UNIT 2.2 NONHOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS
Recap:
Second Order LDE
𝑦ˊˊ + 𝑃(𝑥)𝑦ˊ + 𝑄(𝑥)𝑦 = 𝑔(𝑥) … [2.1]
𝑦ˊˊ + 𝑃(𝑥 )𝑦ˊ + 𝑄(𝑥 )𝑦 = 0 … [2.2]
𝑦 = 𝑐# 𝑦# + 𝑐$ 𝑦$ … [2.3]
Wronskian of two functions:
Linear independent
HSOLE with constant coefficients:
𝑎𝑦ˊˊ + 𝑏𝑦ˊ + 𝑐𝑦 = 0
Characteristic equation associated HSOLE:
𝑎𝑚$ + 𝑏𝑚 + 𝑐 = 0
Solutions of the HSOLE:
Case I:
Case II:
Case III:
Example 1
Find the general solution of HSOLE
𝑦 (%) + 𝑦 !! − 2𝑦 ! = 0
Non-homogeneous equations
𝑎𝑦 !! + 𝑏𝑦 ! + 𝑐𝑦 = 𝑔(𝑥) … [1]
and
𝒂𝒚!! + 𝒃𝒚! + 𝒄𝒚 = 𝟎 … [2]
𝒚 = 𝒚𝒄 + 𝒚𝒑
𝑦' −
Is a general solution of Eq. 1
𝒂𝒚 + 𝒃𝒚 + 𝒄𝒚 = 𝟎 … [2]
𝒚 = 𝒚𝒄 + 𝒚𝒑
Is a general solution of Eq. 1
𝑦' −
Defined by
and
𝑦' = 𝑐# 𝑦# + 𝑐$ 𝑦$ … [3]
𝑦" −
NB:
∗ 𝑦" − must always linearly independent with 𝑦# and 𝑦$
Solving nonhomogeneous equations:
1. From eq. 1, determine the form of 𝑦" by looking at the form of 𝑔(𝑥), where
the coefficients are determined by substituting the form into the DE.
2. Use either case given below to find the precise 𝑦" .
Method of Undetermined coefficients
CASE I: The form of 𝑦" is a linear combination of all linearly independent functions
that are generated by repeated differentiation of 𝑔(𝑥).
CASE II: 𝑦" = 𝑦"# + 𝑦" $ + 𝑦" % + ⋯ + 𝑦" & , 𝑖 = 1, 2, 3, … 𝑛
If any of the 𝑦" terms contains terms that duplicate terms in 𝑦' , then that term 𝑦" & must be
multiplied by 𝑥 ( , where 𝑛 is the smallest positive integer that eliminates that duplication.
Table of Undetermined coefficients
𝑎𝑦 !! + 𝑏𝑦 ! + 𝑐𝑦 = 𝑔(𝑥) … [1]
Form of 𝑔(𝑥) = 𝑎( 𝑥 ( + 𝑎(+# 𝑥 (+# + ⋯ + 𝑎𝑥 + 𝑎,
Values of 𝒏 𝒈(𝒙)
𝒚𝒑
𝑛=0
𝑛=1
𝑔(𝑥) = 𝑎,
𝐴,
𝑔(𝑥) = 𝑎# 𝑥
𝑔(𝑥) = 𝑎# 𝑥 + 𝑎,
𝐴𝑥 + 𝐴,
𝑛=2
$
𝑔(𝑥) = 𝑎$ 𝑥
𝑔(𝑥) = 𝑎$ 𝑥 $ + 𝑎,
𝑔(𝑥) = 𝑎$ 𝑥 $ + 𝑎# 𝑥
𝑔(𝑥) = 𝑎$ 𝑥 $ + 𝑎# 𝑥 + 𝑎,
𝑛=3
𝐴$ 𝑥 $ + 𝐴𝑥 + 𝐴,
𝑔(𝑥) = 𝑎% 𝑥 %
𝐴% 𝑥 % + 𝐴$ 𝑥 $ + 𝐴𝑥 + 𝐴,
%
$
𝑔(𝑥) = 𝑎% 𝑥 + 𝑎$ 𝑥
𝑔(𝑥) = 𝑎% 𝑥 % + 𝑎$ 𝑥 $ + 𝑎# 𝑥
𝑔(𝑥) = 𝑎% 𝑥 % + 𝑎$ 𝑥 $ + 𝑎# 𝑥 + 𝑎,
CASE I: 𝑦" = 𝐴# 𝑔! (𝑥) + 𝐴$ 𝑔!! (𝑥) + ⋯ + 𝐴( 𝑔( (𝑥).
Example 1
Determine the general solution of the following DEs
𝑎. 𝑦 !! − 𝑦 ! − 2𝑦 = 𝑡 $
𝑏. 𝑦 !! +4𝑦 = 𝑡 $ + 4𝑡 + 2
Form of 𝒈(𝒙) = 𝒔𝒊𝒏𝜷𝒙 or 𝒈(𝒙) = 𝒄𝒐𝒔𝜷𝒙
𝑔(𝑥) = 𝑠𝑖𝑛𝛽𝑥
𝑔(𝑥) = 𝑐𝑜𝑠𝛽𝑥
𝐴𝑠𝑖𝑛𝛽𝑥 + 𝐵𝑐𝑜𝑠𝛽𝑥
𝑔(𝑥) = 𝑠𝑖𝑛𝛽𝑥 + 𝑐𝑜𝑠𝛽𝑥
Example 2
Determine the form of 𝑦"
𝑎. 𝑦 !! − 4𝑦 = 𝑐𝑜𝑠3𝑥
𝑏. 𝑦 !! − 9𝑦 = 𝑐𝑜𝑠2𝑥 + 𝑠𝑖𝑛2𝑥
Form of 𝑔(𝑥) = 𝑒 𝛂𝐱
𝑒 𝛂𝐱
𝐴𝑒 𝛂𝐱
Example 3
Determine the general solution of the following DE
𝑎. 𝑦 !! + 4𝑦 ! = 𝑒 +$/
More examples 𝒈(𝒙) is a combination of different functions
Example 4
𝑔(𝑥) = 𝑠𝑖𝑛𝛽𝑥 or 𝑐𝑜𝑠𝛽𝑥 𝑦" = 𝐴𝑠𝑖𝑛𝛽𝑥 + 𝐵𝑐𝑜𝑠𝛽𝑥
𝑔(𝑥) = 𝑥
Determine the form of 𝑦"
𝑎. 𝑦 !! − 9𝑦 = 𝑥𝑠𝑖𝑛2𝑥
𝑦" = 𝐴𝑥 + 𝐴,
Determine the form of 𝑦"
𝑎. 𝑦 !! − 9𝑦 = 𝑥𝑠𝑖𝑛2𝑥
CASE II: 𝑦" = 𝑦"# + 𝑦" $ + 𝑦" % + ⋯ + 𝑦" & , 𝑖 = 1, 2, 3, … 𝑛
If 𝑦" & is a duplicate of 𝑦' , then 𝑥 ( 𝑦" &
Example 5
Determine the form of 𝑦"
𝑎. 𝑦 !! + 4𝑦 = 𝑠𝑖𝑛2𝑥
𝑔(𝑥) = 𝑒 𝛂𝐱
𝑦" = 𝐴𝑒 2/
𝑔(𝑥) = 𝑥 (
𝑦" = 𝐴( 𝑥 ( + 𝐴(+# 𝑥 (+# + ⋯ + 𝐴𝑥 + 𝐴,
𝑏. 𝑦 !! + 4𝑦 ! + 4𝑦 = 𝑥 % 𝑒 +$/ + 𝑥 $ + 𝑥 + 1
𝑔(𝑥) = 𝑒 𝛂𝐱
𝑦" = 𝐴𝑒 2/
𝑔(𝑥) = 𝑠𝑖𝑛𝛽𝑥 or 𝑐𝑜𝑠𝛽𝑥 𝑦" = 𝐴𝑠𝑖𝑛𝛽𝑥 + 𝐵𝑐𝑜𝑠𝛽𝑥
𝑐. 𝑦 !! − 2𝑦 ! + 5𝑦 = 𝑒 / 𝑐𝑜𝑠2𝑥