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[Axiom-developer] 20071006.02.tpd.patch
|
From: |
daly |
|
Subject: |
[Axiom-developer] 20071006.02.tpd.patch |
|
Date: |
Sat, 6 Oct 2007 20:46:50 -0500 |
This patch adds the remaining Kamke ODEs that give results in Axiom.
===================================================================
diff --git a/changelog b/changelog
index c63be68..7c61a45 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,5 @@
+20071005 tpd src/input/Makefile kamke7.input regression test added
+20071005 tpd src/input/kamke7.input ODE regression test added
20071005 tpd src/input/Makefile kamke6.input regression test added
20071005 tpd src/input/kamke6.input ODE regression test added
20071005 tpd src/input/Makefile kamke5.input regression test added
diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet
index 3d92581..f1f937b 100644
--- a/src/input/Makefile.pamphlet
+++ b/src/input/Makefile.pamphlet
@@ -323,7 +323,7 @@ REGRES= algaggr.regress algbrbf.regress algfacob.regress
alist.regress \
iprntpk.regress \
ipftest.regress is.regress isprime.regress kafile.regress \
kamke0.regress kamke1.regress kamke2.regress kamke3.regress \
- kamke4.regress kamke5.regress kamke6.regress \
+ kamke4.regress kamke5.regress kamke6.regress kamke7.regress \
kernel.regress knot2.regress kovacic.regress kuipers.regress \
laplace.regress leg.regress limit.regress lindep.regress \
list.regress lode.regress lodesys.regress lodo1.regress \
@@ -564,7 +564,7 @@ FILES= ${OUT}/algaggr.input ${OUT}/algbrbf.input
${OUT}/algfacob.input \
${OUT}/ipftest.input ${OUT}/is.input ${OUT}/isprime.input \
${OUT}/kafile.input ${OUT}/kamke0.input ${OUT}/kamke1.input \
${OUT}/kamke2.input ${OUT}/kamke3.input ${OUT}/kamke4.input \
- ${OUT}/kamke5.input ${OUT}/kamke6.input \
+ ${OUT}/kamke5.input ${OUT}/kamke6.input ${OUT}/kamke7.input \
${OUT}/kernel.input ${OUT}/knot.input \
${OUT}/kovacic.input ${OUT}/kuipers.input \
${OUT}/laplace.input ${OUT}/leg.input ${OUT}/lexp.input \
@@ -833,7 +833,7 @@ DOCFILES= \
${DOC}/kamke0.input.dvi ${DOC}/kamke1.input.dvi \
${DOC}/kamke2.input.dvi ${DOC}/kamke3.input.dvi \
${DOC}/kamke4.input.dvi ${DOC}/kamke5.input.dvi \
- ${DOC}/kamke6.input.dvi \
+ ${DOC}/kamke6.input.dvi ${DOC}/kamke7.input.dvi \
${DOC}/kernel.input.dvi ${DOC}/knot2.input.dvi \
${DOC}/knot.input.dvi ${DOC}/knownbugs.input.dvi \
${DOC}/kovacic.input.dvi ${DOC}/kuipers.input.dvi \
diff --git a/src/input/kamke7.input.pamphlet b/src/input/kamke7.input.pamphlet
new file mode 100644
index 0000000..eb989d3
--- /dev/null
+++ b/src/input/kamke7.input.pamphlet
@@ -0,0 +1,1538 @@
+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/input kamke7.input}
+\author{Timothy Daly}
+\maketitle
+\begin{abstract}
+This is the remaining ODEs of the Kamke test suite as published by
+E. S. Cheb-Terrab\cite{1}. They have been rewritten using Axiom
+syntax. Where possible we show that the particular solution actually
+satisfies the original ordinary differential equation.
+
+Note that after a certain point Axiom can no longer generate useful
+results. The failures fall into several cases which have been included
+in other regression test files.
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+\section{Generated results}
+<<*>>=
+)spool kamke7.output
+)set break resume
+)set mes auto off
+)clear all
+
+--S 1 of 97
+y:=operator 'y
+--R
+--R
+--R (1) y
+--R Type:
BasicOperator
+--E 1
+
+--S 2 of 97
+f:=operator 'f
+--R
+--R
+--R (2) f
+--R Type:
BasicOperator
+--E 2
+
+--S 3 of 97
+g:=operator 'g
+--R
+--R
+--R (3) g
+--R Type:
BasicOperator
+--E 3
+
+--S 4 of 97
+h:=operator 'h
+--R
+--R
+--R (4) h
+--R Type:
BasicOperator
+--E 4
+
+--S 5 of 97
+fa:=operator 'fa
+--R
+--R
+--R (5) fa
+--R Type:
BasicOperator
+--E 5
+
+--S 6 of 97
+fb:=operator 'fb
+--R
+--R
+--R (6) fb
+--R Type:
BasicOperator
+--E 6
+
+--S 7 of 97
+fc:=operator 'fc
+--R
+--R
+--R (7) fc
+--R Type:
BasicOperator
+--E 7
+
+--S 8 of 97
+fd:=operator 'fd
+--R
+--R
+--R (8) fd
+--R Type:
BasicOperator
+--E 8
+
+--S 9 of 97
+fe:=operator 'fe
+--R
+--R
+--R (9) fe
+--R Type:
BasicOperator
+--E 9
+
+--S 10 of 97
+ff:=operator 'ff
+--R
+--R
+--R (10) ff
+--R Type:
BasicOperator
+--E 10
+
+--S 11 of 97
+ode352 := D(y(x),x)*(cos(y(x))-sin(alpha)*sin(x))*cos(y(x))+(cos(x)-_
+ sin(alpha)*sin(y(x)))*cos(x)
+--R
+--R
+--R (11)
+--R 2 ,
+--R (cos(y(x)) - sin(alpha)sin(x)cos(y(x)))y (x) -
cos(x)sin(alpha)sin(y(x))
+--R
+--R +
+--R 2
+--R cos(x)
+--R Type: Expression
Integer
+--E 11
+
+--S 12 of 97
+yx:=solve(ode352,y,x)
+--R
+--R
+--R (cos(y(x)) - 2sin(alpha)sin(x))sin(y(x)) + cos(x)sin(x) + y(x) + x
+--R (12) ------------------------------------------------------------------
+--R 2
+--R Type: Union(Expression
Integer,...)
+--E 12
+
+--S 13 of 97
+ode352expr := D(yx,x)*(cos(yx)-sin(alpha)*sin(x))*cos(yx)+(cos(x)-_
+ sin(alpha)*sin(yx))*cos(x)
+--R
+--R
+--R (13)
+--R -
+--R 2cos(x)sin(alpha)
+--R *
+--R sin
+--R (cos(y(x)) - 2sin(alpha)sin(x))sin(y(x)) + cos(x)sin(x)
+--R +
+--R y(x) + x
+--R /
+--R 2
+--R +
+--R 2 2 ,
+--R (- sin(y(x)) + cos(y(x)) - 2sin(alpha)sin(x)cos(y(x)) + 1)y (x)
+--R
+--R +
+--R 2 2
+--R - 2cos(x)sin(alpha)sin(y(x)) - sin(x) + cos(x) + 1
+--R *
+--R (cos(y(x)) - 2sin(alpha)sin(x))sin(y(x)) + cos(x)sin(x) + y(x) +
x 2
+--R
cos(------------------------------------------------------------------)
+--R 2
+--R +
+--R 2 2
+--R sin(alpha)sin(x)sin(y(x)) - sin(alpha)sin(x)cos(y(x))
+--R +
+--R 2 2
+--R 2sin(alpha) sin(x) cos(y(x)) - sin(alpha)sin(x)
+--R *
+--R ,
+--R y (x)
+--R
+--R +
+--R 2 3
+--R 2cos(x)sin(alpha) sin(x)sin(y(x)) + sin(alpha)sin(x)
+--R +
+--R 2
+--R (- cos(x) - 1)sin(alpha)sin(x)
+--R *
+--R (cos(y(x)) - 2sin(alpha)sin(x))sin(y(x)) + cos(x)sin(x) + y(x) +
x
+--R
cos(------------------------------------------------------------------)
+--R 2
+--R +
+--R 2
+--R 2cos(x)
+--R /
+--R 2
+--R Type: Expression
Integer
+--E 13
+
+--S 14 of 97
+ode353 := x*D(y(x),x)*cos(y(x))+sin(y(x))
+--R
+--R
+--R ,
+--R (14) x cos(y(x))y (x) + sin(y(x))
+--R
+--R Type: Expression
Integer
+--E 14
+
+--S 15 of 97
+yx:=solve(ode353,y,x)
+--R
+--R
+--R (15) x sin(y(x))
+--R Type: Union(Expression
Integer,...)
+--E 15
+
+--S 16 of 97
+ode353expr := x*D(yx,x)*cos(yx)+sin(yx)
+--R
+--R
+--R 2 ,
+--R (16) sin(x sin(y(x))) + (x cos(y(x))y (x) + x sin(y(x)))cos(x sin(y(x)))
+--R
+--R Type: Expression
Integer
+--E 16
+
+--S 17 of 97
+ode354 := (x*sin(y(x))-1)*D(y(x),x)+cos(y(x))
+--R
+--R
+--R ,
+--R (17) (x sin(y(x)) - 1)y (x) + cos(y(x))
+--R
+--R Type: Expression
Integer
+--E 17
+
+--S 18 of 97
+yx:=solve(ode354,y,x)
+--R
+--R
+--R - sin(y(x)) + x
+--R (18) ---------------
+--R cos(y(x))
+--R Type: Union(Expression
Integer,...)
+--E 18
+
+--S 19 of 97
+ode354expr := (x*sin(yx)-1)*D(yx,x)+cos(yx)
+--R
+--R
+--R (19)
+--R 2 2 2 ,
+--R ((x sin(y(x)) - x sin(y(x)) + x cos(y(x)) )y (x) - x cos(y(x)))
+--R
+--R *
+--R sin(y(x)) - x
+--R sin(-------------)
+--R cos(y(x))
+--R +
+--R 2 sin(y(x)) - x
+--R cos(y(x)) cos(-------------)
+--R cos(y(x))
+--R +
+--R 2 2 ,
+--R (sin(y(x)) - x sin(y(x)) + cos(y(x)) )y (x) - cos(y(x))
+--R
+--R /
+--R 2
+--R cos(y(x))
+--R Type: Expression
Integer
+--E 19
+
+--S 20 of 97
+ode355 := (x*cos(y(x))+cos(x))*D(y(x),x)-y(x)*sin(x)+sin(y(x))
+--R
+--R
+--R ,
+--R (20) (x cos(y(x)) + cos(x))y (x) + sin(y(x)) - y(x)sin(x)
+--R
+--R Type: Expression
Integer
+--E 20
+
+--S 21 of 97
+yx:=solve(ode355,y,x)
+--R
+--R
+--R (21) x sin(y(x)) + y(x)cos(x)
+--R Type: Union(Expression
Integer,...)
+--E 21
+
+--S 22 of 97
+ode355expr := (x*cos(yx)+cos(x))*D(yx,x)-yx*sin(x)+sin(yx)
+--R
+--R
+--R (22)
+--R sin(x sin(y(x)) + y(x)cos(x))
+--R +
+--R 2 ,
+--R ((x cos(y(x)) + x cos(x))y (x) + x sin(y(x)) - x y(x)sin(x))
+--R
+--R *
+--R cos(x sin(y(x)) + y(x)cos(x))
+--R +
+--R 2 ,
+--R (x cos(x)cos(y(x)) + cos(x) )y (x) + (- x sin(x) + cos(x))sin(y(x))
+--R
+--R +
+--R - 2y(x)cos(x)sin(x)
+--R Type: Expression
Integer
+--E 22
+
+--S 23 of 97
+ode356 := (x**2*cos(y(x))+2*y(x)*sin(x))*D(y(x),x)+2*x*sin(y(x))+y(x)**2*cos(x)
+--R
+--R
+--R 2 , 2
+--R (23) (x cos(y(x)) + 2y(x)sin(x))y (x) + 2x sin(y(x)) + y(x) cos(x)
+--R
+--R Type: Expression
Integer
+--E 23
+
+--S 24 of 97
+yx:=solve(ode356,y,x)
+--R
+--R
+--R 2 2
+--R (24) x sin(y(x)) + y(x) sin(x)
+--R Type: Union(Expression
Integer,...)
+--E 24
+
+--S 25 of 97
+ode356expr:=(x**2*cos(yx)+2*yx*sin(x))*D(yx,x)+2*x*sin(yx)+yx**2*cos(x)
+--R
+--R
+--R (25)
+--R 2 2
+--R 2x sin(x sin(y(x)) + y(x) sin(x))
+--R +
+--R 4 2 , 3 2 2
+--R ((x cos(y(x)) + 2x y(x)sin(x))y (x) + 2x sin(y(x)) + x y(x) cos(x))
+--R
+--R *
+--R 2 2
+--R cos(x sin(y(x)) + y(x) sin(x))
+--R +
+--R 4 2 2
+--R (2x sin(x)cos(y(x)) + 4x y(x)sin(x) )sin(y(x))
+--R +
+--R 2 2 2 3 3
+--R 2x y(x) sin(x) cos(y(x)) + 4y(x) sin(x)
+--R *
+--R ,
+--R y (x)
+--R
+--R +
+--R 3 4 2
+--R (4x sin(x) + x cos(x))sin(y(x))
+--R +
+--R 2 2 2 2 4 2
+--R (4x y(x) sin(x) + 4x y(x) cos(x)sin(x))sin(y(x)) + 3y(x) cos(x)sin(x)
+--R Type: Expression
Integer
+--E 25
+
+--S 26 of 97
+ode358 := D(y(x),x)*sin(y(x))*cos(x)+cos(y(x))*sin(x)
+--R
+--R
+--R ,
+--R (26) cos(x)sin(y(x))y (x) + sin(x)cos(y(x))
+--R
+--R Type: Expression
Integer
+--E 26
+
+--S 27 of 97
+yx:=solve(ode358,y,x)
+--R
+--R
+--R (27) - cos(x)cos(y(x))
+--R Type: Union(Expression
Integer,...)
+--E 27
+
+--S 28 of 97
+ode358expr := D(yx,x)*sin(yx)*cos(x)+cos(yx)*sin(x)
+--R
+--R
+--R (28)
+--R 2 ,
+--R (- cos(x) sin(y(x))y (x) - cos(x)sin(x)cos(y(x)))sin(cos(x)cos(y(x)))
+--R
+--R +
+--R sin(x)cos(cos(x)cos(y(x)))
+--R Type: Expression
Integer
+--E 28
+
+--S 29 of 97
+ode361 := (x*sin(x*y(x))+cos(x+y(x))-sin(y(x)))*D(y(x),x)+_
+ y(x)*sin(x*y(x))+cos(x+y(x))+cos(x)
+--R
+--R
+--R (29)
+--R ,
+--R (x sin(x y(x)) - sin(y(x)) + cos(y(x) + x))y (x) + y(x)sin(x y(x))
+--R
+--R +
+--R cos(y(x) + x) + cos(x)
+--R Type: Expression
Integer
+--E 29
+
+--S 30 of 97
+yx:=solve(ode361,y,x)
+--R
+--R
+--R (30)
+--R y(x) 2 y(x) y(x)
+--R 2cos(----) sin(y(x) + x) - 2cos(----)cos(y(x) + x)sin(----) - cos(x
y(x))
+--R 2 2 2
+--R +
+--R cos(y(x))
+--R Type: Union(Expression
Integer,...)
+--E 30
+
+--S 31 of 97
+ode361expr:=(x*sin(x*yx)+cos(x+yx)-sin(yx))*D(yx,x)+_
+ yx*sin(x*yx)+cos(x+yx)+cos(x)
+--R
+--R
+--R (31)
+--R 2 y(x) 2
+--R x sin(x y(x)) - x sin(y(x)) + x cos(y(x) + x)sin(----)
+--R 2
+--R +
+--R y(x) 2
+--R x cos(----) cos(y(x) + x)
+--R 2
+--R *
+--R ,
+--R y (x)
+--R
+--R +
+--R y(x) y(x) y(x) 2
+--R x y(x)sin(x y(x)) + (2x cos(----)sin(----) + 2cos(----) )sin(y(x)
+ x)
+--R 2 2 2
+--R +
+--R y(x) y(x)
+--R - 2cos(----)cos(y(x) + x)sin(----) - cos(x y(x))
+--R 2 2
+--R +
+--R y(x) 2
+--R 2x cos(----) cos(y(x) + x) + cos(y(x))
+--R 2
+--R *
+--R sin
+--R y(x) 2 y(x) y(x)
+--R 2x cos(----) sin(y(x) + x) - 2x cos(----)cos(y(x) + x)sin(----)
+--R 2 2 2
+--R +
+--R - x cos(x y(x)) + x cos(y(x))
+--R +
+--R y(x) 2
+--R - x sin(x y(x)) + sin(y(x)) - cos(y(x) + x)sin(----)
+--R 2
+--R +
+--R y(x) 2
+--R - cos(----) cos(y(x) + x)
+--R 2
+--R *
+--R ,
+--R y (x)
+--R
+--R +
+--R y(x) y(x)
+--R - y(x)sin(x y(x)) - 2cos(----)sin(----)sin(y(x) + x)
+--R 2 2
+--R +
+--R y(x) 2
+--R - 2cos(----) cos(y(x) + x)
+--R 2
+--R *
+--R sin
+--R y(x) 2 y(x) y(x)
+--R 2cos(----) sin(y(x) + x) - 2cos(----)cos(y(x) + x)sin(----)
+--R 2 2 2
+--R +
+--R - cos(x y(x)) + cos(y(x))
+--R +
+--R y(x) 2
+--R x sin(x y(x)) - sin(y(x)) + cos(y(x) + x)sin(----)
+--R 2
+--R +
+--R y(x) 2
+--R cos(----) cos(y(x) + x)
+--R 2
+--R *
+--R ,
+--R y (x)
+--R
+--R +
+--R y(x) y(x)
+--R y(x)sin(x y(x)) + 2cos(----)sin(----)sin(y(x) + x)
+--R 2 2
+--R +
+--R y(x) 2
+--R 2cos(----) cos(y(x) + x) + 1
+--R 2
+--R *
+--R cos
+--R y(x) 2 y(x) y(x)
+--R 2cos(----) sin(y(x) + x) - 2cos(----)cos(y(x) + x)sin(----)
+--R 2 2 2
+--R +
+--R - cos(x y(x)) + cos(y(x)) + x
+--R +
+--R cos(x)
+--R Type: Expression
Integer
+--E 31
+
+--S 32 of 97
+ode363 := (x*D(y(x),x)-y(x))*cos(y(x)/x)**2+x
+--R
+--R
+--R y(x) 2 , y(x) 2
+--R (32) x cos(----) y (x) - y(x)cos(----) + x
+--R x x
+--R Type: Expression
Integer
+--E 32
+
+--S 33 of 97
+yx:=solve(ode363,y,x)
+--R
+--R
+--R y(x) y(x)
+--R x cos(----)sin(----) + 2x log(x) + y(x)
+--R x x
+--R (33) ---------------------------------------
+--R 2x
+--R Type: Union(Expression
Integer,...)
+--E 33
+
+--S 34 of 97
+ode363expr := (x*D(yx,x)-yx)*cos(yx/x)**2+x
+--R
+--R
+--R (34)
+--R y(x) 2 y(x) 2 , y(x) 2
+--R (- x sin(----) + x cos(----) + x)y (x) + y(x)sin(----)
+--R x x x
+--R +
+--R y(x) y(x) y(x) 2
+--R - x cos(----)sin(----) - y(x)cos(----) - 2x log(x) - 2y(x) + 2x
+--R x x x
+--R *
+--R y(x) y(x) 2
+--R x cos(----)sin(----) + 2x log(x) + y(x)
+--R x x
+--R cos(---------------------------------------)
+--R 2
+--R 2x
+--R +
+--R 2
+--R 2x
+--R /
+--R 2x
+--R Type: Expression
Integer
+--E 34
+
+--S 35 of 97
+ode364 := (y(x)*sin(y(x)/x)-x*cos(y(x)/x))*x*D(y(x),x)-_
+ (x*cos(y(x)/x)+y(x)*sin(y(x)/x))*y(x)
+--R
+--R
+--R (35)
+--R y(x) 2 y(x) , 2 y(x) y(x)
+--R (x y(x)sin(----) - x cos(----))y (x) - y(x) sin(----) - x y(x)cos(----)
+--R x x x x
+--R Type: Expression
Integer
+--E 35
+
+--S 36 of 97
+yx:=solve(ode364,y,x)
+--R
+--R
+--R y(x)
+--R (36) - x y(x)cos(----)
+--R x
+--R Type: Union(Expression
Integer,...)
+--E 36
+
+--S 37 of 97
+ode364expr := (yx*sin(yx/x)-x*cos(yx/x))*x*D(yx,x)-_
+ (x*cos(yx/x)+yx*sin(yx/x))*yx
+--R
+--R
+--R (37)
+--R 2 2 y(x) y(x) 3 y(x) 2 ,
+--R (x y(x) cos(----)sin(----) - x y(x)cos(----) )y (x)
+--R x x x
+--R +
+--R 3 y(x) y(x)
+--R - x y(x) cos(----)sin(----)
+--R x x
+--R *
+--R y(x)
+--R sin(y(x)cos(----))
+--R x
+--R +
+--R 2 y(x) 3 y(x) , 2 y(x)
+--R (- x y(x)sin(----) + x cos(----))y (x) + x y(x) sin(----)
+--R x x x
+--R +
+--R 2 y(x)
+--R 2x y(x)cos(----)
+--R x
+--R *
+--R y(x)
+--R cos(y(x)cos(----))
+--R x
+--R Type: Expression
Integer
+--E 37
+
+--S 38 of 97
+ode434 := D(y(x),x)-1
+--R
+--R
+--R ,
+--R (38) y (x) - 1
+--R
+--R Type: Expression
Integer
+--E 38
+
+--S 39 of 97
+ode434a:=solve(ode434,y,x)
+--R
+--R
+--R (39) [particular= x,basis= [1]]
+--RType: Union(Record(particular: Expression Integer,basis: List Expression
Integer),...)
+--E 39
+
+--S 40 of 97
+yx:=ode434a.particular
+--R
+--R
+--R (40) x
+--R Type: Expression
Integer
+--E 40
+
+--S 41 of 97
+ode434expr := D(yx,x)-1
+--R
+--R
+--R (41) 0
+--R Type: Expression
Integer
+--E 41
+
+--S 42 of 97
+ode683 := (D(y(x),x) = y(x)*(-1+log(x*(x+1))*y(x)*x**4-log(x*(x+1))*x**3)/x)
+--R
+--R
+--R 4 2 3 2
+--R , (x y(x) - x y(x))log(x + x) - y(x)
+--R (42) y (x)= ------------------------------------
+--R x
+--R Type: Equation Expression
Integer
+--E 42
+
+--S 43 of 97
+solve(ode683,y,x)
+--R
+--R
+--R - x y(x) + 1
+--R (43) -----------------------------------------------
+--R 3 2 3 2
+--R 6x log(x + x) - 4x + 3x - 6x
+--R -------------------------------
+--R 3+-----+ 18
+--R x y(x)\|x + 1 %e
+--R Type: Union(Expression
Integer,...)
+--E 43
+
+--S 44 of 97
+ode703 := (D(y(x),x) = y(x)*(1-x+y(x)*x**2*log(x)+y(x)*x**3-x*log(x)-x**2)/_
+ (x-1)/x)
+--R
+--R
+--R 2 2 3 2 2
+--R , (x y(x) - x y(x))log(x) + x y(x) + (- x - x + 1)y(x)
+--R (44) y (x)= -------------------------------------------------------
+--R 2
+--R x - x
+--R Type: Equation Expression
Integer
+--E 44
+
+--S 45 of 97
+solve(ode703,y,x)
+--R
+--R
+--R - x y(x) + 1
+--R (45) ----------------------------
+--R 2 - dilog(x) + x
+--R (x - x)y(x)%e
+--R Type: Union(Expression
Integer,...)
+--E 45
+
+--S 46 of 97
+ode714 := (D(y(x),x) = -y(x)*(-log(1/x)+exp(x)+y(x)*x**2*log(x)+_
+ y(x)*x**3-x*log(x)-x**2)/(-log(1/x)+exp(x))/x)
+--R
+--R
+--R (46)
+--R 2 2 1 x 3 2 2
+--R (x y(x) - x y(x))log(x) - y(x)log(-) + y(x)%e + x y(x) - x y(x)
+--R , x
+--R y (x)= ------------------------------------------------------------------
+--R 1 x
+--R x log(-) - x %e
+--R x
+--R Type: Equation Expression
Integer
+--E 46
+
+--S 47 of 97
+solve(ode714,y,x)
+--R
+--R
+--R (47)
+--R -
+--I 1 %I 2
+--I x %I log(%I) + log(--) - %e + %I
+--I ++ %I
+--I | --------------------------------- d%I
+--I ++ 1 %I
+--I %I log(--) - %I %e
+--I %I
+--R y(x)%e
+--R *
+--R INTSIGN
+--R ,
+--R x
+--R ,
+--R 2
+--I - %I log(%I) - %I
+--R
--------------------------------------------------------------
+--I 1 %I 2
+--I %I %I log(%I) + log(--) - %e + %I
+--I ++ %I
+--I | ---------------------------------
d%I
+--I ++ 1 %I
+--I %I log(--) - %I %e
+--I 1 %I %I
+--R (log(--) - %e )%e
+--I %I
+--R *
+--I d%I
+--R +
+--R 1
+--R /
+--I 1 %I 2
+--I x %I log(%I) + log(--) - %e + %I
+--I ++ %I
+--I | --------------------------------- d%I
+--I ++ 1 %I
+--I %I log(--) - %I %e
+--I %I
+--R y(x)%e
+--R Type: Union(Expression
Integer,...)
+--E 47
+
+--S 48 of 97
+ode719 := (D(y(x),x) = y(x)*(-exp(x)+log(2*x)*x**2*y(x)-log(2*x)*x)/x/exp(x))
+--R
+--R
+--R 2 2 x
+--R , (x y(x) - x y(x))log(2x) - y(x)%e
+--R (48) y (x)= -----------------------------------
+--R x
+--R x %e
+--R Type: Equation Expression
Integer
+--E 48
+
+--S 49 of 97
+solve(ode719,y,x)
+--R
+--R
+--R - x y(x) + 1
+--R (49) ----------------------------------
+--I x %I
+--I ++ %I log(2%I) + %e
+--I | ------------------ d%I
+--I ++ %I
+--I %I %e
+--R y(x)%e
+--R Type: Union(Expression
Integer,...)
+--E 49
+
+--S 50 of 97
+ode736 := (D(y(x),x) = (2*x**2+2*x+x**4-2*y(x)*x**2-1+y(x)**2)/(x+1))
+--R
+--R
+--R 2 2 4 2
+--R , y(x) - 2x y(x) + x + 2x + 2x - 1
+--R (50) y (x)= -----------------------------------
+--R x + 1
+--R Type: Equation Expression
Integer
+--E 50
+
+--S 51 of 97
+solve(ode736,y,x)
+--R
+--R
+--R 2 4 3 2
+--R (x + 2x - 2)y(x) - x - 2x + 3x + 2x + 4
+--R (51) -------------------------------------------
+--R 2
+--R 2y(x) - 2x - 2
+--R Type: Union(Expression
Integer,...)
+--E 51
+
+--S 52 of 97
+ode765 := (D(y(x),x) = y(x)*(-1-log((x-1)*(1+x)/x)+_
+ log((x-1)*(1+x)/x)*x*y(x))/x)
+--R
+--R
+--R 2
+--R 2 x - 1
+--R (x y(x) - y(x))log(------) - y(x)
+--R , x
+--R (52) y (x)= ----------------------------------
+--R x
+--R Type: Equation Expression
Integer
+--E 52
+
+--S 53 of 97
+solve(ode765,y,x)
+--R
+--R
+--R - x y(x) + 1
+--R (53) --------------------------------
+--R 2
+--I %I - 1
+--R x log(-------) + 1
+--I ++ %I
+--I | ---------------- d%I
+--I ++ %I
+--R y(x)%e
+--R Type: Union(Expression
Integer,...)
+--E 53
+
+--S 54 of 97
+ode766 := (D(y(x),x) = y(x)*(-log(x)-x*log((x-1)*(1+x)/x)+_
+ log((x-1)*(1+x)/x)*x**2*y(x))/x/log(x))
+--R
+--R
+--R 2
+--R 2 2 x - 1
+--R - y(x)log(x) + (x y(x) - x y(x))log(------)
+--R , x
+--R (54) y (x)= --------------------------------------------
+--R x log(x)
+--R Type: Equation Expression
Integer
+--E 54
+
+--S 55 of 97
+solve(ode766,y,x)
+--R
+--R
+--R (55)
+--R -
+--R 2
+--I %I - 1
+--I x log(%I) + %I log(-------)
+--I ++ %I
+--I | ------------------------- d%I
+--I ++ %I log(%I)
+--R y(x)%e
+--R *
+--R 2
+--I %I - 1
+--I x %I log(-------)
+--I ++ %I
+--I | - --------------------------------------------- d%I
+--R ++ 2
+--I %I - 1
+--I %I log(%I) + %I log(-------)
+--I ++ %I
+--I | ------------------------- d%I
+--I ++ %I log(%I)
+--I log(%I)%e
+--R +
+--R 1
+--R /
+--R 2
+--I %I - 1
+--I x log(%I) + %I log(-------)
+--I ++ %I
+--I | ------------------------- d%I
+--I ++ %I log(%I)
+--R y(x)%e
+--R Type: Union(Expression
Integer,...)
+--E 55
+
+--S 56 of 97
+ode776 := (D(y(x),x) = y(x)*(-log(1/x)-log((x**2+1)/x)*x+_
+ log((x**2+1)/x)*x**2*y(x))/x/log(1/x))
+--R
+--R
+--R 2
+--R 2 2 x + 1 1
+--R (x y(x) - x y(x))log(------) - y(x)log(-)
+--R , x x
+--R (56) y (x)= ------------------------------------------
+--R 1
+--R x log(-)
+--R x
+--R Type: Equation Expression
Integer
+--E 56
+
+--S 57 of 97
+solve(ode776,y,x)
+--R
+--R
+--R - x y(x) + 1
+--R (57) -----------------------------------------
+--R 2
+--I %I + 1 1
+--I x %I log(-------) + log(--)
+--I ++ %I %I
+--I | ------------------------- d%I
+--R ++ 1
+--I %I log(--)
+--I %I
+--R y(x)%e
+--R Type: Union(Expression
Integer,...)
+--E 57
+
+--S 58 of 97
+ode872 := (D(y(x),x) = 1/5*(-30*y(x)*x**3+12*x**6+70*x**(7/2)-30*x**3-_
+ 25*y(x)*x**(1/2)+50*x-25*x**(1/2)-25)/_
+ (-5*y(x)+2*x**3+10*x**(1/2)-5)/x)
+--R
+--R
+--R 3 +-+ 3 6 3
+--R , (- 25y(x) + 70x - 25)\|x - 30x y(x) + 12x - 30x + 50x -
25
+--R (58) y (x)=
--------------------------------------------------------------
+--R +-+ 4
+--R 50x\|x - 25x y(x) + 10x - 25x
+--R Type: Equation Expression
Integer
+--E 58
+
+--S 59 of 97
+solve(ode872,y,x)
+--R
+--R
+--R (59)
+--R +-+ 3 +-+ 2 3
+--R 100log(\|x ) + (100y(x) - 40x + 100)\|x - 25y(x) + (20x - 50)y(x)
+--R +
+--R 6 3
+--R - 4x + 20x - 100x
+--R /
+--R 2
+--R Type: Union(Expression
Integer,...)
+--E 59
+
+--S 60 of 97
+ode555 := sqrt(D(y(x),x)**2+1)+x*D(y(x),x)-y(x)
+--R
+--R
+--R +----------+
+--R | , 2 ,
+--R (60) |y (x) + 1 + xy (x) - y(x)
+--R \|
+--R Type: Expression
Integer
+--E 60
+
+--S 61 of 97
+solve(ode555,y,x)
+--R
+--R
+--R +-----------+
+--R | , 2
+--I x |y (%I) + 1 - y(x)
+--R ++ \|
+--I (61) | --------------------- d%I
+--R ++ 2
+--I %I
+--R Type: Union(Expression
Integer,...)
+--E 61
+
+--S 62 of 97
+ode557 := x*(sqrt(D(y(x),x)**2+1)+D(y(x),x))-y(x)
+--R
+--R
+--R +----------+
+--R | , 2 ,
+--R (62) x |y (x) + 1 + xy (x) - y(x)
+--R \|
+--R Type: Expression
Integer
+--E 62
+
+--S 63 of 97
+solve(ode557,y,x)
+--R
+--R
+--R +-----------+
+--R | , 2
+--I x %I |y (%I) + 1 - y(x)
+--R ++ \|
+--I (63) | ----------------------- d%I
+--R ++ 2
+--I %I
+--R Type: Union(Expression
Integer,...)
+--E 63
+
+--S 64 of 97
+ode558 := a*x*sqrt(D(y(x),x)**2+1)+x*D(y(x),x)-y(x)
+--R
+--R
+--R +----------+
+--R | , 2 ,
+--R (64) a x |y (x) + 1 + xy (x) - y(x)
+--R \|
+--R Type: Expression
Integer
+--E 64
+
+--S 65 of 97
+solve(ode558,y,x)
+--R
+--R
+--R +-----------+
+--R | , 2
+--I x %I a |y (%I) + 1 - y(x)
+--R ++ \|
+--I (65) | ------------------------- d%I
+--R ++ 2
+--I %I
+--R Type: Union(Expression
Integer,...)
+--E 65
+
+--S 66 of 97
+ode562 := a*(D(y(x),x)**3+1)**(1/3)+b*x*D(y(x),x)-y(x)
+--R
+--R
+--R +----------+
+--R | , 3 ,
+--R (66) a 3|y (x) + 1 + b xy (x) - y(x)
+--R \|
+--R Type: Expression
Integer
+--E 66
+
+--S 67 of 97
+solve(ode562,y,x)
+--R
+--R
+--I log(%I) log(%I)
+--R - ------- +-----------+ - -------
+--R b | , 3 b
+--I x a %e 3|y (%I) + 1 - y(x)%e
+--R ++ \|
+--I (67) | --------------------------------------------- d%I
+--I ++ %I
+--R Type: Union(Expression
Integer,...)
+--E 67
+
+--S 68 of 97
+ode563 := log(D(y(x),x))+x*D(y(x),x)+a*y(x)+b
+--R
+--R
+--R , ,
+--R (68) log(y (x)) + xy (x) + a y(x) + b
+--R
+--R Type: Expression
Integer
+--E 68
+
+--S 69 of 97
+solve(ode563,y,x)
+--R
+--R
+--I a log(%I) , a log(%I)
+--I x %e log(y (%I)) + (a y(x) + b)%e
+--R ++
+--I (69) | ------------------------------------------------ d%I
+--I ++ %I
+--R Type: Union(Expression
Integer,...)
+--E 69
+
+--S 70 of 97
+ode564 := log(D(y(x),x))+a*(x*D(y(x),x)-y(x))
+--R
+--R
+--R , ,
+--R (70) log(y (x)) + a xy (x) - a y(x)
+--R
+--R Type: Expression
Integer
+--E 70
+
+--S 71 of 97
+solve(ode564,y,x)
+--R
+--R
+--R ,
+--I x log(y (%I)) - a y(x)
+--R ++
+--I (71) | -------------------- d%I
+--R ++ 2
+--I %I
+--R Type: Union(Expression
Integer,...)
+--E 71
+
+--S 72 of 97
+ode571 := a*x**n*f(D(y(x),x))+x*D(y(x),x)-y(x)
+--R
+--R
+--R n , ,
+--R (72) a x f(y (x)) + xy (x) - y(x)
+--R
+--R Type: Expression
Integer
+--E 72
+
+--S 73 of 97
+solve(ode571,y,x)
+--R
+--R
+--R n ,
+--I x a %I f(y (%I)) - y(x)
+--R ++
+--I (73) | --------------------- d%I
+--R ++ 2
+--I %I
+--R Type: Union(Expression
Integer,...)
+--E 73
+
+--S 74 of 97
+ode573 := f(x*D(y(x),x)**2)+2*x*D(y(x),x)-y(x)
+--R
+--R
+--R , 2 ,
+--R (74) f(x y (x) ) + 2xy (x) - y(x)
+--R
+--R Type: Expression
Integer
+--E 74
+
+--S 75 of 97
+solve(ode573,y,x)
+--R
+--R
+--R , 2
+--I x f(%I y (%I) ) - y(x)
+--R ++
+--I (75) | -------------------- d%I
+--R ++ +--+
+--I %I\|%I
+--R Type: Union(Expression
Integer,...)
+--E 75
+
+--S 76 of 97
+ode683 := (D(y(x),x) = y(x)*(-1+log(x*(x+1))*y(x)*x**4-log(x*(x+1))*x**3)/x)
+--R
+--R
+--R 4 2 3 2
+--R , (x y(x) - x y(x))log(x + x) - y(x)
+--R (76) y (x)= ------------------------------------
+--R x
+--R Type: Equation Expression
Integer
+--E 76
+
+--S 77 of 97
+solve(ode683,y,x)
+--R
+--R
+--R - x y(x) + 1
+--R (77) -----------------------------------------------
+--R 3 2 3 2
+--R 6x log(x + x) - 4x + 3x - 6x
+--R -------------------------------
+--R 3+-----+ 18
+--R x y(x)\|x + 1 %e
+--R Type: Union(Expression
Integer,...)
+--E 77
+
+--S 78 of 97
+ode703 := (D(y(x),x) = y(x)*(1-x+y(x)*x**2*log(x)+y(x)*x**3-x*log(x)-x**2)/_
+ (x-1)/x)
+--R
+--R
+--R 2 2 3 2 2
+--R , (x y(x) - x y(x))log(x) + x y(x) + (- x - x + 1)y(x)
+--R (78) y (x)= -------------------------------------------------------
+--R 2
+--R x - x
+--R Type: Equation Expression
Integer
+--E 78
+
+--S 79 of 97
+solve(ode703,y,x)
+--R
+--R
+--R - x y(x) + 1
+--R (79) ----------------------------
+--R 2 - dilog(x) + x
+--R (x - x)y(x)%e
+--R Type: Union(Expression
Integer,...)
+--E 79
+
+--S 80 of 97
+ode714 := (D(y(x),x) = -y(x)*(-log(1/x)+exp(x)+y(x)*x**2*log(x)+_
+ y(x)*x**3-x*log(x)-x**2)/(-log(1/x)+exp(x))/x)
+--R
+--R
+--R (80)
+--R 2 2 1 x 3 2 2
+--R (x y(x) - x y(x))log(x) - y(x)log(-) + y(x)%e + x y(x) - x y(x)
+--R , x
+--R y (x)= ------------------------------------------------------------------
+--R 1 x
+--R x log(-) - x %e
+--R x
+--R Type: Equation Expression
Integer
+--E 80
+
+--S 81 of 97
+solve(ode714,y,x)
+--R
+--R
+--R (81)
+--R -
+--I 1 %I 2
+--I x %I log(%I) + log(--) - %e + %I
+--I ++ %I
+--I | --------------------------------- d%I
+--I ++ 1 %I
+--I %I log(--) - %I %e
+--I %I
+--R y(x)%e
+--R *
+--R INTSIGN
+--R ,
+--R x
+--R ,
+--R 2
+--I - %I log(%I) - %I
+--R
--------------------------------------------------------------
+--I 1 %I 2
+--I %I %I log(%I) + log(--) - %e + %I
+--I ++ %I
+--I | ---------------------------------
d%I
+--I ++ 1 %I
+--I %I log(--) - %I %e
+--I 1 %I %I
+--R (log(--) - %e )%e
+--I %I
+--R *
+--I d%I
+--R +
+--R 1
+--R /
+--I 1 %I 2
+--I x %I log(%I) + log(--) - %e + %I
+--I ++ %I
+--I | --------------------------------- d%I
+--I ++ 1 %I
+--I %I log(--) - %I %e
+--I %I
+--R y(x)%e
+--R Type: Union(Expression
Integer,...)
+--E 81
+
+--S 82 of 97
+ode719 := (D(y(x),x) = y(x)*(-exp(x)+log(2*x)*x**2*y(x)-log(2*x)*x)/x/exp(x))
+--R
+--R
+--R 2 2 x
+--R , (x y(x) - x y(x))log(2x) - y(x)%e
+--R (82) y (x)= -----------------------------------
+--R x
+--R x %e
+--R Type: Equation Expression
Integer
+--E 82
+
+--S 83 of 97
+solve(ode719,y,x)
+--R
+--R
+--R - x y(x) + 1
+--R (83) ----------------------------------
+--I x %I
+--I ++ %I log(2%I) + %e
+--I | ------------------ d%I
+--I ++ %I
+--I %I %e
+--R y(x)%e
+--R Type: Union(Expression
Integer,...)
+--E 83
+
+--S 84 of 97
+ode736 := (D(y(x),x) = (2*x**2+2*x+x**4-2*y(x)*x**2-1+y(x)**2)/(x+1))
+--R
+--R
+--R 2 2 4 2
+--R , y(x) - 2x y(x) + x + 2x + 2x - 1
+--R (84) y (x)= -----------------------------------
+--R x + 1
+--R Type: Equation Expression
Integer
+--E 84
+
+--S 85 of 97
+solve(ode736,y,x)
+--R
+--R
+--R 2 4 3 2
+--R (x + 2x - 2)y(x) - x - 2x + 3x + 2x + 4
+--R (85) -------------------------------------------
+--R 2
+--R 2y(x) - 2x - 2
+--R Type: Union(Expression
Integer,...)
+--E 85
+
+--S 86 of 97
+ode765 := (D(y(x),x) = y(x)*(-1-log((x-1)*(1+x)/x)+_
+ log((x-1)*(1+x)/x)*x*y(x))/x)
+--R
+--R
+--R 2
+--R 2 x - 1
+--R (x y(x) - y(x))log(------) - y(x)
+--R , x
+--R (86) y (x)= ----------------------------------
+--R x
+--R Type: Equation Expression
Integer
+--E 86
+
+--S 87 of 97
+solve(ode765,y,x)
+--R
+--R
+--R - x y(x) + 1
+--R (87) --------------------------------
+--R 2
+--I %I - 1
+--R x log(-------) + 1
+--I ++ %I
+--I | ---------------- d%I
+--I ++ %I
+--R y(x)%e
+--R Type: Union(Expression
Integer,...)
+--E 87
+
+--S 88 of 97
+ode766 := (D(y(x),x) = y(x)*(-log(x)-x*log((x-1)*(1+x)/x)+_
+ log((x-1)*(1+x)/x)*x**2*y(x))/x/log(x))
+--R
+--R
+--R 2
+--R 2 2 x - 1
+--R - y(x)log(x) + (x y(x) - x y(x))log(------)
+--R , x
+--R (88) y (x)= --------------------------------------------
+--R x log(x)
+--R Type: Equation Expression
Integer
+--E 88
+
+--S 89 of 97
+solve(ode766,y,x)
+--R
+--R
+--R (89)
+--R -
+--R 2
+--I %I - 1
+--I x log(%I) + %I log(-------)
+--I ++ %I
+--I | ------------------------- d%I
+--I ++ %I log(%I)
+--R y(x)%e
+--R *
+--R 2
+--I %I - 1
+--I x %I log(-------)
+--I ++ %I
+--I | - --------------------------------------------- d%I
+--R ++ 2
+--I %I - 1
+--I %I log(%I) + %I log(-------)
+--I ++ %I
+--I | ------------------------- d%I
+--I ++ %I log(%I)
+--I log(%I)%e
+--R +
+--R 1
+--R /
+--R 2
+--I %I - 1
+--I x log(%I) + %I log(-------)
+--I ++ %I
+--I | ------------------------- d%I
+--I ++ %I log(%I)
+--R y(x)%e
+--R Type: Union(Expression
Integer,...)
+--E 89
+
+--S 90 of 97
+ode776 := (D(y(x),x) = y(x)*(-log(1/x)-log((x**2+1)/x)*x+_
+ log((x**2+1)/x)*x**2*y(x))/x/log(1/x))
+--R
+--R
+--R 2
+--R 2 2 x + 1 1
+--R (x y(x) - x y(x))log(------) - y(x)log(-)
+--R , x x
+--R (90) y (x)= ------------------------------------------
+--R 1
+--R x log(-)
+--R x
+--R Type: Equation Expression
Integer
+--E 90
+
+--S 91 of 97
+solve(ode776,y,x)
+--R
+--R
+--R - x y(x) + 1
+--R (91) -----------------------------------------
+--R 2
+--I %I + 1 1
+--I x %I log(-------) + log(--)
+--I ++ %I %I
+--I | ------------------------- d%I
+--R ++ 1
+--I %I log(--)
+--I %I
+--R y(x)%e
+--R Type: Union(Expression
Integer,...)
+--E 91
+
+--S 92 of 97
+ode872 := (D(y(x),x) = 1/5*(-30*y(x)*x**3+12*x**6+70*x**(7/2)-30*x**3-_
+ 25*y(x)*x**(1/2)+50*x-25*x**(1/2)-25)/(-5*y(x)+2*x**3+_
+ 10*x**(1/2)-5)/x)
+--R
+--R
+--R 3 +-+ 3 6 3
+--R , (- 25y(x) + 70x - 25)\|x - 30x y(x) + 12x - 30x + 50x -
25
+--R (92) y (x)=
--------------------------------------------------------------
+--R +-+ 4
+--R 50x\|x - 25x y(x) + 10x - 25x
+--R Type: Equation Expression
Integer
+--E 92
+
+--S 93 of 97
+solve(ode872,y,x)
+--R
+--R
+--R (93)
+--R +-+ 3 +-+ 2 3
+--R 100log(\|x ) + (100y(x) - 40x + 100)\|x - 25y(x) + (20x - 50)y(x)
+--R +
+--R 6 3
+--R - 4x + 20x - 100x
+--R /
+--R 2
+--R Type: Union(Expression
Integer,...)
+--E 93
+
+--S 94 of 97
+ode956 := (D(y(x),x) = 1/(1+log(x))*y(x)*(-1-x**(2/(1+log(x)))*_
+ exp(2/(1+log(x))*log(x)**2)*x**2-x**(2/(1+log(x)))*_
+ exp(2/(1+log(x))*log(x)**2)*x**2*log(x)+x**(2/(1+log(x)))*_
+ exp(2/(1+log(x))*log(x)**2)*x**2*y(x)+2*x**(2/(1+log(x)))*_
+ exp(2/(1+log(x))*log(x)**2)*x**2*y(x)*log(x)+x**(2/(1+log(x)))*_
+ exp(2/(1+log(x))*log(x)**2)*x**2*y(x)*log(x)**2)/x)
+--R
+--R
+--R (94)
+--R ,
+--R y (x) =
+--R
+--R 2 2 2 2 2 2 2 2 2
+--R (x y(x) log(x) + (2x y(x) - x y(x))log(x) + x y(x) - x y(x))
+--R *
+--R 2
+--R 2log(x) 2
+--R ---------- ----------
+--R log(x) + 1 log(x) + 1
+--R %e x
+--R +
+--R - y(x)
+--R /
+--R x log(x) + x
+--R Type: Equation Expression
Integer
+--E 94
+
+--S 95 of 97
+solve(ode956,y,x)
+--R
+--R
+--R - y(x)log(x) - y(x) + 1
+--R (95) -------------------------
+--R 4 4
+--R x x
+--R -- --
+--R 4 4
+--R y(x)%e log(x) + y(x)%e
+--R Type: Union(Expression
Integer,...)
+--E 95
+
+--S 96 of 97
+ode957 := (D(y(x),x) = 1/(1+log(x))*y(x)*(-1-x**3*x**(2/(1+log(x)))*_
+ exp(2/(1+log(x))*log(x)**2)-x**3*x**(2/(1+log(x)))*_
+ exp(2/(1+log(x))*log(x)**2)*log(x)+x**3*x**(2/(1+log(x)))*_
+ exp(2/(1+log(x))*log(x)**2)*y(x)+2*x**3*x**(2/(1+log(x)))*_
+ exp(2/(1+log(x))*log(x)**2)*y(x)*log(x)+x**3*x**(2/(1+log(x)))*_
+ exp(2/(1+log(x))*log(x)**2)*y(x)*log(x)**2)/x)
+--R
+--R
+--R (96)
+--R ,
+--R y (x) =
+--R
+--R 3 2 2 3 2 3 3 2 3
+--R (x y(x) log(x) + (2x y(x) - x y(x))log(x) + x y(x) - x y(x))
+--R *
+--R 2
+--R 2log(x) 2
+--R ---------- ----------
+--R log(x) + 1 log(x) + 1
+--R %e x
+--R +
+--R - y(x)
+--R /
+--R x log(x) + x
+--R Type: Equation Expression
Integer
+--E 96
+
+--S 97 of 97
+solve(ode957,y,x)
+--R
+--R
+--R - y(x)log(x) - y(x) + 1
+--R (97) -------------------------
+--R 5 5
+--R x x
+--R -- --
+--R 5 5
+--R y(x)%e log(x) + y(x)%e
+--R Type: Union(Expression
Integer,...)
+--E 97
+)spool
+)lisp (bye)
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} {\bf http://www.cs.uwaterloo.ca/$\tilde{}$ecterrab/odetools.html}
+\end{thebibliography}
+\end{document}
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