Here are some solutions to the sample problems concerning series solution of differential equations with non-constant coefficients (Chapter 12).
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1 Lecture Appedi B: Some sampe probems from Boas Here are some soutios to the sampe probems cocerig series soutio of differetia equatios with o-costat coefficiets (Chapter ) : Soutio: We wat to cosider the soutio to the foowig equatio (with ocostat coefficiets) usig series soutio (ad other meas) We start with y y First cosider the od stadby techique of separatio of variabes We fid dy y c d y c y ce c e Now we try sovig this equatio usig a series epasio about the origi assumig that it is a reguar poit We have,, y c y c c c c c c c We ca sove for the c by settig the coefficiets of the various powers to zero We fid Physics 8 Lecture Appedi B Witer 9
2 c ucostraied, c c, c c, c c, 4 5 c c4 c, c5 c, c6 c, etc c c, c, c! y c c e! So, as epected, we obtai the same resut as the od techoogy produced : 9 Soutio: Now cosider the foowig equatio (with o-costat coefficiets) usig series soutio (ad other meas) We start with y y y First cosider the od stadby techique of tryig to guess the aswer Ceary a sige costat is ot a soutio but we ca try a b ad a b c We fid y a b, y b, y, y y y b a b y b y a b c, y b c, y c y y y c b 4c a b c c a, b ucostraied y c y y b c Now we try sovig this equatio usig a series epasio about the origi assumig that it is a reguar poit We have Physics 8 Lecture Appedi B Witer 9
3 ,, y c y c y c y y y c c c c c c c So we have succeeded i writig the differetia equatio as a agebraic equatio ivovig the sum To make this usefu we reay wat it i the form d Due to the iear idepedece of the powers of, it the foows that d for a vaues To obtai this form we eed to rewrite the term with More specificay we ote that the = ad = terms vaish (due to the factors i frot of c ) ad we ca write c c c Substitutig this back i above we obtai c c c c Equatig the powers of o both sides of the equatio (ie, a zero o the LHS) we see that oy the first terms ca be ozero ad there are idepedet soutios, Physics 8 Lecture Appedi B Witer 9
4 c c c c c c4 c5 c ucostraied, ucostraied,, etc y c c, c c, y c c c c, y c So agai we deduce the same soutios as we obtaied above : 4 Soutio: Let s ook for the secod (diverget) soutio for the Legedre differetia equatio usig the method of reductio of order from Sectio 87(e) i Boas We start with the equatio y y y, ad the usua Legedre poyomia, 87 we try the Asatz y P v eadig to (sice equatio), P, as a soutio As suggested i Sectio P v P P v P is a soutio of the P P P v P v Pv Pv Thus, as was guarateed, we have just a first order equatio for sove usig separatio of variabes to fid v that we ca Physics 8 Lecture Appedi B 4 Witer 9
5 d P dv P P P d v P v P c v c P Igorig the overa costat (the origia equatio is homogeeous) ad otig that this derivative is we behaved for, we ca write v d P y Q P v P d P where the choice of the sig ad overa ormaizatio is by covetio Net cosider some epicit cases d d d Q P P d c, d d Q P P d d c c, Physics 8 Lecture Appedi B 5 Witer 9
6 Note that if we are carefu there are costats c ad c that correspod to the choice of where to put the brach cuts for the ogarithm as discussed i the Lecture (the costats are typicay imagiary) Fiay cosider the power series epasio of Q We fid Q This is the epected series (from Eq7), which diverges for (as it must sice the ogarithm is siguar there) 4: Soutio: Here we practice with the Rodrigues formua, d! d P to fid the ow order Legedre poyomias We fid, P,! d P,! d d d 4 P,! d 8 d Physics 8 Lecture Appedi B 6 Witer 9
7 d d P! d 48 d 6 4 5, 4 4 d d P ! d 84 d : Soutio: Here we practice usig the recursio reatio, P P P, to fid the higher order Legedre Poyomias from the first We fid Physics 8 Lecture Appedi B 7 Witer 9
8 P P P, P P 5 P P P, 4P 7 P P 4 P P 9 P 4 P , P , P P 5 P P Ceary we coud proceed idefiitey i this fashio 5: 9 Soutio: We wat to epress the poyomia i terms of the Legedre poyomias We proceed essetiay by ispectio of each power We see that we eed both odd ad eve poyomias with maimum degree of We fid Physics 8 Lecture Appedi B 8 Witer 9
9 P ap bp cp a b c a, P P b, P P P P P P c 5: 4 Soutio: Fiay we wat to epress the poyomia 7 i terms of the Legedre poyomias Agai we ca proceed essetiay by ispectio of each power We see that we eed both odd ad eve poyomias with maimum degree of 4 We fid 4 P 7 ap4 bp cp dp ep c 4, P P4 4P P P4 4P, a b c d e a, P P b d P P P P e 5 P P 5 Physics 8 Lecture Appedi B 9 Witer 9
ds xˆ h dx xˆ h dx xˆ h dx.
Lecture : Legedre Poyomias I (See Chapter i Boas) I the previous ectures we have focused o the (commo) case of d differetia equatios with costat coefficiets However, secod order differetia equatios with
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