 s
t d t yt y
22
214
s s sY sY
22241 s s sY 222 4
2
s s s sY
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42
2
s sY
t t y 2sin No. 9
20 d yt y t
s
t d yt y 20
s s
sY sY 21
s s
sY 211 s s
s sY 21
1
2 s
sY
t et y 2
No.13
t t t ted e yet y
02
211
02 s
t t d e yet y
211
02 s
t t t t d e yeet y
211
112
s s sY sY
211
121
s s sY
211
11
s s s sY
11
11
21
111
s s s s sY
t eet y t t sinh21
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No.25
2
362
18
s
s s F -1 t s
6sin3362
18 -1 t s
s 6cos362
-1
t
s
s d t 02362
18 6cos6sin3
t d t t 023 666sin666sin
t d t t 023 612sin6sin
t t t 012123 612cos6sin
t t t t 6cos6cos6sin121
121
23
t t 6sin23
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Solution 6-6
No. 2
t t t f 4sinh3
2
444sinh
t et et
162
4
1628
21
41
41
214sinh
s s s st
2162
24
2162
241624
334sinh3
s
s
s
s
ds
sd
t t ds
sdF t tf
No. 6
t t t f 3sin2
9
23
23
233sin
s s
t
2
92
62323
3sin
s
sds
sd
t t
492
922262
926292
6
3sin2
s
s s s s
ds
s
sd
t t
492
2216424486210846
s
s s s s
392
3218
492
923218
492
4862108418
s
s
s
s s
s
s s
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No. 14
2162
s
s s F ds
sd
s
s
1621
2
1
2162
-1 t
s4sin
41
1621 -1 t t t t
s
s 4sin814sin
41
21
2162
Or
-1
d t t t d t t
s
s
0
444sin444sin81cos0 4sin4
12
162
t
t t d t t t 0
48cos814sin
81
0 48sin4sin81
t t t t t t 4sin814cos814cos814sin81
No.15
2)42( s
s s F
ds s
d
s
s 421
21
2)42(
and2
12
141
421
s s s
-1 t t
see 22
41
421
-1 t t t t ds s
d
eet eet 228122
41
2142
1
21
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Or42
1
422)42( s s s
s
s
21
21
21
2242 s s s s s
s
s
-1 t t s
s ee 2221
42
And -1 t t s
ee 2241
421
-1 d eeeet t t s
s
0)(2)(2
4122
21
242
d eeeet t t t t 0
24242281
t t t t t eeee0
2424142
412
81
t t t t t t teeeeete 22412
412
412
412
81
t t tete 2281
No.17
1
ln s
s s F 1lnln1
ln s s s
s
Set -1 )(
1ln t f
s s
then,
-1)(
1ln
t tf ds s
sd
;
-1
ds s
sd 1
ln
-1 )(111
1 t tf t e s s
t
et f t 1
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No.20
b sa s s F ln b sa s
b sa s lnlnln
Set -1 )(ln t f b sa s then, -1 )(
lnt tf ds
d b s
a s
;
-1
ds
d b sa sln
-1 )(11 t tf at ebt e
a sb s
)(1)( at ebt et
t f
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Solution 6-7
No. 2
00 ,10 ,cos2 ,0 21'212'1 y yt y y y y
Writing ,11 sY y sY y 22
10 211 sY y s sY
12
2221 0
s
s y s sY sY Replace 00,10 21 y y
121 sY s sY 1
122
21 s
s s sY sY 2
1 s 2 12
21
2 1 s
s s sY s
2
12
2
121
s
s
s
s sY
Then from 1 21222
121
21222
122
12 11
s
s s s
s s
s s sY sY
Using (3) of sec.6.6 -1 t t
s
s
sin2222
t t t t t t y t sincossincos22
1 In this case 1
And using (4) of sec.6.6 -1 t t t s
s
cossin21
222
2
t t t t t t t t t t t y coscossinsincossinsin22
2 In this case 1
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No. 4
30 ,00 ,4sin93 ,4cos84 211'22'1 y yt y yt y y
Writing
,11 sY y
sY y 22
162
8211 40
s
s sY y s sY
162
36122 30
s sY y s sY Replace 30,00 21 y y
162
821 4
s
s sY s sY 1
16
236
21 33 s
s sY sY 2
1 s 2 4 162
144
162
281
2 1212 s s
s sY s
162
144192212281
2 12 s
s s sY s 162
1224
1624824
12 12
s
s
s
s sY s
162
41
s sY
1
3 2
s
162
2123
162
3336
16236483324
16236
16224
22 312
s
s s
s
s s
s
s s s s
s
s
s
s s sY s
162
32
s
s sY
t t y 4sin1 t t y 4cos32
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No. 6
30 ,10 ,5 ,5 2121'221'1 y y y y y y y y
Writing
,11 sY y
sY y 22 sY sY y s sY 2111 50 sY sY y s sY 2122 50 Replace 30,10 21 y y 15 211 sY sY s sY 15 21 sY sY s 1 35 221 sY s sY sY 35 21 sY s sY 2
1 ( s 5) 2 3515 12 s sY s
6
1
4
2
24102
8
1
s s s s
s sY
1 +2 ( s 5)
163153151 22 s s sY s
6
14
2
24102163
2
s s s s
s sY
t t eet y 641 2
t t eet y 642 2
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No. 8
30 ,40 ,4 ,32 2121'221'1 y y y y y y y y
Writing
,11 sY y
sY y 22 sY sY y s sY 2111 320 sY sY y s sY 2122 40 Replace 30,40 21 y y 432 211 sY sY s sY 432 21 sY sY s 1
34 221 sY s sY sY 314 21 sY s sY 2 1 ( s 1) 2 3 9441212 1 s sY s s
5
12
3
1032134
1
s s s s
s sY
1 4+2 ( s 2)
22363161212 2 s s sY s s
5
12
4
1032223
2
s s s s
s sY
t t eet y 521 3
t t eet y 522 4
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No. 12
00 ,20 ,00 ,10 ,2 ,54 '22'1121"221"1 y y y y y y y y y y
Writing
,11 sY y
sY y 22
sY sY y sy sY s 21'1112 5400
sY sY y sy sY s 21'2222 200 Replace
00 ,20 ,00 ,10 '22'11 y y y y
s sY sY s 21
2 54 1
s sY s sY 22 221 2
1 22 s 2 5 s s s sY s s 102542 2122
s s sY s s 832 3124 s s sY s s 831 3122
3245
189
189
3245
1249
321283
1
s
s
s s s
s
s
s
s s
s s
sY
1 2 42 s 42245 2222 s s s sY s s s s sY s s 7232 3224
s s sY s s 7231 3222
32
41
189
189
3241
1249
3212732
2
s
s s s s
s
s
s
s s
s s sY
t t t eet y t t 3coscosh3cos45
49
45
89
89
1
t t t eet y t t 3coscosh3cos41
49
41
89
89
2
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No.13
60 ,80 ,60 ,00 ,10sin101 ,10sin101 '22'111"22"1 y y y yt y yt y y
Writing
,11 sY y
sY y 22
1002
10102
'111
2 00 s
sY y sy sY s
1002
10101
'222
2 00 s
sY y sy sY s Replace
60 ,80 ,60 ,00 '22'11 y y y y
1002
101021
2 6 s
sY sY s 1
1002
10102
21 68
s s sY s sY 2
1 2 s 2 1002
10102
10021010
14 6861
s s s s sY s
s s sY s s
s 811 2100241026
14
12
4
124
100210
124
12128
12100241026
1 s
s
s
s
s s s s
s
s s
s sY
124
100210
12
12
12
12
124
100210
124
124
s
s
s s s s s s
s
s s
s
s
124
100210
14
s
s
s s
1 2 2 s 210021010
10021010
24 6861 s s sY s
s s
s s sY s s
8161 21002
10102
4
12
4
100210
14
12128
12100241026
2
s
s
s s s s
s
s s
s sY
t t et y t cos410sin41
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t t et y t cos410sin42
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Solution Ch6 review No.13
No.21
No.25
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No.38
No.39
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No.44