=qcx nana ?t¥¥y=n%iu typical · 2019. 12. 23. · nomi ddi× =gapfcx) ← separable equation...

NOMI ddI× =gapfCx ) ← separable equation

§dIa×tPCxy=QTf← linear equation-⑨

EI Xy'

ty = 2X

⇒ xd£×+y = 2x → bbdoosidfinoiolois separable ege

⑨ did, tpcxy =QCx)

gadoids Mex) n'qrroinwohuvs ⑨ Nona ⑨www.Vcroqnvcsrrcsdsnwmcxl#xtPlxIyJ=ycxlQCx,

Nana @ lay ] '=µWy' typicalni

!?①?t¥¥Y=n%Iu solar

⇒Key =nEH¥µ' Cx) =µCx) PG ) -0←

¥141 =µlx7PCx )

ate, duh = pcxldx

fµ÷dµCx) =

fpcxtdxlnlulxll-fpcxldx-efpcxtdxMIT.noumrom.mn( integrating factors

ïììĊę ÿöÖćøđßĉÜĂîčóĆîíŤĂĆîéĆïĀîċęÜĒúąÖćøðøą÷čÖêŤ

C ÝąĔßšđüúćîćîđìŠćĔéìĊęĕÖŠÜüÜĂïÝąöĊĂčèĀõĎöĉúéúÜđĀúČĂ 100'

ÿöÖćøđßĉÜđÿšî -JOFBS FRVBUJPOT

ÿöÖćøđßĉÜĂîčóĆîíŤĂĆîéĆïĀîċęÜđßĉÜđÿšî öĊøĎðìĆęüĕðÙČĂ

dy

dx+ P (x)y = Q(x)

đöČęĂ P Ēúą Q đðŨîôŦÜÖŤßĆîêŠĂđîČęĂÜ×ĂÜêĆüĒðø x êĆüĂ÷ŠćÜđßŠî xy′ + y = 2x ÝąđĀĘîüŠćëšć x != 0 Ēúšü

ÿöÖćøéĆÜÖúŠćüÝąÿćöćøëđ×Ċ÷îĕéšĔîøĎð

y′ +1

xy = 2

àċęÜĕöŠđðŨîÿöÖćøĒ÷ÖêĆüĒðøĕéšđîČęĂÜÝćÖĕöŠÿćöćøëđ×Ċ÷î y′ ĔîøĎðñúÙĎè×ĂÜôŦÜÖŤßĆî×ĂÜ x Ēúą y ĕéš ĒêŠ

ÝćÖÖãÖćøĀćĂîčóĆîíŤ×ĂÜôŦÜÖŤßĆîñúÙĎè đøćÝąĕéšüŠć

xy′ + y = (xy)′

éĆÜîĆĚî

(xy)′ = 2x

ēé÷ÖćøĂĉîìĉđÖøêìĆĚÜÿĂÜ×šćÜ×ĂÜÿöÖćø Ýąĕéš

xy = x2 + C ĀøČĂ y = x+C

x

ÝćÖêĆüĂ÷ŠćÜìĊę ÖúŠćüöć×šćÜêšî ÝąđĀĘîĕéš üŠćÖćøĀćñúđÞú÷×ĂÜÿöÖćøđßĉÜĂîčóĆîíŤĂĆîéĆïĀîċęÜ đßĉÜ đÿšî

ÿćöćøëìĈĕéšēé÷ÖćøđðúĊę÷îÿöÖćøéšćîàšć÷öČĂĔĀšĂ÷ĎŠĔîøĎðĂîčóĆîíŤ×ĂÜñúÙĎè µ(x)y ēé÷ìĊę µ(x) đðŨî

ôŦÜÖŤßĆî×ĂÜ x ĒúšüÝċÜìĈÖćøĂĉîìĉđÖøêđóČęĂĀćñúđÞú÷ đøćÿćöćøëÿøčð×ĆĚîêĂîÖćøĀćñúđÞú÷×ĂÜÿöÖćø

đßĉÜđÿšîĕéšéĆÜîĊĚ

ÿööêĉüŠćöĊôŦÜÖŤßĆî µ(x) ìĊęàċęÜ

[µ(x)y]′ = µ(x)[y′ + P (x)y] = µ(x)Q(x)


ÝćÖÿöÖćø ÝąĕéšüŠć

µ(x)[y′ + P (x)y] = [µ(x)y]′ = µ(x)y′ + yµ′(x)

îĆęîÙČĂ

µ(x)P (x) = µ′(x)

ÝćÖÿöÖćø ĒúąÖćøĀćñúđÞú÷×ĂÜÿöÖćøĒ÷ÖêĆüĒðøĕéš ÝąĕéšüŠć

µ(x) = e∫P (x)dx

đøĊ÷Ö µ(x) üŠć êĆüðøąÖĂïðøĉóĆîíŤ *OUFHSBUJOH GBDUPS

ēé÷ÖćøĂĉîìĉđÖøêìĆĚÜÿĂÜ×šćÜ×ĂÜÿöÖćø Ýąĕéš

y(x) =1

µ(x)

(∫µ(x)Q(x)dx

)

àċęÜđðŨîñúđÞú÷×ĂÜÿöÖćøđßĉÜĂîčóĆîíŤìĊęđøćêšĂÜÖćø

êĆüĂ÷ŠćÜ ÝÜĀćñúđÞú÷ìĆęüĕð×ĂÜÿöÖćøđßĉÜđÿšîêŠĂĕðîĊĚ

dy

dx+ 3x2y = 6x2

⇒a-×tPy=QCx,PG) -- 3×2 ;QLx7=6xT

① an oiwr.novmso-wninrm.hu/=eSPCx1dX=ef3x2dx=ex3② bowiwr.nouwo-unnsmqrrnorwrrs.es#xt3xyJ--ex?6x2

⇒"y ] -

- ex? 6×2

d (e"y ) = ex? 6×2 d× yU-- X' ;dU=3x2dx

" '¥;' 46×3×17

'se"-wax

-

= 2. feud UZeist C = ze

"+ C

-

'

. /y⇒⇒ e'''te

#


xdy

dx= x2 + 3y

3xy′ − y = lnx+ 1, x > 0

Forme /d¥+Pcx7y=0#7⑦- -

XddI× - zy =x2 / *n'Four . awhday, hints I lose<

daY×-Z# -④"Tiwa'm Integrating factor

① oink. nwmsoisriinsor µ Cx) = ef¥d× = e-3h ×

= elm I'

= ×- 3=1×3

② to Mlt) Nrs⑦ E.own

1×114×-2×4] -- tix

÷÷÷÷÷÷:÷:← Forme ¥×tPcx7y=O

3X - y = lnxt I -

E - ¥y=bhxz① µ (x) = ESP

d'= es

-ztxdx= e-Thx = elm

¥= x

- I=×

② * faux - tail -

- ftp.tenzxxtt-

dad Y] = + 3¥43 ) ¥I%dxd ( ¥34) = (FI, -1¥,)dx =x' +c

C -Itt )Jd y ) =¥dxff¥s/ = x + c

Worm fln×j¥, dx lui u -- thx ; do-

-¥, dx l

du -- txdx ; o-

-- ×-is ? tpayrts = - X

-Jtc

i. ftp.#.zdx---x-"lnx - f - x

"? ¥d×= - x'

"'lnxt f x'

""dx

= -x'"'enx - 3 x'

"'te

¥, y = - x'"'en x - 3143- I"3 + C

y-tnx-4-E.TL/--


êĆüĂ÷ŠćÜ ÝÜĀćñúđÞú÷×ĂÜðŦâĀćÙŠćđøĉęöêšî

x2y′ + xy = 1; x > 0, y(1) = 2

êŠĂĕðđøćÝąýċÖþćëċÜÖćøðøą÷čÖêŤÿöÖćøĂîčóĆîíŤđßĉÜđÿšîÖĆïĒïïÝĈúĂÜ×ĂÜüÜÝøĕôôŜć &MFDUSJD DJSDVJUT

ēé÷ÝąóĉÝćøèćüÜÝøĕôôŜć ĒïïÙČĂ üÜÝø 3- Ēúą üÜÝø 3$

óĉÝćøèćüÜÝøĕôôŜćĂ÷ŠćÜÜŠć÷éĆÜøĎð

øĎðìĊę 3-$JSDVJU

ēé÷ìĊę R đðŨîÙüćöêšćîìćî öĊĀîŠü÷đðŨîēĂĀŤö Ω L đðŨîÙüćöđĀîĊę÷üîĈ öĊĀîŠü÷đðŨîđăîøĊ Ēúą E(t)

đðŨîÙüćöêŠćÜýĆÖ÷Ť öĊĀîŠü÷đðŨîēüúŤê đøĊ÷ÖüÜÝøîĊĚüŠć üÜÝø 3-

đøćóïüŠćÙüćöêŠćÜýĆÖ÷ŤìĊęêĆüêšćîìćî Ēìîéšü÷ ER đðŨîÿĆéÿŠüîÖĆïÖøąĒÿ I(t) öĊĀîŠü÷đðŨîĒĂöĒðøŤ

Ĕî×èąîĆĚî îĆęîÙČĂ

ER = RI(t)

tomfdaftt.MY#7dyq,+zy-- ÷ ③ YC " -- 2 m④µ (x) = estxdx = eh × = X 2 = F- t Cz"o '"¥÷÷÷÷÷÷÷fi÷÷÷⇒fdlxy ) = ftxdx ptarticwlar solution

Xy = In x TC

qy=ln+§L/general solution

-

G

00

=

II E-ER


Ĕî×èąđéĊ÷üÖĆî ÙüćöêŠćÜýĆÖ÷ŤìĊęêĆüđĀîĊę÷üîĈ Ēìîéšü÷ EL ÝąđðŨîÿĆéÿŠüîÖĆïĂĆêøćÖćøđðúĊę÷îĒðúÜ×ĂÜ

ÖøąĒÿĔî×èąîĆĚî

EL = LdI

dt

ēé÷ÖãÙüćöêŠćÜýĆÖ÷Ť×ĂÜ ,JSDIIPGG ìĊęÖúŠćüüŠć ñúïüÖìćÜóĊßÙèĉê×ĂÜÙüćöêŠćÜýĆÖ÷Ť×èąđüúćĀîċęÜ

øĂïüÜÝøðŗéĔéėöĊÙŠćđðŨîýĎî÷Ť ĀøČĂ ÙüćöêŠćÜýĆÖ÷ŤìĊęĔÿŠđ×šćĕðĔîüÜÝøðŗé đìŠćÖĆïñúïüÖ×ĂÜÙüćöêŠćÜ

ýĆÖ÷ŤìĊęÿŠüîðøąÖĂïìĊęđĀúČĂĔîüÜÝø ÝąĕéšüŠć

ER + EL = E(t)

ĀøČĂ

LdI

dt+RI = E(t)

àċęÜđðŨîÿöÖćøđßĉÜĂîčóĆîíŤĂĆîéĆïĀîċęÜđßĉÜđÿšî éĆÜîĆĚîđøćÿćöćøëĀćñúđÞú÷×ĂÜÿöÖćøêćöüĉíĊÖćøìĊęĕéšÖúŠćü

öćÖŠĂîĀîšćîĊĚ

êĆüĂ÷ŠćÜ üÜÝø RL üÜĀîċęÜ öĊÙüćöêšćîìćîđðŨî Ω Ēúą ÙüćöđĀîĊę÷üîĈđðŨî đăîøĊ ÿööêĉüŠćÙüćöêšćîìćîøüöĔîüÜÝøđðŨî ēüúŤê ÝÜĀć

B ÖøąĒÿĔîüÜÝø ëšćìĊęđüúćđøĉęöêšîĕöŠöĊÖøąĒÿĔîüÜÝøđú÷

C ÖøąĒÿĔîüÜÝøđöČęĂđüúćñŠćîĕð üĉîćìĊ

D ÖøąĒÿĔîüÜÝøđöČęĂđüúćñŠćîĕðîćîė

EL xdat

EL=Ld

IT-

IT affect,

12=12 2=4

--

Nah,MfYg E# 60

ICO ) =O

LddtI + RI - Ect ) ←b6rm (2) 0=5 + Ceo[41¥ -1121=60

-

'./C=-5T

DII -131=15.

'

. Ift e- 3T

st (b) IG ) -- 5 - Se-3

me:÷÷÷÷÷÷÷: I "÷÷÷:c:*.d[e'+I] --Best Ido,#d.www.wyns-ollrdvootsfdce3tI) =fnse3tdt Go 5110201105 #e'tI =se3ttCi.IICH=5T


óĉÝćøèćüÜÝøĕôôŜćìĊęöĊÖćøĔßšêĆüđÖĘïðøąÝčĒìîêĆüđĀîĊę÷üîĈ àċęÜđøĊ÷ÖüŠć üÜÝø RC éĆÜøĎð

C đðŨîÙüćöÝč ôćøĆé đøćóïüŠćÙüćöêŠćÜýĆÖ÷ŤìĊęêĆüđÖĘïðøąÝč Ēìîéšü÷ EC đðŨîÿĆéÿŠüîÖĆïðøąÝčĕôôŜć

Q(t) öĊĀîŠü÷đðŨîÙĎúĂöïŤ Ĕî×èąîĆĚî đ×Ċ÷îĕéšđðŨî

EC =1

CQ(t)

ĒêŠđîČęĂÜÝćÖ I(t) =dQ

dtéĆÜîĆĚîēé÷ÖãÙüćöêŠćÜýĆÖ÷Ť×ĂÜ ,JSDIIPGG ÝąĕéšüŠć

RI +1

CQ = E(t)

ĀøČĂ

RdQ

dt+

1

CQ = E(t)

êĆüĂ÷ŠćÜ üÜÝø RC üÜĀîċęÜ öĊÙüćöêšćîìćîđðŨî Ω Ēúą ÙüćöÙüćöÝčđðŨî ôćøĆé ÿööêĉüŠćÙüćöêšćîìćîøüöĔîüÜÝøđðŨî ēüúŤê ĒúąìĊęđüúćđøĉęöêšîĕöŠöĊðøąÝčĕôôŜćĔîüÜÝøđú÷ ÝÜĀćðøąÝčĕôôŜćĔîüÜÝøìĊęđüúćĔéė

OO

F-exact )

F-c-

- Qct)

QCttnodr.pwdnis.it/#-0bbiramsunQIt ,E

Ett QT¥Tract )

Rfef + IQ = Ect) ;

s# t Q --60

""÷:÷÷÷÷÷:/ "÷¥÷÷÷÷÷÷÷.fdCe4tQ) =fize4tdt

e4tQ =ze4t + C

i.QCt7=z+Ce-4TQCO) -- O'

,O =3+ Ceoi.C=-3_

QCtt-3-3e.TL#

=qcx nana ?t¥¥y=n%iu typical · 2019. 12. 23. · nomi ddi× =gapfcx) ← separable equation...

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