Advanced Engineering Mathematics Solutions (2-4~2-7&2-10)

8/13/2019 Advanced Engineering Mathematics Solutions (2-4~2-7&2-10)

1/26

Solution 2-4~2-7and 2-10

Solution 2-4 No. 5

No. 6


2/26

No. 7

No.14

No.17


3/26

Solution2-5

No. 1

Double root 22

212

2122 11 aa mmambmam

The differential equation becomes 0'" 22 12 yaxy y x a (1) x x y a ln21

x

a xaa x x y2121

21 ln'

x x x x x aaaaa ln1ln 2121212121

xa xaaaa x x y

21

21

2123

21 ln1"

232121212123 ln aaaaaa x x x

2321212121 ln aaaaa x x 23

4

21 ln aa x xa

Put y, y and y into (1)

x x xaxx x xa x aaaaaa lnln1ln 2122 12121234212

x x xax x xa aaaaaa lnln1ln 2122 1212121421

2122 1214 12

lnlnln aaaa x x xaa xa

x x aaaaaa ln214

122

2

2

412

x x aaaaaa ln214

12222212

0


4/26

No. 40'4" y xy

Multiply the original equation by x

0'4"2

xy y x

Auxiliary equation: 030141 222 mmmmbmam

0332 mmmm

3,0 21 mm

3

2

0

1,1 x y x y

General sol.: 321 xC C x y

No. 7

05'4"22 y xy y x

Auxiliary equation: 05225422 222 mmmmcmbaam

im i 5.15.02

312

91

2

25211

x x y x x y B A ln5.1sin,ln5.1cos 5.05.0

General sol.: x B x A x ln5.1sinln5.1cos5.0


5/26

No. 9

036.02.022 y I xD D x

Auxiliary equation: 036.02.1)1(22

mmbmam

06.0 2m

6.021 mm double root

x x y x y ln, 6.026.0

1

General sol.: 6.0216.026.01 lnln x xC C x xC xC x y

No. 12

01',4.01 ,06'4"2 y y y xy y x


032652 mmmm

3 ,2 21 mm

32

21 , x y x y

General sol.: 3221 xC xC x y

221 32' xC xC x y As the initial values are 01',4.01 y y

4.021 C C 032 21 C C

And 8.0,2.1 21 C C

Particular sol.: 32 8.02.1 x x x y


6/26

No. 16

21',1 ,04322 y y y I xD D x

Auxiliary equation: 0444131222

mmmmbmam

02 2m

221 mm double root

x x y x y ln, 222

1

General sol.: 2212221 lnln x xC C x xC xC x y

212122122 ln22ln2ln2' C x x x xC x xC C xC x xC C x x y xC

Then the initial values 21',1 y y are inserted

1221 11ln1 C C C y

221ln121121' 2121 C C C C y And 4, 21 C C

Particular sol.: 2ln4 x x x y

No. 19

5.11',5.01 ,06'2"2 y y y xy y x


02362 mmmm 2,3 21 mm

22

31 , x y x y

General sol.: 22312211 xC xC yC yC x y

xC xC x y 241 23'

Then the initial values 5.11',5.01 y y are inserted


7/26

5.01 21 C C y 5.1231' 21 C C y

And 6.0,1.0 21 C C

Particular sol.: 23

6.01.0 x x x y


8/26

Solution2-6

No. 5

No. 7


9/26

No. 9


10/26

No.11


11/26

No.13


12/26

No.15


13/26

Solution 2-7 No. 5

xe y y y x

cos4'4"

The corresponding homogeneous equation 04'4" y y y

Auxiliary equation 0442 02 2

221 x x xe ye y 22

21 ,

x xh xeeC yC yC y

22

212211 C

In the nonhomogeneous equation xe xr x cos

We set x M x K e y x p sincos

x M x K e x M x K e y x x p cossinsincos'

x M K e x M K e x x sincos

x M K e x M K e x M K e x M K e y x x x x p cossinsincos"

x M K M K e x M K M K e x x sincos

x Ke x Me x x sin2cos2

Substitute p y p y p y ,'," into the nonhomogeneous equation

x M x K e x M K e x M K e x Ke x Me x x x x x sincos4sin4cos4sin2cos2

xe x cos

xe x M M K K e x K M K M e x x x cossin4442cos4442

xe x Ke x Me x x x cossin2cos2

21

;12 M M


14/26

0 K

xe y x p sin21

General sol. xe xeeC y y x y x x x

ph sinC 212

22

1

No. 7

xe y I D D x292 34


Auxiliary equation 0342 031

3 ,1 21 x x e ye y 321 ,

x xh eC eC yC yC y

3212211

In the nonhomogeneous equation xe xr x 29

Since xe y 1 is the same as xe , we modify p y as o

x K x K Cxe 1

1' K xeeC y x x p x x x x x p xeeC xeeeC y 2"


xe y I D D x292 34

xe K x K Cxe K xeeC xeeC xo x x x x x 2911 333442

xe K K x K xe xee xeeC xo x x x x x

29

11 3433442

xe K K x K Ce xo x

29

11 3432

Equalize each term on both sides.

C;1221 C


15/26

23

129

1 :3 K K

2 ;034 1 oo K K K

223

21 x xe y x p

General sol. 223

213

21 x xeeC eC y y x y x x x

ph

No. 11

00' ,30 ,84" 2 y y x y y

The corresponding homogeneous equation 04" y y

Auxiliary equation 042 i2

x y x y B A 2sin ,cos

x B x A By Ay y B Ah 2sin2cos

In the nonhomogeneous equation 28 x xr

Seto p

K x K x K y 1

2

2

122' K x K y p 22" K y p


284" x y y 2

12

22 84442 x K x K x K K o 2

212

2 84244 x K K x K x K o

Equalize the coefficients ahead terms on both sides.

2 ;84 22 K K

0 ;04 11 K K

1 ;042 2 oo K K K

12 2 x y p

General sol. 122sin2cos 2 x x B x A y y x y ph x x B x A x y 42cos22sin2'


16/26

Substitute the initial values, 00' ,30 y y into the above equations ofgeneral solution and its derivative.

311020sin0cos0 A B A y

02040cos20sin20' B B A y 0B ,2 A are obtained.

Particular solution 122cos2 2 x x x y

No. 14

10' ,10 ,2cos9'6" y y xe y y y x



3 21 (double root) x x xe ye y 323

1 ,

xC C e yC yC y xh 2132211

In the nonhomogeneous equation xe xr x 2cos

We set x M x K e y x p 2sin2cos

x M x K e x M x K e y x x p 2cos22sin22sin2cos'

x M K x M K e x 2s i n22c o s2

x M K x M K e x M K x M K e y x x p 2cos222sin222sin22cos2"

x M K M K x M K M K e x 2s i n4222c o s242 x M K x M K e x 2sin342cos43

Substitute p p y y ,y'," p into the nonhomogeneous equation

x M K x M K e x M K x M K e x x 2sin22cos262sin342cos43

xe x M x K e x x 2cos2sin2cos9


17/26

xe x M M K M K x K M K M K e x x 2cos2sin9612342cos912643

xe x K x M e x x 2cos2sin82cos8


81 ;18 M M

0 K

xe y x p 2sin81

General sol. xe xC C e y y x y x x ph 2sin81

213

xe xeeC xC C e x y x x x x 2cos2sin3' 41

813

2213

x xe xC C C e x x 2sin2cos33 81412213 Substitute the initial values, 10' ,10 y y into the above equations ofgeneral solution and its derivative.

1 ;10sin0 11081

10 C C eC e y

1330sin0cos30' 4121412181410210 C C C C eC C e y 1.253 21 C C 75.1C ,1 21 C are obtained.

Particular solution xe xe x y x x 2sin75.11 813

No. 17

5.00' ,5.00 ,25.24.04.0 25.02 y ye y I D D x

The corresponding homogeneous equation 04.0'4.0" y y y

Auxiliary equation 04.04.02

i6.00.24.00.20.2 2

ii 6.00.2 ,6.02.0 21

xe y xe y x B x

A 6.0sin ,6.0cos2.02.0

x B x Ae By Ay y x B Ah 6.0sin6.0cos2.0


18/26

In the nonhomogeneous equation xe xr 25.025.2

We set x p Ce y25.0

x p Ce y

25.025.0'

x p Ce y

25.00625.0"


25.24.025.04.00625.0 25.025.025.025.0 x x x x eCeCeCe 25.24.01.00625.0 25.025.025.025.0 x x x x eCeCeCe

25.25625.025.025.0 x x

eCe 4C ;25.25625.0

5625.025.25.05.0 x x eCe

x p e y

25.04

General sol. x x ph e x B x Ae y y x y 25.02.0 46.0sin6.0cos

x x x e x B x Ae x B x Ae x y 25.02.02.0 6.0cos6.06.0sin6.06.0sin6.0cos2.0'

x x e x B A x B Ae 25.02.0 6.0sin2.06.06.0cos6.02.0

Substitute the initial values, 5.00' ,5.00 y y into the above equations ofgeneral solution and its derivative.

5.3A ;5.0440sin0cos0 00 Ae B Ae y

5.016.02.00sin2.06.00cos6.02.00' 00 B Ae B A B Ae y

5.16.02.0 B A

B ,5.33

11 A

Particular solution x x e x xe x y 25.03112.0 46.0sin6.0cos5.3

No. 18


19/26

2.20' ,6.60 ,3sin37sin171022 y y x x y I D D

The corresponding homogeneous equation 010'2" y y

Auxiliary equation 01022

i311011

ii 31 ,31 21

xe y xe y x B x

A 3sin ,3cos

x B x Ae y xh 3sin3cos

In the nonhomogeneous equation x x xr 3sin37sin17

We set x M x K x M x K y p 3sin3cossincos 2211 x M x K x M x K y p 3cos33sin3cossin' 2211

x M x K x M x K y p 3sin93cos9sincos" 2211


x M x K x M x K x M x K x M x K 3cos33sin3cossin23sin93cos9sincos 22112211

x x x M x K x M x K 3sin37sin173sin3cossincos10 2211 x K M K x M M K x K M K 3cos1069sin102cos102 222111111

x x x M K M 3sin37sin173sin1069 222 x x x M K x M K x M K x M K 3sin37sin173sin63cos6sin92cos29 22221111


029 11 M K

1792 11

M K 06 22 M K

376 22 M K

1 ,6,8.1 ,4.0 2211 M K M K

x x x x y p 3sin3cos6sin8.1cos4.0

General sol.

x x x x x B x Ae y y x y x ph 3sin3cos6sin8.1cos4.03sin3cos


20/26

x x x x x B x Ae x B x Ae x y x x 3cos33sin18cos8.1sin4.03cos33sin33sin3cos'

x x x x x B A x B Ae x 3cos33sin18cos8.1sin4.03sin33cos3 Substitute the initial values, 2.20' ,6.60 y y into the above equations ofgeneral solution and its derivative.

0sin0cos60sin8.10cos4.00sin0cos0 0 B Ae y

6.66.564.0 A A

0cos30sin180cos8.10sin4.00sin30cos30' 0 B A B Ae y

0.13 ;2.22.1338.13 B A B A B A And 0,1 B A

Particular solution x x x x xe x y x

3sin3cos6sin8.1cos4.03cos


21/26

Solution 2-10 No. 5

x x y y sincos"

Find the general solution of homogeneous equation 0" y y

Auxiliary equation ii 212 , ;01

x y x y B A sin ,cos x B x A By Ay y B Ah sincos

1sincoscossin

sincos

' '

, 22 x x

x x

x x

y y

y y y yW

B A

B A B A

x x xr sincos

dx x x xdxdxu x x x

B y A yW xr B y 2

1sincossin

,sincossin

42sin242cos2 2cos122sin x x x x x dx

dx x x xdxdxv x x x

B y A yW xr A y sincoscos 2

1sincoscos

,

42cos

42sin

222sin

22cos1 x x x x x dx

x xvyuy y x x x x x x B A p sincos 42cos

42sin

242sin

242cos

4sin2cos

4sin2sin

2sin

4cos2sin

2cos

4cos2cos x x x x x x x x x x x x

4sin2coscos2sin

2sin

2cos

4sin2sincos2cos x x x x x x x x x x x x

4

2sin2

sin2

cos42cos x x x x x x x x

4sin

4cos

2sin

2cos x x x x x x

General sol. 4sin

4cos

2sin

2cossincos x x x x x x ph x B x A y y x y

x y may be expressed as 2sin2cos4141 sincos x x x x x B x A

2sin

2cos

21 sincos x x x x xc xc


22/26

No. 7

xe x y I D D 22 62

Find the general solution of homogeneous equation 0'2" y y y


121 (double root) x x xe ye y 21 , xh e xcc yc yc y 212211

x x x x x x x

x xee xe xe

xeee

xee

y y

y y y yW 2222

21

2121 ' '

,

xe x xr 26

x x x x x xe

xe x x xe

y yW xr y

e xee xe xdxe xdxdxu 2492

2922

292323

2

26

2,12 36

x x x x xe

xe x xe

y yW xr y

e xee xdxe xdxdxv 22322222

2

26

2,11 336

x x x x x x x x x p xee xee xee xee xe xvyuy y 22322224922922292321 333

xe x x x x x x 2323

49

292

293 333

xe x x 492

23 3

General sol. x x ph e x xe xcc y y x y

492

23

21 3


23/26

No. 9

xe x y I D D 232 352


Auxiliary equation 0122 0112 22 121 (double root)

x x xe ye y 21 , xh e xC C yC yC y 212211

In the nonhomogeneous equation, xe x xr 2335

x x x x x x x

x xe xe xee

xe ee

xe e

y y

y y y yW 2222

21

2121

' '

,

dx xdxdxu xe

xe x x xe

y yW xr y

252

2335

2,12 35

2710 x

25232

2335

2,

1

1 1435 xdx xdxdxv x

e

xe x xe

y yW xr y

x x p xe xe xvyuy y

252721 1410

xe x 274

General sol. x x ph e xe xcc y y x y 2721 4

No. 10


24/26

xe y I D D x 32 sec422


Auxiliary equation 0222 i1211 2

ii 1 ,1 21

xe y xe y x B x

A sin ,cos x B x Ae By Ay y x B Ah sincos

In the nonhomogeneous equation, xe xr x 3sec4

cosxsinxsinxcosx

sinx cosx

' '

,

x x x x

x x

B A

B A B A

eeee

ee

y y

y y y yW

x x x x x e xe x xe xe x xe 2222222 sinsincoscossincos

dx x xdxdxu xe

x xe x xe

B y A yW xr B y

32

3sec4sin

,-

secsin4

xdx x x 22 tan2sectan4 or x2sec2

dx x xdxdxv xe

x xe x xe

B y A yW xr A y

32

3sec4cos

, seccos4

xdx x tan4sec4 2

x xe x xevyuy y x x B A p sintan4costan22

x xe x xe x xe x x x sintan2sintan4sintan2

Or x xe x xevyuy y x x

B A p sintan4cossec22

x x

x x x x x e x x xe x xe xe

cos

2sincos

2 4tansin4sec2sintan4sec2

x x x x x x ee cos2coscos2sin21 22

General sol. x xe x B x Ae y y x y x x ph sintan2sincos

Or x x x x e x B x Ae x y cos 2cos2sincos


25/26

No. 11

422 2164 x y I xD D x

Find the general solution of homogeneous equation 06'4"2 y xy y x

(Euler-Cauchy equation)

Auxiliary equation 06142 mm 0652 mm

032 mm 3 ,2 21 mm 3

22

1 , x y x y 3

22

12211 xc xc yc yc yh

In the nonhomogeneous equation, 62 421 21 x xr x x

4442

32

21

2121 23

3 2

' '

, x x x

x x

x x y y

y y y yW

6276

6217

4

6213

2,12 21 x xdx xdxdxu

x

x x y yW xr y

784

6212

2,11 321 xdx xdxdxv

x

x x y yW xr y

42144

273726

27

21 33 x x x x x x xvyuy y p

General sol. 4213

22

1 x xc xc y y x y ph

No. 12

x y I D sinh12

Find the general solution of homogeneous equation 0" y y

Auxiliary equation 012 1 ,1 21

x x e ye y 21 , x x

h ecec yc yc y 212211

In the nonhomogeneous equation, xe xe x xr 2

sinh1

211 ' '

,

21

2121

x x

x x

ee

ee

y y

y y y yW


26/26

dxdxdxdxu xe

xe xe xe

xe xe xe

xe y yW xr y

12

2

2

22,1

2

Set x

eu

x

eu22

dx x

edu

udu

du x

edx

dudududx uuuu

u

uu

u xe

xe1

11

121

1112

2

12

2

1ln1ln1ln1ln 21212121 x x eeuu

1ln1ln 2121122

x x

xe

xe eedxu

dxdxdxdxv xe xe xe

xe xe xe

xe y yW xr y

1212

22,11

Set xeu xeu 22 dx xedu u

dudu xedx

dudududx uuuuuuuu xe 1

121

121

111

121

121

1ln1ln11lnln1ln1lnln1ln 212121212121 x x x x x eeeeeuuu

1ln1ln1 2121 x x eev

x x x x x x p eeeeeevyuy y 1ln1ln11ln1ln 2121212121 General sol.

x x x x x x x x

ph eeeeeeecec y y x y

1ln1ln11ln1ln 2121212121

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Advanced Engineering Mathematics Solutions (2-4~2-7&2-10)

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