Series Solutions to ODEs
theFrobeniusMetht
A generic inFFin Ésradeoroliaryoligantiaquain one hastheform
W tpeewitglow e X
with twolinearlyindependent solutions Walz andWalz and generalsolution
W AW E BWalz
for constant A B
Weknow how tosolve when
look
Pigconstants forsolutions waemz
characteristicpolynomial matpmtq o with roots me ma
Aemz Bent me ma
w At Bzems ma ma
a
Put z ex Wax ween w z Wa exw zw Wa zaw tew
Z'wtzwitazw zu't bw
zaw tazw't bw z w Ew'tbzw o
Wxxt a s Wxtbw o
a
CA Bloge em ma ma
in
The idea ofthe Frobenius METHOD forgeneral p and q lookforseries solutionsexpandedaboutapoi
z z
we E ace za
with CER constant a o acer
coefficients
Isothat firstterminseriesexpansion is aczz.sc
WLOG we can scale zo o
bychange of variables
we will return to discusswhenthis methodwill worklater
EXAMPLE
Findseries solutionabout z o
of wi't w t w o
solution try WE.az't w E aceticzetasand w Eaccctidactica z't's
equation becomes
É ÉÉÉtgÉÉÉJÉn a
E gazer o
s
aceticCork a galacticthars za
or LaoCcc 2
tza.cz't o
, a Ccc s
24 0 INDIGALEQUATION ie
acctk atk a
za can ya RECURRENCERELATION RR
so IE C
zc o c ya c o c o or C Ya
i c o ar ke s
zack g ar act
Iggy
LEI
s
check e byinduction
as
af an
15 15.2 etc ak GIFT g
Solution is wa aoÉ.cz zk aeEEIIcrz a cost
in C Ya a
GEE LEEK
find that ax
ELIF
wa he E 2kt
Solution is
81 a sine
The generalsolution is W AcosetBsintz
Note in general it is not possible tospot a closedformexpressionfor the infinite sum
DEFINITION
A point ordinary PointYjÉn 5ÉÉÉFÉw'tacew e x pazos and
to is an if
qczo are analytic Ci e theyhave a Taylorseriesexpansion
The point z zo is a REGULAR SINGULARPOINT if Z Zo plz and Z Zo2pCz
are analytic at Z zo
but 2plz
e
g in previous example plz Iz qz Iz whichbothhavesinglepolesat z o
and 22g z are bothanalytic infinitelydifferentiable at 2 0 2 0 is a regular
singular point
THEOREM Foch'sTheorem
Thegeneral solution to x can befound as a generalisedcomplexpowerseriesexpanded around
Z Zo provided Ezo is a regularsingularpointof
If
Z Zo is also an ordinarypoint then solutions will beanalytic at z z i.e powerseries will be a
Taylorseries and c o
Note generalisedheremeansthat it may containsome logarithmicterms see later
e g Find seriessolutionabout X e C1 x2 y 2xy thy
for D
and aex
Identify pix
Iya Iya
se p and q are analytic at z o 2 0 is an ordinarypoint
Canimmediatelyset co and y E ax note he isnot constrainedhere
theFrobeniusMetht
A generic inFFin Ésradeoroliaryoligantiaquain one hastheform
W tpeewitglow e X
with twolinearlyindependent solutions Walz andWalz and generalsolution
W AW E BWalz
for constant A B
Weknow how tosolve when
look
Pigconstants forsolutions waemz
characteristicpolynomial matpmtq o with roots me ma
Aemz Bent me ma
w At Bzems ma ma
a
Put z ex Wax ween w z Wa exw zw Wa zaw tew
Z'wtzwitazw zu't bw
zaw tazw't bw z w Ew'tbzw o
Wxxt a s Wxtbw o
a
CA Bloge em ma ma
in
The idea ofthe Frobenius METHOD forgeneral p and q lookforseries solutionsexpandedaboutapoi
z z
we E ace za
with CER constant a o acer
coefficients
Isothat firstterminseriesexpansion is aczz.sc
WLOG we can scale zo o
bychange of variables
we will return to discusswhenthis methodwill worklater
EXAMPLE
Findseries solutionabout z o
of wi't w t w o
solution try WE.az't w E aceticzetasand w Eaccctidactica z't's
equation becomes
É ÉÉÉtgÉÉÉJÉn a
E gazer o
s
aceticCork a galacticthars za
or LaoCcc 2
tza.cz't o
, a Ccc s
24 0 INDIGALEQUATION ie
acctk atk a
za can ya RECURRENCERELATION RR
so IE C
zc o c ya c o c o or C Ya
i c o ar ke s
zack g ar act
Iggy
LEI
s
check e byinduction
as
af an
15 15.2 etc ak GIFT g
Solution is wa aoÉ.cz zk aeEEIIcrz a cost
in C Ya a
GEE LEEK
find that ax
ELIF
wa he E 2kt
Solution is
81 a sine
The generalsolution is W AcosetBsintz
Note in general it is not possible tospot a closedformexpressionfor the infinite sum
DEFINITION
A point ordinary PointYjÉn 5ÉÉÉFÉw'tacew e x pazos and
to is an if
qczo are analytic Ci e theyhave a Taylorseriesexpansion
The point z zo is a REGULAR SINGULARPOINT if Z Zo plz and Z Zo2pCz
are analytic at Z zo
but 2plz
e
g in previous example plz Iz qz Iz whichbothhavesinglepolesat z o
and 22g z are bothanalytic infinitelydifferentiable at 2 0 2 0 is a regular
singular point
THEOREM Foch'sTheorem
Thegeneral solution to x can befound as a generalisedcomplexpowerseriesexpanded around
Z Zo provided Ezo is a regularsingularpointof
If
Z Zo is also an ordinarypoint then solutions will beanalytic at z z i.e powerseries will be a
Taylorseries and c o
Note generalisedheremeansthat it may containsome logarithmicterms see later
e g Find seriessolutionabout X e C1 x2 y 2xy thy
for D
and aex
Identify pix
Iya Iya
se p and q are analytic at z o 2 0 is an ordinarypoint
Canimmediatelyset co and y E ax note he isnot constrainedhere
