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2 Second equation Homogeneous equation Wisor

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1- First order differential equation 2 ! Second order differential equation linear - Homogeneous equation bums been Wisor ay " t by ' t Cy = 0 In a ,b , C bedrock soto Gains Games Iou axe + b X t C a O
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Page 1: 2 Second equation Homogeneous equation Wisor

1- First order differential equation2! Second order differential equation

linear - Homogeneous equationbums been Wisoray"t by

't Cy = 0

In a ,b ,C bedrock soto

Gains Games Iou

axe + b X t C a O

Page 2: 2 Second equation Homogeneous equation Wisor

axe + b x t C ' O

x = - b ± THAT-

za

r.

b'- Aac > o X - X, ,X,6860460 wood,

7

Y = C,@

"'''

+ qe.AEniiomnoinunj

a-2.

b! Aac =o Xl XX

y . C,

×

tqxeI =

3.

b'- da CLO parts i,

i - Fi

y , er" [ c. cos @ x > tqsincsxl]I ←

Page 3: 2 Second equation Homogeneous equation Wisor

IN.

unnioiioiw

ay"

+ by'

+ Cy -o

ycxo ) = Yo

y'

exo ) '

- Y,

adam b✓oundn C,bir : Cz

Page 4: 2 Second equation Homogeneous equation Wisor

farms 6dm dis ion Whoo,

'

Bardo : I noted , Swain, rio

a ex) y"t b Cx ) y

'

t CCA y = O--

.

- --

aunt Kero wow u Ti Yean En w- laputa fore

Ya = 4) I e-I bad×d×(y

,y'

own burner riots no, Gums

y s C,Y

,

t Ce Yz

Page 5: 2 Second equation Homogeneous equation Wisor

Gamo bfsrfwhsiionw.ws!acx ) y

' 't bcxiy 't can y a rcx)

r Cx ) fo

- g : ironed bingo Endo : Ind'

eine,

Undetermined Coefficients

- me 66 Weeks nosoio HIS iron carny

Variation of parameters

Page 6: 2 Second equation Homogeneous equation Wisor

-O:

'

.

Page 7: 2 Second equation Homogeneous equation Wisor

-0

-

-

Page 8: 2 Second equation Homogeneous equation Wisor

2h p py y y=

-- -

Yep. -Yp= Cil . '

-Citi. Yu

Page 9: 2 Second equation Homogeneous equation Wisor

h py y y= +run

{no!organized booth-ericson :

' croissant'scrosswind lonartwo,

r

voofsrmsTIionntw.es"homogeneous Yparticular

Page 10: 2 Second equation Homogeneous equation Wisor

mm

-run

Page 11: 2 Second equation Homogeneous equation Wisor

'' 2 ' xy y y e + =จงหาผลเฉลยทัวไปของสมการ

เมือพบว่าผลเฉลยหนึงของสมการคือ2

2

xx ey =Games conNissin inevo Fooy"- 2 y

't y =o

farm 8%0×2- 2x th = 0

( X - a) 2=0X = I

Yh = C,e"

t q x ex

Page 12: 2 Second equation Homogeneous equation Wisor

y"- ay 't y = ex

y= x'I2

y'-- da.

'

in. Ed + EDI= XII + ×e×

y"

.-adf.ci#+xe5=xII+xeIxdaEteidff=xzI+xetxe'+ ex-

= xI§+zxe×te×

Page 13: 2 Second equation Homogeneous equation Wisor

yll-zytty.ee#y = x'I

2

y'

: XIE x x etY'!Eef + exet t et

fait exEt ex - z ( Eee"

+ xex) t'II

=EEte - e¥+E set

woo burner riot duo, Samo7I ion Wu siY = Th t Yp = c. e'

'

t CexE t XII-

Page 14: 2 Second equation Homogeneous equation Wisor

'' 2 ' 2cos( )y y y x + =

เมือพบว่าผลเฉลยหนึงของสมการคือ siny x= oy"- Ey 't -1=0 cossrmocomovoininieo

Ya :c,e×tqxe×Too)

uol.aoouriohdoesoswmolsiionnkso.rs/=YntYp=qextqxe- sin x

Page 15: 2 Second equation Homogeneous equation Wisor

mmmm

rumrunner

Es

Page 16: 2 Second equation Homogeneous equation Wisor
Page 17: 2 Second equation Homogeneous equation Wisor

woos-ooo woo Gamo conversing favs : Infedna's avoid

y ace the"

xx

y = c. et't Czxe"

= Cqtqxle

y : er×[ c. cos Csx) tqsincs xDe" ? C

, tcaxtcgxt . - ten, ,x"

,cos Csk)

,sin Csx)

--- -

Page 18: 2 Second equation Homogeneous equation Wisor
Page 19: 2 Second equation Homogeneous equation Wisor

mmmm

m

E A

Page 20: 2 Second equation Homogeneous equation Wisor
Page 21: 2 Second equation Homogeneous equation Wisor
Page 22: 2 Second equation Homogeneous equation Wisor

ay"+ by

'+ C y a- rcx)

A,

b,C lov a'insane

Donson Gomes tonnes sin in'

voto,

ay"

t by'

toy so

Yh = C.Yet Czyz

h 2

- aotanxt - - tant ly Btwn Yp orsoojnnf,- COS

- sin Qiu Kay,

Iu- n x' Yp .

-

Page 23: 2 Second equation Homogeneous equation Wisor
Page 24: 2 Second equation Homogeneous equation Wisor
Page 25: 2 Second equation Homogeneous equation Wisor

Go mo e on Wrs si ni hier o fo,> y

''t 2 y'- z y = o

i::::*: ::X = 1

,

- 3

- 3 X

Y =

C.e"t Cz eh

Page 26: 2 Second equation Homogeneous equation Wisor

Y"t Ey'- 3 Y = 9 cos Csx)

Yh " C.e''+ c. e'

×

=

Gandhi(Yp . A cosczex) t Bsinezxyy

Page 27: 2 Second equation Homogeneous equation Wisor

Y "t2y ' - 3Y=5E3Yi ai't

"

g , yj.AE" wi'#

bdnienyp-h.IE?j

Page 28: 2 Second equation Homogeneous equation Wisor

Y"t Ey'- 3 Y = x' cos CITI

= - - i - - - -

Yh " C.e'' tcze"

tu yp=CA-113×+4%05 (Tx) e- ( Dt Ext Ext)Sin CITI )- = -

T

Page 29: 2 Second equation Homogeneous equation Wisor

ya . c.ex.ge. .!"+ 'Y'- BY = e'

'

six -eEos±

E-

Efa, aid cos × c- cbatbzx) sin x)taint

=

Page 30: 2 Second equation Homogeneous equation Wisor

Y"t Ey'- 3 Y = x'ex t3xe×

- 3 X X

Yh " C.e''

the = ( 5+3×7 e←

q 9 x

Wynn e9iy÷I¥÷.ee?tBxe'tcx2ecooi,×

belote's Yp =XCAtBxtcx3e×=CAxtBx4c×3e

= AxItBx2e×tc×3e×✓

- = I

Page 31: 2 Second equation Homogeneous equation Wisor

Y"t 2y'- 3 Y = tan x

-=

Tsi Grano fo : idea IS sinew four : Ind'

% mo un woo laPeta

Page 32: 2 Second equation Homogeneous equation Wisor

'' ( )y r x=เมือ r(x) มีค่าเป็น

1. 1

2. 3x2

3. x2ex

mm

Page 33: 2 Second equation Homogeneous equation Wisor

62 MS 1On Wu o,

'

n' Inigo for

y"= o

Ba mo toaXZ = O

X --O

ero inner riot N no, Gum or I onnow sinned fo,ya .

. c,et t czxeo

y'

- C,t Czx

h

Page 34: 2 Second equation Homogeneous equation Wisor

y"= I

y'

- C,t Czx f

" rssionfuyp.AT @isb^yp=A± coots

burials Yp = A XY

Page 35: 2 Second equation Homogeneous equation Wisor

y''= 3×2

y'

- C,t Czx f

"

tax' voi,

w¥÷¥¥:*. wi,

i. daters yp=x(AxtBx2tCx3)= A- x' + Bx't Cx4✓

Page 36: 2 Second equation Homogeneous equation Wisor

y"= x' ex

y'

- C,t Czx

" Tei i. "E.exterior

Page 37: 2 Second equation Homogeneous equation Wisor

'' 2 ' ( )y y y r x + =เมือ r(x) มีค่าเป็น

1. ex

2. xex

3. exsin x

Page 38: 2 Second equation Homogeneous equation Wisor

Bar M 8 Con Whoo,

' ni b ni ee Foo

y"- Ey't y = 0

Gums bowth - 2 X t I = 0

( X - c )h

= O

X = I

was now riot J woo Gamo tonnes sin inn of

Yn = c. e"

t qx ex

Page 39: 2 Second equation Homogeneous equation Wisor

Y'try 'ty=e×

yn -- c.Etcieos.ooihiy.ee @is

Ti¥¥wi.iodous yp=x( Axel )

=Ax2e×=/

Page 40: 2 Second equation Homogeneous equation Wisor

y'tay 'ty=xe×

xyn . c.It Mi y,=CAtBx)e

eri ,

iodous Yp = x ( AtBx7e×

:X:*: wi.bedders Yp = x ( Axt Bx

' )e×= (Ax 't 13×31 ex

= A- x' e't 13×38 ✓

Page 41: 2 Second equation Homogeneous equation Wisor

y''- ey 't y

-

- ex sin ×-

a = a ± i

Ny = etcc.cos xtgsiuyyi. c. E. '- cigar

,; ELA I=[↳Ae×cos × t Bets in x✓X = 1

,I

Page 42: 2 Second equation Homogeneous equation Wisor

Gamo bots to Ewha bondinginhomogeneous equation

a ca y''

t b ex) y'

tax) y = rcx )

Dois nos Dum d ion who sinned To,a ex) y

"t b Cx) y 't can y = O

un woo riot duos fund lonnksoinineo Too

4h = C,Y,

t Czyz

Un w Nih ou loan : eroiodous's ro, Barn 8)si ionWu O,

'

Ypwoo ca

poniard no, Gum odd 'onWwf y ' Yntyp

Page 43: 2 Second equation Homogeneous equation Wisor

ME UP tmrw town : Ypt

.Mrs briber LINE : 2nd

'

- ay''t by

'

te y = rcxl

a. b. C oiooufsoinnsoio

- red 8qduuw5niae'×

, a.+ a. xtqx't . - thnx

"

,coscaxl

,sdncbxl

( got - - tax"

le>×

,Corot - -tax

"

I coscaxl::i::n→.D-- -

Page 44: 2 Second equation Homogeneous equation Wisor

Gowri yp annie assure , rex)

n

Yp - WU.

how aota.lt - - tan x-- - -

-

yp- oiofnrssroi cos can sincaxl

oioosnniiw- noooo

.

Inari iv yn refoldIn Vicious ypotovmoa.mx

orooowtonsouunnniioiosa.mx

Page 45: 2 Second equation Homogeneous equation Wisor

กาํหนดให้โดยระเบยีบวิธีเทียบสมัประสิทธิจะต้องสมมติ

ให้ผลเฉลยเฉพาะ อยู่ในรปูใดถ้า

4 4 ( )y y y r x + + =

py

(1) (2)

(3) (4)

(5)

2( ) 2007 xr x e=2x

py Ae= 2xpy Axe=

20 1( ) x

py A A x e= + 2 2xpy Ax e=

3 2xpy Ax e=

-

Page 46: 2 Second equation Homogeneous equation Wisor

กาํหนดให้โดยระเบยีบวิธีเทียบสมัประสิทธิจะต้องสมมติ

ให้ผลเฉลยเฉพาะ อยู่ในรปูใดถ้า

4 4 ( )y y y r x + + =

py

(1) (2)

(3) (4)

(5)

2( ) 2550 xr x e=2x

py Ae= 2xpy Axe=

20 1( ) x

py A A x e= + 2 2xpy Ax e=

3 2xpy Ax e=

-

Page 47: 2 Second equation Homogeneous equation Wisor

กาํหนดให้โดยระเบยีบวิธีเทียบสมัประสิทธิจะต้องสมมติ

ให้ผลเฉลยเฉพาะ อยู่ในรปูใดถ้า

4 4 ( )y y y r x + + =

py

(1) (2)

(3) (4)

(5)

2( ) 555 xr x xe=2

0 1( ) xpy A A x e= + 2x

py Axe=2 2

0 1( ) xpy A x A x e= + 2 2x

py Ax e=

2 3 20 1( ) x

py A x A x e= +

Page 48: 2 Second equation Homogeneous equation Wisor

กาํหนดให้โดยระเบยีบวิธีเทียบสมัประสิทธิจะต้องสมมติ

ให้ผลเฉลยเฉพาะ อยู่ในรปูใดถ้า

4 4 ( )y y y r x + + =

py

(1) (2)

(3) (4)

(5)

20 1( ) x

py A A x e= + 2xpy Axe=

2 20 1( ) x

py A x A x e= + 2 2xpy Ax e=

2 3 20 1( ) x

py A x A x e= +

2( ) 7,000,000 xr x xe=

Page 49: 2 Second equation Homogeneous equation Wisor

y''

tay 't dy = 2009£"

hi Yp = ( n A e-×

-

Y"

t dy 't dy = 2550 e-2×

- 2X

%" Yp = can, Ake-Y"

t dy 't dy = 555 xe"

2X

%iYp=Y"

t dy 't dy = 7,000,000 XE"

%" Yp = ( 5 ) ( A.x'+ Aisle"


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