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