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Course Notes for Differential Equations
Remkes Kooistra
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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\,}$}}} \)
Front Matter
1
Preliminaries
1.1
Functions
1.2
Linear Algebra
1.3
Taylor Series
1.4
Recurrence Relations
1.5
Complex Numbers
1.6
A Note on Variables
2
First Order DEs
2.1
What is a Differential Equation?
2.1.1
Equations and Questions
2.1.2
Notations and Definitions
2.1.3
The Most Important Example
2.1.4
Archtypical Examples
2.1.5
Pure and Applied Perspectives
2.1.6
The Scarcity of Solutions
2.2
Qualitative Methods for First Order DEs
2.2.1
Autonomous DEs and Phase-Line Analysis
2.2.2
Direction Fields
2.3
Stability and Linearization
2.3.1
Stability
2.3.2
Linearization
2.4
Separable Equations
2.5
Existence and Uniqueness of Solutions
2.5.1
Existence
2.5.2
Lipschitz Continuity
2.5.3
Uniqueness
2.6
Linear Equations and Integrating Factors
2.6.1
Homogeneous Solutions
2.6.2
Linear Operators and Superposition
2.6.3
Integrating Factors
2.6.4
Examples
2.7
Substitutions
2.7.1
Homogeneous DEs
2.7.2
Bernoulli Equations
2.8
Approximation Methods and Applications
2.9
Activities for First Order ODEs
2.9.1
Phase Lines
2.9.2
Direction Fields
2.9.3
Separable Equations
2.9.4
First Order Linear Equations
2.9.5
Substitution - Homogeneous Equations
2.9.6
Substitution - Bernoulli Equations
3
Second Order DEs
3.1
Linear DEs with Constant Coefficients
3.1.1
Harmonic Motion
3.1.2
Alternating Current Circuits
3.1.3
Solving SOCCLDEs
3.1.4
Case 1: Two Real Roots
3.1.5
Case 2: Repeated Real Roots
3.1.6
Case 3: Complex Roots
3.2
The Method of Undetermined Coefficients
3.3
Resonance
3.3.1
Situation 1: No Friction, No Forcing
3.3.2
Situation 2: No Friction, Forcing
3.3.3
Section 3: Friction, No Forcing
3.3.4
Situation 4: Friction and Forcing
3.4
Variation of Parameters
3.4.1
Wronskians
3.4.2
The Technique of Variations of Parameters
3.5
Activities for Second Order ODEs
3.5.1
Homogeneous Equations
3.5.2
Undetermined Coefficients
3.5.3
Variation of Parameters
4
Series Solutions
4.1
Series Solutions to Linear Differential Equations
4.2
Ordinary Points
4.2.1
Existence of Solutions
4.2.2
The Method for Solutions at Ordinary Points
4.3
Legendre Functions
4.4
Issues and Next Directions
4.5
Regular Singular Points
4.5.1
Examples of Regular Singular Points
4.5.2
The Method of Frobenius
4.5.3
Examples
4.5.4
Multiple Solutions in the Method of Frobenius
4.6
Bessel Functions
4.6.1
The
\(\Gamma\)
Function
4.6.2
Bessel’s Equation
4.6.3
The Aging Spring
4.6.4
Properties of the Bessel Functions
4.7
Activities for Series Solutions
4.7.1
4.7.2
Method of Frobenius
5
Laplace Transforms
5.1
Definitions
5.2
Properties of the Laplace Transform
5.2.1
First Properties
5.2.2
Shifts
5.3
Distributions
5.4
Laplace Transforms and Derivatives
5.5
Inverse Laplace Transforms
5.6
Using Laplace Transforms to Solve Linear DEs
5.7
Laplace Transforms of Periodic Functions
5.8
Algebraic Properties of the Laplace Transform
5.9
More Examples of Solving DEs with Laplace Transforms
5.10
Activities for Laplace Transforms
5.10.1
Using Laplace Transforms to Solve DEs
6
Systems of Differential Equations
6.1
Systems of Differential Equations
6.1.1
Definitions
6.1.2
Key Examples
6.2
Qualitative Methods
6.3
Linear Systems
6.3.1
Qualitative Analysis
6.3.2
Laplace Transforms for Linear Systems
6.3.3
Second Order Systems
6.3.4
General Theory via Eigenvalues
6.3.5
Stability via Eigenvalues
6.4
Activities for Systems of ODEs
6.4.1
Phase Plane Diagrams
6.4.2
Linear Systems
7
Partial Differential Equations
7.1
Partial Differential Equations
7.2
The Heat Equation
7.2.1
Initial and Boundary Conditions
7.2.2
Solving the Heat Equation
7.3
Fourier Series
7.3.1
Series and Bases for Functions
7.3.2
Scope of Fourier Series
7.3.3
Properties of Fourier Series
7.3.4
Finishing the Heat Equation Solution
7.4
Boundary Conditions for the Heat Equation
Backmatter
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