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\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/input exprode.input}
\author{The Axiom Team}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
<<*>>=
)clear all
)set streams calculate 7
-- We will solve y''' = sin(y'') * exp(y) + cos(x)
-- subject to y(0) = 1, y'(0) = 0, y''(0) = 0
y := operator 'y
eq := differentiate(y x, x, 3) - sin differentiate(y x, x, 2) * exp y x
= cos x
seriesSolve(eq, y, x = 0, [1, 0, 0])
-- Airy, isn't it?
airy := differentiate(y x, x, 2) - x * y x
seriesSolve(airy, y, x = 0, [a0, a1])
-- We can solve around other points than x = 0
seriesSolve(airy, y, x = 1, [a0, a1])
-- System of equations for tan(t) and sec(t)
x := operator 'x
eq1 := differentiate(x t, t) = 1 + x(t)**2
eq2 := differentiate(y t, t) = x(t) * y(t)
seriesSolve([eq2, eq1], [x, y], t = 0, [y 0 = 1, x 0 = 0])
-- System of equations for a damped pendulum of mass and length 1:
eq1 := differentiate(x t, t) = y t
eq2 := differentiate(y t, t) = - g * sin(x t) - c * y t
seriesSolve([eq1, eq2], [x, y], t = 0, [y 0 = a, x 0 = b])
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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