diff --git a/changelog b/changelog index dcdf237..28ead41 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,6 @@ +20091004 tpd src/axiom-website/patches.html 20091004.02.tpd.patch +20091004 tpd src/input/Makefile +20091004 tpd src/input/zimmbron.input 20091004 tpd src/axiom-website/patches.html 20091004.01.tpd.patch 20091004 tpd src/interp/i-map.lisp cleanup 20091003 tpd src/axiom-website/patches.html 20091003.03.tpd.patch diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html index 30c6bfc..db7386c 100644 --- a/src/axiom-website/patches.html +++ b/src/axiom-website/patches.html @@ -2095,5 +2095,7 @@ src/interp/i-resolv.lisp cleanup
src/interp/i-output.lisp cleanup
20091004.01.tpd.patch src/interp/i-map.lisp cleanup
+20091004.02.tpd.patch +src/input/zimmbron.input added
diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet index 7b8eb90..5abd2ce 100644 --- a/src/input/Makefile.pamphlet +++ b/src/input/Makefile.pamphlet @@ -240,7 +240,8 @@ SKIP= as-eg1.output as-eg2.output \ s21bcf.output s21bdf.output saddle.output \ sininv.output sinsin2.output sinsin.output spiral.output \ tetra.output tknot.output tschirn.output typo.output \ - vectors.output wester.output wiggle.output zimmer.output + vectors.output wester.output wiggle.output zimmbron.output \ + zimmer.output # Error ASEC is invalid as a function. ASEC=errortrap.output @@ -390,7 +391,7 @@ REGRES= algaggr.regress algbrbf.regress algfacob.regress alist.regress \ triglim.regress tsetcatvermeer.regress tutchap1.regress \ typetower.regress void.regress uniseg.regress \ unittest1.regress unittest2.regress unittest3.regress unittest4.regress \ - unit-i-funsel.regress + unit-i-funsel.regress zimmbron.regress IN= ${SRC}/input MID= ${INT}/input @@ -712,7 +713,7 @@ FILES= ${OUT}/algaggr.input ${OUT}/algbrbf.input ${OUT}/algfacob.input \ ${OUT}/vector.input ${OUT}/vectors.input ${OUT}/viewdef.input \ ${OUT}/void.input ${OUT}/wiggle.input \ ${OUT}/wutset.input \ - ${OUT}/xpoly.input ${OUT}/xpr.input \ + ${OUT}/xpoly.input ${OUT}/xpr.input ${OUT}/zimmbron.input \ ${OUT}/zdsolve.input ${OUT}/zimmer.input ${OUT}/zlindep.input FILES2=${OUT}/arith.input ${OUT}/bugs.input \ @@ -1077,6 +1078,7 @@ DOCFILES= \ ${DOC}/wester.input.dvi ${DOC}/wiggle.input.dvi \ ${DOC}/wutset.input.dvi \ ${DOC}/xpoly.input.dvi ${DOC}/xpr.input.dvi \ + ${DOC}/zimmbron.input.dvi \ ${DOC}/zdsolve.input.dvi ${DOC}/zimmer.input.dvi \ ${DOC}/zlindep.input.dvi diff --git a/src/input/zimmbron.input.pamphlet b/src/input/zimmbron.input.pamphlet new file mode 100644 index 0000000..5c06bfb --- /dev/null +++ b/src/input/zimmbron.input.pamphlet @@ -0,0 +1,1836 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/src/input zimmbron.input} +\author{Paul Zimmermann and Manuel Bronstein} +\maketitle +\begin{abstract} +\end{abstract} +\eject +\tableofcontents +\eject +\begin{chunk}{*} +)set break resume +)sys rm -f zimmbron.output +)spool zimmbron.output +)set message test on +)set message auto off +)clear all + +--S 1 of 143 +y := operator y +--R +--R +--R (1) y +--R Type: BasicOperator +--E 1 + +--S 2 of 143 +y' := D(y x, x) +--R +--R +--R , +--R (2) y (x) +--R +--R Type: Expression Integer +--E 2 + +--S 3 of 143 +y'' := D(y', x) +--R +--R +--R ,, +--R (3) y (x) +--R +--R Type: Expression Integer +--E 3 + +--S 4 of146 +y1 := operator y1 +--R +--R +--R (4) y1 +--R Type: BasicOperator +--E 4 + +--S 5 of 143 +y2 := operator y2 +--R +--R +--R (5) y2 +--R Type: BasicOperator +--E 5 + +--S 6 of 143 +y1' := D(y1 t, t) +--R +--R +--R , +--R (6) y1 (t) +--R +--R Type: Expression Integer +--E 6 + +--S 7 of 143 +y2' := D(y2 t, t) +--R +--R +--R , +--R (7) y2 (t) +--R +--R Type: Expression Integer +--E 7 + +\end{chunk} +\subsection{First Order 1 } +\begin{chunk}{*} +--S 8 of 143 +eq := (x^4 - x^3) * y' + 2 * x^4 * y x = x^3/3 + C +--R +--R +--R 3 +--R 4 3 , 4 x + 3C +--R (8) (x - x )y (x) + 2x y(x)= ------- +--R 3 +--R Type: Equation Expression Integer +--E 8 + +--S 9 of 143 +solve(eq,y,x) +--R +--R +--R 3 2 - 2x +--R 2x - 3x + 6C %e +--R (9) [particular= ------------------,basis= [-----------]] +--R 4 3 2 2 +--R 12x - 24x + 12x x - 2x + 1 +--RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) +--E 9 + +\end{chunk} +\subsection{First Order 2 } +\begin{chunk}{*} +--S 10 of 143 +eq := - y' / 2 + y x = sin x +--R +--R +--R , +--R - y (x) + 2y(x) +--R +--R (10) ---------------= sin(x) +--R 2 +--R Type: Equation Expression Integer +--E 10 + +--S 11 of 143 +solve(eq,y,x) +--R +--R +--R 4sin(x) + 2cos(x) 2x +--R (11) [particular= -----------------,basis= [%e ]] +--R 5 +--RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) +--E 11 + +\end{chunk} +\subsection{First Order 3 } +\begin{chunk}{*} +--S 12 of 143 +eq := y' = y x / (y x * log y x + x) +--R +--R +--R , y(x) +--R (12) y (x)= ----------------- +--R y(x)log(y(x)) + x +--R Type: Equation Expression Integer +--E 12 + +--S 13 of 143 +solve(eq,y,x) +--R +--R +--R 2 +--R y(x)log(y(x)) - 2x +--R (13) ------------------- +--R 2y(x) +--R Type: Union(Expression Integer,...) +--E 13 + +\end{chunk} +\subsection{First Order 4 } +\begin{chunk}{*} +--S 14 of 143 +eq := 2*(y x)*y'**2-2*x*y'-y x=0 +--R +--R +--R , 2 , +--R (14) 2y(x)y (x) - 2xy (x) - y(x)= 0 +--R +--R Type: Equation Expression Integer +--E 14 + +--S 15 of 143 +solve(eq,y,x) +--R +--R +--RDaly Bug +--R >> Error detected within library code: +--R getlincoeff: not an appropriate ordinary differential equation +--R +--R Continuing to read the file... +--R +--E 15 + +\end{chunk} +\subsection{Bernoulli } +\begin{chunk}{*} +--S 16 of 143 +eq := y' + y x = y(x)^3 * sin x +--R +--R +--R , 3 +--R (15) y (x) + y(x)= y(x) sin(x) +--R +--R Type: Equation Expression Integer +--E 16 + +--S 17 of 143 +solve(eq,y,x) +--R +--R +--R 2 2 +--R - 4y(x) sin(x) - 2y(x) cos(x) + 5 +--R (16) --------------------------------- +--R 2 2x +--R 5y(x) %e +--R Type: Union(Expression Integer,...) +--E 17 + +\end{chunk} +\subsection{Bernoulli 2 } +\begin{chunk}{*} +--S 18 of 143 +P := operator P +--R +--R +--R (17) P +--R Type: BasicOperator +--E 18 + +--S 19 of 143 +Q := operator Q +--R +--R +--R (18) Q +--R Type: BasicOperator +--E 19 + +--S 20 of 143 +eq := y' + P x * y x = Q x * y(x)^n +--R +--R +--R , n +--R (19) y (x) + P(x)y(x)= Q(x)y(x) +--R +--R Type: Equation Expression Integer +--E 20 + +--S 21 of 143 +solve(eq,y,x) +--R +--R +--R (20) "failed" +--R Type: Union("failed",...) +--E 21 + +\end{chunk} +In the original text this gave the wrong answer. +The answer should contain Q. The original text read: +\begin{verbatim} + + 1 + (19) -------------------- + x + ++ + | P(%L)d%L + ++ + y(x)%e + +\end{verbatim} +\begin{chunk}{*} +--S 22 of 143 +eq := y' + P x * y x = Q x * y(x)^2 +--R +--R +--R , 2 +--R (21) y (x) + P(x)y(x)= Q(x)y(x) +--R +--R Type: Equation Expression Integer +--E 22 + +--S 23 of 143 +solve(eq,y,x) +--R +--R +--RDaly Bug +--R >> Error detected within library code: +--R Function not supported by Risch d.e. +--R +--R Continuing to read the file... +--R +--E 23 + +\end{chunk} +We cannot solve for n rational +\begin{chunk}{*} +--S 24 of 143 +eq := y' + P x * y x = Q x * y(x)^(2/3) +--R +--R +--R , 3+----+2 +--R (22) y (x) + P(x)y(x)= Q(x)\|y(x) +--R +--R Type: Equation Expression Integer +--E 24 + +--S 25 of 143 +solve(eq,y,x) +--R +--R +--R (23) "failed" +--R Type: Union("failed",...) +--E 25 + +\end{chunk} +Again, the previous answer was wrong. It should contain Q. +The original result was: +\begin{verbatim} + + 1 + (31) ---------------------- + x + ++ + | 2P(%L)d%L + ++ + 2 + y(x) %e + +\end{verbatim} +\begin{chunk}{*} +--S 26 of 143 +eq := y' + P x * y x = Q x * y(x)^3 +--R +--R +--R , 3 +--R (24) y (x) + P(x)y(x)= Q(x)y(x) +--R +--R Type: Equation Expression Integer +--E 26 + +--S 27 of 143 +solve(eq,y,x) +--R +--R +--RDaly Bug +--R >> Error detected within library code: +--R Function not supported by Risch d.e. +--R +--R Continuing to read the file... +--R +--E 27 + +\end{chunk} +\subsection{Clairaut} +\begin{chunk}{*} +--S 28 of 143 +eq := (x^2-1)*y'^2-2*x*y(x)*y'+y(x)^2-1=0 +--R +--R +--R 2 , 2 , 2 +--R (25) (x - 1)y (x) - 2x y(x)y (x) + y(x) - 1= 0 +--R +--R Type: Equation Expression Integer +--E 28 + +--S 29 of 143 +solve(eq,y,x) +--R +--R +--RDaly Bug +--R >> Error detected within library code: +--R getlincoeff: not an appropriate ordinary differential equation +--R +--R Continuing to read the file... +--R +--E 29 + +\end{chunk} +\subsection{Clairaut2} +\begin{chunk}{*} +--S 30 of 143 +f := operator f +--R +--R +--R (26) f +--R Type: BasicOperator +--E 30 + +--S 31 of 143 +g := operator g +--R +--R +--R (27) g +--R Type: BasicOperator +--E 31 + +--S 32 of 143 +eq := f(x*y'-y(x))=g(y') +--R +--R +--R , , +--R (28) f(xy (x) - y(x))= g(y (x)) +--R +--R Type: Equation Expression Integer +--E 32 + +--S 33 of 143 +solve(eq,y,x) +--R +--R +--RDaly Bug +--R >> Error detected within library code: +--R parseODE: equation has order 0 +--R +--R Continuing to read the file... +--R +--E 33 + +\end{chunk} +\subsection{Exact First} +\begin{chunk}{*} +--S 34 of 143 +eq := y' = (3 * x^2 - y(x)^2 - 7) / (exp y x + 2 * x * y x + 1) +--R +--R +--R 2 2 +--R , - y(x) + 3x - 7 +--R (29) y (x)= -------------------- +--R y(x) +--R %e + 2x y(x) + 1 +--R Type: Equation Expression Integer +--E 34 + +--S 35 of 143 +solve(eq,y,x) +--R +--R +--R y(x) 2 3 +--R (30) %e + x y(x) + y(x) - x + 7x +--R Type: Union(Expression Integer,...) +--E 35 + +\end{chunk} +\subsection{Homogeneous} +\begin{chunk}{*} +--S 36 of 143 +eq := y' = (2 * x^3 * y x - y(x)^4) / (x^4 - 2 * x * y(x)^3) +--R +--R +--R 4 3 +--R , y(x) - 2x y(x) +--R (31) y (x)= --------------- +--R 3 4 +--R 2x y(x) - x +--R Type: Equation Expression Integer +--E 36 + +--S 37 of 143 +solve(eq,y,x) +--R +--R +--R (32) "failed" +--R Type: Union("failed",...) +--E 37 + +\end{chunk} +\subsection{Factor} +\begin{chunk}{*} +--S 38 of 143 +eq := y'*(y'+y(x))=x*(x+y(x)) +--R +--R +--R , 2 , 2 +--R (33) y (x) + y(x)y (x)= x y(x) + x +--R +--R Type: Equation Expression Integer +--E 38 + +--S 39 of 143 +solve(eq,y,x) +--R +--R +--RDaly Bug +--R >> Error detected within library code: +--R getlincoeff: not an appropriate ordinary differential equation +--R +--R Continuing to read the file... +--R +--E 39 + +\end{chunk} +\subsection{Interchange} +\begin{chunk}{*} +--S 40 of 143 +eq := y' = x / (x^2 * y(x)^2 + y(x)^5) +--R +--R +--R , x +--R (34) y (x)= --------------- +--R 5 2 2 +--R y(x) + x y(x) +--R Type: Equation Expression Integer +--E 40 + +--S 41 of 143 +solve(eq,y,x) +--R +--R +--R 3 +--R 2y(x) +--R - ------ +--R 3 2 3 +--R (- 2y(x) - 2x - 3)%e +--R (35) ------------------------------ +--R 4 +--R Type: Union(Expression Integer,...) +--E 41 + +\end{chunk} +\subsection{Lagrange} +\begin{chunk}{*} +--S 42 of 143 +eq := y(x)=2*x*y'-a*y'^3 +--R +--R +--R , 3 , +--R (36) y(x)= - a y (x) + 2xy (x) +--R +--R Type: Equation Expression Integer +--E 42 + +--S 43 of 143 +solve(eq,y,x) +--R +--R +--RDaly Bug +--R >> Error detected within library code: +--R getlincoeff: not an appropriate ordinary differential equation +--R +--R Continuing to read the file... +--R +--E 43 + +\end{chunk} +\subsection{Lagrange 2} +\begin{chunk}{*} +--S 44 of 143 +eq := y(x)=2*x*y'-y'^2 +--R +--R +--R , 2 , +--R (37) y(x)= - y (x) + 2xy (x) +--R +--R Type: Equation Expression Integer +--E 44 + +--S 45 of 143 +solve(eq,y,x) +--R +--R +--RDaly Bug +--R >> Error detected within library code: +--R getlincoeff: not an appropriate ordinary differential equation +--R +--R Continuing to read the file... +--R +--E 45 + +\end{chunk} +\subsection{Riccati} +\begin{chunk}{*} +--S 46 of 143 +eq := y' = exp x * y(x)^2 - y x + exp(-x) +--R +--R +--R , 2 x - x +--R (38) y (x)= y(x) %e + %e - y(x) +--R +--R Type: Equation Expression Integer +--E 46 + +--S 47 of 143 +solve(eq,y,x) +--R +--R +--R (39) "failed" +--R Type: Union("failed",...) +--E 47 + +\end{chunk} +\subsection{Riccati 2} +\begin{chunk}{*} +--S 48 of 143 +eq := y' = y(x)^2 - x * y x + 1 +--R +--R +--R , 2 +--R (40) y (x)= y(x) - x y(x) + 1 +--R +--R Type: Equation Expression Integer +--E 48 + +--S 49 of 143 +solve(eq,y,x) +--R +--R +--R 2 +--R x +--R - -- x +--R 2 ++ 1 +--R (- y(x) + x)%e | - ------- d%M + 1 +--R ++ 2 +--R %M +--R - --- +--R 2 +--R %e +--R (41) ----------------------------------------- +--R 2 +--R x +--R - -- +--R 2 +--R (y(x) - x)%e +--R Type: Union(Expression Integer,...) +--E 49 + +\end{chunk} +\subsection{Separable} +\begin{chunk}{*} +--S 50 of 143 +eq := y' = (9 * x^8 + 1) / (y(x)^2 + 1) +--R +--R +--R 8 +--R , 9x + 1 +--R (42) y (x)= --------- +--R 2 +--R y(x) + 1 +--R Type: Equation Expression Integer +--E 50 + +--S 51 of 143 +solve(eq,y,x) +--R +--R +--R 3 9 +--R y(x) + 3y(x) - 3x - 3x +--R (43) ------------------------ +--R 3 +--R Type: Union(Expression Integer,...) +--E 51 + +\end{chunk} +\subsection{Solvable x} +\begin{chunk}{*} +--S 52 of 143 +eq := y(x) = 2*x*y'+y(x)*y'^2 +--R +--R +--R , 2 , +--R (44) y(x)= y(x)y (x) + 2xy (x) +--R +--R Type: Equation Expression Integer +--E 52 + +--S 53 of 143 +solve(eq,y,x) +--R +--R +--RDaly Bug +--R >> Error detected within library code: +--R getlincoeff: not an appropriate ordinary differential equation +--R +--R Continuing to read the file... +--R +--E 53 + +\end{chunk} +\subsection{Solvable y} +\begin{chunk}{*} +--S 54 of 143 +eq := x = y(x)*y'-x*y'^2 +--R +--R +--R , 2 , +--R (45) x= - x y (x) + y(x)y (x) +--R +--R Type: Equation Expression Integer +--E 54 + +--S 55 of 143 +solve(eq,y,x) +--R +--R +--RDaly Bug +--R >> Error detected within library code: +--R getlincoeff: not an appropriate ordinary differential equation +--R +--R Continuing to read the file... +--R +--E 55 + +\end{chunk} +\subsection{Second Order Change of Variable} +\begin{chunk}{*} +--S 56 of 143 +eq := y''*(a*x+b)^2+4*y'*(a*x+b)*a+2*y(x)*a^2 = 0 +--R +--R +--R 2 2 2 ,, 2 , 2 +--R (46) (a x + 2a b x + b )y (x) + (4a x + 4a b)y (x) + 2a y(x)= 0 +--R +--R Type: Equation Expression Integer +--E 56 + +--S 57 of 143 +solve(eq,y,x) +--R +--R +--R x 2a x + b +--R (47) [particular= 0,basis= [------------------,------------------]] +--R 2 2 2 2 2 2 +--R a x + 2a b x + b a x + 2a b x + b +--RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) +--E 57 + +\end{chunk} +\subsection{Adjoint} +\begin{chunk}{*} +--S 58 of 143 +eq := (x^2-x)*y'' + (2*x^2+4*x-3)*y' + 8*x*y(x) = 1 +--R +--R +--R 2 ,, 2 , +--R (48) (x - x)y (x) + (2x + 4x - 3)y (x) + 8x y(x)= 1 +--R +--R Type: Equation Expression Integer +--E 58 + +--S 59 of 143 +solve(eq,y,x) +--R +--R +--R 3 2 - 2x +--R 2x - 3x + 53 1 %e +--R (49) [particular= ------------------,basis= [-------------,-----------]] +--R 4 3 2 4 3 2 2 +--R 12x - 24x + 12x x - 2x + x x - 2x + 1 +--RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) +--E 59 + +\end{chunk} +\subsection{Second Order} +\begin{chunk}{*} +--S 60 of 143 +eq := (x^2-x)*y'' + (1-2*x^2)*y' + (4*x-2)*y(x) = 0 +--R +--R +--R 2 ,, 2 , +--R (50) (x - x)y (x) + (- 2x + 1)y (x) + (4x - 2)y(x)= 0 +--R +--R Type: Equation Expression Integer +--E 60 + +--S 61 of 143 +solve(eq,y,x) +--R +--R +--R 2 2x +--R (51) [particular= 0,basis= [x ,%e ]] +--RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) +--E 61 + +\end{chunk} +\subsection{Autonomous} +\begin{chunk}{*} +--S 62 of 143 +eq := y''-y' = 2*y(x)*y' +--R +--R +--R ,, , , +--R (52) y (x) - y (x)= 2y(x)y (x) +--R +--R Type: Equation Expression Integer +--E 62 + +--S 63 of 143 +solve(eq,y,x) +--R +--R +--RDaly Bug +--R >> Error detected within library code: +--R getfreelincoeff: not a linear ordinary differential equation +--R +--R Continuing to read the file... +--R +--E 63 + +\end{chunk} +\subsection{Autonomous 2} +\begin{chunk}{*} +--S 64 of 143 +eq := y''/y(x)-y'^2/y(x)^2-1+1/y(x)^3 = 0 +--R +--R +--R 2 ,, , 2 3 +--R y(x) y (x) - y(x)y (x) - y(x) + 1 +--R +--R (53) ------------------------------------= 0 +--R 3 +--R y(x) +--R Type: Equation Expression Integer +--E 64 + +--S 65 of 143 +solve(eq,y,x) +--R +--R +--RDaly Bug +--R >> Error detected within library code: +--R parseLODE: not a linear ordinary differential equation +--R +--R Continuing to read the file... +--R +--E 65 + +\end{chunk} +\subsection{y missing} +\begin{chunk}{*} +--S 66 of 143 +eq := y'' + 2 * x * y' = 2 * x +--R +--R +--R ,, , +--R (54) y (x) + 2xy (x)= 2x +--R +--R Type: Equation Expression Integer +--E 66 + +--S 67 of 143 +solve(eq,y,x) +--R +--R +--R (55) [particular= x,basis= [1,erf(x)]] +--RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) +--E 67 + +\end{chunk} +\subsection{diff} +\begin{chunk}{*} +--S 68 of 143 +eq := 2*y(x)*y''-y'^2 = (y'-x*y'')^2/3 +--R +--R +--R 2 ,, 2 , ,, , 2 +--R x y (x) - 2xy (x)y (x) + y (x) +--R ,, , 2 +--R (56) 2y(x)y (x) - y (x) = ---------------------------------- +--R 3 +--R Type: Equation Expression Integer +--E 68 + +--S 69 of 143 +solve(eq,y,x) +--R +--R +--RDaly Bug +--R >> Error detected within library code: +--R getlincoeff: not an appropriate ordinary differential equation +--R +--R Continuing to read the file... +--R +--E 69 + +\end{chunk} +\subsection{Equidimensional x} +\begin{chunk}{*} +--S 70 of 143 +eq := x*y'' = 2*y(x)*y' +--R +--R +--R ,, , +--R (57) xy (x)= 2y(x)y (x) +--R +--R Type: Equation Expression Integer +--E 70 + +--S 71 of 143 +solve(eq,y,x) +--R +--R +--RDaly Bug +--R >> Error detected within library code: +--R getfreelincoeff: not a linear ordinary differential equation +--R +--R Continuing to read the file... +--R +--E 71 + +\end{chunk} +\subsection{Equidimensional y} +\begin{chunk}{*} +--S 72 of 143 +eq := (1-x)*(y(x)*y''-y'^2)+x^2*y(x)^2 = 0 +--R +--R +--R ,, , 2 2 2 +--R (58) (- x + 1)y(x)y (x) + (x - 1)y (x) + x y(x) = 0 +--R +--R Type: Equation Expression Integer +--E 72 + +--S 73 of 143 +solve(eq,y,x) +--R +--R +--RDaly Bug +--R >> Error detected within library code: +--R parseLODE: not a linear ordinary differential equation +--R +--R Continuing to read the file... +--R +--E 73 + +\end{chunk} +\subsection{Exact 2nd Order} +\begin{chunk}{*} +--S 74 of 143 +eq := x*y(x)*y''+x*y'^2+y(x)*y'=0 +--R +--R +--R ,, , 2 , +--R (59) x y(x)y (x) + x y (x) + y(x)y (x)= 0 +--R +--R Type: Equation Expression Integer +--E 74 + +--S 75 of 143 +solve(eq,y,x) +--R +--R +--RDaly Bug +--R >> Error detected within library code: +--R parseLODE: not a linear ordinary differential equation +--R +--R Continuing to read the file... +--R +--E 75 + +\end{chunk} +\subsection{Factoring} +\begin{chunk}{*} +--S 76 of 143 +eq := y''^2-2*y'*y''+2*y(x)*y'-y(x)^2=0 +--R +--R +--R ,, 2 , ,, , 2 +--R (60) y (x) - 2y (x)y (x) + 2y(x)y (x) - y(x) = 0 +--R +--R Type: Equation Expression Integer +--E 76 + +--S 77 of 143 +solve(eq,y,x) +--R +--R +--RDaly Bug +--R >> Error detected within library code: +--R getlincoeff: not an appropriate ordinary differential equation +--R +--R Continuing to read the file... +--R +--E 77 + +\end{chunk} +\subsection{Liouvillian} +The original document gave a different answer. +\begin{verbatim} + + 1 + 1 -- + - - x %L + x ++ %e + 1 %e | ------------- d%L + - - ++ +----------+ + x | 4 3 + %e \|%L - 2%L + (65) [particular= 0,basis= [----------,----------------------------]] + +-------+ +-------+ + | 2 | 2 + \|x - 2x \|x - 2x + +\end{verbatim} +\begin{chunk}{*} +--S 78 of 143 +eq := (x^3/2-x^2)*y'' + (2*x^2-3*x+1)*y' + (x-1)*y(x) = 0 +--R +--R +--R 3 2 ,, 2 , +--R (x - 2x )y (x) + (4x - 6x + 2)y (x) + (2x - 2)y(x) +--R +--R (61) -----------------------------------------------------= 0 +--R 2 +--R Type: Equation Expression Integer +--E 78 + +--S 79 of 143 +solve(eq,y,x) +--R +--R +--R (62) +--R [particular= 0, +--R 1 1 1 +--R - - +-------+ - - +-------+ x -- +----------+ +--R x | 1 x | 1 ++ %M | 1 +--R basis= [%e |------- ,%e |------- | %e |---------- d%M ]] +--R | 2 | 2 ++ | 4 3 +--R \|x - 2x \|x - 2x \|%M - 2%M +--RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) +--E 79 + +\end{chunk} +\subsection{Reduction} +\begin{chunk}{*} +--S 80 of 143 +eq := y'' - 2 * x * y' + 2 * y x = 3 +--R +--R +--R ,, , +--R (63) y (x) - 2xy (x) + 2y(x)= 3 +--R +--R Type: Equation Expression Integer +--E 80 + +--S 81 of 143 +solve(eq,y,x) +--R +--R +--R 2 +--R x %M +--R 3 ++ %e +--R (64) [particular= -,basis= [x,x | ----- d%M ]] +--R 2 ++ 2 +--R %M +--RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) +--E 81 + +\end{chunk} +\subsection{Integer Factors} +\begin{chunk}{*} +--S 82 of 143 +eq := sqrt(x) * y'' + 2 * x * y' + 3 * y x = 0 +--R +--R +--R +-+ ,, , +--R (65) \|x y (x) + 2xy (x) + 3y(x)= 0 +--R +--R Type: Equation Expression Integer +--E 82 + +--S 83 of 143 +solve(eq,y,x) +--R +--R +--R (66) [particular= 0,basis= []] +--RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) +--E 83 + +\end{chunk} +\subsection{Scale Invariant} +\begin{chunk}{*} +--S 84 of 143 +eq := x^2*y''+3*x*y'+2*y(x)=1/y(x)^3/x^4 +--R +--R +--R 2 ,, , 1 +--R (67) x y (x) + 3xy (x) + 2y(x)= ------- +--R 4 3 +--R x y(x) +--R Type: Equation Expression Integer +--E 84 + +--S 85 of 143 +solve(eq,y,x) +--R +--R +--RDaly Bug +--R >> Error detected within library code: +--R getlincoeff: not an appropriate ordinary differential equation +--R +--R Continuing to read the file... +--R +--E 85 + +\end{chunk} +\subsection{undet} +\begin{chunk}{*} +--S 86 of 143 +eq := y'' - 2 / x^2 * y x = 7 * x^4 + 3 * x^3 +--R +--R +--R 2 ,, +--R x y (x) - 2y(x) +--R 4 3 +--R (68) ----------------= 7x + 3x +--R 2 +--R x +--R Type: Equation Expression Integer +--E 86 + +--S 97 of 143 +solve(eq,y,x) +--R +--R +--R 7 6 3 3 3 +--R 3x + 2x - 7x + 14 x - 1 x + 2 +--R (69) [particular= --------------------,basis= [------,------]] +--R 12x x x +--RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) +--E 97 + +\end{chunk} +\subsection{variation} +\begin{chunk}{*} +--S 88 of 143 +eq := y'' + y x = csc x +--R +--R +--R ,, +--R (70) y (x) + y(x)= csc(x) +--R +--R Type: Equation Expression Integer +--E 88 + +--S 89 of 143 +solve(eq,y,x) +--R +--R +--R (71) +--R sin(x) 2 +--R [particular= sin(x)log(----------) - sin(x)log(----------) - x cos(x), +--R cos(x) + 1 cos(x) + 1 +--R basis= [cos(x),sin(x)]] +--RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) +--E 89 + +\end{chunk} +\subsection{Constant Coefficients} +\begin{chunk}{*} +--S 90 of 143 +eq := D(y x,x,7) - 14 * D(y x,x,6) + 80 * D(y x,x,5) - _ + 242 * D(y x,x,4) + 419 * D(y x,x,3) - 416 * y'' + _ + 220 * y' - 48 * y x = 0 +--R +--R +--R (72) +--R (vii) (vi) (v) (iv) ,,, ,, +--R y (x) - 14y (x) + 80y (x) - 242y (x) + 419y (x) - 416y (x) +--R +--R + +--R , +--R 220y (x) - 48y(x) +--R +--R = +--R 0 +--R Type: Equation Expression Integer +--E 90 + +--S 91 of 143 +solve(eq,y,x) +--R +--R +--R 4x 3x 2x 2x x x 2 x +--R (73) [particular= 0,basis= [%e ,%e ,%e ,x %e ,%e ,x %e ,x %e ]] +--RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) +--E 91 + +\end{chunk} +\subsection{Euler} +\begin{chunk}{*} +--S 92 of 143 +eq := D(y x,x,4) - 4/x^2 * y'' + 8/x^3 * y' -8/x^4 * y x = 0 +--R +--R +--R 4 (iv) 2 ,, , +--R x y (x) - 4x y (x) + 8xy (x) - 8y(x) +--R +--R (74) ----------------------------------------= 0 +--R 4 +--R x +--R Type: Equation Expression Integer +--E 92 + +--S 93 of 143 +solve(eq,y,x) +--R +--R +--R (75) +--R [particular= 0, +--R +--R basis = +--R 5 3 2 5 3 2 5 3 2 +--R x - 5x + 5x - 1 x + 5x - 10x + 4 x - 10x + 5x + 4 +--R [------------------, -------------------, -------------------, +--R x x x +--R 5 3 2 +--R x - 10x + 20x + 4 +--R --------------------] +--R x +--R ] +--RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) +--E 93 + +\end{chunk} +\subsection{Exact nth} +The original document gave a different answer for the particular: +\begin{verbatim} + + 2 + x - 4 1 x + 1 + (85) [particular= ------------,basis= [----------,----------,1]] + 2 2 2 + 4x + 4x + 4 x + x + 1 x + x + 1 + +\end{verbatim} +\begin{chunk}{*} +--S 94 of 143 +eq := (1+x+x^2)*D(y x,x,3) + (3+6*x)*y'' + 6*y' = 6*x +--R +--R +--R 2 ,,, ,, , +--R (76) (x + x + 1)y (x) + (6x + 3)y (x) + 6y (x)= 6x +--R +--R Type: Equation Expression Integer +--E 94 + +--S 95 of 143 +solve(eq,y,x) +--R +--R +--R 4 +--R x - 4 1 x + 1 +--R (77) [particular= ------------,basis= [----------,----------,1]] +--R 2 2 2 +--R 4x + 4x + 4 x + x + 1 x + x + 1 +--RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) +--E 95 + +\end{chunk} +\subsection{Circle} +\begin{chunk}{*} +--S 96 of 143 +eq := (y'^2+1)*D(y(x),x,3)-3*y'*y''^2 = 0 +--R +--R +--R , 2 ,,, , ,, 2 +--R (78) (y (x) + 1)y (x) - 3y (x)y (x) = 0 +--R +--R Type: Equation Expression Integer +--E 96 + +--S 97 of 143 +solve(eq,y,x) +--R +--R +--RDaly Bug +--R >> Error detected within library code: +--R parseLODE: not a linear ordinary differential equation +--R +--R Continuing to read the file... +--R +--E 97 + +\end{chunk} +\subsection{transfBernoulli} +\begin{chunk}{*} +--S 98 of 143 +eq := 3*y''*D(y(x),x,4)-5*D(y(x),x,3)^2 = 0 +--R +--R +--R ,, (iv) ,,, 2 +--R (79) 3y (x)y (x) - 5y (x) = 0 +--R +--R Type: Equation Expression Integer +--E 98 + +--S 99 of 143 +solve(eq,y,x) +--R +--R +--RDaly Bug +--R >> Error detected within library code: +--R parseLODE: not a linear ordinary differential equation +--R +--R Continuing to read the file... +--R +--E 99 + +\end{chunk} +\subsection{Delay} +\begin{chunk}{*} +--S 100 of 143 +eq := y'+a*y(x-1) = 0 +--R +--R +--R , +--R (80) y (x) + a y(x - 1)= 0 +--R +--R Type: Equation Expression Integer +--E 100 + +--S 101 of 143 +solve(eq,y,x) +--R +--R +--R x +--R ++ +--R (81) | a y(%M - 1)d%M + y(x) +--R ++ +--R Type: Union(Expression Integer,...) +--E 101 + +\end{chunk} +\subsection{Several} +\begin{chunk}{*} +--S 102 of 143 +eq := D(y(x,a),x,1) = a*y(x,a) +--R +--R +--R (82) y (x,a)= a y(x,a) +--R ,1 +--R Type: Equation Expression Integer +--E 102 + +--S 103 of 143 +solve(eq,y,x) +--R +--R +--RDaly Bug +--R >> Error detected within library code: +--R parseODE: equation has order 0 +--R +--R Continuing to read the file... +--R +--E 103 + +\end{chunk} +\subsection{Nth order} +\begin{chunk}{*} +--S 104 of 143 +eq := D(y x,x,4) = sin x +--R +--R +--R (iv) +--R (83) y (x)= sin(x) +--R +--R Type: Equation Expression Integer +--E 104 + +--S 105 of 143 +ini := [0, 0, 0, 0] +--R +--R +--R (84) [0,0,0,0] +--R Type: List NonNegativeInteger +--E 105 + +--S 106 of 143 +solve(eq,y,x=0,ini) +--R +--R +--R 3 +--R 6sin(x) + x - 6x +--R (85) ----------------- +--R 6 +--R Type: Union(Expression Integer,...) +--E 106 + +\end{chunk} +\subsection{BesselJ} +\begin{chunk}{*} +--S 107 of 143 +eq := x * y'' + y' + 2 * x * y x = 0 +--R +--R +--R ,, , +--R (86) xy (x) + y (x) + 2x y(x)= 0 +--R +--R Type: Equation Expression Integer +--E 107 + +--S 108 of 143 +solve(eq,y,x=0,[1,0]) +--R +--R +--R (87) "failed" +--R Type: Union("failed",...) +--E 108 + +\end{chunk} +\subsection{Separable} +\begin{chunk}{*} +--S 109 of 143 +eq := x*y'^2-y(x)^2+1=0 +--R +--R +--R , 2 2 +--R (88) x y (x) - y(x) + 1= 0 +--R +--R Type: Equation Expression Integer +--E 109 + +--S 110 of 143 +solve(eq,y,x=0,[1]) +--R +--R +--RDaly Bug +--R >> Error detected within library code: +--R getlincoeff: not an appropriate ordinary differential equation +--R +--R Continuing to read the file... +--R +--E 110 + +--S 111 of 143 +eq := y' = sqrt((y(x)^2-1)/x) +--R +--R +--R +---------+ +--R | 2 +--R , |y(x) - 1 +--R (89) y (x)= |--------- +--R \| x +--R Type: Equation Expression Integer +--E 111 + +--S 112 of 143 +solve(eq,y,x=0,[1]) +--R +--R +--RDaly Bug +--R >> Error detected within library code: +--R catdef: division by zero +--R +--R Continuing to read the file... +--R +--E 112 + +\end{chunk} +\subsection{ic2} +\begin{chunk}{*} +--S 113 of 143 +eq := y''+y(x)*y'^3 = 0 +--R +--R +--R ,, , 3 +--R (90) y (x) + y(x)y (x) = 0 +--R +--R Type: Equation Expression Integer +--E 113 + +--S 114 of 143 +solve(eq,y,x=0,[0,2]) +--R +--R +--RDaly Bug +--R >> Error detected within library code: +--R getlincoeff: not an appropriate ordinary differential equation +--R +--R Continuing to read the file... +--R +--E 114 + +)clear all + + +\end{chunk} +\subsection{intcomb} +\begin{chunk}{*} +--S 115 of 143 +x := operator x +--R +--R +--R (1) x +--R Type: BasicOperator +--E 115 + +--S 116 of 143 +y := operator y +--R +--R +--R (2) y +--R Type: BasicOperator +--E 116 + +--S 117 of 143 +z := operator z +--R +--R +--R (3) z +--R Type: BasicOperator +--E 117 + +--S 118 of 143 +x' := D(x(t),t) +--R +--R +--R , +--R (4) x (t) +--R +--R Type: Expression Integer +--E 118 + +--S 119 of 143 +y' := D(y(t),t) +--R +--R +--R , +--R (5) y (t) +--R +--R Type: Expression Integer +--E 119 + +--S 120 of 143 +z' := D(z(t),t) +--R +--R +--R , +--R (6) z (t) +--R +--R Type: Expression Integer +--E 120 + +--S 121 of 143 +sys:=[x'=-3*y(t)*z(t),y'=3*x(t)*z(t),z'=-x(t)*y(t)] +--R +--R +--R , , , +--R (7) [x (t)= - 3y(t)z(t),y (t)= 3x(t)z(t),z (t)= - x(t)y(t)] +--R +--R Type: List Equation Expression Integer +--E 121 + +--S 122 of 143 +solve(sys,[x,y,z],t) +--R +--R +--RDaly Bug +--R >> Error detected within library code: +--R getfreelincoeff: not a linear ordinary differential equation +--R +--R Continuing to read the file... +--R +--E 122 + +\end{chunk} +\subsection{MRiccatti} +\begin{chunk}{*} +--S 123 of 143 +a := operator a +--R +--R +--R (8) a +--R Type: BasicOperator +--E 123 + +--S 124 of 143 +b := operator b +--R +--R +--R (9) b +--R Type: BasicOperator +--E 124 + +--S 125 of 143 +eq1 := x'=a(t)*(y(t)^2-x(t)^2)+2*b(t)*x(t)*y(t)+2*c*x(t) +--R +--R +--R , 2 2 +--R (10) x (t)= a(t)y(t) + 2b(t)x(t)y(t) - a(t)x(t) + 2c x(t) +--R +--R Type: Equation Expression Integer +--E 125 + +--S 126 of 143 +eq2 := y'=b(t)*(y(t)^2-x(t)^2)-2*a(t)*x(t)*y(t)+2*c*y(t) +--R +--R +--R , 2 2 +--R (11) y (t)= b(t)y(t) + (- 2a(t)x(t) + 2c)y(t) - b(t)x(t) +--R +--R Type: Equation Expression Integer +--E 126 + +--S 127 of 143 +solve([eq1,eq2],[x,y],t) +--R +--R +--RDaly Bug +--R >> Error detected within library code: +--R getlincoeff: not an appropriate ordinary differential equation +--R +--R Continuing to read the file... +--R +--E 127 + +\end{chunk} +\subsection{Triangular} +\begin{chunk}{*} +--S 128 of 143 +eq1 := x'=x(t)*(1+cos(t)/(2+sin(t))) +--R +--R +--R , x(t)sin(t) + x(t)cos(t) + 2x(t) +--R (12) x (t)= ------------------------------- +--R sin(t) + 2 +--R Type: Equation Expression Integer +--E 128 + +--S 129 of 143 +eq2 := y'=x(t)-y(t) +--R +--R +--R , +--R (13) y (t)= - y(t) + x(t) +--R +--R Type: Equation Expression Integer +--E 129 + +--S 130 of 143 +solve([eq1,eq2],[x,y],t) +--R +--R +--R (14) "failed" +--R Type: Union("failed",...) +--E 130 + +\end{chunk} +\subsection{Vector} +\begin{chunk}{*} +--S 131 of 143 +eq1 := x' = 9*x(t)+2*y(t) +--R +--R +--R , +--R (15) x (t)= 2y(t) + 9x(t) +--R +--R Type: Equation Expression Integer +--E 131 + +--S 132 of 143 +eq2 := y' = x(t)+8*y(t) +--R +--R +--R , +--R (16) y (t)= 8y(t) + x(t) +--R +--R Type: Equation Expression Integer +--E 132 + +--S 133 of 143 +solve([eq1,eq2],[x,y],t) +--R +--R +--R 10t +--R 10t %e 7t 7t +--R (17) [particular= [0,0],basis= [[%e ,-----],[%e ,- %e ]]] +--R 2 +--RType: Union(Record(particular: Vector Expression Integer,basis: List Vector Expression Integer),...) +--E 133 + +\end{chunk} +\subsection{High Order} +\begin{chunk}{*} +--S 134 of 143 +eq1 := x'-x(t)+2*y(t)=0 +--R +--R +--R , +--R (18) x (t) + 2y(t) - x(t)= 0 +--R +--R Type: Equation Expression Integer +--E 134 + +--S 135 of 143 +eq2 := D(x(t),t,2)-2*y'=2*t-cot(2*t) +--R +--R +--R ,, , +--R (19) x (t) - 2y (t)= - cot(2t) + 2t +--R +--R Type: Equation Expression Integer +--E 135 + +--S 136 of 143 +solve([eq1,eq2],[x,y],t) +--R +--R +--RDaly Bug +--R >> Error detected within library code: +--R solve: not a first order linear system +--R +--R Continuing to read the file... +--R +--E 136 + +\end{chunk} +\subsection{Inhomogeneous} +\begin{chunk}{*} +--S 137 of 143 +eq1 := x' = -x(t)/t/(t^2+1)+y(t)/t^2/(t^2+1)+1/t +--R +--R +--R 3 +--R , y(t) - t x(t) + t + t +--R (20) x (t)= ---------------------- +--R 4 2 +--R t + t +--R Type: Equation Expression Integer +--E 137 + +--S 138 of 143 +eq2 := y' = -t^2*x(t)/(t^2+1)+(2*t^2+1)*y(t)/t/(t^2+1)+1 +--R +--R +--R 2 3 3 +--R , (2t + 1)y(t) - t x(t) + t + t +--R (21) y (t)= ------------------------------- +--R 3 +--R t + t +--R Type: Equation Expression Integer +--E 138 + +--S 139 of 143 +solve([eq1,eq2],[x,y],t) +--R +--R +--R 1 2 +--R (22) [particular= [log(t) - 1,t log(t) - t],basis= [[-,- t ],[1,t]]] +--R t +--RType: Union(Record(particular: Vector Expression Integer,basis: List Vector Expression Integer),...) +--E 139 + +)clear all + +\end{chunk} +\subsection{Bronstein} +\begin{chunk}{*} +--S 140 of 143 +s := sqrt(-222) +--R +--R +--R +-----+ +--R (1) \|- 222 +--R Type: AlgebraicNumber +--E 140 + +--S 141 of 143 +a0 := 104/25*x^10+(274/25-22/15*s)*x^8+(7754/75-68/15*s)*x^6 +--R +--R +--R +-----+ +-----+ +--R 104 10 - 110\|- 222 + 822 8 - 340\|- 222 + 7754 6 +--R (2) --- x + ------------------- x + -------------------- x +--R 25 75 75 +--R Type: Polynomial AlgebraicNumber +--E 141 + +--S 142 of 143 +a0 := a0+(11248/75-194/15*s)*x^4+(29452/75-296/5*s)*x^2-10952/5-148/3*s +--R +--R +--R (3) +--R +-----+ +-----+ +--R 104 10 - 110\|- 222 + 822 8 - 340\|- 222 + 7754 6 +--R --- x + ------------------- x + -------------------- x +--R 25 75 75 +--R + +--R +-----+ +-----+ +-----+ +--R - 970\|- 222 + 11248 4 - 4440\|- 222 + 29452 2 - 740\|- 222 - 32856 +--R --------------------- x + ---------------------- x + --------------------- +--R 75 75 15 +--R Type: Polynomial AlgebraicNumber +--E 142 + +--S 143 of 143 +a2 := x^12+2*x^10+151/3*x^8+296/3*x^6+5920/9*x^4+10952/9*x^2+5476/9 +--R +--R +--R 12 10 151 8 296 6 5920 4 10952 2 5476 +--R (4) x + 2x + --- x + --- x + ---- x + ----- x + ---- +--R 3 3 9 9 9 +--R Type: Polynomial Fraction Integer +--E 143 + +\end{chunk} +This code loops trying to factor the formula. +\begin{chunk}{*) +--y := operator y +--y' := D(y x, x) +--y'' := D(y', x) +--eq := a2*y'' - a0*y(x) +--solve(eq,y,x) +--R +--R WARNING (genufact): No known algorithm to factor +--R 8 43808 6 827062184 4 44060595526144 2 27067141014651136 +--R ? + --------- ? - --------- ? - -------------- ? - ----------------- +--R +-----+ 5625 +-----+ 31640625 +--R 5\|- 222 253125\|- 222 +--R , trying square-free. +--R WARNING (genufact): No known algorithm to factor +--R 8 4292 6 16006348 4 58334426144 2 94037902336 +--R ? + ---- ? + -------- ? + ----------- ? + ----------- +--R 25 1875 421875 140625 +--R , trying square-free. +--R WARNING (genufact): No known algorithm to factor +--R 8 1776 6 151034 4 108731456 2 902641936 +--R ? - --------- ? - ------ ? + ------------ ? - --------- +--R +-----+ 625 +-----+ 390625 +--R 5\|- 222 9375\|- 222 +--R , trying square-free. +--R WARNING (genufact): No known algorithm to factor +--R 8 7 3426 6 14364 5 903369 4 277938 3 31725044 2 +--R ? - 18? + ---- ? - ----- ? + ------ ? - ------ ? + -------- ? +--R 25 25 625 125 15625 +--R + +--R 15594264 3118752 +--R - -------- ? + ------- +--R 15625 15625 +--R , trying square-free. +--R + +\end{chunk} +\subsection{Moussiaux} +This equation crashes Axiom silently and is a major bug. +\begin{chunk}{*} +--solve(15*D(y(x),x)+24*y(x)^2=7*x^(-8/3),y,x) + +)spool +)lisp (bye) + +\end{chunk} +\eject +\begin{thebibliography}{99} +\bibitem{1} Zimmermann, Paul +``http://www.loria.fr/~zimmerma/ComputerAlgebra/ode_comp.axiom'' +\end{thebibliography} +\end{document}