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1 4.4) Undetermined Coefficients – Superposition Approach

1. The general solution of nth-order non-homogeneous linear DE:

)()()()(...)()( 012)1(

1)(

 x g  x ya x ya x ya x ya x yan

n

n

n  =+′+′′+++

  −

−

(1)

on an interval I  is given by: c p y y y= +

2.   c y  is the complementary function of (1) whih is the general solution of the

associated homogeneous equation:

( ) ( 1)1 2 1 0... 0n n

n na y a y a y a y a y−−

  ′′ ′+ + + + + =

3.   p y is any particular solution of (1).

4.   p y an be determined by the method of  undetermined coefficients.

This method is! however! limited to non-homogeneous linear e"uations (1) where:

•the oeffiients

! 0!1! 2!...!i

a i n=

are constants and• ( ) g x  is a constant ! e#am$le: ( ) % g x   = !

• or a polynomial funtion e#am$le: 2( ) & '! g x x= −

• or an eponential funtion e#am$le: 2( )   x g x e−=

• or a sine or cosine funtion e#am$le: ( ) sin or ( ) os ! g x x g x x= =

• or finite sums and products of these funtions

e#am$le: &( ) & 2 sin  x g x x x e x= − + +

!. "o determine the particular solution#  p y # $y the superposition approach%

*ine ( ) g x have derivatives of a form similar to ( ) g x itself! then we hoose a trial

 $artiular solution   p y ! that is similar to that of ( ) g x ! and involving un+nown

oeffiients to be determined by substituting that hoie for  p y  into (1).

&ample% *olve the given DE by undetermined oeffiients.

) ' ,, % 1 ) ,, 10 , 2 &0 &a y y b y y y x+ = − + = +

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2 4.4) Undetermined Coefficients – Superposition Approach

"a$le 1% Some &amples of trial particular solutions.

'. (inding the trial particular solution!  p y :

• Case 1: onsider the ase when the  p y  is also a solution of the assoiated

homogeneous DE. (ie i p y  ontains terms that du$liate terms in c y )

• ule 1: Then that i p y  must be multi$lied by n x ! where n is the smallest

 $ositive integer that eliminate that du$liation.

• Case 2: onsider the ase when ( ) g x onsists of m terms of the +ind listed

in the table! let: 1 2( ) ( ) ( ) ... ( )m g x g x g x g x= + + +

• ule 2: Then! 1 2...

m p p p y y y y= + + +  

&ample% *Case 1# ule 1) 

*olve the given DE by undetermined oeffiients: ',, 1 2   x y y e− =

&ample% *Case 2# ule 2) 

*olve the given DE by undetermined oeffiients: 2,, 2 , 2   x y y x e−+ = + −

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3 4.4) Undetermined Coefficients – Superposition Approach

&ample% inding the form of $artiular solution  p y

c y * ) g x (orm of  p y

1 2& &cos sinc x c x+   2 &'/  x

 x e−

2 &* )   x Ax Bx C e+ +

 o du$liation between c y

and   p y

1 2 &

 xc c x c e+ +  &   cos x− * cos sin ) A B x C x+ +

Du$liation between c y and

 p y  in  A

Therefore:

2 p y Ax B x C x= + +* cos sin )

' '1 2

 x xc e c e−+

' 22 2 xe x x+ +' 2

* ) x Ae Bx Cx D+ + +

Du$liation between c y and

 p y  in ' x Ae

Therefore:

' 2* ) x p y Axe Bx Cx D= + + +

21 2

 xc c e−+

22   x x e−+ −

2* )

  x Ax B Ce−+ +

Du$liation between c y and

 p y  in * ) Ax B+ and 2 xCe−

Therefore:

2 2* )

  x p y x Ax B Cxe−

= + +

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4 4.4) Undetermined Coefficients – Superposition Approach

&ample% inding the form of $artiular solution  p y (D T *3E)

2 21 ' 2 )  x x y y x e xe−

′′ − = − +

22 2 2 )   x y y x e−′′ ′+ = + −

2& / 20 100 2)  x y y y x xe′′ ′− + = −

'& ' 2* ))

  x y y x xe−′′− = +