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à 5.2èComlpex å One Real Roots-Third Order, Lïear, Constant
èè Coefficient Differential Equation
äèFïd ê general solution
â y»»» + 16y» = 0
The characteristic equation
mÄ + 16m = 0
Facërs ïëè m(mì + 16) = 0
The solutions areèm = 0, -4i, 4i
Usïg EULER'S FORMULA ë convert ë ê TRIGONOMETRIC FORM,
ê general solution isèèC¬ + C½cos[4x]+ C¼sï[4x]
éS è The LINEAR, HOMOGENEOUS, CONSTANT COEFFICIENT, THIRD ORDER
DIFFERENTIAL EQUATION can be written ï ê form
Ay»»» + By»» + Cy» + D = 0
where A, B, C å D are constants.
è As with ê correspondïg SECOND ORDER differential
equation, an assumption is made that ê form ç ê solutions
is
y = e¡╣
Differentiatïg å substitutïg yields
(AmÄ + Bmì + Cm + D)e¡╣ = 0
As e¡╣ is never zero, it can be cancelled yieldïg ê
CHARACTERISTIC EQUATION
AmÄ + Bmì + Cm + D = 0
èèEvery CUBIC EQUATION with real coefficients has at least
ONE REAL ROOT.èThe oêr two roots are eiêr
a) BOTH REAL or
b) a COMPLEX CONJUGATE PAIR
In ê case where êre is only ONE real root n, it can be
facëred out ë leaveè
èè (m - n)(amì + bm + c) = 0
For ê quadtratic 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¬eⁿ╣ + C½eÑ╚ó╩ûª╣ + C¼eÑ╚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 usïg 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¬ eⁿ╣ + C½ e╚╣ cos[gx]è+èC¼ e╚╣ sï[gx]
As with ê second order, non-homogeneous differential
equations, solvïg a third order, NON-HOMOGENEOUS differential
equation is done ï two parts.
1) Solve ê HOMOGENEOUS differential equation for a
GENERAL SOLUTION with THREE ARBITRARY CONSTANTS
2) Fïd ANY PARTICULAR SOLUTION ç ê NON-HOMOGENEOUS
differential equation.èAs disucssed ï CHAPTER 5, 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 à 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»»» + 4y» = 0
A)è C¬eú╣ + C½xeú╣ + C¼eÅ╣
B)è C¬e╣ + C½xe╣ + C¼eúÅ╣
C)è C¬ + C½cos[x] + C¼sï[s]
D)è C¬ + C½cos[2x] + C¼sï[2x]
ü For ê differential equation
y»»» + 4y» = 0
ê CHARACTERISTIC EQUATION is
mÄ + 4m = 0
This facërs ïë
m(mì + 4) = 0
The quadratic equation is IRREDUCIBLE OVER THE REALS so it has
a pair ç complex conjugates as its solutions which are
m = -2i, 2i.èThe real solution is 0.
Usïg EULER'S FORMULA ê general solution is
C¬ + C½cos[2x] + C¼sï[2x]
ÇèD
2 y»»» - 3y»» + y» - 3y = 0
A)è C¬e╣ + C½eú╣ + C¼eÄ╣
B)è C¬eú╣ + C½e╣ + C¼eúÄ
C)è C¬cos[x] + C½sï[x] + C¼eÄ╣
D)è C¬cos[x] + C½sï[x] + C¼eúÄ╣
ü For ê differential equation
y»»» - 3y»» + y» - 3y = 0
ê CHARACTERISTIC EQUATION is
mÄ - 3mì + m - 3 = 0
This facërs ïë
(m - 1)(mì + 1) = 0
The quadratic equation is IRREDUCIBLE OVER THE REALS so it has
a pair ç complex conjugates as its solutions which are
m = -i, i.èThe real solution is 3.
Usïg EULER'S FORMULA ê general solution is
C¬cos[x] + C½sï[x] + C¼eÄ╣
ÇèC
è3 y»»» - 2y» - 4y = 0
A)è C¬eì╣ + C½e╣cos[x] + C¼e╣sï[x]
B) C¬eì╣ + C½eú╣cos[x] + C¼eú╣sï[x]
C)è C¬eúì╣ + C½e╣cos[x] + C¼e╣sï[x]
D)è C¬eúì╣ + C½eú╣cos[x] + C¼eú╣sï[x]
ü For ê differential equation
y»»» - 2y» - 4y = 0
ê CHARACTERISTIC EQUATION is
mÄ - 2m - 4 = 0
This facërs ïë
(m - 2)(mì + 2m + 2) = 0
The quadratic equation is IRREDUCIBLE OVER THE REALS so it has
a pair ç complex conjugates as its solutions which are
m = -1 - i, -1 + i.èThe real solution is 2.
Usïg EULER'S FORMULA ê general solution is
C¬eì╣ + C½eú╣cos[x] + C¼eú╣sï[x]
ÇèB
4 y»»» + 8y = 0
A)è C¬eì╣ + C½e╣cos[√3 x] + C¼e╣sï[√3 x]
è B)è C¬eúì╣ + C½e╣cos[√3 x] + C¼e╣sï[√3 x]
C)è C¬eì╣ + C½eú╣cos[√3 x] + C¼eú╣sï[√3 x]
D)è C¬eúì╣ + C½eú╣cos[√3 x] + C¼eú╣sï[√3 x]
ü èèFor ê differential equation
y»»» + 8y = 0
ê CHARACTERISTIC EQUATION is
mÄ + 8è= 0
This facërs (by SUM OF CUBES) ïë
(m + 2)(mì - 2m + 4) = 0
The quadratic equation is IRREDUCIBLE OVER THE REALS so it has
a pair ç complex conjugates as its solutions which are
m = 1 - √3i, 1 + i√3.èThe real solution is -2.
Usïg EULER'S FORMULA ê general solution is
C¬eúì╣ + C½e╣cos[√3 x] + C¼e╣sï[√3 x]
ÇèB
S 5 8y»»» - 12y»» + 2y» - 3 = 0
A)è C¬eÄ╣»ì + C½cos[2x] + C¼sï[2x]
B)è C¬eì╣»Ä + C½cos[2x] + C¼sï[2x]
C)è C¬eÄ╣»ì + C½cos[x/2] + C¼sï[x/2]
D)è C¬eì╣»Ä + C½cos[x/2] + C¼sï[x/2]
ü For ê differential equation
8y»»» - 12y»» + 2y» - 3y = 0
ê CHARACTERISTIC EQUATION is
8mÄ - 12mì + 2m - 3 = 0
This facërs ïë
(2m - 3)(4mì + 1) = 0
The quadratic equation is IRREDUCIBLE OVER THE REALS so it has
a pair ç complex conjugates as its solutions which are
m =è-1/2 i, +1/2 i.èThe real solution is 3/2.
Usïg EULER'S FORMULA ê general solution is
C¬eÄ╣»ì + C½cos[x/2] + C¼sï[x/2]
ÇèC
6 y»»» - 10y»» + 37y» - 52y = 0
A) C¬eÅ╣ + C½eÄ╣cos[2x] + C¼eÄ╣sï[2x]
B) C¬eì╣ + C½eÅ╣cos[3x] + C¼eÅ╣sï[3x]
C) C¬eÄ╣ + C½eì╣cos[4x] + C¼eì╣sï[4x]
D) C¬eúÄ╣ + C½eúì╣cos[4x] + C¼eúì╣sï[4x]
ü èèFor ê differential equation
y»»» - 10y»» + 37y» - 52y = 0
ê CHARACTERISTIC EQUATION is
mÄ - 10mì + 37m - 52è= 0
This facërs ïë
(m - 4)(mì - 6m + 13) = 0
The quadratic equation is IRREDUCIBLE OVER THE REALS so it has
a pair ç complex conjugates as its solutions which are
m = 3 - 2i, 3 + 2i.èThe real solution is 4.
Usïg EULER'S FORMULA ê general solution is
C¬eÅ╣ + C½eÄ╣cos[2x] + C¼eÄ╣sï[2x]
ÇèA
äèSolve ê ïitial value problem
â è For ê Initial Value Problem,
y»»» + y» = 0
y(0) = 3, y»(0) = 3, y»»(0) = 3
The general solution isè C¬ + C½cos[x] + C¼sï[x]
Differentiatïg å substitutïg 0 for x produces a system ç
three equations ï ê three constants.èSolvïg this system
gives ê solutionèè y = 3 + 3cos[s] + 4sï[x]
éSèèAs ê GENERAL SOLUTON ç a THIRD ORDER differential
equation has THREE ARBITRARY CONSTANTS, for an Initial Value
Problem ë completely specify which member ç this three
parameter family ç curves requires INITAL VALUES.
è The ståard ïitial values problem for a third order,
lïear, constant coefficient differential equation is
Ay»»» + By»» + Cy» + Dy = g(x)
èèèy(x╠) =è y╠
èè y»(x╠) =èy»╠
èèy»»(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 three arbitrary constants.
2)èèCalculate ê first å second derivatives ç ê general
solution.èThen substitue ê ïitial value ç ïdependent
variable, x╠ , ïë ê general solution å its first two
derivatives.èThis will produce a system ç 3 equations ï
ê three arbitrary constants.èSolvïg this system gives ê
values ç ê three constants which gives ê specific
solution ç ê ïitial value problem.
7è y»»» + 9y» = 0
y(0) = -5èy»(0) = 12èy»»(0) = -18
A) 3 + 2cos[3x] + 4sï[3x]
B) 3 + 2cos[3x] - 4sï[3x]
C) 3 - 2cos[3x] + 4sï[3x]
D) -3 - 2cos[3x] + 4sï[3x]
ü èèFor ê differential equation
y»»» + 9y» = 0
ê CHARACTERISTIC EQUATION is
mÄ + 9m = 0
This facërs ïë
m(mì + 9) = 0
The quadratic equation is IRREDUCIBLE OVER THE REALS so it has
a pair ç complex conjugates as its solutions which are
m =è-3i, 3i.èThe real solution is 0.
Usïg EULER'S FORMULA ê general solution is
èy = C¬ + C½cos[3x] + C¼sï[3x]
Differentiatïg
y» = -3C½sï[3x] + 3C¼cos[3x]
y»» =è9C½cos[3x] - 9C¼cos[3x]
Substitutïg ê ïital value ç ê dependent variable 0
èy(0) =è-5 = C¬ +èC½
y»(0) =è12 =èèèèè 3C¼
y»»(0) = -18 =èèè9C½
Sovlïg this system ç equations yields
C¬ = -3è C½ = -2èC¼ = 4
Thus ê solution ç ê ïitial value problem is
y = -3 - 2cos[3x] + 4sï[3x]
ÇèD
8 y»»» - 3y»» + 4y» - 2y = 0
y(0) = -3èy»(0) = 0èy»»(0) = 5
A) e╣ + 2e╣cos[x] + 3e╣sï[x]
B) e╣ + 2e╣cos[x] - 3e╣sï[x]
C) -e╣ + 2e╣cos[x] - 3e╣sï[x]
D) -e╣ - 2e╣cos[x] + 3e╣sï[x]
ü èèFor ê differential equation
y»»» - 3y»» + 4y» - 2y = 0
ê CHARACTERISTIC EQUATION is
mÄ - 3mì + 4mè- 2 = 0
This facërs ïë
(m - 1)(mì -2m + 2) = 0
The quadratic equation is IRREDUCIBLE OVER THE REALS so it has
a pair ç complex conjugates as its solutions which are
m =è1 - i, 1 + i.èThe real solution is 1.
Usïg EULER'S FORMULA ê general solution is
èy = C¬e╣ + C½e╣cos[x] + C¼e╣sï[x]
Differentiatïg
y» = C¬e╣ + C½{-e╣sï[x] + e╣cos[x]}
+ C¼{e╣cos[x] + e╣sï[x]}
y»» = C¬e╣ + C½{-2e╣sï[x]} + C¼{2e╣cos[x]}
Substitutïg ê ïital value ç ê dependent variable 0
èy(0) =è-3 = C¬ + C½
y»(0) =è 0 = C¬ + C½ +èC¼
y»»(0) =è 5 = C¬èèè+ 2C¼
Sovlïg this system ç equations yields
C¬ = -1è C½ = -2èC¼ = 3
Thus ê solution ç ê ïitial value problem is
y = -e╣ - 2e╣cos[x] + 3e╣sï[x]
ÇèD