UNDETERMIEND COEFFICIENT AND ITS EXAMPLES AND APPLICATIONS>

2. Presented By:
Amenah Gondal
Presented To:
Ma’am Mehwish
Class:
BS.ED(IV)
ORDINARY DIFFERENTIAL
EQUATIONS

3. We will discuss:
What is method of Undetermined
Coefficient?
Its detail
The table to find solution of P.I.
Three Roots to find solution of C.F.
Applications of Differential Equation
Example
OBJECTIVES:

4. In mathematics, the method of
undetermined coefficient is an approach to
finding a particular solution to certain
nonhomogeneous ordinary differential
equations.
METHOD OF UNDETERMINED
COEFFICIENT

5. The method of undetermined coefficients which
can prove simpler in finding the particular integral
of the equation
𝒇 𝑫 𝒚 = 𝑭(𝒙)
when 𝐹 𝑥 is
 an exponential function: (𝑒 𝑎𝑥 )
 a polynomial: (𝑏0 𝑥 𝑛 + 𝑏1 𝑥 𝑛−1 + ⋯ + 𝑏 𝑛)
 sinusoidal function (𝑠𝑖𝑛𝛼𝑥 𝑜𝑟 𝑐𝑜𝑠𝛼𝑥)
 the more general case in which 𝐹(𝑥) I sum of a
product of terms of the above types, such as
𝐹 𝑥 = 𝑒 𝑎𝑥 (𝑏0 𝑥 𝑛 + 𝑏1 𝑥 𝑛−1 + ⋯ … … + 𝑏 𝑛)
𝑠𝑖𝑛𝛼𝑥
𝑐𝑜𝑠𝛼𝑥
UNDETERMINED CO-EFFICIENT

6. 𝑭 𝒙 is of the
form
Take 𝒚 𝒑 as
1. 𝑎 𝐴𝑥 𝑘
1. 𝑎𝑥 𝑛
(𝑛 𝑖𝑠 𝑎 +ve integer)
𝑥 𝑘
(𝐴0 𝑥 𝑛
+ 𝐴1 𝑥 𝑛−1
+ ⋯ … … . 𝐴 𝑛−1 𝑥 + 𝐴 𝑛)𝑒 𝑟𝑥
1. 𝑎𝑥 𝑛 𝑒 𝑟𝑥
(𝑛 𝑖𝑠 𝑎 +ve integer)
𝑥 𝑘(𝐴0 𝑥 𝑛 + 𝐴1 𝑥 𝑛−1 + ⋯ … … . 𝐴 𝑛−1 𝑥 + 𝐴 𝑛)𝑒 𝑟𝑥
1. C𝑥 𝑛 𝑐𝑜𝑠𝑎𝑥
2. C𝑥 𝑛
𝑠𝑖𝑛𝑎𝑥
𝑥 𝑘(𝐴𝑐𝑜𝑠𝑎𝑥 + 𝐵𝑠𝑖𝑛𝑎𝑥)
1. C𝑥 𝑛
𝑒 𝑟𝑥
𝑐𝑜𝑠𝑎𝑥
2. C𝑥 𝑛
𝑒 𝑟𝑥
𝑠𝑖𝑛𝑎𝑥
𝑥 𝑘
𝐴0 𝑥 𝑛
+ 𝐴1 𝑥 𝑛−1
+ ⋯ … … . 𝐴 𝑛−1 𝑥 + 𝐴 𝑛 𝑒 𝑟𝑥
𝑐𝑜𝑠
+ (𝐵0 𝑥 𝑛 + 𝐵1 𝑥 𝑛−1 + ⋯ … … . 𝐵 𝑛−1 𝑥
+ 𝐵𝑛)𝑒 𝑟𝑥
𝑠𝑖𝑛𝑎𝑥
THE P.I. WILL BE CONSTRUCTED ACCORDING TO
THE FOLLOWING TABLE:

7.  In 𝑥 𝑘 , k is the smallest +ve integer which will ensure that
no terms in 𝒚 𝒑 is already in the C.F.
 If 𝐹(𝑥) is sum of several terms, write 𝒚 𝒑 for each term
individually and then add up all of them.
 The 𝒚 𝒑 and its derivative will be substituted into the
equation 𝑓 𝐷 𝑦 = 𝐹 𝑥 and coefficients of like terms on
the left hand and right hand sides will be equated to
determine the U.C. 𝐴 𝑜, 𝐴1, … … 𝐴 𝑛 , 𝐵0, 𝐵1, … … 𝐵 𝑛.
PROPERTIES

8.  Real and Distinct
𝑦 𝑐 = 𝑐1 𝑒 𝑚1𝑥 + 𝑐2 𝑒 𝑚2𝑥 + ⋯ 𝑐 𝑛 𝑒 𝑚𝑛 𝑥
 Repeated Real Roots
𝑦 𝑐 = (𝑐1 + 𝑐2 𝑥 + ⋯ 𝑐 𝑛)𝑥 𝑒 𝑚1𝑥
 Complex Roots
𝑦 𝑐 = 𝑒 𝑎 𝑥
[𝑐1 sin 𝑏𝑥 + 𝑐2 cos 𝑏𝑥]
THREE TYPES OF ROOTS

9.  In medicine for modelling cancer growth or the
spread of disease
 In engineering for describing the movement of
electricity
 In chemistry for modelling chemical reactions.
 In economics to find optimum investment strategies
 In physics to describe the motion of waves,
pendulums or chaotic systems. It is also used in
physics with Newton's Second Law of Motion and the
Law of Cooling.
APPLICATIONS OF DIFFERENTIAL
EQUATIONS

10. Solve:
𝒚′′
− 𝟑𝒚′
+ 𝟐𝒚 = 𝒙 𝟑
𝒆 𝒙
---(1)
Solution:
The characteristic eq. is
𝐷2 − 3𝐷 + 2 = 𝑥3 𝑒 𝑥
For C.F.
𝐷2 − 2𝐷 − 𝐷 + 2 = 0
𝐷 𝐷 − 2 − 1 𝐷 − 2 = 0
𝐷 − 1 𝐷 − 2 = 0
𝐷 = 2 , 𝐷 = 1
So 𝑦 𝑐 is given by,
𝑦 𝑐 = 𝑐1 𝑒 𝑥 + 𝑐2 𝑒2𝑥
EXAMPLE:

11. For P.I.
Using the table, 𝑦 𝑝 is given by
𝑦 𝑝 = 𝑥 𝑘
(𝐴𝑥2
+ 𝐵𝑥 + 𝐶)𝑒 𝑥
Since 𝑒 𝑥 is already present in C.F. so we take k=1 so that no
term of C.F. is in 𝑦 𝑝.Hence the modified P.I. is
𝑦 𝑝 = (𝐴𝑥3
+ 𝐵𝑥2
+ 𝐶𝑥)𝑒 𝑥
Differentiate w.r.t x
𝑦 𝑝
′ = 𝐴𝑥3 + 𝐵 𝑥2 + 𝐶𝑥 𝑒 𝑥 + (3𝐴𝑥2+2𝐵𝑥 + 𝐶)𝑒 𝑥
Differentiate again w.r.t x
𝑦 𝑝
′′
= 𝐴𝑥3 + 𝐵𝑥2 + 𝐶𝑥 𝑒 𝑥 + 2(3𝐴𝑥2 + 2𝐵𝑥 + 𝐶)𝑒 𝑥 + (6𝐵𝑥 + 2𝐵)𝑒 𝑥
CONTD….

12. Substituting for 𝑦 𝑝, 𝑦 𝑝
′ , 𝑦 𝑝
′′ into (1), we have
𝐴𝑥3
+ 𝐵𝑥2
+ 𝐶𝑥 𝑒 𝑥
+ 2(3𝐴𝑥2
+ 2𝐵𝑥 + 𝐶)𝑒 𝑥
+ 6𝐵𝑥 + 2𝐵 𝑒 𝑥
−
3 𝐴𝑥3 + 𝐵𝑥2 + 𝐶𝑥 𝑒 𝑥 − 3(3𝐴𝑥2 + 2𝐵𝑥 + 𝐶)𝑒 𝑥 +2(𝐴𝑥3 + 𝐵𝑥2 +
𝐶𝑥)𝑒 𝑥 = 𝑥3 𝑒 𝑥
[−3𝐴𝑥2 6𝐴 − 2𝐵 𝑥 + 𝐶] = 𝑥3 𝑒 𝑥
Equating coefficients of like terms, we obtain
Coeff. of 𝑥3:
−3𝐴 = 1 𝑜𝑟 𝐴 = −
1
3
Coeff. of x:
6𝐴 − 2𝐵 = 0 𝑜𝑟 𝐵 = −1
Coeff. of 𝑥0:
𝐶 = 0
CONTD……

13. So,
𝑦 𝑝 = −
1
3
𝑥3 𝑒 𝑥 − 𝑥2 𝑒 𝑥
The general solution is given by,
𝑦 = 𝑦 𝑐 + 𝑦 𝑝
𝒚 = 𝒄 𝟏 𝒆 𝒙 + 𝒄 𝟐 𝒆 𝟐𝒙 −
𝟏
𝟑
𝒙 𝟑 𝒆 𝒙 − 𝒙 𝟐 𝒆 𝒙
CONTD….