Series Solutions of ODEs 2 the Frobenius method. The basic idea of the Frobenius method is to look for solutions of the form 3

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1 Royal Holloway Unversty of London Department of Physs Seres Solutons of ODEs the Frobenus method Introduton to the Methodology The smple seres expanson method works for dfferental equatons whose solutons are well-behaved at the expanson pont x =. The method works well for many funtons, but there are some whose behavour preludes the smple seres method. The Bessel Y funton s one suh example. And learly any funtons nvolvng negatve or fratonal powers would not be amenable to a smple power seres expanson. The Frobenus method extends the smple power seres method to nlude negatve and fratonal powers, and t also allows a natural extenson nvolvng logarthm terms. The bas dea of the Frobenus method s to look for solutons of the form yx = ax + ax + ax + ax = x a + a x+ a x + a x +... = x = a x. The extenson of the smple power seres method s all n the fator x. The power must now be determned, as well as the oeffents a. Sne may be negatve, postve, and possbly non-ntegral, ths extends onsderably the range of funtons whh may be treated. Note that a s the lowest non-zero oeffent, so by defnton t annot be zero. A smple example We an demonstrate, wth the followng equaton, how the Frobenus method works n prate d y dy 4x + + y =. We would dvde by 4x to get the equaton nto standard form: d y d + y x + 4x y = but we wll work dretly wth the orgnal equaton. The tral soluton s gven by so dfferentatng ths gves and dfferentatng agan: 6= yx = ax + dy = a + x 6 + = PH3 Mathematal Methods

And learly any funtons nvolvng negatve or fratonal powers would not be amenable to a smple power seres expanson.

2 Royal Holloway Unversty of London Department of Physs d + = a x = We substtute these expressons nto the dfferental equaton: y = = = x a + + x + a + x + a x = Next we norporate the x fator n the frst term, gvng a slght smplfaton = = = a + + x + a + x + a x = and, as n the smple seres ase, we alter the summaton ndes to obtan a ommon power of x for all three terms. In ths ase ths means nreasng by n the frst two terms. The dfferental equaton then beomes + < 4a a aa x =. For the expresson to be zero, eah power of x must vansh, thus 4a a a = for all (allowed) Ths equaton gves a reurrene relaton for the oeffents, but before examnng that we must determne the values of whh are allowed. We know that a s the lowest order non-vanshng oeffent; a = by defnton. So puttng = nto the above expresson gves 4a 6 + a = or a 6 =. Now we know that a annot be zero, so to satsfy ths equaton ether = or = /. We have found two possble values for the ndex. For ths reason the equaton obtaned by settng the lowest power of x equal to zero s alled the ndal equaton. Ths s a quadrat equaton. It thus gves two possble values for that suggests there are two seres that satsfy the equaton. Ths seems to be orret for a seond order equaton. Now let us return to the reurrene relaton. The equaton relatng general oeffents may be wrtten a a = or a a+ = We have two ases now to onsder, orrespondng to the two dfferent values taken by the ndex. The reurrene relaton s dfferent for these two ases. PH3 Mathematal Methods

summaton ndes to obtan a ommon power of x for all three terms. In ths ase ths means nreasng by n the frst two terms. The dfferental equaton then beomes + < 4a + + + + + a+ + + + aa x =.

3 Royal Holloway Unversty of London Department of Physs Frst we onsder the ase =. The reurrene relaton s then a a = Set = : a = a. Now set = : a = a 3.. = a 3.. = a. 4! Terms may be bult up n ths way. In general the th term may be wrtten 6 a. 6! 6 = a x = 6! % x x x a& ' 4 x 6 = os. a = Consequently we have obtaned the soluton yx 3 = !!! You mght observe that ths an be expressed n terms of the osne funton yx a x Now we onsder the ase = /. The reurrene relaton s then a a = / 6 Set = : a = a 3.. / =. a. 3! Now set = : a = a 45../ = a 45../ 3! = a. 5! ( ) * PH3 Mathematal Methods 3

a = Consequently we have obtaned the soluton yx 3 = + +.

4 Royal Holloway Unversty of London Department of Physs Terms may be bult up n ths way. In general the th term may be wrtten 6 a = a. + 6! So the other soluton s yx / = a x +! 3 / 5 / / x x = a x ' 3! 5! * You mght observe that ths an be expressed n terms of the sne funton yx a x % & = 6 6 = sn. The role of the a n eah ase s that of an arbtrary onstant and t s not fxed by the dfferental equaton. The general soluton of the ODE s thus yx 6 = Aos x+ Bsn x. In general one annot sum the seres n terms of standard funtons. ( ) PH3 Mathematal Methods 4

/ / x x = a x +.... ' 3! 5! * You mght observe that ths an be expressed n terms of the sne funton yx a x % & = 6 6 = sn.

5 Royal Holloway Unversty of London Department of Physs The mportant onepts of ths seton are: The general expresson for a soluton s x multpled by a smple power seres. The ndex may be non-ntegral, postve or negatve. The seres s substtuted nto the dfferental equaton and a slght jugglng of the summaton ndex asts the equaton nto the form ;@x... + =. Eah power of x must equate to zero. By defnton a, whle a for negatve vansh. The equaton obtaned by settng the lowest power of x equal to zero s the ndal equaton. Ths (for a nd order ODE) s a quadrat equaton; t gves two values for the ndex. Eah value of gves a dfferene reurrene relaton and a dfferent power seres soluton to the ODE. PH3 Mathematal Methods 5

.. + =. Eah power of x must equate to zero. By defnton a, whle a for negatve vansh. The equaton obtaned by settng the lowest power of x equal to zero s the ndal equaton.

6 Royal Holloway Unversty of London Department of Physs Convergene and exstene of solutons The Frobenus method extends the range of equatons for whh a soluton may be expressed n terms of power seres (by extendng/generalsng what we mean by a power seres). We wll gve wthout proof a theorem whh tells us somethng about the valdty of the Frobenus method. Frst we need some defntons. These wll refer to the general seond order homogeneous dfferental equaton expressed n standard form: d y dy + px 6 + qxy 6 =. Defntons The pont x s alled an ordnary pont of the equaton f px 6 and qx 6both have Taylor expansons about x n powers of x x 6. Thus for well behaved funtons px and qx 6all ponts n the range wll be ordnary ponts; n ths ase the ODE an be solved wth the smple seres method. A pont whh s not ordnary s alled a sngular pont or a sngularty. Sngulartes are bad behavour; but there s bad and really bad! 6 6 and x x 6 qx6 both have Taylor seres about If x s a sngular pont but x x p x x, then x s alled a regular sngular pont or a regular sngularty. Here the behavour s not so bad, and at least one seres soluton about x (Frobenus type) s possble. Theorem If xp6and x xqx 6an be expressed as a power seres n x then the Frobenus seres solutons to d y dy + px + qxy= 6 6 obtaned from a root of the ndal equaton onverges for x mnmum radus of onvergene of xp x 6and xqx 6. < Rwhere R s the 6 Classfaton of equaton types If the orgn s a regular sngular pont then we an see that the ndal equaton wll be a quadrat n. Ths, n general, has two roots but f these dffer by an nteger then falure ours; further onsderaton s needed to fnd the seond soluton. We an lassfy the equatons to be solved by the Frobenus method nto four types. PH3 Mathematal Methods 6

These wll refer to the general seond order homogeneous dfferental equaton expressed n standard form: d y dy + px 6 + qxy 6 =.

7 Royal Holloway Unversty of London Department of Physs I II III IV Roots of ndal equaton unequal and not dfferng by an nteger. In ths ase we get two ndependent solutons by substtutng the two values of the ndex nto the seres for yx 6. Roots of ndal equaton zero. In ths ase we get two ndependent solutons by substtutng the value for nto the seres for yx y/. 6 and Roots of ndal equaton dfferng by an nteger, makng a oeffent nfnte Roots of ndal equaton dfferng by an nteger makng a oeffent ndetermnate. PH3 Mathematal Methods 7

Roots of ndal equaton zero. In ths ase we get two ndependent solutons by substtutng the value for nto the seres for yx y/.

8 Royal Holloway Unversty of London Department of Physs The mportant onepts of ths seton are: For well behaved funtons px 6 and qx 6the expanson pont (here x = ) wll be an ordnary pont; n ths ase the ODE an be solved wth the smple seres method. For expansons about a regular sngular pont the Frobenus method may be used. In ths ase the ndal equaton wll be a quadrat n the ndex ; n general ths wll gve the two solutons of the ODE. There are, however, speal ases. PH3 Mathematal Methods 8

method. For expansons about a regular sngular pont the Frobenus method may be used.

n + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2)

n + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2) MATH 16T Exam 1 : Part I (In-Class) Solutons 1. (0 pts) A pggy bank contans 4 cons, all of whch are nckels (5 ), dmes (10 ) or quarters (5 ). The pggy bank also contans a con of each denomnaton. The total