WTW256: Lu 2.2: NON-HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS: Lecture Notes

2. 2. Non -
linear Homo D. E.

Not equal
to taro
↳ where RHS is

Non homo PE 's with
-
constant coeff

'M 'm )
⑧
Any dn ly uqyliqoy -0
'

n -
1
. .
.




⑤ is the general solution of
@
L complimentary solution


⑧ is particular solution of the non hombf ④
-




a



A general solution • 1-
④ is
y= yciyp
( campy part)

IM-eth-od-ot-un-detirm-ine-dioei-f.it"
derivatives
- -
- -




and
-
- -




-
1. consider glx) ( RHS) to
guess a general form of yp
2. Then consider yc to check for duplication of terms

in yp

3 If NO DUPLICATIONS of terms in yp and
go
=
.




•

plug yp into DE ④ and compare coeff .

, Examples
1. Use method to solve :



2nd order
72g -3g guy
"
'
Y
.
-




with
27C
1. 1. gin)= @
N2
1. 2
gla)
=




91×1=9
"
1.3 and ya)=1 y
'



, (a) =4
"
In) :c
LU
g.
2 "
Trial :




µµ
1- 1
yo
'



.




-




2"
Yp A @
:

M 'T 2m -3
= 0


( Mt 3) ( m 1)
-
-0
'

NO duplication
m ,
, , m ,
,
, ,, a ]
g.f- ZAC
'"
ypl
=

-3 " a


¥
'
Cle i
Cz @
"
4Ae
"

yp
:




% Plug into Df :




Zyp 3yp=e2
" ' "
'
Yp
-




general solution
3AAe"=e2
" "

4AAe '^4☆e
"
"
-




Cle "nCze"uY5e
-3
y lm
-
_




22
4 4 -3 @
=
1
~

yc YP 5AAe =e
④ see that yp is the 5nA = I

solution that AA 45
only >
has value for the
,y}
a
.

'
coeff .
I 15cm