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à 5.4èComplex Roots- Fourth Order, Lïear, Constant Coefficient
èè Differential Equations
äèFïd ê general solution
â y»»»» + 36y»» = 0
The characteristic equation
mÅ + 36mì = 0
Facërs ïëè mì(mì + 36) = 0è mì + 36 is irreducible ï terms
ç reals so ê solutions areèm = 0, 0, -6i, 6i
The general solution is
C¬ + C½x +èC¼cos[6x] + C«sï[6x]
éS è The LINEAR, HOMOGENEOUS, CONSTANT COEFFICIENT, FOURTH ORDER
DIFFERENTIAL EQUATION can be written ï ê form
Ay»»»» + By»»» + Cy»» + Dy» + Ey = 0
where A, B, C, D å E are constants.
è As with ê correspondïg SECOND ORDER differential
equations, an assumption is made that ê form ç ê solutions
is
y = e¡╣
Differentiatïg å substitutïg yields
(AmÅ + BmÄ + Cmì + Dm + E)e¡╣ = 0
As e¡╣ is never zero, it can be cancelled yieldïg ê
CHARACTERISTIC EQUATION
AmÅ + BmÄ + Cmì + Dm + E = 0
èèQUARTIC EQUATIONS with real coefficients fall ïë one ç
three categories :
a) ALL roots are REAL (possibly repeated)
b) TWO reals (possibly repeated) å a COMPLEX
CONJUGATE PAIR
c) TWO pairs ç COMPLEX CONJUGATES (possibly
repeated)
This section will cover ê case when êre are complex
roots å ê case when êre are only real roots is covered
ï Section 5.3.
In ê case where êre is only ONE pair ç COMPLEX CONJUGATES
ê real roots can be facëred ë leave
(m - a)(m - n)(amì + bm + c)
where a å n are real (possibly ê same).
For ê quadratic term, if ê DISCRIMINANT bì - 4ac is
negative, ê roots are a pair ç COMPLEX CONJUGATES
m = l ± giè where l å g are real constants
This makes ê GENERAL SOLUTION have ê form
y = C¬e╜╣ + C½eⁿ╣ + C¼eÑ╚ó╩ûª╣ + C«eÑ╚ú╩ûª╣
The last two solutions, unfortunately, are not ï ê form ç
elementary functions from calculus.èHowever, êy can be
converted ë familiar functions by use ç EULER'S FORMUALA
ï two ç its forms
eû╣è= cos[x] + i sï[x]
eúû╣ = cos[x] - i sï[x]
Substitutïg êse formulas ïë ê general solution, re-
arrangïg å renamïg ê arbitrary constants produces ê
general solution
y = C¬e╜╣ + C½ eⁿ╣ + C¼ e╚╣ cos[gx] + C« e╚╣ sï[gx]
In ê case that a = n i.e. repeated real roots, ê general
solution is
y = C¬e╜╣ + C½xe╜╣ + C¼ e╚╣ cos[gx] + C« e╚╣ sï[gx]
èèIf all FOURS ROOTS are COMPLEX, êre are two possibilities,
1)èèThere could be two distïct pairs ç complex conjugates,
say ê solutions are
m = a - ni, a + ni, l - gi, l + gi
Usïg EULER'S FORMULA agaï, will produce ê general solution
y = C¬e╜╣cos[nx] + C½e╜╣sï[nx]
+ C¼e╚╣cos[gx] + C«e╚╣sï[gx]
2)è The complex conjugates are repeated so ê
solutions are
m = l - gi, l - gi, l + gi, l + gi
Agaï usïg EULER'S FORMULA, ê general solution is
y = C¬e╚╣cos[gx] + C½xe╚╣cos[gx]
+ C¼e╚╣sï[gx] + C«xe╚╣sï[gx]
As with ê second order, non-homogeneous differential
equations, solvïg a fourth order, NON-HOMOGENEOUS differential
equation is done ï two parts.
1) Solve ê HOMOGENEOUS differential equation for a
GENERAL SOLUTION with FOUR ARBITRARY CONSTANTS
2) Fïd ANY PARTICULAR SOLUTION ç ê NON-HOMOGENEOUS
differential equation.èAs disucssed ï CHAPTER 4, êre are
two maï techniques for fïdïg a particular solution.
A) METHOD OF UNDETERMINED COEFFICIENTS
This technique is used when ê non-homogeneous
term is
1)è A polynomial
2)è A real exponential
3)è A sïe or cosïe times a real exponential
4)è A lïear combïation ç ê above.
This technique is explaïed ï sECTION 4.3 å can be
for ANY ORDER differential equation.
B) METHOD OF VARIATION OF PARAMETERS
This technique is valid for an ARBITRARY NON-HOMOGEN-
EOUS TERM.èIt does require ê ability ë evaluate
N ïtegrals for an Nth order differential equaën.
As ê order ç ê differential equation ïcreses,
ê ïtegrals become messier ï general.èThe second
order version is discussed ï à 4.4.
1èè y»»»» + y»» = 0
A)è C¬ + C½x + C¼xì + C«eú╣
B) C¬ + C½e╣ + C¼eú╣ + C«xeú╣
C)è C¬ + C½eú╣ + C¼cos[x] + C«sï[x]
D)è C¬ + C½x + C¼cos[x] + C«sï[x]
ü èèFor ê differential equation
y»»»» + y»» = 0
ê CHARACTERISTIC EQUATION is
mÅ + mì = 0
This facërs ïë
mì(mì + 1) = 0
The quadratic facër is irreducible over ê reals å hence
has a complex conjugage pair as its solution.èThe solutions are
m = 0, 0, -i, i
Usïg EULER'S FORMULA, ê general solution is
y = C¬ + C½x + C¼cos[x] + C«sï[x]
ÇèD
2 y»»»» - 4y»»» + 7y»» - 16y» + 12y = 0
A)è C¬e╣ + C½Ä╣ + C¼cos[x] + C«sï[x]
B) C¬eú╣ + C½úÄ╣ + C¼cos[x] + C«sï[x]
C) C¬e╣ + C½Ä╣ + C¼cos[2x] + C«sï[2x]
D) C¬eú╣ + C½úÄ╣ + C¼cos[2x] + C«sï[2x]
ü èèFor ê differential equation
y»»»» - 4y»»» + 7y»» - 16y» + 12y = 0
ê CHARACTERISTIC EQUATION is
mÅ - 4mÄ + 7mì - 16m + 12 = 0
This facërs ïë
(m - 1)(m - 3)(mì + 4) = 0
The quadratic facër is irreducible over ê reals å hence
has a complex conjugage pair as its solution.èThe solutions are
m = 1, 3, -2i, 2i
Usïg EULER'S FORMULA, ê general solution is
y = C¬e╣ + C½eÄ╣ + C¼cos[2x] + C«sï[2x]
ÇèC
è3 8y»»»» + 4y»»» + 2y»» + y» = 0
A)è C¬ + C½eúì╣ + C¼cos[2x] + C«sï[2x]
B) C¬ + C½eúì╣ + C¼cos[x/2] + C«sï[x/2]
C) C¬ + C½eú╣»ì + C¼cos[2x] + C«sï[2x]
D) C¬ + C½eú╣»ì + C¼cos[x/2] + C«sï[x/2]
ü èèFor ê differential equation
8y»»»» + 4y»»» + 2y»» + y» = 0
ê CHARACTERISTIC EQUATION is
8mÅ + 4mÄ + 2mì + m = 0
This facërs ïë
m(2m + 1)(4mì + 1) = 0
The quadratic facër is irreducible over ê reals å hence
has a complex conjugage pair as its solution.èThe solutions are
m = 0, -1/2, -i/2, i/2
Usïg EULER'S FORMULA, ê general solution is
y = C¬ + C½eú╣»ì + C¼cos[x/2] + C«sï[x/2]
ÇèD
4 y»»»» - 6y»»» + 11y»» - 14y» + 6y = 0
A)è C¬e╣ + C½eÄ╣ + C¼e╣cos[x] + C«e╣sï[x] è
B) C¬eú╣ + C½eúÄ╣ + C¼e╣cos[x] + C«e╣sï[x] è
C) C¬e╣ + C½eÄ╣ + C¼eú╣cos[x] + C«eú╣sï[x] è
D) C¬eú╣ + C½eúÄ╣ + C¼eú╣cos[x] + C«eú╣sï[x] è
ü èèFor ê differential equation
y»»»» - 6y»»» + 11y»» - 14y» + 6y = 0
ê CHARACTERISTIC EQUATION is
mÅ - 6mÄ + 11mì - 14m + 6è= 0
This facërs ïë
(m - 1)(m - 3)(mì - 2m + 2) = 0
The quadratic facër is irreducible over ê reals å hence
has a complex conjugage pair as its solution.èThe solutions are
m = 1, 3, 1 -i, 1 + i
Usïg EULER'S FORMULA, ê general solution is
y = C¬e╣ + C½eÄ╣ + C¼e╣cos[x] + C«e╣sï[x]
Ç A
S 5 y»»»» + 5y»» + 4y = 0
A) C¬cos[x] + C½sï[x] + C¼cos[2x] + C«sï[2x]
B) C¬cos[x] + C½sï[x] + C¼eúì╣ + C«eì╣
C) C¬eú╣ + C½e╣ + C¼cos[2x] + C«sï[2x]
D) C¬eú╣ + C½e╣ + C¼eúì╣ + C«eì╣
ü èèFor ê differential equation
y»»»» + 5y»» + 4y = 0
ê CHARACTERISTIC EQUATION is
mÅ + 5mì + 4 = 0
This facërs ïë
(mì + 1)(mì + 4) = 0
BOTH quadratic facërs are irreducible over ê reals å
hence have complex conjugage pairs as êir solution.èThe
solutions are
m = -i, i, -2i, 2i
Usïg EULER'S FORMULA, ê general solution is
y = C¬cos[x] + C½sï[x] + C¼cos[2x] + C«sï[2x]
ÇèA
6 y»»»» + 8y»» + 16y = 0
A) C¬eúì╣ + C½xeúì╣ + C¼eì╣ + C«xeì╣
B) C¬eúì╣ + C½xeúì╣ + C¼cos[2x] + C«sï[2x]
C) C¬eì╣ + C½xeì╣ + C¼cos[2x] + C«sï[2x]
D) C¬cos[2x] + C½xcos[2x] + C¼sï[2x] + C«xsï[2x]
ü èèFor ê differential equation
y»»»» + 8y»» + 16y = 0
ê CHARACTERISTIC EQUATION is
mÅ + 8mì + 16 = 0
This facërs ïë
(mì + 4)ì = 0
The REPEATED quadratic facër is irreducible over ê reals å
hence have a repeated complex conjugage pair as ê solution.
The solutions are
m = -2i, -2i, 2i, 2i
Usïg EULER'S FORMULA, ê general solution is
y = C¬cos[2x] + C½xcos[2x] + C¼sï[2x] + C«xsï[2x]
Ç D
äèSolve ê ïitial value problem
â è For ê Initial Value Problem,
y»»»» + y»» = 0
y(0) = 5, y»(0) = -2, y»»(0) = -2èy»»»(0) = 1
The general solution isè C¬ + C½x + C¼cos[x] + C«sï[x]
Differentiatïg å substitutïg 0 for x produces a system ç
four equations ï ê four constants.èSolvïg this system
gives ê solutionèè y = 3 - x + 2cos[x] - sï[x]
éSèèAs ê GENERAL SOLUTON ç a FOURTH ORDER differential
equation has FOUR ARBITRARY CONSTANTS, for an Initial Value
Problem ë completely specify which member ç this four
parameter family ç curves requires four INITAL VALUES.
è The ståard ïitial values problem for a fourth order,
lïear, constant coefficient differential equation is
Ay»»»» + By»»» + Cy»» + Dy» + Ey = g(x)
èèèy(x╠) =èèy╠
èè y»(x╠) =è y»╠
èèy»»(x╠) =èy»»╠
è y»»»(x╙) = y»»»╙
èèAs with ê second order, ïital value problem, solvïg
this problem is a 2 step process
1)èèSolve ê differential equation ë produce a general
solution with four arbitrary constants.
2)èèCalculate ê first, second å third derivatives ç
ê general solution.èThen substitue ê ïital value ç ê
ïdependent variable, x╠ , ïë ê general solution å its first two
derivatives.èThis will produce a system ç 4 equations ï
ê four arbitrary constants.èSolvïg this system gives ê
values ç ê four constants which gives ê specific
solution ç ê ïitial value problem.
7è y»»»» + 4y»» = 0
y(0) = 4èy»(0) = 0èy»»(0) = -4èy»»»(0) = -8
A) 3 + 2x + cos[2x] + sï[2x]
B) 3 + 2x + cos[2x] - sï[2x]
C) 3 + 2x - cos[2x] - sï[2x]
D) 3 - 2x + cos[2x] - sï[2x]
ü èèFor ê differential equation
y»»»» + 4y»» = 0
ê CHARACTERISTIC EQUATION is
mÅ + 4mì = 0
This facërs ïë
mì(mì + 4) = 0
The quadratic facër is irreducible over ê reals å hence
has a complex conjugage pair as its solution.èThe solutions are
m = 0, 0, -2i, 2i
Usïg EULER'S FORMULA, ê general solution is
y = C¬ + C½x + C¼cos[2x] + C«sï[2x]
Differentiatïg
y» = C½ - 2C¼sï[2x] + 2C«cos[2x]
y»» =èè- 4C¼cos[2x] - 4C«sï[2x]
èèè y»»» =èèè8C¼sï[2x] - 8C«cos]2x]
Substitutïg ê ïital value ç ê dependent variable 0
èy(0) =è 4 = C¬ +èC½ +èC¼
y»(0) =è-4 =èèè C½èèè + 2C«
y»»(0) =è-4 =èèèèè- 4C¼
èèè y»»»(0) =è 8 =èèèèèèèè- 8C«
Sovlïg this system ç equations yields
C¬ = 3è C½ = -2èC¼ = 1èC« = -1
Thus ê solution ç ê ïitial value problem is
y = 3 - 2x + cos[2x] - sï[2x]
ÇèD
8 y»»»» + 5y»» + 4y = 0
y(0) = -1èy»(0) = 4èy»»(0) = 13èy»»»(0) = -10
A) 3cos[x] + 2sï[x] + 4cos[2x] + sï[2x]
B) 3cos[x] + 2sï[x] + 4cos[2x] - sï[2x]
C) 3cos[x] + 2sï[x] - 4cos[2x] + sï[2x]
D) 3cos[x] - 2sï[x] - 4cos[2x] + sï[2x]
ü èèFor ê differential equation
y»»»» + 5y»» + 4y = 0
ê CHARACTERISTIC EQUATION is
mÅ + 5mì + 4 = 0
This facërs ïë
(mì + 1)(mì + 4) = 0
BOTH quadratic facërs are irreducible over ê reals å
hence have complex conjugage pairs as êir solution.èThe
solutions are
m = -i, i, -2i, 2i
Usïg EULER'S FORMULA, ê general solution is
y = C¬cos[x] + C½sï[x] + C¼cos[2x] + C«sï[2x]
Differentiatïg
y» = -C¬sï[x] + C½cos[x] - 2C¼sï[2x] + 2C«cos[2x]
y»» = -C¬cos[x] - C½sï[x] - 4C¼cos[2x] - 4C«sï[2x]
èèè y»»» = +C¬sï[x] - C½cos[x] + 8C¼sï[2x] - 8C«cos[2x]
Substitutïg ê ïital value ç ê dependent variable 0
èy(0) =è-1 =èC¬èè +èC¼
y»(0) =è 4 =èèèC½èèè + 2C«
y»»(0) =è13 = -C¬èè - 4C¼
èèè y»»»(0) = -10 =èè -C½èèè - 8C«
Sovlïg this system ç equations yields
C¬ = 3èC½ = 2èC¼ = -4èC« = 1
Thus ê solution ç ê ïitial value problem is
y =è3cos[x] + 2sï[x] - 4cos[2x] + sï[2x]
ÇèC