%% 1. Introduction
%
% \setcounter{page}{1}
% What if all you had to do to solve an ODE were just to write it
% down?\footnote{ODE stands for {\em ordinary differential equation},
% as the reader presumably knows already.}
% That is the line we will follow in this book.
% Our emphasis is
% not just on the mathematics of ODE\kern .5pt s, but on how the solutions behave.
% Do they blow up, decay, oscillate? Are there rapid transitions
% where they flip from one state to another?
% Does the behavior change if
% a coefficient is perturbed or a new term is added?
% And how can such variety be deployed to explain
% the world around us?
% We shall not just talk about these matters but explore them in action.
%
%%
%
% ODE\kern .5pt s are among the core topics of mathematics,
% with applications so ubiquitous that
% listing examples almost seems inappropriate.
% (Heat conduction, chemical reactions,
% chaos, population dynamics, deformations
% of a beam, radioactivity, bifurcation theory, stability
% theory, differential geometry, quantum mechanics, economics, finance,
% infectious diseases, nerve signals, vibrations, optics,
% waves, dynamics of networks, special functions, ballistics, planetary
% dynamics$,\dots.$)\ \ ODE\kern .5pt s are everywhere.
%
%%
%
% To solve ODE\kern .5pt s by writing them down, we will use Chebfun, an open-source
% Matlab package that is freely
% available at {\tt www.chebfun.org}.
% In Matlab, you type \verb|x = A\b| to solve the system
% of equations ${\bf A}{\bf x}={\bf b}$,
% where ${\bf A}$ is a matrix and ${\bf x}$ and ${\bf b}$ are vectors.
% In Chebfun, analogously, you type \verb|y = L\f| to solve
% the ODE $Ly=f$, where $L$ is a linear or nonlinear
% differential operator with initial or
% boundary conditions and $y$ and $f$ are functions.
% We will obtain solutions this way on nearly every page,
% presenting them with hundreds of computer-generated plots
% without discussing the algorithms Chebfun uses
% to make this possible.\footnote{To
% learn about Chebfun and its ODE algorithms, see
% Appendix~A and {\tt www.chebfun.org.}}\ \ This
% is not a book about numerical analysis or computer science.
% It is a book about ODE\kern .5pt s.
%
%%
%
% Many textbooks on ODE\kern .5pt s concentrate on linear problems,
% because nonlinear ones are rarely analytically solvable.
% Here, with analytical solutions playing a lesser role, we
% will be able to give a more balanced treatment and fully appreciate
% the remarkable effects that come with nonlinearity.
%
%%
%
% Let's get started with the most basic
% {\bf initial value problem (IVP)}\footnote{Throughout this
% book, some terms are set in italics, while other
% particularly important ones are set in bold face.
% To review a chapter, a good way to start is to read the items in
% italics and bold face and also the chapter summary at the end. Almost all
% emphasized terms can be found in the index.},
% $$ y'+ y = 0, \quad t\in [\kern .3pt 0,3], \quad y(0) = 1. \eqno (1.1) $$
% We have written this in the standard notation of this book, with
% $t$ as the independent variable, $y$ the dependent variable,
% and $y' = dy/dt$.\footnote{For boundary value problems, which are
% usually associated with space instead of time, we will
% change $t$ to $x$. Some books also make a distinction between
% $y'$ for a space derivative and $\dot y$ for a time derivative,
% but we shall not do this.}
% Note that although $y$ and $y'$ are functions of $t$, we usually
% do not write them out fully as $y(t)$ and $y'(t)$.
% You can quickly check that a solution to (1.1)
% is the function $y(t)=\exp(-t)$, and moreover, it is
% easily proved that this solution is unique (see Chapter 2).
% To calculate it with Chebfun, we make a ``chebop'' that encodes the
% differential operator of (1.1), which we call $L$.
% First we prescribe the interval.
%
ODEformats, L = chebop(0,3);
%%
%
% \noindent
% Next we prescribe the differential operator $y \mapsto y' + y$, which is
% written in the form of a Matlab anonymous function of $t$ and $y$,
% with {\tt diff} denoting differentiation with respect to $t$.
%
L.op = @(t,y) diff(y) + y;
%%
%
% \noindent
% Finally we set the {\em initial condition} with \verb|L.lbc|, which
% stands for ``left boundary condition.''
%
L.lbc = 1;
%%
%
% \noindent
% (A boundary condition at the right would be specified with {\tt L.rbc.})
% We can now solve (1.1).
%
y = L\0;
%%
%
% \noindent
% As expected, the solution is $e^{-t}$.\footnote{Throughout
% this book, the Chebfun code segments listed are sufficient to
% reproduce the mathematical essence of each figure, but
% Matlab formatting commands like {\tt title},
% {\tt axis}, and {\tt 'linewidth'} have been removed. Users
% wanting to see formatting details can download the M-files for
% each chapter of the book from {\tt www.chebfun.org}.}
%
plot(y,CO,ivp)
title(['Fig.~1.1.~~The basic linear ODE (1.1): ' ...
'exponential decay'],FS,11)
%%
% \vskip 1.02em
%%
% The problem we
% have just solved can be classified by the following properties.
%%
%
% \parindent=0pt
% \mbox{\hbox to 1.5em{\textsf{\textbf{\hss F\hss}}}} {\bf First-order} (the highest-order derivative is $y'$)
% \par\vskip 2pt
% \mbox{\hbox to 1.5em{\textsf{\textbf{\hss L\hss}}}} {\bf Linear} (there are no terms
% like $y^2,$ $\exp(y)$, or $yy'$)
% \par\vskip 2pt
% \mbox{\hbox to 1.5em{\textsf{\textbf{\hss A\hss}}}} {\bf Autonomous} (the equation, though
% not the solution, is independent of $t\kern .5pt$)
% \par\vskip 2pt
% \mbox{\hbox to 1.5em{\textsf{\textbf{\hss S\hss}}}} {\bf Scalar} (there is just one dependent variable $y$ rather
% than $u,v,w,\dots$)
% \par\vskip 2pt
% \mbox{\hbox to 1.5em{\textsf{\textbf{\hss H\hss}}}} {\bf Homogeneous}
% (the right-hand side is
% zero\footnote{When we say ``the right-hand side is zero,'' we really mean that
% the equation contains no nonzero terms that do not involve $y$.
% Of course it
% doesn't matter mathematically whether a term appears on the
% left or the right of an equation. For nonlinear ODE\kern .5pt s, the
% property of homogeneity does not always have much meaning.})
% \par\vskip 2pt
% \mbox{\hbox to 1.5em{\textsf{\textbf{\hss I\hss}}}} {\bf Initial-value problem}
% (not a boundary-value or eigenvalue problem)
% \vskip 1.02em
%
%%
%
% \noindent Note that these
% letters spell the easily remembered word {\sf FLASH\kern .3pt I}.\ \ In this
% book, we will use this word occasionally to encode some of the
% properties of a problem.
% If we state that a problem is, say,
% of type {\sf Fl\kern .7pt aShi\kern .7pt}, that means that it is a first-order
% scalar problem, since the letters {\sf F} and {\sf S} are capitalized.
% On the other hand since {\sf l}, {\sf a}, {\sf h}, and {\sf i\kern .7pt}
% are in lower case,
% the problem is nonlinear, nonautonomous, inhomogeneous, and
% a boundary-value or eigenvalue problem rather than an IVP.\ \ In
% the chapters ahead, we will vary all of these properties and
% explore in the process a great variety of phenomena.
% The {\sf FLASH\kern .3pt I} classification will
% help add structure to the discussion.
% Sometimes one considers an ODE without specifying initial or
% boundary conditions, and in this case the {\sf I}
% drops away to give just {\sf FLASH}.
%
%%
%
% Here is a {\sf FLASH} classification of the theorems in this book.
%
%%
%
% \obeylines\parindent=0pt
% \vskip .7em
% {\em Linear, first-order, scalar}
% \mbox{\hbox to .45in{\sf FLASH:}} Thm.~2.1 (separation of variables)
% \mbox{\hbox to .45in{\sf FLASh:}} Thm.~2.4 (integrating factor)
% \mbox{\hbox to .45in{\sf FLaSH:}} Thm.~2.2 (separation of variables)
% \mbox{\hbox to .45in{\sf FLaSh:}} Thm.~2.3 (integrating factor), Thm.~19.1 (periodic)
%
%%
%
% \obeylines\parindent=0pt
% \vskip .7em
% {\em Nonlinear, first-order, scalar}
% \mbox{\hbox to .45in{\sf FlASH:}} Thm.~3.2 (separation of variables)
% \mbox{\hbox to .45in{\sf FlaSH:}} Thm.~3.1 (separation of variables)
% \mbox{\hbox to .45in{\sf FlaSh:}} Thms.~11.1 and 11.3 (Picard iteration)
%
%%
%
% \obeylines\parindent=0pt
% \vskip .7em
% {\em Linear, first-order, system}
% \mbox{\hbox to .45in{\sf FLAsH:}} Thm.~14.1 (matrix exponential)
% \mbox{\hbox to .45in{\sf FLAsh:}} Thm.~14.2 (matrix exponential), Thm.~14.3 (variation of parameters)
% \mbox{\hbox to .45in{\sf FLash:}} Thm.~19.2 (periodic)
%
%%
%
% \obeylines\parindent=0pt
% \vskip .7em
% {\em Nonlinear, first-order, system}
% \mbox{\hbox to .45in{\sf FlAsH:}} Thm.~14.4 (linearization), Thm.~15.1 (stability)
% \mbox{\hbox to .45in{\sf Flash:}} Thms.~11.2 and 11.3 (Picard iteration)
%
%%
%
% \obeylines\parindent=0pt
% \vskip .7em
% {\em Second-order}
% \mbox{\hbox to .45in{\sf f\kern 1pt LASH:}} Thm.~4.1 (solution formula)
% \mbox{\hbox to .45in{\sf f\kern 1pt LaSH:}} Thm.~7.1 (eigenproblems), Thm.~19.3 (Hill's eq.)
% \mbox{\hbox to .45in{\sf f\kern 1pt lASH:}} Thm.~14.5 (linearization)
% \vskip .7em
%
%%
%
% As an illustration of a nonlinear ODE, here is an example of type
% {\sf f\kern 1pt lASHI}, the second-order
% equation known as the {\em van der Pol equation,}
% $$ 0.3y''-(1-y^2)y'+y = 0, \quad t\in [\kern .3pt 0,20\kern .3pt], \quad y(0)=1, ~y'(0)=0.
% \eqno (1.2) $$
% (The coefficient $0.3$ is included to make the solution more interesting.)
% Because $y''$ is present, the equation is of second rather than
% first order (hence {\sf f} not~{\sf F}), and it is nonlinear because of the
% coefficient $1-y^2$ multiplying $y'$ (hence {\sf l} not~{\sf L}).
% Here are the interval and the operator,
% which we name \verb|N| instead of \verb|L|
% as a reminder that it is nonlinear.
%
N = chebop(0,20);
N.op = @(t,y) 0.3*diff(y,2) - (1-y^2)*diff(y) + y;
%%
%
% \noindent
% Problem (1.1) had just a single boundary condition, but in (1.2),
% since it is a second-order equation, there
% are two. For a simple scalar problem like this, Chebfun
% permits one to prescribe $y$ and $y'$ at a point by supplying
% a vector of two numbers.
%
N.lbc = [1;0];
%%
%
% \noindent
% Here is the solution.\footnote{Readers viewing these
% pages in color will note that solutions to linear
% IVPs are usually plotted in green and solutions to nonlinear
% ones in dark green. Starting in Chapter 5, we will
% likewise plot solutions to linear BVPs in blue and
% solutions to nonlinear ones in dark blue. The distinctions between
% linear and nonlinear equations and between IVPs and BVPs are
% important, and the colors will serve as a quiet reminder.}
%
y = N\0; plot(y,CO,ivpnl)
title('Fig.~1.2.~~Van der Pol equation (1.2)',FS,11)
%%
% \vskip 1.02em
%%
%
% Chebfun gives us ready access to
% the properties of a computed solution $y$ (which is represented
% as a ``chebfun,'' with a lower-case c since this is a function
% rather than the name of the software system).
% For example, here are the positions of the local maxima of $y$.
%
[mval,mpos] = max(y,'local'); mpos'
%%
%
% \noindent
% By taking differences of successive maxima,
% we see that this van der Pol oscillation is
% settling down to a periodic function with period 4.0725,
%
diff(mpos)'
%%
%
% This book is aimed at everyone who is interested in ODE\kern .5pt s. If you are an
% undergraduate taking a course from one of the big texts like
% Boyce and DiPrima or Edwards and Penney, this is your lightweight
% companion. (The hardcopy from SIAM is inexpensive, and the online version
% is free.) If you are a graduate
% student working in any of the mathematical sciences, this
% may be just the book to take your
% understanding to the next level.
% Whoever you are, we aim to increase your
% appreciation of this fundamental subject.
%
%%
%
% What does it mean to ``solve'' an ODE\kern .3pt?\ \ One kind of solution
% would be an exact explicit formula,
% also known as an {\em analytical solution.}
% Advantages of analytical
% solutions include perfect accuracy, generality, explicit dependence
% on parameters, theoretical insight, and the absence
% of restriction to a particular range of values of\/ $t$. The trouble is, most
% ODE problems, including almost all nonlinear ones, can't be solved
% analytically. Another kind of solution is a numerical one obtained on a
% computer. The great advantage of numerical solutions is that they can be
% obtained for virtually any ODE.\ \ That's not the only advantage, however.
% Another is that numbers are a compact and
% universal currency, so that by examining results obtained
% numerically, one can apply one's analysis to a problem at
% hand or check how one ODE solution compares to
% another. It is not always obvious how to compare two exact
% formulas, but we always know how to compare two numbers, whether
% explicitly as numbers or visually in a plot.
%
%%
%
% Among the huge variety of interesting ODE\kern .5pt s, there
% are five that keep reappearing over and over in this book:
% {\vskip 6pt \obeylines\parindent=0pt
% {\bf 2nd-order linear oscillator.} Simple harmonic motion, or with damping.
% {\bf Van der Pol equation.} Nonlinear oscillator, with solutions on a limit cycle.
% {\bf Nonlinear pendulum.} Large-amplitude, with a periodic phase space.
% {\bf Lorenz equations.} Archetypical chaotic system, with three coupled variables.
% {\bf Linear system.} ${\bf y}' = {\bf Ay}$, with solutions $\exp(t{\bf A}) {\bf y}_0^{}$.
% \vskip 6pt
% \par}
% \noindent
% Many other equations will also be mentioned, including the
% Airy, Belousov--Zhabotinsky, Bernoulli, Bessel, Blasius, Bratu, brusselator,
% Carrier, Duffing, Emden, H\'enon--Heiles, Hill, logistic, Lotka--Volterra,
% Mathieu, $n$-body, oregonator, Painlev\'e, and R\"ossler equations.
%
%%
%
% Easy computer exploration brings new perspectives on ODE\kern .5pt s,
% and we believe we have found distinctive treatments
% of most of the topics presented in this book, which we hope
% will blend clarity for beginners with unexpected insights
% for experts. Here are some items worthy of note
% in each chapter.
% \vskip 1.02em
%
%%
%
% \parindent=0pt
% {\em 1. Introduction.}
% The ``{\sf FLASHI}'' classification.\hfill\break
% {\em 2. First-order scalar linear ODE\kern .5pt s.}
% Fig.~2.3: smooth vs.\ bang-bang forcing.
% \hfill\break
% {\em 3. First-order scalar nonlinear ODE\kern .5pt s.}
% Scalarization by complex arithmetic.\hfill\break
% {\em 4. Second-order equations and damping.}
% Ex.~4.1: elliptical/non-elliptical orbits.\hfill\break
% {\em 5. Boundary value problems.}
% Fig.~5.9: side conditions.\hfill\break
% {\em 6. Eigenvalues of linear BVPs.}
% Fig.~6.2: eigenvalues via response curve.\hfill\break
% {\em 7. Variable coefficients and adjoints.}
% Automated computation of adjoints.\hfill\break
% {\em 8. Resonance.}
% Periodic forcing gives periodic solutions, {\em if\/} there
% is damping.\hfill\break
% {\em 9. Second-order equations in the phase plane.}
% BVPs as well as IVPs. \hfill\break
% {\em 10. Systems of equations.}
% SIR epidemiology models in 6 lines of code.\hfill\break
% {\em 11. The fundamental existence theorem.}
% Picard iteration on the computer.\hfill\break
% {\em 12. Random functions and random ODE\kern .5pt s.}
% Stochastics via smooth functions.\hfill\break
% {\em 13. Chaos.}
% Transient chaos in the 3-body problem.\hfill\break
% {\em 14. Linearization.}
% Figures showing that locally, any ODE behaves linearly.
% \hfill\break
% {\em 15. Stable and unstable fixed points.}
% Application to transition to turbulence.
% \hfill\break
% {\em 16. Multiple solutions of nonlinear BVPs.}
% Shooting to find multiple solutions.\hfill\break
% {\em 17. Bifurcation.}
% Fig.~17.18: tracking hysteresis as a parameter varies.\hfill\break
% {\em 18. Continuation and path-following.}
% Fig.~18.13: numerical bifurcation.
% \hfill\break
% {\em 19. Periodic ODE\kern .5pt s.}
% Application showing the origin of band gaps.
% \hfill\break
% {\em 20. Boundary and interior layers.}
% Plots showing asymptotics in action.\hfill\break
% {\em 21. Into the complex plane.}
% Analytic continuation of real solutions.\hfill\break
% {\em 22. Time-dependent PDE\kern .5pt s.}
% How ODE\kern .5pt s arise from PDE\kern .5pt s as $t\to\infty$.
% \vskip 1.02em
%
%%
%
% \noindent
% Readers will also find many phenomena explored in the
% exercises that do not appear in other textbooks.
%
%%
%
% Any book on ODE\kern .5pt s faces the question of how much space to give to applications.
% As a structure that we hope will prove appealing, we
% follow the pattern that each chapter ends with a 2--4 page
% item designated as an Application.
% This is followed in turn by
% a few sentences about history, a mention of our favorite reference,
% and the exercises for that chapter.
%
%%
%
% With the help of Chebfun, can we really solve
% any ODE just by writing it down? No, of course not. Examples
% can readily be devised that defeat this method of computing
% for reasons including singularities, scaling,
% stiffness, positivity constraints, or sheer computational scale,
% and a page about such challenges can be found at the
% end of Appendix~A.\ \ Nevertheless,
% we have been gratified in writing this book to find how easy it has
% been to explore almost any topic.
%
%%
%
% \begin{center}
% \hrulefill\\[1pt]
% {\sc Applications in this book}\\[-3pt]
% \hrulefill
% \end{center}
%
%%
%
% \parindent=0pt\parskip = 0pt
% \def\app#1.#2/{\par\noindent\kern 1.2in
% \llap{\em Chapter #1.\kern 1pt} #2}
% \app 2. Elimination of caffeine from the bloodstream./
% \app 3. Classic pursuit problems./
% \app 4. Skydiver./
% \app 5. Beam theory and the strength of spaghetti./
% \app 6. Eigenstates of the Schr\"odinger equation./
% \app 7. Adjoints and optimization./
% \app 8. Moon, sun, and tides./
% \app 9. Nonlinear pendulum./
% \app 10. SIR model for epidemics./
% \app 11. Designer nonuniqueness./
% \app 12. Metastability, radioactivity, and tunneling./
% \app 13. Chaos in a food web./
% \app 14. Linearized Lorenz trajectories./
% \app 15. Transition to turbulence in a pipe./
% \app 16. Sending a spacecraft to a destination./
% \app 17. FitzHugh--Nagumo equations of neural signals./
% \app 18. Arrhenius chemical reaction./
% \app 19. Band gaps and forbidden frequencies./
% \app 20. Why is New York hotter than San Francisco?/
% \app 21. Jacobi sine function./
% \app 22. Solitons and the KdV equation./
% \vskip 1.02em
%
%%
%
% {\sc History.}
% Many of the great mathematicians of the past were
% involved in establishing the subject of ODE\kern .5pt s, starting
% around 1670, including Newton,
% Leibniz, Johann and Jacob Bernoulli, Riccati,
% Clairaut, Euler, d'Alembert, Lagrange, Gauss, and Cauchy. Stepping forward to
% 100 or so years ago, some other key
% figures were Poincar\'e, Picard, Lyapunov, Painlev\'e, and Goursat.
%
%%
%
% \smallskip
% {\sc Our favorite reference.} For a charismatic tour of the whole subject
% of ODE\kern .5pt s with a historical emphasis, see Chapter I of Hairer,
% N\o rsett, and Wanner, {\em Solving Ordinary Differential
% Equations I,} Second Revised Edition, Springer-Verlag, 1993.
% \smallskip
%
%%
%
% {\sc Acknowledgments.}
% Many people have helped us in preparing this book, and we
% can acknowledge just a few: David Allwright, Patrick Farrell,
% Abdul-Lateef Haji-Ali, Nick Hale, Des Higham, Hrothgar,
% and Tadashi Tokieda.
% Michael Rawson was a lively participant in a course at NYU that
% class-tested the book in autumn 2016.
% We are particularly grateful to Abinand Gopal, Matt Meyers,
% Niels M\o ller, and Adam Stinchcombe
% for reading the entire manuscript
% at a late stage and offering many helpful suggestions.
%
%%
%
% \def\equals{\kern -1.5pt =\kern 1.5pt}
% \begin{displaymath}
% \framebox[4.7in][c]{\parbox{4.5in}{\vspace{2pt}\sl
% {\sc Summary of Chapter 1.}
% ODE\kern .5pt s can rarely be solved analytically, but they can always
% be solved numerically, and Chebfun provides a convenient tool for
% doing this. This book explores all kinds of ODE problems, from
% the elementary to the advanced, both initial-value
% problems (IVPs) and boundary-value problems (BVPs). ODE problems can be
% classified by a schema with mnemonic {\sf FLASH\kern .3pt I}:\/
% \kern .5pt{\sf F} \equals first-order,
% \mbox{{\sf L} \equals linear,}
% \mbox{{\sf A} \equals autonomous,}
% {\sf S} \equals scalar,
% {\sf H} \equals homogeneous,
% {\sf I} \equals IVP.
% \vspace{2pt}}}
% \end{displaymath}
%
%%
%
% \smallskip\small\parskip=1pt
% \parindent=0pt
% {\em A note on exercises.}
% The exercises in this book mix the theoretical and the
% computational, and the labels of computational
% exercises are {\em\underline{underlined}\kern .3pt.}\ \ If
% you are asked, say, to
% ``find a value'' of a solution, a computational result is
% usually expected unless it is explicitly
% stated that it should be analytical.
% As a rule, give computed results to~6 digits of accuracy.
% For theoretical problems, though your final solution should be
% analytical, you may sometimes find computational explorations
% helpful along the way.
% \par\vskip 4pt
% {\em \underline{Exercise $1.1$}. Local extrema of van der Pol oscillation.}
% In the van der Pol example (1.2), you can find the local maxima
% with {\tt max(y,'local')}, and similarly for minima with
% {\tt min}; you can find both minima and maxima at once with
% {\tt minandmax}.
% How close is the first local minimum value (at $t\approx 1.2$) to
% its asymptotic value for $t\to \infty$? Likewise for
% the first local maximum (at $t\approx 3.2$)?
% \par
% {\em Exercise $1.2$. Classification of ODE problems.}
% Classify the following ODE problems according to the
% {\sf FLASH\kern .3pt I} scheme.
% \par
% {\em (a)} $y' = \sin(t) - y$, $t\in [\kern .3pt 0,100\kern .3pt]$, $y(0) = 1$.
% \par
% {\em (\kern .7pt b)} $y' = \sin(t) - y^3$, $t\in [\kern .3pt 0,100\kern .3pt]$, $y(0) = 1$.
% \par
% {\em (c)} (Nonlinear pendulum equation)
% $y'' = -\sin(y)$, $t\in [\kern .3pt 0,10\kern .3pt]$, $y(0) = y(10) = 2$.
% \par
% {\em (d)} (Advection-diffusion equation)
% $0.02 y'' +y' + y = 0$, $t\in [\kern .3pt 0,1]$, $y(0) =0$, $y(1) = 1$.
% \par
% {\em (e)} (Airy equation)
% $0.02 y'' -t y = 0$, $t\in [-5,5]$, $y(-5) =1$, $y(5) = 0$.
% \par
% {\em (f)} (Harmonic oscillator)
% $u' = v,$ $v' = -u$, $t\in [\kern .3pt 0,100\kern .3pt]$, $u(0) = 1$,
% $v(0) = 0$.
% \par
% {\em (g)} $u' = u^2v,$ $v' = -uv^2$,
% $t\in [\kern .3pt 0,2\kern .3pt]$, $u(0) = 1$, $v(0) = 0$.
% \par
% {\em (h)} (Bessel equation) $t^2y'' + t\kern .3pt y' + (t^2-4)y = 0$,
% $t\in [\kern .3pt 0,8]$, $y(0) = 0$, $y(8) = 1$.
% \par
% {\em (i)} $0.1y'' + yy' = y$, $t\in [-1,1]$, $y(-1) = -2$, $y(1) = 1$.
% \par
% {\em (j)} (Lotka--Volterra equations) $u' = u(1-v)$,
% $v' = v(u-1)$, $t\in [\kern .3pt 0,10\kern .3pt]$, $u(0) = v(0) =1$.
% \par
% {\em (k)} (Blasius equation) $y''' + 0.5 y y'' = 0$,
% $t\in[\kern .3pt 0,10\kern .3pt]$, $y(0)=y'(0) = 0$, $y'(10) = 1$.
% \par
% {\em \underline{Exercise $1.3$}. Airy equation.}
% Plot the solution $y$ of the problem of Exercise 1.2{\em (e)} and
% report its maximum value.
% Letting $k$ denote the coefficient $0.02$, do the same with
% $k=0.002$. Now make a plot of $\max (y)$ as a function
% of $k$ for the values $k= 0.001, 0.002, \dots, 0.039, 0.040$.
% (We shall explore such effects in Chapter 6.)
% \par
% {\em \underline{Exercise $1.4$}. Solution with rapid transient.}
% Plot the solution $y$ of the problem of Exercise 1.2{\em (i)}, and give its maximum
% slope $s = \max_{t\in [-1,1]}^{} y'(t)$.
% Do the same with the coefficient $0.1$ reduced to $1/20$, $1/40$, and
% $1/80$.
% \par
% {\em Exercise $1.5$. Reduction to first-order system.}
% {\em (a)} Show how the IVP of Exercise 1.2{\em (f)} can be rewritten as a second-order
% scalar IVP involving just the dependent variable~$u$. What are
% the initial conditions for this IVP?
% {\em (\kern .7pt b)} Conversely, show how the third-order IVP of
% Exercise 1.2{\em (k)} can be rewritten as a first-order system
% involving three variables $u$, $v$, and $w$. Any higher-order
% ODE problem can be rewritten as a first-order problem like this,
% and numerical software for IVPs often requires the problem to be
% expressed in first-order form.
% \par
% {\em \underline{Exercise $1.6$}. How Chebfun represents functions.}
% Chebfun normally represents solutions $y(t)$ to ODE\kern .5pt s by polynomial approximations,
% typically of accuracy about 10 digits, whose degree~$n$
% may be quite high. The polynomials can be interpreted as
% interpolants through a sufficiently large number $n+1$ of samples
% at {\em Chebyshev points} defined by
% $t_j = \cos(\pi j/n)$, $0\le j \le n$ for $t\in [-1,1]$, or linearly
% transplanted to a different interval $[a,b]$.
% {\em (a)} Let {\tt y} be the computed solution of
% the problem (1.1). Execute {\tt length(y)} to find the number $n+1$ for
% this function, and \verb|plot(y,'.-')| to see the associated Chebyshev
% points.
% {\em (\kern .7pt b)} Do the same for the computed solution of the van der Pol problem (1.2).
% Approximately speaking (say, to within $10\%$), how many interpolation points
% are there on average per wavelength?
% \par
% {\em \underline{Exercise $1.7$}. How Chebfun represents periodic functions.}
% Chebfun also has a representation for periodic functions that takes
% advantage of the periodicity, based on trigonometric polynomials (i.e.,
% Fourier series) rather
% than ordinary algebraic polynomials. Periodic solutions arise
% naturally in ODE\kern .5pt s with periodic coefficients (see Chapter~19).
% {\em (a)} Construct an ordinary chebfun for $f(t) = (1.1-\cos(\pi t))^{-1}$,
% $t\in [-1, 1]$ with the command
% \verb|chebfun('1/(1.1-cos(pi*x))')|. What is its length?
% {\em (\kern .7pt b)} How does the length change if you use
% \verb|chebfun('1/(1.1-cos(pi*x))','trig')|?
% (In an appropriate limit, the ratio of the two
% lengths approaches $\pi/2$.)
% \par
%