The particular integral :

2

2 1 02 d)

d(

d

d

f fLf a a a f

xh

xx+ == +

Second order linear equation with constant coefficients

2 1 0cos oso c sh H x Lf a f a f a f H x!! != " + + =

Sinusoidal h

i

2 1 0( ) '' ' e xLg x a g a g a g H! + + =

i ie( ) [ e( )] e( e ) e(e ) cosx xLg L g H H H x! = ! =! = ! =

iex

g P= i

2 1 0

2 1 0

( i ) ei

x HLg a a a P P

a a a= ! + + " = .

! + +

i

0 2 1

0 2 1

2 2

0 2 1

ee( )( ) i

( ) cos sin

( )

xHf

a a a

a a x a xH

a a a

=!" +

" += .

" +

2

2 1 02

d d( )

d d

f fLf a a a f h x

x x= + + = .

e( )f g=! is the solution to the real equation

Solution:

3 2 cosf f f x!! !+ + = .Ex 5

i'' 3 ' 2 e

xg g g+ + = .

i 1e where

1 3i 2

xg P P= = .

! + +

i

11 10

ee( ) (cos 3sin )1 3i

x

f x x=! = + .+

PI

CF2

0

x xf Ae Be! != +

2 110(cos 3sin )x xf Ae Be x x! !

= + + +General solution

(0) 4, '(0) 1x x= =

3 2 cosf f f x!! !+ + = .

2 110(cos 3sin )x xf Ae Be x x! !

= + + +

17 23

2 5A B= = !

3 2 cosf f f x!! !+ + = .Ex 5

i'' 3 ' 2 e

xg g g+ + = .

i 1e where

1 3i 2

xg P P= = .

! + +

i

1101

em( ) (sin 3cos )1 3i

x

x xf ! "= =+

.PI

CF2

0

mx mxf Ae Be! != +

110

2 (sin 3cos )mx mxf Ae Be x x! != + + !

General solution

What if cos(x) sin(x)?!

m( )f g= !

3 2 3cos 4sinf f f x x!! !+ + = + .Ex 6

i( )5cos( ) 5 e(e )xx

!! += + = "

where arctan( 4 3)! = " /

cos( ) cos cos sin sinx x x! ! !+ = "

2 2

2 2 2 2

2 2

cos sin ( cos sin )

cos( )

A BA x B x A B x x

A B A B

A B x !

+ = + ++ +

= + + ,

2 2cos ,A A B! = / +

and

2 2sin B A B! = " / +

.

Proof:

tan B A! = " /

3 2 3cos 4sinf f f x x!! !+ + = + .Ex 6

i( )5cos( ) 5 e(e )xx

!! += + = "

where arctan( 4 3)! = " /

i( )Trial solution : g e xP

!+=

5 5

1 3i 2 1 3iP = = ,

! + + +

i( )

11 2

e5 e( ) [cos( ) 3sin( )]

1 3i

x

f x x!

! !+

= " = + + + .+

i( )'' 3 ' 2 5e xg g g

!++ + =

icos '' e

xf f x g g!!+ = " + =Ex 7

id dP.I. ( i)( i) e

d d

xg

x x+ ! = .

iTry g= e xPx

i i i i

i

12

d d de ( i)( i) e ( i) e 2i e

d d d

1 ee( ) sin

2i 2i

x x x x

x

Px P Px x x

xP f x x

= + ! = + =

" = " =# =

Then

id dC.F. ( i)( i) e

d d

xg

x x+ ! =

ixg=Ce , f=Acos(x)+Bsin(x)

e (3cos 4sin )xf f x x!""+ = +Ex 8

(i 1) ie xg P

!" +=

Trial function

2

5 5

(i 1) 1 1 2iP = = .

! + !

(i 1) i

1

ePI 5 e( ) e [cos( ) 2sin( )]

1 2i

xxf x x

!

! !" +

"= # = + " + .

"

-x ( )5 e(e )i xe

!+= "

where arctan( 4 3)! = " /

2

2 1 0 1 22

d d( ) ( )

d d

f fLf a a a f h x h x

x x= + + = +

2

1 11 2 1 0 1 12

d d( )

d d

f fLf a a a f h x

x x= + + =

2

2 22 2 1 0 2 22

d d( )

d d

f fLf a a a f h x

x x= + + =

Since the equation is linear a solution to the original equation is given by

1 2f f f= +

Solutions with combinations of driving functions