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à 4.3 Non-Homogeneous Equations; Undetermïed Coefficients
äèFïd ê particular solution, given ê fundmental
èèèèèèèset ç solutions, ë ê non-homogeneous equation.
â è Forèy»» - 5y» + 6y = 2x + 3
The fundamental solutions set is eì╣ å eÄ╣.
Thus assume a particular solution ç ê formèAx + B
Substiutuion ïë ê differential equation yields
0 - 5A + 6(Ax + B) = 2x + 3
Solvïg yieldsèA = 1/3 å B = 7/9
The particular solution isèx/3 + 7/9
éS è The problem ç solvïg a lïear, NON-HOMOGENEOUS, second
order differential equation can be split ïë two parts
1) Solve ê homogeneous differential equation which will
produce two ïdependent solutions (ê fundamental set
ç solutions) say y¬ å y½.èThe general solution
ç ê homogeneous differential equation is
C¬y¬ + C½y½
2) Fïd any PARTICULAR SOLUTION ë ê NON-HOMGENEOUS
differential equation, say y╞.
èèThe general solution ë ê NON-HOMOGENEOUS differential
equation is ê sum ç êse two solutions
C¬y¬ + C½y½ + y╞
èè In this section, ê METHOD OF UNDETERMINED COEFFICIENTS
will be used ë fïd PARTICULAR SOLUTIONS ç ê NON-HOMOGEN-
EOUS differential equation.èThis method assumes ê form ç
ê particular solution based on ê form ç g(x), ê non-
homogeneous term.èIt is not a general method but håles a
large group ç common non-homogeneous terms.èIf this method
does not work, ê method ç VARIATION OF PARAMETERS (à
4.4) can be used.
èè The assumed particular solution is substituted ïë ê
non-homegeneous differential equation.èThis should produce
a system ç as many lïear equations as êre are UNDETERMINED
COEFFICIENTS.
èè For several choices ç g(x), a suggested form ç ê
particular solution are given below.
1. g(x) is a polynomial ç degree n.èThe assumed solution
is ê general polynomial ç degree n.èFor example, if
è g(x) = x║ - 3èênèy╞ = Axì + Bx + CèSubstiutution ïë
ê non-homogeneous solution should produce 3 equations for ê
3 undetermïed coefficients A, B å C.èIf x¡ is a member ç
ê fundamental solution set ç ê homogeneous differential
equation, multiply each term ç ê particular solution by
x¡ prior ë substitution.
2. g(x) = e¡╣ where m is real.èIf m is NOT a solution ç
ê characteristic equation ç ê homogeneous differential
equation, useèy╞ = Ae¡╣.è If m is a solution ç ê charact-
eristic equation ç ê homogeneous differential equation ç
MULTIPLICITY n, useèy╞ = Axⁿe¡╣.èFor example, if
g(x) = 3eÉ╣ å 5 is a root (not repeated) ç ê characteristic
equation, useèy╞ = AxeÉ╣.
3. g(x) = e¡╣sï[nx] or e¡╣cos[nx].èIf m ± ni are NOT
roots ç ê characteristic equation ç ê homogeneous
differential equation, useèy╞ = Ae¡╣sï[nx] + Be¡╣cos[nx]
If m ± ni are roots ç multiplicity l, use
èy╞ = Ax╚e¡╣sï[nx] + Bx╚e¡╣cos[nx]è For example, if 2 ± i
is NOT a solution ç ê characteristic equation, use
èy╞ = Aeì╣sï[x] + Beì╣cos[x].
If g(x) = g¬(x) + g½(x)èwhere g¬ å g½ are ç
different forms ç ê above types, use ê above techniques
ë fïd particular soltuions y¬╞ å y½╞ ë ê respective
non-homogeneous differential equation with g¬(x) å g½(x) as
êir right hå sides.èThe sumèy¬╞ + y½╞ will be a
particular solution ë ê overall problem with g(x) as its
non-homogeneous term.
1èèy»» + 4y» + 3y = 2xèFundamental set are eú╣ å eúÄ╣
A) 2/3 x + 8/9 B) 2/3 x - 8/9
C) -2/3 x + 8/9 D) -2/3 x - 8/9
ü èèAs g(x) = 2x is a first degree polynomial, ê assumed
particular solution isè
y╞ = Ax + B
Differentiatïg
y╞»è= A
y╞»» = 0
Substiutïg
0 + 4A + 3(Ax + B) = 2x
Rearrangïg
3Ax + 4A + 3B = 2x
For ê two sides ë be equal, ê coefficients ç each power
ç x must be ê same, êrefore
3Aèèè= 2
4A + 3B = 0
The solutions are
A =è2/3
B = -8/9
The particular solution is
y╞ = 2/3 xè- 8/9
The general solution is
C¬eú╣ + C½eúÄ╣ + 2/3 xè- 8/9
ÇèB
2è y»» - 4y» + 3y = 3xì ;èFundamental set are e╣ å eÄ╣
A) xì + 8x/3 + 26/9 B) xì + 8x/3 - 26/9
C) xì - 8x/3 + 26/9 D) xì - 8x/3 - 26/9
ü èèAs g(x) = 3xì is a second degree polynomial, ê assumed
particular solution isè
y╞ = Axì + Bx + C
Differentiatïg
y╞»è= 2Ax + B
y╞»» = 2A
Substiutïg
2A - 4(2Ax + B) + 3(Axì + Bx + C) = 3xì
Rearrangïg
3Axì + (-8A + 3B)x + (2A -4B + 3C) = 3xì
For ê two sides ë be equal, ê coefficients ç each power
ç x must be ê same, êrefore
3Aèèè = 3
-8A + 3B = 0
2A - 4B + 3C = 0
The solutions are
A =è1
B = 8/3
C = 26/9
The particular solution is
y╞ = xì + 8/3 xè+ 26/9
The general solution is
C¬e╣ + C½eÄ╣ + xì + 8/3 xè+ 26/9
ÇèA
3 y»» - 4y» + 3y = 2eì╣; Fundamental set are e╣ å eÄ╣
A) -1/2 eì╣ B) 1/2eúì╣
C) -2eì╣ D) 2eúì╣
ü èèAs g(x) = 2eì╣ is a real exponential å is not ï ê
fundamental set, ê assumed particular solution isè
y╞ = Aeì╣
Differentiatïg
y╞»è= 2Aeì╣
y╞»» = 4Aeì╣
Substiutïg
4Aeì╣ - 4(2Aeì╣) + 3Aeì╣ = 2eì╣
Rearrangïg
Aeì╣(4 - 8 + 3) = 2eì╣
è-Aeì╣ = 2eì╣
Dividïg both sides by eì╣ yields
-A = 2è orèA = -2
The particular solution is
y╞ = -2eì╣
The general solution is
C¬e╣ + C½eÄ╣ - 2eì╣
ÇèC
è4è y»» + 4y» + 3y = -3eúÅ╣ Fundamental set are eú╣ å eúÄ╣
A) 4eúÅ╣ B) eúÅ╣
C) -4eúÅ╣ D) -eúÅ╣
ü èèAs g(x) = -3eúÅ╣ is a real exponential å is not ï ê
fundamental set, ê assumed particular solution isè
y╞ = AeúÅ╣
Differentiatïg
y╞»è= -4AeúÅ╣
y╞»» = 16AeúÅ╣
Substiutïg
16AeúÅ╣ + 4(-4AeúÅ╣) + 3AeúÅ╣ = -3eúÅ╣
Rearrangïg
Aeì╣(16 - 16 + 3) = -3eúÅ╣
è3AeúÅ╣ = -3eúÅ╣
Dividïg both sides by eúÅ╣ yields
3A = -3è orèA = -1
The particular solution is
y╞ = -eúÅ╣
The general solution is
C¬eú╣ + C½eúÄ╣ - eúÅ╣
ÇèD
5èy»» - 4y» + 3y = 5eÄ╣ ; Fundamental set are e╣ å eÄ╣
A) 5/2 xeÄ╣ B) 5/2 eÄ╣
C) -5/2 xeÄ╣ D) -5/2 eÄ╣
ü èèAs g(x) = 5eÄ╣ is a real exponential å is ï ê
fundamental set, ê assumed particular solution isè
y╞ = AxeÄ╣
Differentiatïg
y╞»è= 3AxeÄ╣ + AeÄ╣
y╞»» = 9AxeÄ╣ + 6AeÄ╣
Substiutïg
9AeÄ╣ + 6AeÄ╣ - 4(3AeÄ╣ + AeÄ╣) + 3AeÄ╣ = 5eÄ╣
Rearrangïg
AeÄ╣(9 + 6 - 12 - 4 + 3) = 5eÄ╣
è2AeúÅ╣ = 5eÄ╣
Dividïg both sides by eÄ╣ yields
2A = 5è orèA = 5/2
The particular solution is
y╞ = 5/2 xeÄ╣
The general solution is
C¬e╣ + C½eÄ╣ + 5/2 xeÄ╣
ÇèA
6è y»» - 6y» + 9 = 2eúÄ╣ ; Fundamental set are eÄ╣ å xeÄ╣
A) 1/18 eúÄ╣ B) xìeúÄ╣
C) -1/18 eúÄ╣ D) -xìeúÄ╣
ü èèAs g(x) = 2eúÄ╣ is a real exponential å is not ï ê
fundamental set, ê assumed particular solution isè
y╞ = AeúÄ╣
Differentiatïg
y╞»è= -3AeúÄ╣
y╞»» = 9AeúÄ╣
Substiutïg
9AeúÄ╣ - 6(-3AeúÄ╣) + 9AeúÄ╣ = 2eúÄ╣
Rearrangïg
AeúÄ╣(9 + 18 + 9) = 2eúÄ╣
36AeúÄ╣ = 2eúÄ╣
Dividïg both sides by eúÄ╣ yields
18A = 2è orèA = 1/18
The particular solution is
y╞ = 1/18 eúÄ╣
The general solution is
C¬e3╣ + C½xeÄ╣ + 1/18 eúÄ╣
ÇèA
7è y»» - 6y» + 9 = 2eÄ╣ ; Fundamental set are eÄ╣ å xeÄ╣
A) 1/18 eÄ╣ B) xìeÄ╣
C) -1/18 eÄ╣ D) -xìeÄ╣
ü èèAs g(x) = 5eÄ╣ is a real exponential å is repeated ï
ê fundamental set, ê assumed particular solution isè
y╞ = AxìeÄ╣
Differentiatïg
y╞»è= 3AxìeÄ╣ + 2AxeÄ╣
y╞»» = 9AxìeÄ╣ + 12AxeÄ╣ + 2AeÄ╣
Substiutïg
9AxìeÄ╣ + 12AxeÄ╣ +2AeÄ╣- 6(3AxìeÄ╣ + 2AxeÄ╣)
è + 9AxìeÄ╣ = 2eÄ╣
Rearrangïg
AeÄ╣[ xì(9 - 18 + 9) + x(12 - 12) + 2) = 2eÄ╣
è2AeÄ╣ = 2eÄ╣
Dividïg both sides by eÄ╣ yields
2A = 2è orèA = 1
The particular solution is
y╞ = xìeÄ╣
The general solution is
C¬e╣ + C½xeÄ╣ + xìeÄ╣
ÇèB
8èy»» - 4y» + 3y = sï[x]èFundamental set are e╣ å eÄ╣
A)èè1/5 cos[x] + 1/10 sï[x]
B)èè1/5 cos[x] - 1/10 sï[x]
C)èè-1/5 cos[x] + 1/10 sï[x]
D)èè-1/5 cos[x] - 1/10 sï[x]
ü èèAs g(x) = sï[x] results from a complex conjugate pair
as ê solution ç ê characteristic equation but it is not
ï ê fundamental set, ê assumed particular solution isè
y╞ = Acos[x] + Bsï[x]
Differentiatïg
y╞»è= -Asï[x] + Bcos[x]
y╞»» = -Acos[x] - Bsï[x]
Substiutïg
-Acos[x] - Bsï[x] - 4{-Asï[x] + Bcos[x]}
+ 3{Acos[x] + Bsï[x]} = sï[x]
Rearrangïg
cos[x]{-A - 4B + 3A} + sï[x]{-B + 4A + 3B} = sï[x]
è2AeÄ╣ = 2eÄ╣
For this ë be true for all x, ê coefficients ç both cos[x]
å sï[x] must match yieldïg ê two equations
2A - 4B = 0
4A + 2B = 1
Solvïg this system ç lïear equations yields
A = 1/5
B = 1/10
The particular solution is
y╞ = 1/5 cos[x] + 1/10 sï[x]
The general solution is
C¬e╣ + C½xeÄ╣ + 1/5 cos[x] + 1/10 sï[x]
ÇèA
9èy»» + 4y» + 3y = 2e╣cos[2x]èFundamental set are eú╣, eúÄ╣
A) 3/20 e╣cos[2x] + 1/20 e╣sï[2x]
B) 1/20 e╣cos[2x] + 3/20 e╣sï[2x]
C) 3/20 e╣cos[2x] - 1/20 e╣sï[2x]
D) 1/20 e╣cos[2x] - 3/20 e╣sï[2x]
ü èèAs g(x) = e╣cos[2x] results from a complex conjugate pair
as ê solution ç ê characteristic equation but it is not
ï ê fundamental set, ê assumed particular solution isè
y╞ = Ae╣cos[2x] + Be╣sï[2x]
Differentiatïg
y╞»è= Ae╣cos[2x] - 2Ae╣sï[2x]
+ Be╣sï[2x] + 2Be╣cos[2x]
y╞»» = Ae╣cos[2x] - 2Ae╣sï[2x] - 2Ae╣sï[2x]
- 4Ae╣cos[2x] + Be╣sï[2x] + 2Be╣cos[2x]
+ 2Be╣cos[2x] - 4Be╣sï[2x]
Substiutïg
-3Ae╣cos[2x] - 4Ae╣sï[2x] - 3Be╣sï[2x]
+ 4Be╣cos[2x] + 4{Ae╣cos[2x] - 2Ae╣sï[2x]
+ Be╣sï[2x] + 2Be╣cos[2x]}
+ 3{Ae╣cos[2x] + Be╣sï[2x]}è=è2e╣cos[2x]
Rearrangïg
e╣cos[2x]{-3A + 4B + 4A + 8B + 3A}
+ e╣sï[2x]{-4A - 3B - 8A + 4B + 3B} = 2e╣cos[2x]
{4A + 12B}e╣cos[2x] + {-12A + 4b}e╣sï[2x] = 2e╣cos[2x]
For this ë be true for all x, ê coefficients ç both e╣cos[x]
å e╣sï[x] must match yieldïg ê two equations
è4A + 12B = 2
-12A +è4B = 0
Solvïg this system ç lïear equations yields
A = 1/20
B = 3/20
The particular solution is
y╞ = 1/20 e╣cos[2x] + 3/20 e╣sï[2x]
The general solution is
C¬e╣ + C½xeÄ╣ + 1/20 e╣cos[2x] + 3/20 e╣sï[2x]
ÇèB
10è y»» + 4y = 3cos[2x]èFundamental set are cos[2x],sï[2x]
A) 3/4 x cos[2x] èB) -3/4 x cos[2x]
C) 3/4 x sï[2x] èD) -3/4 x sï[2x]
ü èèAs g(x) = 3cos[2x] results from a complex conjugate pair
as ê solution ç ê characteristic equation å it is ï
ê fundamental set, ê assumed particular solution isè
y╞ = Axcos[2x] + Bxsï[2x]
Differentiatïg
y╞»è= Acos[2x] - 2Axsï[2x] + Bsï[2x] + 2Bxcos[2x]
y╞»» = -2Asï[2x] - 2Asï[2x] - 4Axcos[2x]
+ 2Bcos[2x] + 2Bcos[2x] - 4Bxsï[2x]
Substiutïg
-4Asï[2x] - 4Axcos[2x] + 4Bcos[2x] - 4Bxsï[2x]
+ 4{Axcos[2x] + Bxsï[2x]} = 3cos[2x]
Rearrangïg
x cos[2x]{-4A + 4A} + cos[2x]{4B} + sï[2x]{-4a}
èx sï[2x]{-4B + 4B} = 3 cos[2x]
or 4Bcos[2x] - 4Asï[2x] = 3 cos[2x]
For this ë be true for all x, ê coefficients ç both e╣cos[x]
å e╣sï[x] must match yieldïg ê two equations
è 4B = 3
è-4A = 0
Solvïg this system ç lïear equations yields
A = 0
B = 3/4
The particular solution is
y╞ = 3/4 x sï[2x]
The general solution is
C¬e╣ + C½xeÄ╣ + 3/4 x sï[2x]
ÇèC