Issues and Next Directions

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\(\def\vs{{\it vs. }} \def\cf{{\it cf. }} \def\viz{{\it viz. }} \def\ie{{\it i.e. }} \def\etc{{\it etc. }} \def\eg{{\it e.g. }} \def\etal{{\it et al .}} \def\via{{\it via }} \def\adhoc{{\it ad hoc }} \def\apriori{{\it apriori }} \def\Afrak{\mathfrak{A}} \def\Bfrak{\mathfrak{B}} \def\Cfrak{\mathfrak{C}} \def\Dfrak{\mathfrak{D}} \def\Efrak{\mathfrak{E}} \def\Ffrak{\mathfrak{F}} \def\Gfrak{\mathfrak{G}} \def\Hfrak{\mathfrak{H}} \def\Ifrak{\mathfrak{I}} \def\Jfrak{\mathfrak{J}} \def\Kfrak{\mathfrak{K}} \def\Lfrak{\mathfrak{L}} \def\Mfrak{\mathfrak{M}} \def\Nfrak{\mathfrak{N}} \def\Ofrak{\mathfrak{O}} \def\Pfrak{\mathfrak{P}} \def\Qfrak{\mathfrak{Q}} \def\Rfrak{\mathfrak{R}} \def\Sfrak{\mathfrak{S}} \def\Tfrak{\mathfrak{T}} \def\Ufrak{\mathfrak{U}} \def\Vfrak{\mathfrak{V}} \def\Wfrak{\mathfrak{W}} \def\Xfrak{\mathfrak{X}} \def\Yfrak{\mathfrak{Y}} \def\Zfrak{\mathfrak{Z}} \def\afrak{\mathfrak{a}} \def\bfrak{\mathfrak{b}} \def\cfrak{\mathfrak{c}} \def\dfrak{\mathfrak{d}} \def\efrak{\mathfrak{e}} \def\ffrak{\mathfrak{f}} \def\gfrak{\mathfrak{g}} \def\hfrak{\mathfrak{h}} \def\ifrak{\mathfrak{i}} \def\jfrak{\mathfrak{j}} \def\kfrak{\mathfrak{k}} \def\lfrak{\mathfrak{l}} \def\mfrak{\mathfrak{m}} \def\nfrak{\mathfrak{n}} \def\ofrak{\mathfrak{o}} \def\pfrak{\mathfrak{p}} \def\qfrak{\mathfrak{q}} \def\rfrak{\mathfrak{r}} \def\sfrak{\mathfrak{s}} \def\tfrak{\mathfrak{t}} \def\ufrak{\mathfrak{u}} \def\vfrak{\mathfrak{v}} \def\wfrak{\mathfrak{w}} \def\xfrak{\mathfrak{x}} \def\yfrak{\mathfrak{y}} \def\zfrak{\mathfrak{z}} \def\AA{\mathbb{A}} \def\BB{\mathbb{B}} \def\CC{\mathbb{C}} \def\DD{\mathbb{D}} \def\EE{\mathbb{E}} \def\FF{\mathbb{F}} \def\GG{\mathbb{G}} \def\HH{\mathbb{H}} \def\II{\mathbb{I}} \def\JJ{\mathbb{J}} \def\KK{\mathbb{K}} \def\LL{\mathbb{L}} \def\MM{\mathbb{M}} \def\NN{\mathbb{N}} \def\OO{\mathbb{O}} \def\PP{\mathbb{P}} \def\QQ{\mathbb{Q}} \def\RR{\mathbb{R}} \def\SS{\mathbb{S}} \def\TT{\mathbb{T}} \def\UU{\mathbb{U}} \def\VV{\mathbb{V}} \def\WW{\mathbb{W}} \def\XX{\mathbb{X}} \def\YY{\mathbb{Y}} \def\ZZ{\mathbb{Z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\Ap{A^\prime} \def\Bp{B^\prime} \def\Cp{C^\prime} \def\Dp{D^\prime} \def\Ep{E^\prime} \def\Fp{F^\prime} \def\Gp{G^\prime} \def\Hp{H^\prime} \def\Ip{I^\prime} \def\Jp{J^\prime} \def\Kp{K^\prime} \def\Lp{L^\prime} \def\Mp{M^\prime} \def\Mp{N^\prime} \def\Op{O^\prime} \def\Pp{P^\prime} \def\Qp{Q^\prime} \def\Rp{R^\prime} \def\Sp{S^\prime} \def\Tp{T^\prime} \def\Up{U^\prime} \def\Vp{V^\prime} \def\Wp{W^\prime} \def\Xp{X^\prime} \def\Yp{Y^\prime} \def\Zp{Z^\prime} \def\ap{a^\prime} \def\bp{b^\prime} \def\cp{c^\prime} \def\dprime{d^\prime} \def\ep{e^\prime} \def\fp{f^\prime} \def\gp{g^\prime} \def\hp{h^\prime} \def\ip{i^\prime} \def\jp{j^\prime} \def\kp{k^\prime} \def\lp{l^\prime} \def\mp{m^\prime} \def\np{n^\prime} \def\op{o^\prime} \def\pp{p^\prime} \def\qp{q^\prime} \def\rp{r^\prime} \def\sp{s^\prime} \def\tp{t^\prime} \def\up{u^\prime} \def\vp{v^\prime} \def\wp{w^\prime} \def\xp{x^\prime} \def\yp{y^\prime} \def\zp{z^\prime} \def\App{A^{\prime\prime}} \def\Bpp{B^{\prime\prime}} \def\Cpp{C^{\prime\prime}} \def\Dpp{D^{\prime\prime}} \def\Epp{E^{\prime\prime}} \def\Fpp{F^{\prime\prime}} \def\Gpp{G^{\prime\prime}} \def\Hpp{H^{\prime\prime}} \def\Ipp{I^{\prime\prime}} \def\Jpp{J^{\prime\prime}} \def\Kpp{K^{\prime\prime}} \def\Lpp{L^{\prime\prime}} \def\Mpp{M^{\prime\prime}} \def\Mpp{N^{\prime\prime}} \def\Opp{O^{\prime\prime}} \def\Ppp{P^{\prime\prime}} \def\Qpp{Q^{\prime\prime}} \def\Rpp{R^{\prime\prime}} \def\Spp{S^{\prime\prime}} \def\Tpp{T^{\prime\prime}} \def\Upp{U^{\prime\prime}} \def\Vpp{V^{\prime\prime}} \def\Wpp{W^{\prime\prime}} \def\Xpp{X^{\prime\prime}} \def\Ypp{Y^{\prime\prime}} \def\Zpp{Z^{\prime\prime}} \def\app{a^{\prime\prime}} \def\bpp{b^{\prime\prime}} \def\cpp{c^{\prime\prime}} \def\dpp{d^{\prime\prime}} \def\epp{e^{\prime\prime}} \def\fpp{f^{\prime\prime}} \def\gpp{g^{\prime\prime}} \def\hpp{h^{\prime\prime}} \def\ipp{i^{\prime\prime}} \def\jpp{j^{\prime\prime}} \def\kpp{k^{\prime\prime}} \def\lpp{l^{\prime\prime}} \def\mpp{m^{\prime\prime}} \def\npp{n^{\prime\prime}} \def\opp{o^{\prime\prime}} \def\ppp{p^{\prime\prime}} \def\qpp{q^{\prime\prime}} \def\rpp{r^{\prime\prime}} \def\spp{s^{\prime\prime}} \def\tpp{t^{\prime\prime}} \def\upp{u^{\prime\prime}} \def\vpp{v^{\prime\prime}} \def\wpp{w^{\prime\prime}} \def\xpp{x^{\prime\prime}} \def\ypp{y^{\prime\prime}} \def\zpp{z^{\prime\prime}} \def\abar{\overline{a}} \def\bbar{\overline{b}} \def\cbar{\overline{c}} \def\dbar{\overline{d}} \def\ebar{\overline{e}} \def\fbar{\overline{f}} \def\gbar{\overline{g}} \def\ibar{\overline{i}} \def\jbar{\overline{j}} \def\kbar{\overline{k}} \def\lbar{\overline{l}} \def\mbar{\overline{m}} \def\nbar{\overline{n}} \def\obar{\overline{o}} \def\pbar{\overline{p}} \def\qbar{\overline{q}} \def\rbar{\overline{r}} \def\sbar{\overline{s}} \def\tbar{\overline{t}} \def\ubar{\overline{u}} \def\vbar{\overline{v}} \def\wbar{\overline{w}} \def\xbar{\overline{x}} \def\ybar{\overline{y}} \def\zbar{\overline{z}} \def\Abar{\overline{A}} \def\Bbar{\overline{B}} \def\Cbar{\overline{C}} \def\Dbar{\overline{D}} \def\Ebar{\overline{E}} \def\Fbar{\overline{F}} \def\Gbar{\overline{G}} \def\Hbar{\overline{H}} \def\Ibar{\overline{I}} \def\Jbar{\overline{J}} \def\Kbar{\overline{K}} \def\Lbar{\overline{L}} \def\Mbar{\overline{M}} \def\Nbar{\overline{N}} \def\Obar{\overline{O}} \def\Pbar{\overline{P}} \def\Qbar{\overline{Q}} \def\Rbar{\overline{R}} \def\Sbar{\overline{S}} \def\Tbar{\overline{T}} \def\Ubar{\overline{U}} \def\Vbar{\overline{V}} \def\Wbar{\overline{W}} \def\Xbar{\overline{X}} \def\Ybar{\overline{Y}} \def\Zbar{\overline{Z}} \def\aunder{\underline{a}} \def\bunder{\underline{b}} \def\cunder{\underline{c}} \def\dunder{\underline{d}} \def\eunder{\underline{e}} \def\funder{\underline{f}} \def\gunder{\underline{g}} \def\hunder{\underline{h}} \def\iunder{\underline{i}} \def\junder{\underline{j}} \def\kunder{\underline{k}} \def\lunder{\underline{l}} \def\munder{\underline{m}} \def\nunder{\underline{n}} \def\ounder{\underline{o}} \def\punder{\underline{p}} \def\qunder{\underline{q}} \def\runder{\underline{r}} \def\sunder{\underline{s}} \def\tunder{\underline{t}} \def\uunder{\underline{u}} \def\vunder{\underline{v}} \def\wunder{\underline{w}} \def\xunder{\underline{x}} \def\yunder{\underline{y}} \def\zunder{\underline{z}} \def\Aunder{\underline{A}} \def\atilde{\widetilde{a}} \def\btilde{\widetilde{b}} \def\ctilde{\widetilde{c}} \def\dtilde{\widetilde{d}} \def\etilde{\widetilde{e}} \def\ftilde{\widetilde{f}} \def\gtilde{\widetilde{g}} \def\htilde{\widetilde{h}} \def\itilde{\widetilde{i}} \def\jtilde{\widetilde{j}} \def\ktilde{\widetilde{k}} \def\ltilde{\widetilde{l}} \def\mtilde{\widetilde{m}} \def\ntilde{\widetilde{n}} \def\otilde{\widetilde{o}} \def\ptilde{\widetilde{p}} \def\qtilde{\widetilde{q}} \def\rtilde{\widetilde{r}} \def\stilde{\widetilde{s}} \def\ttilde{\widetilde{t}} \def\utilde{\widetilde{u}} \def\vtilde{\widetilde{v}} \def\wtilde{\widetilde{w}} \def\xtilde{\widetilde{x}} \def\ytilde{\widetilde{y}} \def\ztilde{\widetilde{z}} \def\Atilde{\widetilde{A}} \def\Btilde{\widetilde{B}} \def\Ctilde{\widetilde{C}} \def\Dtilde{\widetilde{D}} \def\Etilde{\widetilde{E}} \def\Ftilde{\widetilde{F}} \def\Gtilde{\widetilde{G}} \def\Htilde{\widetilde{H}} \def\Itilde{\widetilde{I}} \def\Jtilde{\widetilde{J}} \def\Ktilde{\widetilde{K}} \def\Ltilde{\widetilde{L}} \def\Mtilde{\widetilde{M}} \def\Ntilde{\widetilde{N}} \def\Otilde{\widetilde{O}} \def\Ptilde{\widetilde{P}} \def\Qtilde{\widetilde{Q}} \def\Rtilde{\widetilde{R}} \def\Stilde{\widetilde{S}} \def\Ttilde{\widetilde{T}} \def\Utilde{\widetilde{U}} \def\Vtilde{\widetilde{V}} \def\Wtilde{\widetilde{W}} \def\Xtilde{\widetilde{X}} \def\Ytilde{\widetilde{Y}} \def\Ztilde{\widetilde{Z}} \def\Alphatilde{\widetilde{\Alpha}} \def\Betatilde{\widetilde{\Beta}} \def\Gammatilde{\widetilde{\Gamma}} \def\Deltatilde{\widetilde{\Delta}} \def\Epsilontilde{\widetilde{\Epsilon}} \def\Zetatilde{\widetilde{\Zeta}} \def\Etatilde{\widetilde{\Eta}} \def\Thetatilde{\widetilde{\Theta}} \def\Iotatilde{\widetilde{\Iota}} \def\Kappatilde{\widetilde{\Kappa}} \def\Lambdatilde{\widetilde{\Lamdba}} \def\Mutilde{\widetilde{\Mu}} \def\Nutilde{\widetilde{\Nu}} \def\Xitilde{\widetilde{\Xi}} \def\Omicrontilde{\widetilde{\Omicron}} \def\Pitilde{\widetilde{\Pi}} \def\Rhotilde{\widetilde{\Rho}} \def\Sigmatilde{\widetilde{\Sigma}} \def\Tautilde{\widetilde{\Tau}} \def\Upsilontilde{\widetilde{\Upsilon}} \def\Phitilde{\widetilde{\Phi}} \def\Chitilde{\widetilde{\Chi}} \def\Psitilde{\widetilde{\Psi}} \def\Omegatilde{\widetilde{\Omega}} \def\alphatilde{\widetilde{\alpha}} \def\betatilde{\widetilde{\beta}} \def\gammatilde{\widetilde{\gamma}} \def\deltatilde{\widetilde{\delta}} \def\epsilontilde{\widetilde{\epsilon}} \def\zetatilde{\widetilde{\zeta}} \def\etatilde{\widetilde{\eta}} \def\thetatilde{\widetilde{\theta}} \def\iotatilde{\widetilde{\iota}} \def\kappatilde{\widetilde{\kappa}} \def\lambdatilde{\widetilde{\lamdba}} \def\mutilde{\widetilde{\mu}} \def\nutilde{\widetilde{\nu}} \def\xitilde{\widetilde{\xi}} \def\omicrontilde{\widetilde{\omicron}} \def\pitilde{\widetilde{\pi}} \def\rhotilde{\widetilde{\rho}} \def\sigmatilde{\widetilde{\sigma}} \def\tautilde{\widetilde{\tau}} \def\upsilontilde{\widetilde{\upsilon}} \def\phitilde{\widetilde{\phi}} \def\chitilde{\widetilde{\chi}} \def\psitilde{\widetilde{\psi}} \def\omegatilde{\widetilde{\omega}} \def\Alphabar{\bar{\Alpha}} \def\Betabar{\bar{\Beta}} \def\Gammabar{\bar{\Gamma}} \def\Deltabar{\bar{\Delta}} \def\Epsilonbar{\bar{\Epsilon}} \def\Zetabar{\bar{\Zeta}} \def\Etabar{\bar{\Eta}} \def\Thetabar{\bar{\Theta}} \def\Iotabar{\bar{\Iota}} \def\Kappabar{\bar{\Kappa}} \def\Lambdabar{\bar{\Lamdba}} \def\Mubar{\bar{\Mu}} \def\Nubar{\bar{\Nu}} \def\Xibar{\bar{\Xi}} \def\Omicronbar{\bar{\Omicron}} \def\Pibar{\bar{\Pi}} \def\Rhobar{\bar{\Rho}} \def\Sigmabar{\bar{\Sigma}} \def\Taubar{\bar{\Tau}} \def\Upsilonbar{\bar{\Upsilon}} \def\Phibar{\bar{\Phi}} \def\Chibar{\bar{\Chi}} \def\Psibar{\bar{\Psi}} \def\Omegabar{\bar{\Omega}} \def\alphabar{\bar{\alpha}} \def\betabar{\bar{\beta}} \def\gammabar{\bar{\gamma}} \def\deltabar{\bar{\delta}} \def\epsilonbar{\bar{\epsilon}} \def\zetabar{\bar{\zeta}} \def\etabar{\bar{\eta}} \def\thetabar{\bar{\theta}} \def\iotabar{\bar{\iota}} \def\kappabar{\bar{\kappa}} \def\lambdabar{\bar{\lamdba}} \def\mubar{\bar{\mu}} \def\nubar{\bar{\nu}} \def\xibar{\bar{\xi}} \def\omicronbar{\bar{\omicron}} \def\pibar{\bar{\pi}} \def\rhobar{\bar{\rho}} \def\sigmabar{\bar{\sigma}} \def\taubar{\bar{\tau}} \def\upsilonbar{\bar{\upsilon}} \def\phibar{\bar{\phi}} \def\chibar{\bar{\chi}} \def\psibar{\bar{\psi}} \def\omegabar{\bar{\omega}} \def\del{\partial} \def\delbar{\overline{\partial}} \def\Cech{\check{C}} \def\half{\frac{1}{2}} \def\defeq{\mathrel{\mathop:}=} \def\alg{\mathrm{alg}} \def\Alt{\mathrm{Alt}} \def\Amp{\mathrm{Amp}} \def\Arg{\mathrm{Arg}} \def\an{\mathrm{an}} \def\anti{\mathrm{anti}} \def\Ap{\mathrm{Ap}} \def\arcsinh{\mathrm{arcsinh\hspace{0.07cm}}} \def\arccosh{\mathrm{arccosh\hspace{0.07cm}}} \def\arctanh{\mathrm{arctanh\hspace{0.07cm}}} \def\arccsch{\mathrm{arccsch\hspace{0.07cm}}} \def\arcsech{\mathrm{arcsech\hspace{0.07cm}}} \def\arccoth{\mathrm{arccoth\hspace{0.07cm}}} \def\arccsc{\mathrm{arccsc\hspace{0.07cm}}} \def\arcsec{\mathrm{arcsec\hspace{0.07cm}}} \def\arccot{\mathrm{arccot\hspace{0.07cm}}} \def\arg{\mathrm{arg}} \def\BC{\mathrm{BC}} \def\Bel{\mathrm{Bel}} \def\calCH{\mathcal{CH}} \def\csch{\mathrm{csch}\hspace{0.07cm}} \def\CH{\mathrm{CH}} \def\ch{\mathrm{ch}} \def\closed{\mathrm{closed}} \def\codim{\mathrm{codim}} \def\coth{\mathrm{coth}\hspace{0.07cm}} \def\Coh{\mathfrak{Coh}} \def\Coker{\mathrm{Coker}} \def\Cone{\mathrm{Cone}} \def\darg{d\mathrm{arg}} \def\Db{\mathrm{Db}} \def\dclosed{\mathrm{d-closed}} \def\deg{\mathrm{deg}} \def\dim{\mathrm{dim}} \def\divisor{\mathrm{div}} \def\dlog{d\mathrm{log}} \def\DNE{\mathrm{DNE}} \def\DR{\mathrm{DR}} \def\DST{\mathrm{DST}} \def\exp{\mathrm{exp}} \def\FLB{\mathrm{FLB}} \def\FLS{\mathrm{FLS}} \def\Gr{\mathrm{Gr}} \def\Hzar{H_{\mathrm{Zar}}} \def\Hol{\mathrm{Hol}} \def\Id{\mathrm{Id}} \def\Image{\mathrm{Im}} \def\Ka{\mathcal{K}_A} \def\Ker{\mathrm{Ker}} \def\kod{\mathrm{kod}} \def\Kx{\mathcal{K}_X} \def\Kz{\mathcal{K}_Z} \def\log{\mathrm{log}} \def\Log{\mathrm{Log}} \def\Li{\mathrm{Li}} \def\min{\mathrm{min}} \def\Mon{\mathrm{Mon}} \def\Nef{\mathrm{Nef}} \def\NS{\mathrm{NS}} \def\Oa{\mathcal{O}_A} \def\Ox{\mathcal{O}_X} \def\Oz{\mathcal{O}_Z} \def\Perp{\mathrm{Perp}} \def\Pic{\mathrm{Pic}} \def\Proj{\mathrm{Proj}} \def\rank{\mathrm{rank}} \def\Rat{\mathrm{Rat}} \def\Real{\mathrm{Re}} \def\reg{\mathrm{reg}} \def\Res{\mathrm{Res}} \def\res{\mathrm{res}} \def\Ric{\mathrm{Ric}} \def\sech{\mathrm{sech}\hspace{0.07cm}} \def\Span{\mathrm{Span}} \def\Spec{\mathrm{Spec}} \def\sing{\mathrm{sing}} \def\Singx{\mathrm{Sing}(X)} \def\sheafKer{\mathcal{\Ker}} \def\sheafIm{\mathcal{\Im}} \def\Span{\mathrm{Span}} \def\Spin{\mathrm{Spin}} \def\Str{\mathrm{Str}} \def\td{\mathrm{td}} \def\tr{\mathrm{tr}} \def\Todd{\mathrm{Todd}} \def\tor{\mathrm{tor}} \def\trdeg{\mathrm{trdeg}} \def\Zar{\mathrm{Zar}} \def\ZFLS{\mathrm{ZFLS}} \usepackage{tikz} \usepackage{tkz-graph} \usepackage{tkz-euclide} \usetikzlibrary{patterns} \usetikzlibrary{positioning} \usetikzlibrary{matrix,arrows} \usetikzlibrary{calc} \usetikzlibrary{shapes} \usetikzlibrary{through,intersections,decorations,shadows,fadings} \usepackage{pgfplots} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)

Section 4.4 Issues and Next Directions

After the initial examples of series solutions, two of questions seem natural. First, I might observe that \(P\) and \(Q\) were, so far, only rational functions, which means that I only dealt with polynomials in the calculation. What if \(P\) and \(Q\) were other analytic functions? In this cases, we would have to expand \(P\) and \(Q\) as series about the ordinary point and then multiply them by the series for \(y\) in the calculation. This is possible, but miserable.

I might also wonder why I restricted myself to homogeneous cases. What happens if I add a forcing term \(f(t)\text{?}\) I can certainly do this is \(f\) is analytic on the same domain as \(P\) and \(Q\text{.}\) In all our examples so far, I compared the coefficients to \(0\text{.}\) If there is a forcing terms, I would instead compare the coefficients to the coefficients of \(f\) instead of \(0\text{.}\) This leads to more complications, since the relationship between the coefficients may no longer be a linear recurrence relation.

In both of these situations (non-polynomial \(P\) and \(Q\) or non-homogeneous DE with forcing), I can easily end up in a situation were the challenges of computation prevent me from finding a nice form for the coefficients of \(y\text{.}\) Quite frequently, I will only decide to calculate the first few terms. The result is a Taylor polynomial approximation to the solution, instead of a complete Taylor series solution. However, approximate solutions (if they have sufficient precision) are often sufficient. One of the advantages of working with Taylor series solutions is the easy availability of Taylor polynomial approximations.