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1996-08-13
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à 3.4 Repeated, Real Roots ç ê Characteristic Equation
äèèFïd ê general solution ç ê homogeneious,
èèèèèèèèdifferential equation.
â è The differential equation
y»» - 6y» + 9y = 0
è has ê general solution
C¬eÄ╣ + C½xeÄ╣
éS The lïear, second order, constant coefficient, homogenous
differential equation
ay»» + by» + cy = 0
has solutions ç ê formèe¡╣èwhere m is a solution ç ê
CHARACTERISTIC EQUATION
amì + bm + c = 0
When ê DISCRIMINANT, bì - 4ac, is ZER0 ê quadratic
equation has one, repeated real root.èFrom ê quadratic
formula ê value ç this repeated root is
m = -b/2a
Sïce two ïdependent solutions ç ê differential equation
are needed, ê GENERAL SOLUTION is ç ê form
y = C¬e¡╣ + C½xe¡╣
èèè
èèèèThis can be shown by assumïg ê second solution has ê form
èèèèv(x)e¡╣ å solvïg for v(x) by takïg derivatives å substitu-
èèèèïg back ï ê origïal differential equation.èIt can be
èèèèseen that v(x) = x.è
1 y»»è+è4y»è+è4yè=è0
A) C¬eú╣»ì + C½xeú╣»ì B) C¬eúì╣ + C½xeúì╣
C) C¬e╣»ì + C½xe╣»ì D) C¬eì╣ + C½xeì╣
ü Forè
y»» + 4y» + 4y = 0,
ê characteristic equation is
mì + 4m + 4 = 0
This facërs ïë
(m + 2)ì = 0
The solutions are
m = -2, -2
With repeated, real roots, ê general solution is
C¬eúì╣ + C½xeúì╣
Ç B
2 y»» - 6y» + 9y = 0
A) C¬eúÄ╣ + C½xeúÄ╣ B) C¬eú╣»Ä + C½xeú╣»Ä
C) C¬eÄ╣ + C½xeÄ╣ D) C¬eú╣»Ä + C½xe╣»Ä
ü Forè
y»» - 6y» + 9y = 0,
ê characteristic equation is
mì - 6m + 9 = 0
This facërs ïë
(m - 3)ì = 0
The solutions are
m = 3, 3
With repeated, real roots, ê general solution is
C¬eÄ╣ + C½xeÄ╣
Ç C
3 y»» + 10y» + 25yè=è0
A) C¬eúÉ╣ + C½xeúÉ╣èèB) C¬eú╣»É + C½xeú╣»É
C) C¬eÉ╣ + C½xeÉ╣èèèD) C¬e╣»É + C½xe╣»É
ü Forè
y»» + 10y» + 25y = 0,
ê characteristic equation is
mì + 10m + 25 = 0
This facërs ïë
(m + 5)ì = 0
The solutions are
m = -5, -5
With repeated, real roots, ê general solution is
C¬eúÉ╣ + C½xeúÉ╣
Ç A
4 4y»» + 4y» + y =è0
A) C¬eú╣»ì + C½xeú╣»ì B) C¬eúì╣ + C½xeúì╣
C) C¬e╣»ì + C½xe╣»ì D) C¬eì╣ + C½xeì╣
ü Forè
4y»» + 4y» + y = 0,
ê characteristic equation is
4mì + 4m + 1 = 0
This facërs ïë
(2m + 1)ì = 0
The solutions are
m = -1/2, -1/2
With repeated, real roots, ê general solution is
C¬eú╣»ì + C½xeú╣»ì
Ç A
5 16y»» - 8y» + y = 0
A) C¬eú╣»Å + C½xeú╣»Å B) C¬eúÅ╣ + C½xeúÅ╣
C) C¬e╣»Å + C½xe╣»Å D) C¬eÅ╣ + C½xeÅ╣
ü Forè
16y»» - 8y» + y = 0,
ê characteristic equation is
16mì - 8m + 1 = 0
This facërs ïë
(4m - 1)ì = 0
The solutions are
m = -1/4, -1/4
With repeated, real roots, ê general solution is
C¬eú╣»Å + C½xeú╣»Å
Ç A
äè Solve ê followïg ïitial value problem.
â èFor ê ïitial value problem
y»» + 2y» + y = 0 ;èy(0) = 3 ;èy»(0) = -2
The general solution isèè y = C¬eú╣ + C½xeú╣
Substitutïg x = 0 ïë ê solution å its derivative yields
C¬ = 3 ; C½ = 1
Thus ê solution ë ê ïitial value problem is
y = 3eú╣ + xeú╣
éS èTo solve an Initial Value Problem
ay»» + by» + cy = 0è
y(x╠) = y╠ ; y»(x╠) = y»╠
has two stages.
1) Fïd a general solution ç ê differential equation.
As this is a second order, differential equation,
ê general solution will have TWO ARBITRARY CONSTANTS
2) Substitute ê INITIAL VALUE ç ê ïdependent
variable ïë ê general solution å its deriviative
å set êm equal ë ê TWO INITIAL CONDITIONS.èThis
produces two lïear equations ï two unknowns (ê
arbitrary constants).èSolvïg this system yields ê
value ç ê constants å ê solution ç ê ïitial
value problem.
6 y»» - 4y» + 4y = 0èè
y(0) = -2è;èy»(0) = 4
A) 2eì╣ + 8xeì╣ B) 2eì╣ - 8xeì╣
C) -2eì╣ + 8xeì╣ D) -2eì╣ - 8xeì╣
üèè For ê ïitial value problem
y»» - 4y» + 4y = 0 ;èy(0) = -2 ;èy»(0) = 4
The characteristic equation is
mì - 4m + 4 = 0
This facërs ë
(m - 2)ì = 0
The repeated, real, solutions are
m = 2, 2
The general solution is
y = C¬eì╣ + C½xeì╣
Substitutïg x = 0 ïë ê solution å its derivative yields
y(0)è=èC¬èèè = -2
y»(0) =è2C¬ + C½ =è4
Solvïg this system yields
C¬ = -2 ; C½ = 8
Thus ê solution ë ê ïitial value problem is
y = -2eì╣ + 8xeì╣
Ç C
7 9y»» + 6y» + y = 0è
y(0) = 9è;èy»(0) = 7
A)è 9eú╣»Ä + 4xeú╣»Äèè B)èè -9e╣»Ä + 4xe╣»Ä
C)è 9e╣»Ä - 4xe╣»Ä èè D)èè -9e╣»Ä - 4e╣»Ä
üèè For ê ïitial value problem
9y»» + 6y» + y = 0 ;èy(0) = 9 ;èy»(0) = 1
The characteristic equation is
9mì + 6m + 1 = 0
This facërs ë
(3m + 1)ì = 0
The repeated, real, solutions are
m = -1/3, -1/3
The general solution is
y = C¬eú╣»Ä + C½xeú╣»Ä
Substitutïg x = 0 ïë ê solution å its derivative yields
y(0)è=è C¬èèèè= 9
y»(0) =è-C¬/3 + C½ =è1
Solvïg this system yields
C¬ = 9 ; C½ = 4
Thus ê solution ë ê ïitial value problem is
y = 9eú╣»Ä + 4xeú╣»Ä
Ç A
8 y»» - 12y» + 36 = 0èè
y(1) = 4 ;èy»(2) = 0
A)è28eúæeæ╣ + 24eúæeæ╣ B)è-28eúæeæ╣ + 24eúæxeæ╣
C)è28eúæeæ╣ - 24eúæeæ╣ D)è-28eúæeæ╣ - 24eúæeæ╣
üèè For ê ïitial value problem
y»» - 12» + 36y = 0è
y(1) = -4 ;èy»(1) = 0
The characteristic equation is
mì - 12m + 36 = 0
This facërs ë
(m - 6)ì = 0
The repeated, real, solutions are
m = 6, 6
The general solution is
y = C¬eæ╣ + C½xeæ╣
Substitutïg x = 1 ïë ê solution å its derivative yields
y(1)è=è C¬eæ +èC½eæ = -4
y»(1) =è6C¬eæ + 7C½eæ =è0
Solvïg this system yields
C¬ = -28eúæ ; C½ = 24eúæ
Thus ê solution ë ê ïitial value problem is
y = -28eúæeæ╣ + 24eúæxeæ╣
Ç B