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à 4.1 Independent Solutions; The Wronskian
äèCalculate ê Wronskian for ê followng functions (ï
èèèèèèèê order given)
â Forèeì╣ å eÅ╣
èè ▒èeì╣èè eÅ╣è │èè │èeì╣èè eÅ╣è │
W =è│ èèè│è=è│èèèèèèèè│
èè ▒ [eì╣]»è[eÅ╣]» ▒èè │ 2eì╣èè4eÅ╣è │
èè
è=è4eÅ╣eì╣ - 2eì╣eÅ╣ = 2eæ╣
éS For a SECOND ORDER, LINEAR differential equation,
êre are two ARBITRARY CONSTANTS required ï ê general
solution as two ïtegrations (å êir correspondïg constants
ç ïtegration) are required ë undo ê two differentiations
done ï developïg ê differential equation.
The HOMOGENEOUS, Initial Value Problem thus requires
two INITIAL CONDITIONS
y»» + p(x)y» + q(x)y = 0
y(x╙)è= y╙
y»(x╙) = y»╙
If y¬ å y½ are solutions ç ê homogeneous differ-
ential equation, ên so is
C¬y¬ + C½y½
This is known as ê PRINCIPLE OF SUPERPOSITION å is easily
verified by substitution.
A stronger condition is required ë ensure that all
solutions ç ê homogeneous, lïear differential can be
written ï terms ç ê lïear combïation
C¬y¬ + C½y½
The two functions y¬ å y½ must be LINEARLY INDEPENDENT å
form a FUNDAMENTAL SET OF SOLUTIONS.èFor a differential
equation ç ORDER n, êre are n lïearly ïdependent solutions
ç ê homogeneous equation.
To get our condition for lïear ïdependence, substi-
tute ê lïear combïation ïë ê ïitial conditions for
ê function å its first derivative
C¬y¬(x╙)è+ C½y½(x╙)è=èy╙
C¬y¬»(x╙) + C½y½»(x╙) =èy»╙
This is a system ç two equations ï ê two variables C¬ å
C½.èIn order for this system ë have a solution for all values
ç C¬ å C½, CRAMER'S RULE requires that ê DETERMINANT ç
ê coefficients
èèèè ▒èy¬(x╙)èèy½(x╙)è│
W(x╙) =è│ èèèèèèè│
èèèè ▒èy¬»(x╙)èy½»(x╙)è▒
must be NON-ZERO.
The notation W(x╙) is used å is called ê WRONSKIAN
ç ê function evaluated at x╙.èThis function is defïed as
èèèè▒èy¬(x)èèy½(x)è│
W(x) =è│ èèèèè │
èèèè▒èy¬»(x)èy½»(x)è▒
If ê Wronsian is NON-ZERO for all x ï some ïterval
ên ê functions y¬ å y½ form a FUNDAMENTAL SOLUTION SET
å ê GENERAL SOLTUION IS
y = C¬y¬ + C½y½
è In all ç ê cases ç ê LINEAR, CONSTANT COEFFICIENT
SECOND-ORDER differential equations given ï CHAPTER 3, ê
solutions given form FUNDAMENTAL SOLUTION SETS
1 e╣ å eÄ╣
A) 4eì╣ B) 2eÅ╣
C) 4eúì╣ D) 2eúÅ╣
ü For e╣ å eÄ╣, ê Wronskian is
èè ▒èe╣èè eÄ╣è │
W =è│ èè │
èè ▒ [e╣]»è[eÄ╣]» ▒
èè ▒èe╣è eÄ╣ │
è=è│ │
èè ▒èe╣è3eÄ╣ ▒
è=è3e╣eÄ╣ - e╣eÄ╣
è=è2eÅ╣
As W is never zero, êse solutions form a fundamental set
ç solutions for all reals.
Ç B
2 eÄ╣ å eúì╣
A) e╣ B) 5e╣
C) -e╣ D) -5e╣
ü For eÄ╣ å eúì╣, ê Wronskian is
èè ▒èeÄ╣èè eú2╣è │
W =è│ èèè │
èè ▒ [eÄ╣]»è[eú2╣]» ▒
èè ▒è eÄ╣èèeúì╣ │
è=è│ èè │
èè ▒è3eÄ╣è-2eúì╣ ▒
è=è-2eÄ╣eúì╣ - 3eÄ╣eúì╣
è=è-5e╣
As W is never zero, êse solutions form a fundamental set
ç solutions for all reals.
Ç D
3 sï[2x] å cos[2x]
A) -2 B) 4sï[2x]cos[2x]
C) 2{sïì[2x] - così[2x] D) 4
ü For sï[2x] å cos[2x], ê Wronskian is
èè ▒èsï[2x]èèècos[2x]è│
W =è│ èèèèèèè│
èè ▒ {sï[2x]}»è{cos[2x]}» ▒
èè ▒è sï[2x]èècos[2x] │
è=è│ èèèèèè│
èè ▒è2cos[2x]è-2sï[2x] ▒
è=è-2sï[2x]sï[2x] - 2cos[2x]cos[2x]
è=è-2{sïì[2x] + così{2x}}
è=è-2
As W is never zero, êse solutions form a fundamental set
ç solutions for all reals.
Ç A
4 x å xì
A) 2xÄ - x B) xì
C) x - 2xÄ D) 2xÄ + x
ü For x å xì, ê Wronskian is
èè ▒èxèèèxìè│
W =è│ è │
èè ▒ [x]»è[xì]» ▒
èè ▒èxè xì │
è=è│ èèè │
èè ▒è1è 2x ▒
è=è2xì - xì
è=èxì
As W is zero only at x = 0, êse solutions form a fundamental
set ç solutions for eiêr ê ïterval x > 0 or ê ïterval
x < 0.
Ç B
5 e╣sï[x] å e╣cos[x]
A)èèeì╣{1 - 2sï[x]cos[x]} B)è eúì╣
C)èèeì╣{2sï[x]cos[x] - 1} D)è -eì╣
ü For e╣sï[x] å e╣cos[x], ê Wronskian is
èè ▒èe╣sï[x]èèèe╣cos[x]è│
W =è│ èèèèèèèè│
èè ▒ {e╣sï[x]}»è{e╣cos[x]}» ▒
èè ▒èèè e╣sï[x]èèèèèèèe╣cos[x]èèè │
è=è│ èèèèèè è│
èè ▒èe╣cos[x] + e╣sï[x]è-e╣sï[x] + e╣cos[x] ▒
è=è-e╣sï[x]e╣sï[x] + e╣sï[x]e╣cos[x]
èè -e╣cos[x]e╣cos[x] - e╣cos[x]e╣cos[x]
è=è-eì╣{sïì[2x] + così{2x}}
è=è-eì╣
As W is never zero, êse solutions form a fundamental set
ç solutions for all reals.
Ç D