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à 1.1 Basic Defïitions
äèèDetermïe if ê followïg is an ordïary or a
èèèèèèèèpartial differential equation.
â dìyèèèè dy
─── + sï[x]── - 6yì = 3e╣ is an ordïary differential equation.
dxìèèèè dx
è ┤║uèè ┤║u
a║ ───è=è───èis a partial differential equation.
è ┤x║èè ┤t║
éSèA DIFFERENTIAL EQUATION is any equation that contaïs at least
one derivative.èThis derivative may be eiêr a ëtal derivative ç
a function ç a sïgle variable or it may be a partial derivative ç
a function ç two or more variables.
A differential equation ç a function ç a sïgle variable is
called an ORDINARY differential equation.èExamples are
1) d║yèèè dy
───è-è4 ──è+ 3 yè=è7eú╣
dx║èèè dx
Generally, ê PRIME notation for a derviative will be used so this
ordïary differential equation could also be written
y»» - 4y» + 3y = 7eú╣
2) 2x + y║ + 2xyy» = 0
3) P(x)y»» + Q(x)y» + R(x)y = 0èwhere P, Q å R are polynomials.
A differential equation that ïvolves a function that has two or
more variables requires partial derivatives å is called a PARTIAL
differential equation.èExamples are
1) è┤║uèèè┤u
y ───è- x ──è= 0
è┤y║èèè┤x
Often, partial derivatives can be written by use ç SUBSCRIPTS. This same differential equation would be
yu╤╤ - xu╨ = 0
2) a║u╨╨è=èu▌▌èThis is ê wave equation.
3) u╨╨ + u╤╤ + u╓╓ = 0èThis is LaPlace's equation
This program will only cover techniques for solvïg ordïary
differential equations.
1 dÄyèèèdìyèèè dy
───è- 7 ───è+ 13 ──è- 25 y = tanúî[x] + cosh[3x]
dxÄèèèdxìèèè dx
A)èOrdïary B)èPartial
ü As all ç ê derivatives are ëtal derivatives, this is an
ordïary differential equation.
Ç A
2 ┤║uèèèèè ┤║u
───è+ (x - y)───è=è0
┤x║ èèè┤yì
A)èOrdïary B)èPartial
üèThis differential equation contaïs partial derivatives with
respect ë both x å y meanïg that u is a function ç those
two variables.èThus this is a partial differential equation.
ÇèB
3
èèèè u╨╨ + u╤╤ - u╓╓ = 0
A)èOrdïary B)èPartial
ü This differential equation contaïs partial derivatives with
respect ë x, ë y å ë z which means that u is a function
ç êse three variables.èHence, this is a partial
differential equation.
ÇèB
4
èèèè y»» - 3[y»]ì + sï[y] = 0
A)èOrdïary B)èPartial
ü This differential equation only has ëtal derivatives ç ê
function y å hence is an ordïary differential equation.
Ç A
äèèGive ê order ç ê followïg differential equations.
â y»»» - 5y»» + 7y» - 3y = sï[x]
has a third derivative as its highest order derivative å
hence is ç order 3.
éS There are a number ç ways ç classifyïg differential
equations.èThe ORDER ç a partial differential equation is defïed as
ê order ç ê highest derivative present ï ê differential
equation.èFor example
1) dy
──è= sï[x]èis a first order differential equaën.
dx
2) y»» - 4y» + 3y = e╣èis a second order differential equation.
3) y»»»» - 16y = 5sï[2x] is a fourth order differential equation.
5
èèèè y»»» - 4y»» + y║ = 0
A)èèè1 B) 2
C) 3 D) 4
üèAs a third derivative is present å no higher order derivatives
are present this is a differential equation ç order 3.
Ç C
6 dyèè2
──è+ ─ y = tan[x]
dxèèx
A) 1 B) 2
C) 3 D) 4
ü The only derivative present is a first derivative so this
differential equation is ç order 1.
Ç A
7è èd║xèèè dx
m ───è+èb ──è +èw║ yè=èF╠cos[w╠t]
èdt║èèè dt
A) 1 B) 2
C) 3 D) 4
ü This differential equation contaïs both a first å a second
derviative å so ê order is 2.
Ç B
8 xÄy»» + [y»]É - 9yÅ = xÆ
A) 1 B) 2
C) 3 D) 4
ü This differential equation contaïs both a second derivative
a first derivative (raised ë ê fifth power).èThus ê
highest derivative is ê second å this is ç order 2.
ÇèB
äèèDetermïe which is a solution ç ê given differential
èèèèèèèèequation.
â For ê differential equation
y»» - 4y» + 3y = 0
y = 2eÄ╣ is a solution
éS A SOLUTION ç a differential equation is any function y = f(x)
which when substituted ïë ê differential equation produces
a true statement.
For example, for ê differential equation
y»» - 4y» + 3y = 0
The functionèy = 4eÄ╣ is a solution as
[4eÄ╣]»» - 4[4eÄ╣]» + 3[4eÄ╣] =
36eÄ╣ - 48eÄ╣ + 12eÄ╣èèèè = 0
It can also be shown that -5e╣ is a solution as is 26.84eÄ╣.
Most differential equations have what is known as a GENERAL
SOLUTION which contaï one or more (dependïg on ê order)
arbitrary constants.èThis means that any substituion for ê
constants will produce a solution ë ê differential equation.
For ê differential equation
y»» -è4y» + 3y = 0
The general solution is
y = C¬e╣ + C½eÄ╣
The previous solution
y = 4eÄ╣ fits ïë ê general solution with C¬ = 0 å C½ = 4
as does ê solution y = -5e╣ with C¬ = -5 å C½ = 0.
Pickïg C¬ = -8 å C½ = 7 produces ê solution
y = -8e╣ + 7eÄ╣.
9 dyèè 4
──è+è─ yè=èxÄ
dxèè x
A) xÄèèè2 B) xÅèèè2
──è-è── ──è-è──
8èè xÉ 8èè xÅ
C) xÉèèè2 D) All ç ê above
──è-è──
8èè xÄ
ü For y = xÅ/8 - 2xúÅè y»è =èxÄ/2 + 8xúÉ
4y/x =èxÄ/2 - 8xúÉ
So y» + 4y/x = xÄ å this is a solution.
Ç B
10 y»» - 4y»è+ 3y = 0
A) 3e╣ B) -0.25eÄ╣
C) 5e╣ - 4e╣ D) All ç ê above
ü Consider yè = C¬e╣ + C½eÄ╣
y»è= C¬e╣ + 3C½eÄ╣
y»» = C¬e╣ + 9C½eÄ╣
y»» - 4y» + 3y = C¬e╣ + 9C½eÄ╣ - 4(C¬e╣ + 3C½eÄ╣)
+ 3(C¬e╣ + C½eÄ╣)
èè = C¬e╣(1 - 4 + 3) + C½e╣(9 - 12 + 3)
èè = 0
Thus yè = C¬e╣ + C½eÄ╣ is a general solution ç this
differential equation.
Answer A corresponds ë C¬ = 3 å C½ = 0,
Answer B corresponds ë C¬ = 0 å C½ = -0.25 å
Answer C corresponds ë C¬ = 5 å C½ = -4.
Thus all are solutions.
Ç D
è11 y»» + 6y» + 8yè=è2x + 4
A) 7eúì╣ + 5eúÅ╣ + x/4 + 5/16
B) 7eúì╣ - 5eúÅ╣ + x/4 - 5/16
C) -7eúì╣ + 5eúÅ╣ - x/4 + 5/16
D) -7eúì╣ - 5eúÅ╣ - x/4 - 5/16
ü If y = 7eúì╣ + 5eúÅ╣ + x/4 + 5/16
èèèy»» =è28eúì╣è+ 80eúÅ╣
èèè6y» = -84eúì╣ - 120eúÅ╣ + 3/2
èèè 8y =è56eúì╣ +è40eúÅ╣ + 2x + 5/2
Thus y»» + 6y» + 8y = 2x + 4 which shows that this is a
solution.èThe oêr answers do not produce a solution.
Ç A
äèèDetermïe if ê followïg differential equation is
èèèèèèèèlïear or non-lïear.
â
èèè y»» + xy» + e╣y = cosh[x] is lïear.
y»» + [y»]║ + tan[y] = 0èis non-lïear.
éS If a differential equation can be written ï ê form
a╠(x)yÑⁿª + a¬(x)yÑⁿúî) + ∙∙∙ + a┬▀¬(x)y» + a┬(x)yè=èg(x)
it is a LINEAR differential equation.èIf it cannot be written
ï this form it is a NON-LINEAR differential equation.
For example
1)è7y»» + x║y» + eú╣y = cot[x] is lïear as ê coefficients
èèç ê derivatives are functions ç x alone å no
èèderivative is raised ë any power.
2)èy»» - [y»]Ä + cosh[y] = e╣èis non-lïear for two reasons.
èèFirst, ê first derivative is raised ë ê third power
èèå second, y is ê argument ç ê hyperbolic cosïe
èèfunction.
12
èèèèè y»»è+èxy»è+èsecì[x]yè=èeúÄ╣
A)èLïear B)èNon-lïear
ü The coefficients ç ê derivatives are functions ç x å ê
derivatives are not raised ë any power.èThe right hå side
is a function x alone.èThus, this is a lïear differential
equation.
Ç A
13
èèèèè y»»è+èyy»è+èsec║[x]yè=èeúÄ╣
A)èLïear B)èNon-lïear
ü The second term yy» is not ç ê required form å hence this
is a non-lïear differential equation.
Ç B
14
èèèèè y»»è+ètan[y]è=è7xÄ
A)èLïear B)èNon-lïear
ü The second term tan[y] is not ç ê required form å hence
this is a non-lïear differential equation.
Ç B
15
èèèèè xìy»»è-è3xy»è+èsï[x]yè=ècosh[3x]
A)èLïear B)èNon-lïear
ü The coefficients ç ê derivatives are functions ç x å ê
derivatives are not raised ë any power.èThe right hå side
is a function x alone.èThus, this is a lïear differential
equation.
Ç A