Chapter 8 and 9 OPE

OBE ordinary differential equation

equation involving derivatives ofa function

yet Y'HI y t's y it yday't II yet hiya

D dat y HI Dyy t D y y It Day

Linear differential equations

Ex Y Its 12Y'Hityet O

Y t 12g Its yet Sint

General form A It Y it t anti y it 1

divide dolt t an Iti YIM Fitson both sides if a Hito

1Goal Solve this equation

With initial condition

y it C Y Ito Ct yotsuPThis is called initial value problem

Usually we focus on a HI 1

Def Linear differential operator of order nL Dnt a HID t ants D t tanit

y y i a

X al solve Ly FH ftwhen a It Arlt an ft

are constants

Want to study 1A from the point of viewof linear algebra

Compare A with Ax b

Fiat view L as a linear transformation

V 1 functions oft with derivatives ofall orders y I smooth functions

L is a linear transformation from to

D 1212 11

L V V

in yetYu r LlyHD Y Iti 12g

Check L is a linear transformation

Lly eye L'HthyrLlc y c Ly

V Vpe trsduIytIty

o

Lyn fit I

7hm If y.lt is a particular solution to

then all the solutions to CA hasthe form yet you txH

Xlt is a solution to homogeneous

equation 1 147 0in Kerl

Two steps

Find her L Chaplin 8.2

Solve homogeneous equation

Ly o

Find one particular solution Yott chapter 8.38 4

7 day homogeneous equation

Ly o

Ex D 12171 1 1 154417 3 2

Y it y t TY't tY o

Black Box Theorem 1 Ly FitsExistence 1 uniqueness

has a unique solution for any fixedinitial conditions

yih1It Ci y Holtz

y Holt Ca

Ex of the theorem

Y Lt 12445 0initial condition

ay YH I

i Y'Ho 2t

From the black box them

7hm for Ly O L has order n

Dnt all D t i Tanit

herl is a subspace of V

herl has dim n

7hm tells us we should findn linearly independent solutions to ly p

basis of tarLall the solutions are linear combinations

Ex L D Ltp a it

Ly o 4M

D 4 y O y sty p

multiply both sides by ef2t eth

e fy it g o

e thy't e They o

TdEt

e y o

e try C

Y c et

bool is t din't with basis fetyb L D 4

Ly Y Iti 4gal o

Assumeylts e.tt I linedi

by rent D2y r ert

r ert yert o r 412 y H e't

Ynet ett

Y y linearly indepent a yheiniebasis of herl

any solution has the form

yet C e't the It

Wronshian Method to prove functions are

linearly in dependent

Y HI yah

Cry.lt thyaltj o

C y T iz yd derivative

vii is IiWronskian

LYYf Wly yayigi

If W y Yu 1 0 then 49 yay islinearly independent for some t

Ex y it e't y eit

wing hee t z e

IO

Y HI grits y its

c y they 1473 0.2 derivativec yituyitsyi

ee.namec y taya toys o

f

my.mnftp gl to for

some t then by ya yY is linearlyindependent

c L D 14

Ly y 14g o

fAisnmeyH e

Y r ett

Pert y ert o

r2t4 r X y imaginarynumber

r tf 12ft i

Euler's formula µ0

usOtftsin

y.lt et

f tfisinyah e

IF'tist Lt t ft Sintzt

Ft finreal iii

giant YETNext Wliasat sing f

4 sink

fSiaa wht

2412T t 215in't 2.76nosh linity linearly independent

general solutiontuna basis of her L

Y H C wht Tca sin 2T

Prove dim kerl order of L

L has constant toefficients

how to find n linearly independent g lupus

Pf We constrict a linear transformation

from her L IR

T herl an

gifts tip f'tlyin it

Check T is a linear transformation

He Nullity 7hm O n

dim kerL di T t dim Image 1

dim KI her 1 44 4 o

Ly o

f yttot o.ytt.to y it

black boy theorem uniquenessy It p

her 1 404 din o

dim Image 1

black box thm Existence

For any initial conditions

b Ho y Hoy y Ho

there existsYHI.LY 0

Image 1 1124 din

0