WTW256: LU 2.1: SECOND AND HIGHER ORDER HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS Lecture notes

2. 1 .
Higher order linear DE 's
Has the form :




an Ix ) Yh i
any Ln )Y
"^
} .
.
.iq , Cody 'eqoxy=gC c) >




y(xo)=yo- Not power to

but order of derivative
④ ✗a) =


y, -



1) ( xo ) Yh -1
ya :
-




Theorem 4.1.1 .




The IVD ⑦ solution yl ≥) on an
has a unique
interval 1 it the functions 9h / 9h I -




. . .
go
,


and g 2 and
are continuous on the interval

dhlx ) ¥0 for d 11 kt 1

DE it
Homogeneous gl a) a
=




↳ not like previous homo this is d HIGHER
ORDER HOMO ! !
,
⑤ any.hn .
.
.to ,y'yqoy=o
Theorem 4.1.2 .




If y , , yz . .

.yn are solutions of ② on an interval
2 , any linear combination of the solutions

GY It
CZY 24
=

y . . .
4
Cnyh
will also be a solution oh I CCER ,n=1, 2
,
. -



yn)

, linearly dependent functions
it constants 4,4 , Cs ,
. . .
,Cn , not all equal to Zero
,
can be found such that

Cifl Y Czfzt . . -
t Cnfn =
0



linearly dependent functions

If C , fly Czfzt Cnfn -0 implies that
'

. . -
t



4=02 =

.
. .
= Ch =
0




G)121 Czk 4 =
0

↓ ↓
positive hrs
zero if 4 :( 2=0
can
only be
i.

=
linearly independent

" 2" 0
☐ e Cz @
=
n
ex ≠ cel
"


↓ ↓ Cmu / tiple)
always positive .




?
Wronski ah of functions
findep
dep ?
criteria hest

•
mm
linearly Ep it W{ fi.fr ,
. . _
fn } =/ 0 for
all 71 C- 1

Fundamental set

solutions
set of linearly indep