diff --git a/changelog b/changelog index 46dc9dc..44f04bc 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,5 @@ +20071004 tpd src/input/Makefile kamke1.input regression test added +20071004 tpd src/input/kamke1.input ODE regression test added 20071004 tpd src/input/Makefile kamke0.input regression test added 20071004 tpd src/input/kamke0.input ODE regression test added 20071001 tpd src/algebra/exposed.lisp add (|AxiomServer| . AXSERV) to basic diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet index c1ade3e..159377a 100644 --- a/src/input/Makefile.pamphlet +++ b/src/input/Makefile.pamphlet @@ -322,7 +322,7 @@ REGRES= algaggr.regress algbrbf.regress algfacob.regress alist.regress \ intmix2.regress intmix.regress int.regress intrf.regress \ iprntpk.regress \ ipftest.regress is.regress isprime.regress kafile.regress \ - kamke0.regress \ + kamke0.regress kamke1.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 \ @@ -561,7 +561,7 @@ FILES= ${OUT}/algaggr.input ${OUT}/algbrbf.input ${OUT}/algfacob.input \ ${OUT}/intg0.input ${OUT}/intheory.input ${OUT}/int.input \ ${OUT}/intlf.input ${OUT}/intmix.input ${OUT}/intrf.input \ ${OUT}/ipftest.input ${OUT}/is.input ${OUT}/isprime.input \ - ${OUT}/kafile.input ${OUT}/kamke0.input \ + ${OUT}/kafile.input ${OUT}/kamke0.input ${OUT}/kamke1.input \ ${OUT}/kernel.input ${OUT}/knot.input \ ${OUT}/kovacic.input ${OUT}/kuipers.input \ ${OUT}/laplace.input ${OUT}/leg.input ${OUT}/lexp.input \ @@ -827,7 +827,7 @@ DOCFILES= \ ${DOC}/intmix.input.dvi ${DOC}/intrf.input.dvi \ ${DOC}/ipftest.input.dvi ${DOC}/is.input.dvi \ ${DOC}/isprime.input.dvi ${DOC}/kafile.input.dvi \ - ${DOC}/kamke0.input.dvi \ + ${DOC}/kamke0.input.dvi ${DOC}/kamke1.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/kamke1.input.pamphlet b/src/input/kamke1.input.pamphlet new file mode 100644 index 0000000..2b5237a --- /dev/null +++ b/src/input/kamke1.input.pamphlet @@ -0,0 +1,2124 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/src/input kamke1.input} +\author{Timothy Daly} +\maketitle +\begin{abstract} +This is the next 50 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. +\end{abstract} +\eject +\tableofcontents +\eject +<<*>>= +)spool kamke1.output +)set break resume +)set mes auto off +)clear all + +--S 1 of 123 +y:=operator 'y +--R +--R +--R (1) y +--R Type: BasicOperator +--E 1 + +--S 2 of 123 +f := operator 'f +--R +--R (2) f +--R Type: BasicOperator +--E 2 + +--S 3 of 123 +g := operator 'g +--R +--R (3) g +--R Type: BasicOperator +--E 3 + +--S 4 of 123 +h := operator 'h +--R +--R (4) h +--R Type: BasicOperator +--E 4 + +--S 5 of 123 +ode51 := D(y(x),x) - (y(x)-f(x))*(y(x)-g(x))*(y(x)-(a*f(x)+b*g(x))/(a+b))*h(x)_ + - D(f(x),x)*(y(x)-g(x))/(f(x)-g(x)) _ + - D(g(x),x)*(y(x)-f(x))/(g(x)-f(x)) +--R +--R (5) +--R , , +--R ((b + a)g(x) + (- b - a)f(x))y (x) + ((- b - a)y(x) + (b + a)f(x))g (x) +--R +--R + +--R , +--R ((b + a)y(x) + (- b - a)g(x))f (x) +--R +--R + +--R 3 +--R ((- b - a)g(x) + (b + a)f(x))h(x)y(x) +--R + +--R 2 2 2 +--R ((2b + a)g(x) + (- b + a)f(x)g(x) + (- b - 2a)f(x) )h(x)y(x) +--R + +--R 3 2 2 3 +--R (- b g(x) + (- b - 2a)f(x)g(x) + (2b + a)f(x) g(x) + a f(x) )h(x)y(x) +--R + +--R 3 2 2 3 +--R (b f(x)g(x) + (- b + a)f(x) g(x) - a f(x) g(x))h(x) +--R / +--R (b + a)g(x) + (- b - a)f(x) +--R Type: Expression Integer +--E 5 + +--S 6 of 123 +ode51a:=solve(ode51,y,x) +--R +--R (6) "failed" +--R Type: Union("failed",...) +--E 6 + +--S 7 of 123 +ode52 := D(y(x),x) - a*y(x)**n - b*x**(n/(1-n)) +--R +--R n +--R - ----- +--R , n n - 1 +--R (7) y (x) - a y(x) - b x +--R +--R Type: Expression Integer +--E 7 + +--S 8 of 123 +ode52a:=solve(ode52,y,x) +--R +--R (8) "failed" +--R Type: Union("failed",...) +--E 8 + +--S 9 of 123 +ode53 := D(y(x),x) - f(x)**(1-n)*D(g(x),x)*y(x)**n/(a*g(x)+b)**n _ + - D(f(x),x)*y(x)/f(x) - f(x)*D(g(x),x) +--R +--R (9) +--R n , +--R f(x)(a g(x) + b) y (x) +--R +--R + +--R - n + 1 n 2 n , n , +--R (- f(x)f(x) y(x) - f(x) (a g(x) + b) )g (x) - y(x)(a g(x) + b) f (x) +--R +--R / +--R n +--R f(x)(a g(x) + b) +--R Type: Expression Integer +--E 9 + +--S 10 of 123 +ode53a:=solve(ode53,y,x) +--R +--R (10) "failed" +--R Type: Union("failed",...) +--E 10 + +--S 11 of 123 +ode54 := D(y(x),x) - a**n*f(x)**(1-n)*D(g(x),x)*y(x)**n - _ + D(f(x),x)*y(x)/f(x) - f(x)*D(g(x),x) +--R +--R , n - n + 1 n 2 , , +--R f(x)y (x) + (- f(x)a f(x) y(x) - f(x) )g (x) - y(x)f (x) +--R +--R (11) --------------------------------------------------------------- +--R f(x) +--R Type: Expression Integer +--E 11 + +--S 12 of 123 +ode54a:=solve(ode54,y,x) +--R +--R (12) "failed" +--R Type: Union("failed",...) +--E 12 + +--S 13 of 123 +ode55 := D(y(x),x) - f(x)*y(x)**n - g(x)*y(x) - h(x) +--R +--R , n +--R (13) y (x) - f(x)y(x) - g(x)y(x) - h(x) +--R +--R Type: Expression Integer +--E 13 + +--S 14 of 123 +ode55a:=solve(ode55,y,x) +--R +--R (14) "failed" +--R Type: Union("failed",...) +--E 14 + +--S 15 of 123 +ode56 := D(y(x),x) - f(x)*y(x)**a - g(x)*y(x)**b +--R +--R , b a +--R (15) y (x) - g(x)y(x) - f(x)y(x) +--R +--R Type: Expression Integer +--E 15 + +--S 16 of 123 +ode5a:=solve(ode56,y,x) +--R +--R (16) "failed" +--R Type: Union("failed",...) +--E 16 + +--S 17 of 123 +ode57 := D(y(x),x) - sqrt(abs(y(x))) +--R +--R +---------+ , +--R (17) - \|abs(y(x)) + y (x) +--R +--R Type: Expression Integer +--E 17 + +--S 18 of 123 +yx:=solve(ode57,y,x) +--R +--R +---------+ +--R - x\|abs(y(x)) + 2y(x) +--R (18) ----------------------- +--R +----+ +--R \|y(x) +--R Type: Union(Expression Integer,...) +--E 18 + +--S 19 of 123 +ode57expr := D(yx,x) - sqrt(abs(yx)) +--R +--R (19) +--R +--------------------------+ +--R | +---------+ +--R +----+ +---------+ | x\|abs(y(x)) - 2y(x) , +---------+ +--R - \|y(x) \|abs(y(x)) |abs(---------------------) + y (x)\|abs(y(x)) +--R | +----+ +--R \| \|y(x) +--R + +--R - abs(y(x)) +--R / +--R +----+ +---------+ +--R \|y(x) \|abs(y(x)) +--R Type: Expression Integer +--E 19 + +--S 20 of 123 +ode58 := D(y(x),x) - a*sqrt(y(x)) - b*x +--R +--R , +----+ +--R (20) y (x) - a\|y(x) - b x +--R +--R Type: Expression Integer +--E 20 + +--S 21 of 123 +ode58a:=solve(ode58,y,x) +--R +--R (21) "failed" +--R Type: Union("failed",...) +--E 21 + +-- this never finishes +-- ode59 := D(y(x),x) - a*sqrt(y(x)**2+1) - b +-- + +--S 22 of 123 +ode60 := D(y(x),x) - sqrt(y(x)**2-1)/sqrt(x**2-1) +--R +--R +------+ +---------+ +--R | 2 , | 2 +--R \|x - 1 y (x) - \|y(x) - 1 +--R +--R (22) ----------------------------- +--R +------+ +--R | 2 +--R \|x - 1 +--R Type: Expression Integer +--E 22 + +--S 23 of 123 +ode60a:=solve(ode60,y,x) +--R +--R (23) "failed" +--R Type: Union("failed",...) +--E 23 + +--S 24 of 123 +ode61 := D(y(x),x) - sqrt(x**2-1)/sqrt(y(x)**2-1) +--R +--R +---------+ +------+ +--R | 2 , | 2 +--R \|y(x) - 1 y (x) - \|x - 1 +--R +--R (24) ----------------------------- +--R +---------+ +--R | 2 +--R \|y(x) - 1 +--R Type: Expression Integer +--E 24 + +--S 25 of 123 +yx:=solve(ode61,y,x) +--R +--R (25) +--R +------+ +---------+ +--R | 2 2 | 2 +--R (4x y(x)\|x - 1 + (- 4x + 2)y(x))\|y(x) - 1 +--R + +--R +------+ +--R 2 | 2 2 2 2 +--R (- 4x y(x) + 2x)\|x - 1 + (4x - 2)y(x) - 2x + 1 +--R * +--R +---------+ +--R | 2 +--R log(\|y(x) - 1 - y(x)) +--R + +--R +------+ +------+ +--R | 2 2 | 2 +--R (- 4x y(x)\|x - 1 + (4x - 2)y(x))log(\|x - 1 - x) +--R + +--R +------+ +--R 3 3 | 2 2 3 +--R (- 4x y(x) + 4x y(x))\|x - 1 + (4x - 2)y(x) +--R + +--R 4 2 +--R (- 4x + 2x + 1)y(x) +--R * +--R +---------+ +--R | 2 +--R \|y(x) - 1 +--R + +--R +------+ +------+ +--R 2 | 2 2 2 2 | 2 +--R ((4x y(x) - 2x)\|x - 1 + (- 4x + 2)y(x) + 2x - 1)log(\|x - 1 - x) +--R + +--R +------+ +--R 4 3 2 3 | 2 2 4 +--R (4x y(x) + (- 4x - 2x)y(x) + 2x - x)\|x - 1 + (- 4x + 2)y(x) +--R + +--R 4 2 4 2 +--R (4x - 2)y(x) - 2x + 2x +--R / +--R +------+ +---------+ +--R | 2 2 | 2 +--R (8x y(x)\|x - 1 + (- 8x + 4)y(x))\|y(x) - 1 +--R + +--R +------+ +--R 2 | 2 2 2 2 +--R (- 8x y(x) + 4x)\|x - 1 + (8x - 4)y(x) - 4x + 2 +--R Type: Union(Expression Integer,...) +--E 25 + +--S 26 of 123 +ode61expr := D(yx,x) - sqrt(x**2-1)/sqrt(yx**2-1) +--R +--R (26) +--R 4 2 5 4 2 3 +--R (- 64x + 64x - 8)y(x) + (96x - 96x + 12)y(x) +--R + +--R 4 2 +--R (- 32x + 32x - 4)y(x) +--R * +--R +------+ +--R | 2 +--R \|x - 1 +--R + +--R 5 3 5 5 3 3 +--R (64x - 96x + 32x)y(x) + (- 96x + 144x - 48x)y(x) +--R + +--R 5 3 +--R (32x - 48x + 16x)y(x) +--R * +--R +---------+ +--R | 2 +--R \|y(x) - 1 +--R + +--R 4 2 6 4 2 4 +--R (64x - 64x + 8)y(x) + (- 128x + 128x - 16)y(x) +--R + +--R 4 2 2 4 2 +--R (72x - 72x + 9)y(x) - 8x + 8x - 1 +--R * +--R +------+ +--R | 2 +--R \|x - 1 +--R + +--R 5 3 6 5 3 4 +--R (- 64x + 96x - 32x)y(x) + (128x - 192x + 64x)y(x) +--R + +--R 5 3 2 5 3 +--R (- 72x + 108x - 36x)y(x) + 8x - 12x + 4x +--R * +--R , +--R y (x) +--R +--R + +--R 5 3 4 5 3 2 5 +--R (64x - 96x + 32x)y(x) + (- 64x + 96x - 32x)y(x) + 8x +--R + +--R 3 +--R - 12x + 4x +--R * +--R +------+ +--R | 2 +--R \|x - 1 +--R + +--R 6 4 2 4 6 4 2 2 +--R (- 64x + 128x - 72x + 8)y(x) + (64x - 128x + 72x - 8)y(x) +--R + +--R 6 4 2 +--R - 8x + 16x - 9x + 1 +--R * +--R +---------+ +--R | 2 +--R \|y(x) - 1 +--R + +--R 5 3 5 5 3 3 +--R (- 64x + 96x - 32x)y(x) + (96x - 144x + 48x)y(x) +--R + +--R 5 3 +--R (- 32x + 48x - 16x)y(x) +--R * +--R +------+ +--R | 2 +--R \|x - 1 +--R + +--R 6 4 2 5 6 4 2 3 +--R (64x - 128x + 72x - 8)y(x) + (- 96x + 192x - 108x + 12)y(x) +--R + +--R 6 4 2 +--R (32x - 64x + 36x - 4)y(x) +--R * +--R ROOT +--R +------+ +--R 3 3 3 | 2 +--R ((64x - 32x)y(x) + (- 32x + 16x)y(x))\|x - 1 +--R + +--R 4 2 3 4 2 +--R (- 64x + 64x - 8)y(x) + (32x - 32x + 4)y(x) +--R * +--R +---------+ +--R | 2 +--R \|y(x) - 1 +--R + +--R 3 4 3 2 3 +--R ((- 64x + 32x)y(x) + (64x - 32x)y(x) - 8x + 4x) +--R * +--R +------+ +--R | 2 +--R \|x - 1 +--R + +--R 4 2 4 4 2 2 4 +--R (64x - 64x + 8)y(x) + (- 64x + 64x - 8)y(x) + 8x +--R + +--R 2 +--R - 8x + 1 +--R * +--R +---------+ 2 +--R | 2 +--R log(\|y(x) - 1 - y(x)) +--R + +--R +------+ +--R 3 3 3 | 2 +--R ((- 128x + 64x)y(x) + (64x - 32x)y(x))\|x - 1 +--R + +--R 4 2 3 4 2 +--R (128x - 128x + 16)y(x) + (- 64x + 64x - 8)y(x) +--R * +--R +------+ +--R | 2 +--R log(\|x - 1 - x) +--R + +--R 3 5 5 3 +--R (- 128x + 64x)y(x) + (128x - 48x)y(x) +--R + +--R 5 3 +--R (- 64x + 48x )y(x) +--R * +--R +------+ +--R | 2 +--R \|x - 1 +--R + +--R 4 2 5 +--R (128x - 128x + 16)y(x) +--R + +--R 6 4 2 3 +--R (- 128x + 64x + 64x - 16)y(x) +--R + +--R 6 4 2 +--R (64x - 80x + 16x + 2)y(x) +--R * +--R +---------+ +--R | 2 +--R \|y(x) - 1 +--R + +--R 3 4 3 2 3 +--R (128x - 64x)y(x) + (- 128x + 64x)y(x) + 16x +--R + +--R - 8x +--R * +--R +------+ +--R | 2 +--R \|x - 1 +--R + +--R 4 2 4 4 2 2 +--R (- 128x + 128x - 16)y(x) + (128x - 128x + 16)y(x) +--R + +--R 4 2 +--R - 16x + 16x - 2 +--R * +--R +------+ +--R | 2 +--R log(\|x - 1 - x) +--R + +--R 3 6 5 3 4 +--R (128x - 64x)y(x) + (- 128x - 64x + 80x)y(x) +--R + +--R 5 3 2 5 3 +--R (128x - 64x - 16x)y(x) - 16x + 16x - 2x +--R * +--R +------+ +--R | 2 +--R \|x - 1 +--R + +--R 4 2 6 6 2 4 +--R (- 128x + 128x - 16)y(x) + (128x - 128x + 24)y(x) +--R + +--R 6 4 2 6 4 2 +--R (- 128x + 128x - 8)y(x) + 16x - 24x + 8x +--R * +--R +---------+ +--R | 2 +--R log(\|y(x) - 1 - y(x)) +--R + +--R +------+ +--R 3 3 3 | 2 +--R ((64x - 32x)y(x) + (- 32x + 16x)y(x))\|x - 1 +--R + +--R 4 2 3 4 2 +--R (- 64x + 64x - 8)y(x) + (32x - 32x + 4)y(x) +--R * +--R +------+ 2 +--R | 2 +--R log(\|x - 1 - x) +--R + +--R 3 5 5 3 +--R (128x - 64x)y(x) + (- 128x + 48x)y(x) +--R + +--R 5 3 +--R (64x - 48x )y(x) +--R * +--R +------+ +--R | 2 +--R \|x - 1 +--R + +--R 4 2 5 +--R (- 128x + 128x - 16)y(x) +--R + +--R 6 4 2 3 +--R (128x - 64x - 64x + 16)y(x) +--R + +--R 6 4 2 +--R (- 64x + 80x - 16x - 2)y(x) +--R * +--R +------+ +--R | 2 +--R log(\|x - 1 - x) +--R + +--R 3 7 5 3 5 +--R (64x - 32x)y(x) + (- 128x + 32x + 32x)y(x) +--R + +--R 7 5 3 3 +--R (64x + 32x - 320x + 128x)y(x) +--R + +--R 7 5 3 +--R (- 32x + 32x + 128x - 66x)y(x) +--R * +--R +------+ +--R | 2 +--R \|x - 1 +--R + +--R 4 2 7 6 4 2 5 +--R (- 64x + 64x - 8)y(x) + (128x - 96x - 32x + 12)y(x) +--R + +--R 8 4 2 3 +--R (- 64x + 344x - 280x + 28)y(x) +--R + +--R 8 6 4 2 +--R (32x - 48x - 116x + 132x - 16)y(x) +--R * +--R +---------+ +--R | 2 +--R \|y(x) - 1 +--R + +--R 3 4 3 2 3 +--R ((- 64x + 32x)y(x) + (64x - 32x)y(x) - 8x + 4x) +--R * +--R +------+ +--R | 2 +--R \|x - 1 +--R + +--R 4 2 4 4 2 2 4 +--R (64x - 64x + 8)y(x) + (- 64x + 64x - 8)y(x) + 8x +--R + +--R 2 +--R - 8x + 1 +--R * +--R +------+ 2 +--R | 2 +--R log(\|x - 1 - x) +--R + +--R 3 6 5 3 4 +--R (- 128x + 64x)y(x) + (128x + 64x - 80x)y(x) +--R + +--R 5 3 2 5 3 +--R (- 128x + 64x + 16x)y(x) + 16x - 16x + 2x +--R * +--R +------+ +--R | 2 +--R \|x - 1 +--R + +--R 4 2 6 6 2 4 +--R (128x - 128x + 16)y(x) + (- 128x + 128x - 24)y(x) +--R + +--R 6 4 2 6 4 2 +--R (128x - 128x + 8)y(x) - 16x + 24x - 8x +--R * +--R +------+ +--R | 2 +--R log(\|x - 1 - x) +--R + +--R 3 8 5 6 +--R (- 64x + 32x)y(x) + (128x - 48x)y(x) +--R + +--R 7 5 3 4 +--R (- 64x - 96x + 344x - 116x)y(x) +--R + +--R 7 5 3 2 7 5 3 +--R (64x - 32x - 280x + 132x)y(x) - 8x + 12x + 28x - 16x +--R * +--R +------+ +--R | 2 +--R \|x - 1 +--R + +--R 4 2 8 6 4 2 6 +--R (64x - 64x + 8)y(x) + (- 128x + 64x + 64x - 16)y(x) +--R + +--R 8 6 4 2 4 +--R (64x + 64x - 400x + 272x - 23)y(x) +--R + +--R 8 6 4 2 2 8 6 4 +--R (- 64x + 64x + 272x - 272x + 31)y(x) + 8x - 16x - 23x +--R + +--R 2 +--R 31x - 4 +--R / +--R +------+ +--R 3 3 3 | 2 +--R ((256x - 128x)y(x) + (- 128x + 64x)y(x))\|x - 1 +--R + +--R 4 2 3 4 2 +--R (- 256x + 256x - 32)y(x) + (128x - 128x + 16)y(x) +--R * +--R +---------+ +--R | 2 +--R \|y(x) - 1 +--R + +--R 3 4 3 2 3 +--R ((- 256x + 128x)y(x) + (256x - 128x)y(x) - 32x + 16x) +--R * +--R +------+ +--R | 2 +--R \|x - 1 +--R + +--R 4 2 4 4 2 2 4 +--R (256x - 256x + 32)y(x) + (- 256x + 256x - 32)y(x) + 32x +--R + +--R 2 +--R - 32x + 4 +--R + +--R 5 3 4 5 3 2 5 +--R (64x - 96x + 32x)y(x) + (- 64x + 96x - 32x)y(x) + 8x +--R + +--R 3 +--R - 12x + 4x +--R * +--R +------+ +--R | 2 +--R \|x - 1 +--R + +--R 6 4 2 4 6 4 2 2 +--R (- 64x + 128x - 72x + 8)y(x) + (64x - 128x + 72x - 8)y(x) +--R + +--R 6 4 2 +--R - 8x + 16x - 9x + 1 +--R * +--R +---------+ +--R | 2 +--R \|y(x) - 1 +--R + +--R 5 3 5 5 3 3 +--R (- 64x + 96x - 32x)y(x) + (96x - 144x + 48x)y(x) +--R + +--R 5 3 +--R (- 32x + 48x - 16x)y(x) +--R * +--R +------+ +--R | 2 +--R \|x - 1 +--R + +--R 6 4 2 5 6 4 2 3 +--R (64x - 128x + 72x - 8)y(x) + (- 96x + 192x - 108x + 12)y(x) +--R + +--R 6 4 2 +--R (32x - 64x + 36x - 4)y(x) +--R / +--R 4 2 4 4 2 2 4 2 +--R (64x - 64x + 8)y(x) + (- 64x + 64x - 8)y(x) + 8x - 8x +--R + +--R 1 +--R * +--R +------+ +--R | 2 +--R \|x - 1 +--R + +--R 5 3 4 5 3 2 5 3 +--R (- 64x + 96x - 32x)y(x) + (64x - 96x + 32x)y(x) - 8x + 12x +--R + +--R - 4x +--R * +--R +---------+ +--R | 2 +--R \|y(x) - 1 +--R + +--R 4 2 5 4 2 3 +--R (- 64x + 64x - 8)y(x) + (96x - 96x + 12)y(x) +--R + +--R 4 2 +--R (- 32x + 32x - 4)y(x) +--R * +--R +------+ +--R | 2 +--R \|x - 1 +--R + +--R 5 3 5 5 3 3 +--R (64x - 96x + 32x)y(x) + (- 96x + 144x - 48x)y(x) +--R + +--R 5 3 +--R (32x - 48x + 16x)y(x) +--R * +--R ROOT +--R +------+ +--R 3 3 3 | 2 +--R ((64x - 32x)y(x) + (- 32x + 16x)y(x))\|x - 1 +--R + +--R 4 2 3 4 2 +--R (- 64x + 64x - 8)y(x) + (32x - 32x + 4)y(x) +--R * +--R +---------+ +--R | 2 +--R \|y(x) - 1 +--R + +--R +------+ +--R 3 4 3 2 3 | 2 +--R ((- 64x + 32x)y(x) + (64x - 32x)y(x) - 8x + 4x)\|x - 1 +--R + +--R 4 2 4 4 2 2 4 2 +--R (64x - 64x + 8)y(x) + (- 64x + 64x - 8)y(x) + 8x - 8x +--R + +--R 1 +--R * +--R +---------+ 2 +--R | 2 +--R log(\|y(x) - 1 - y(x)) +--R + +--R +------+ +--R 3 3 3 | 2 +--R ((- 128x + 64x)y(x) + (64x - 32x)y(x))\|x - 1 +--R + +--R 4 2 3 4 2 +--R (128x - 128x + 16)y(x) + (- 64x + 64x - 8)y(x) +--R * +--R +------+ +--R | 2 +--R log(\|x - 1 - x) +--R + +--R 3 5 5 3 +--R (- 128x + 64x)y(x) + (128x - 48x)y(x) +--R + +--R 5 3 +--R (- 64x + 48x )y(x) +--R * +--R +------+ +--R | 2 +--R \|x - 1 +--R + +--R 4 2 5 +--R (128x - 128x + 16)y(x) +--R + +--R 6 4 2 3 +--R (- 128x + 64x + 64x - 16)y(x) +--R + +--R 6 4 2 +--R (64x - 80x + 16x + 2)y(x) +--R * +--R +---------+ +--R | 2 +--R \|y(x) - 1 +--R + +--R 3 4 3 2 3 +--R ((128x - 64x)y(x) + (- 128x + 64x)y(x) + 16x - 8x) +--R * +--R +------+ +--R | 2 +--R \|x - 1 +--R + +--R 4 2 4 4 2 2 +--R (- 128x + 128x - 16)y(x) + (128x - 128x + 16)y(x) +--R + +--R 4 2 +--R - 16x + 16x - 2 +--R * +--R +------+ +--R | 2 +--R log(\|x - 1 - x) +--R + +--R 3 6 5 3 4 +--R (128x - 64x)y(x) + (- 128x - 64x + 80x)y(x) +--R + +--R 5 3 2 5 3 +--R (128x - 64x - 16x)y(x) - 16x + 16x - 2x +--R * +--R +------+ +--R | 2 +--R \|x - 1 +--R + +--R 4 2 6 6 2 4 +--R (- 128x + 128x - 16)y(x) + (128x - 128x + 24)y(x) +--R + +--R 6 4 2 6 4 2 +--R (- 128x + 128x - 8)y(x) + 16x - 24x + 8x +--R * +--R +---------+ +--R | 2 +--R log(\|y(x) - 1 - y(x)) +--R + +--R +------+ +--R 3 3 3 | 2 +--R ((64x - 32x)y(x) + (- 32x + 16x)y(x))\|x - 1 +--R + +--R 4 2 3 4 2 +--R (- 64x + 64x - 8)y(x) + (32x - 32x + 4)y(x) +--R * +--R +------+ 2 +--R | 2 +--R log(\|x - 1 - x) +--R + +--R 3 5 5 3 +--R (128x - 64x)y(x) + (- 128x + 48x)y(x) +--R + +--R 5 3 +--R (64x - 48x )y(x) +--R * +--R +------+ +--R | 2 +--R \|x - 1 +--R + +--R 4 2 5 +--R (- 128x + 128x - 16)y(x) +--R + +--R 6 4 2 3 +--R (128x - 64x - 64x + 16)y(x) +--R + +--R 6 4 2 +--R (- 64x + 80x - 16x - 2)y(x) +--R * +--R +------+ +--R | 2 +--R log(\|x - 1 - x) +--R + +--R 3 7 5 3 5 +--R (64x - 32x)y(x) + (- 128x + 32x + 32x)y(x) +--R + +--R 7 5 3 3 +--R (64x + 32x - 320x + 128x)y(x) +--R + +--R 7 5 3 +--R (- 32x + 32x + 128x - 66x)y(x) +--R * +--R +------+ +--R | 2 +--R \|x - 1 +--R + +--R 4 2 7 6 4 2 5 +--R (- 64x + 64x - 8)y(x) + (128x - 96x - 32x + 12)y(x) +--R + +--R 8 4 2 3 +--R (- 64x + 344x - 280x + 28)y(x) +--R + +--R 8 6 4 2 +--R (32x - 48x - 116x + 132x - 16)y(x) +--R * +--R +---------+ +--R | 2 +--R \|y(x) - 1 +--R + +--R +------+ +--R 3 4 3 2 3 | 2 +--R ((- 64x + 32x)y(x) + (64x - 32x)y(x) - 8x + 4x)\|x - 1 +--R + +--R 4 2 4 4 2 2 4 2 +--R (64x - 64x + 8)y(x) + (- 64x + 64x - 8)y(x) + 8x - 8x +--R + +--R 1 +--R * +--R +------+ 2 +--R | 2 +--R log(\|x - 1 - x) +--R + +--R 3 6 5 3 4 +--R (- 128x + 64x)y(x) + (128x + 64x - 80x)y(x) +--R + +--R 5 3 2 5 3 +--R (- 128x + 64x + 16x)y(x) + 16x - 16x + 2x +--R * +--R +------+ +--R | 2 +--R \|x - 1 +--R + +--R 4 2 6 6 2 4 +--R (128x - 128x + 16)y(x) + (- 128x + 128x - 24)y(x) +--R + +--R 6 4 2 6 4 2 +--R (128x - 128x + 8)y(x) - 16x + 24x - 8x +--R * +--R +------+ +--R | 2 +--R log(\|x - 1 - x) +--R + +--R 3 8 5 6 +--R (- 64x + 32x)y(x) + (128x - 48x)y(x) +--R + +--R 7 5 3 4 +--R (- 64x - 96x + 344x - 116x)y(x) +--R + +--R 7 5 3 2 7 5 3 +--R (64x - 32x - 280x + 132x)y(x) - 8x + 12x + 28x - 16x +--R * +--R +------+ +--R | 2 +--R \|x - 1 +--R + +--R 4 2 8 6 4 2 6 +--R (64x - 64x + 8)y(x) + (- 128x + 64x + 64x - 16)y(x) +--R + +--R 8 6 4 2 4 +--R (64x + 64x - 400x + 272x - 23)y(x) +--R + +--R 8 6 4 2 2 8 6 4 +--R (- 64x + 64x + 272x - 272x + 31)y(x) + 8x - 16x - 23x +--R + +--R 2 +--R 31x - 4 +--R / +--R +------+ +--R 3 3 3 | 2 +--R ((256x - 128x)y(x) + (- 128x + 64x)y(x))\|x - 1 +--R + +--R 4 2 3 4 2 +--R (- 256x + 256x - 32)y(x) + (128x - 128x + 16)y(x) +--R * +--R +---------+ +--R | 2 +--R \|y(x) - 1 +--R + +--R 3 4 3 2 3 +--R ((- 256x + 128x)y(x) + (256x - 128x)y(x) - 32x + 16x) +--R * +--R +------+ +--R | 2 +--R \|x - 1 +--R + +--R 4 2 4 4 2 2 4 +--R (256x - 256x + 32)y(x) + (- 256x + 256x - 32)y(x) + 32x +--R + +--R 2 +--R - 32x + 4 +--R Type: Expression Integer +--E 26 + +--S 27 of 123 +ode62 := D(y(x),x) - (y(x)-x**2*sqrt(x**2-y(x)**2))/_ + (x*y(x)*sqrt(x**2-y(x)**2)+x) +--R +--R +------------+ +------------+ +--R | 2 2 , 2 | 2 2 +--R (x y(x)\|- y(x) + x + x)y (x) + x \|- y(x) + x - y(x) +--R +--R (27) ----------------------------------------------------------- +--R +------------+ +--R | 2 2 +--R x y(x)\|- y(x) + x + x +--R Type: Expression Integer +--E 27 + +--S 28 of 123 +ode62a:=solve(ode62,y,x) +--R +--R (28) "failed" +--R Type: Union("failed",...) +--E 28 + +--S 29 of 123 +ode63 := D(y(x),x) - (1+ y(x)**2)/(abs(y(x)+sqrt(1+y(x)))*sqrt(1+x)**3) +--R +--R +-----+ , +--------+ 2 +--R (x + 1)\|x + 1 y (x)abs(\|y(x) + 1 + y(x)) - y(x) - 1 +--R +--R (29) ------------------------------------------------------- +--R +-----+ +--------+ +--R (x + 1)\|x + 1 abs(\|y(x) + 1 + y(x)) +--R Type: Expression Integer +--E 29 + +--S 30 of 123 +ode63a:=solve(ode63,y,x) +--R +--R (30) "failed" +--R Type: Union("failed",...) +--E 30 + +--S 31 of 123 +ode64 := D(y(x),x) - sqrt((a*y(x)**2+b*y(x)+c)/(a*x**2+b*x+c)) +--R +--R +--------------------+ +--R | 2 +--R , |a y(x) + b y(x) + c +--R (31) y (x) - |-------------------- +--R | 2 +--R \| a x + b x + c +--R Type: Expression Integer +--E 31 + +--S 32 of 123 +yx:=solve(ode64,y,x) +--R +--R (32) +--R log +--R +--------------------+ +--R | 2 +--R 2 2 +-+ |a y(x) + b y(x) + c +--R (2a x + 2a b x + 2a c)\|a |-------------------- +--R | 2 +--R \| a x + b x + c +--R * +--R +--------------------+ +--R | 2 +--R \|a y(x) + b y(x) + c +--R + +--R 3 3 2 2 2 2 +--R (- 2a x - 2a b x - 2a c x)y(x) +--R + +--R 2 3 2 2 2 3 2 +--R (- 2a b x - 2a b x - 2a b c x)y(x) - 2a c x - 2a b c x +--R + +--R 2 +--R - 2a c x +--R * +--R +-------------------------+ +--R | 2 2 +--R \|a c y(x) + b c y(x) + c +--R + +--R 3 4 2 3 2 2 2 3 2 +--R (- a x - a b x - 2a c x - a b c x - a c - a )y(x) +--R + +--R 2 4 2 3 2 2 2 2 +--R (- a b x - a b x - 2a b c x - b c x - b c - a b)y(x) +--R + +--R 2 4 3 2 2 2 3 2 +--R - a c x - a b c x - 2a c x - b c x - c - a c +--R * +--R +--------------------+ +--R +-+ | 2 +--R \|a \|a y(x) + b y(x) + c +--R + +--R 4 3 3 2 3 2 +--R (2a x + 2a b x + 2a c x)y(x) +--R + +--R 3 3 2 2 2 2 3 3 2 2 +--R (2a b x + 2a b x + 2a b c x)y(x) + 2a c x + 2a b c x +--R + +--R 2 2 +--R 2a c x +--R * +--R +--------------------+ +--R | 2 +--R |a y(x) + b y(x) + c +--R |-------------------- +--R | 2 +--R \| a x + b x + c +--R / +--R +--------------------+ +--R | 2 +--R 2 2 |a y(x) + b y(x) + c +--R (2a x + 2a b x + 2a c) |-------------------- +--R | 2 +--R \| a x + b x + c +--R * +--R +-------------------------+ +--R | 2 2 +--R \|a c y(x) + b c y(x) + c +--R + +--R 3 4 2 3 2 3 2 +--R (a x + a b x - a b c x - a c - a )y(x) +--R + +--R 2 4 2 3 2 2 2 2 4 3 +--R (a b x + a b x - b c x - b c - a b)y(x) + a c x + a b c x +--R + +--R 2 3 2 +--R - b c x - c - a c +--R + +--R log +--R +--------------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a y(x))\|a y(x) + b y(x) + c + 2a y(x)\|c +--R + +--R 2 +-+ +--R (- 2a y(x) - b y(x) - 2c)\|a +--R / +--R +--------------------+ +--R +-+ | 2 +--R 2\|c \|a y(x) + b y(x) + c - b y(x) - 2c +--R / +--R +-+ +--R \|a +--R Type: Union(Expression Integer,...) +--E 32 + +@ +The results of this substitution are too long to include. +It should be zero but Axiom cannot simplify it. +<<*>>= +--S 33 of 123 +ode64expr := D(yx,x) - sqrt((a*yx**2+b*yx+c)/(a*x**2+b*x+c)); +--E 33 + +--S 34 of 123 +ode65 := D(y(x),x) - sqrt((y(x)**3+1)/(x**3+1)) +--R +--R +---------+ +--R | 3 +--R , |y(x) + 1 +--R (34) y (x) - |--------- +--R | 3 +--R \| x + 1 +--R Type: Expression Integer +--E 34 + +--S 35 of 123 +ode65a:=solve(ode65,y,x) +--R +--R +---------+ +--R | 3 +--R |y(x) + 1 +--R |--------- +--R x | 3 y(x) +--I ++ \| %P + 1 ++ 1 +--I (35) | - ------------ d%P + | ---------- d%P +--R ++ +---------+ ++ +-------+ +--R | 3 | 3 +--I \|y(x) + 1 \|%P + 1 +--R Type: Union(Expression Integer,...) +--E 35 + +--S 36 of 123 +ode66 := D(y(x),x) - sqrt(abs(y(x)*(1-y(x))*(1-a*y(x))))/_ + sqrt(abs(x*(1-x)*(1-a*x))) +--R +--R (36) +--R +------------------------------------+ +--R | 3 2 +--R - \|abs(a y(x) + (- a - 1)y(x) + y(x)) +--R + +--R +---------------------------+ +--R | 3 2 , +--R \|abs(a x + (- a - 1)x + x) y (x) +--R +--R / +--R +---------------------------+ +--R | 3 2 +--R \|abs(a x + (- a - 1)x + x) +--R Type: Expression Integer +--E 36 + +--S 37 of 123 +ode66a:=solve(ode66,y,x) +--R +--R (37) "failed" +--R Type: Union("failed",...) +--E 37 + +--S 38 of 123 +ode67 := D(y(x),x) - sqrt(1-y(x)**4)/sqrt(1-x**4) +--R +--R +--------+ +-----------+ +--R | 4 , | 4 +--R \|- x + 1 y (x) - \|- y(x) + 1 +--R +--R (38) --------------------------------- +--R +--------+ +--R | 4 +--R \|- x + 1 +--R Type: Expression Integer +--E 38 + +--S 39 of 123 +ode67a:=solve(ode67,y,x) +--R +--R (39) "failed" +--R Type: Union("failed",...) +--E 39 + +--S 40 of 123 +ode68 := D(y(x),x) - sqrt((a*y(x)**4+b*y(x)**2+1)/(a*x**4+b*x**2+1)) +--R +--R +---------------------+ +--R | 4 2 +--R , |a y(x) + b y(x) + 1 +--R (40) y (x) - |--------------------- +--R | 4 2 +--R \| a x + b x + 1 +--R Type: Expression Integer +--E 40 + +--S 41 of 123 +ode68a:=solve(ode68,y,x) +--R +--R +--R (41) +--R +---------------------+ +--R | 4 2 +--R |a y(x) + b y(x) + 1 +--R |--------------------- +--R x | 2 4 y(x) +--I ++ \| %N b + %N a + 1 ++ 1 +--I | - ------------------------ d%N + | ------------------ d%N +--R ++ +---------------------+ ++ +---------------+ +--R | 4 2 | 2 4 +--I \|a y(x) + b y(x) + 1 \|%N b + %N a + 1 +--R Type: Union(Expression Integer,...) +--E 41 + +--S 42 of 123 +ode69 := D(y(x),x) - sqrt((b4*y(x)**4+b3*y(x)**3+b2*y(x)**2+b1*y(x)+b0)*_ + (a4*x**4+a3*x**3+a2*x**2+a1*x+a0)) +--R +--R +--R (42) +--R , +--R y (x) +--R +--R + +--R - +--R ROOT +--R 4 3 2 4 +--R (a4 b4 x + a3 b4 x + a2 b4 x + a1 b4 x + a0 b4)y(x) +--R + +--R 4 3 2 3 +--R (a4 b3 x + a3 b3 x + a2 b3 x + a1 b3 x + a0 b3)y(x) +--R + +--R 4 3 2 2 +--R (a4 b2 x + a3 b2 x + a2 b2 x + a1 b2 x + a0 b2)y(x) +--R + +--R 4 3 2 4 +--R (a4 b1 x + a3 b1 x + a2 b1 x + a1 b1 x + a0 b1)y(x) + a4 b0 x +--R + +--R 3 2 +--R a3 b0 x + a2 b0 x + a1 b0 x + a0 b0 +--R Type: Expression Integer +--E 42 + +--S 43 of 123 +ode69a:=solve(ode69,y,x) +--R +--R +--R >> Error detected within library code: +--R PFO::possibleOrder: more than 1 algebraic constant +--R +--R Continuing to read the file... +--R +--E 43 + +--S 44 of 123 +ode70 := D(y(x),x) - sqrt((a4*x**4+a3*x**3+a2*x**2+a1*x+a0)/_ + (b4*y(x)**4+b3*y(x)**3+b2*y(x)**2+b1*y(x)+b0)) +--R +--R +---------------------------------------------+ +--R | 4 3 2 +--R , | a4 x + a3 x + a2 x + a1 x + a0 +--R (43) y (x) - |--------------------------------------------- +--R | 4 3 2 +--R \|b4 y(x) + b3 y(x) + b2 y(x) + b1 y(x) + b0 +--R Type: Expression Integer +--E 44 + +--S 45 of 123 +ode70a:=solve(ode70,y,x) +--R +--R +--R >> Error detected within library code: +--R PFO::possibleOrder: more than 1 algebraic constant +--R +--R Continuing to read the file... +--R +--E 45 + +--S 46 of 123 +ode71 := D(y(x),x) - sqrt((b4*y(x)**4+b3*y(x)**3+b2*y(x)**2+b1*y(x)+b0)/_ + (a4*x**4+a3*x**3+a2*x**2+a1*x+a0)) +--R +--R +---------------------------------------------+ +--R | 4 3 2 +--R , |b4 y(x) + b3 y(x) + b2 y(x) + b1 y(x) + b0 +--R (44) y (x) - |--------------------------------------------- +--R | 4 3 2 +--R \| a4 x + a3 x + a2 x + a1 x + a0 +--R Type: Expression Integer +--E 46 + +--S 47 of 123 +ode71a:=solve(ode71,y,x) +--R +--R +--R (45) +--R +---------------------------------------------+ +--R | 4 3 2 +--R |b4 y(x) + b3 y(x) + b2 y(x) + b1 y(x) + b0 +--R |--------------------------------------------- +--R x | 4 3 2 +--I ++ \| %N a4 + %N a3 + %N a2 + %N a1 + a0 +--I | - ------------------------------------------------ d%N +--R ++ +---------------------------------------------+ +--R | 4 3 2 +--R \|b4 y(x) + b3 y(x) + b2 y(x) + b1 y(x) + b0 +--R + +--R y(x) +--R ++ 1 +--I | ------------------------------------- d%N +--R ++ +----------------------------------+ +--R | 4 3 2 +--I \|%N b4 + %N b3 + %N b2 + %N b1 + b0 +--R Type: Union(Expression Integer,...) +--E 47 + +--S 48 of 123 +R1:=operator 'R1 +--R +--R (46) R1 +--R Type: BasicOperator +--E 48 + +--S 49 of 123 +R2:=operator 'R2 +--R +--R (47) R2 +--R Type: BasicOperator +--E 49 + +--S 50 of 123 +ode72 := D(y(x),x) - R1(x,sqrt(a4*x**4+a3*x**3+a2*x**2+a1*x+a0))*_ + R2(y(x),sqrt(b4*y(x)**4+b3*y(x)**3+b2*y(x)**2+b1*y(x)+b0)) +--R +--R (48) +--R - +--R +---------------------------------+ +--R | 4 3 2 +--R R1(x,\|a4 x + a3 x + a2 x + a1 x + a0 ) +--R * +--R +---------------------------------------------+ +--R | 4 3 2 +--R R2(y(x),\|b4 y(x) + b3 y(x) + b2 y(x) + b1 y(x) + b0 ) +--R + +--R , +--R y (x) +--R +--R Type: Expression Integer +--E 50 + +--S 51 of 123 +ode72a:=solve(ode72,y,x) +--R +--R >> Error detected within library code: +--R Function not supported by Risch d.e. +--R +--R Continuing to read the file... +--R +--E 51 + +--S 52 of 123 +ode73 := D(y(x),x) - ((a3*x**3+a2*x**2+a1*x+a0)/_ + (a3*y(x)**3+a2*y(x)**2+a1*y(x)+a0))**(2/3) +--R +--R +----------------------------------+2 +--R | 3 2 +--R , | a3 x + a2 x + a1 x + a0 +--R (49) y (x) - |---------------------------------- +--R 3| 3 2 +--R \|a3 y(x) + a2 y(x) + a1 y(x) + a0 +--R Type: Expression Integer +--E 52 + +@ +Attempting to solve this problem fails with the error: +\begin{verbatim} +ode73a:=solve(ode73,y,x) + >> Error detected within library code: + Table construction failed in MLIFT +\end{verbatim} +<<*>>= + +--S 53 of 123 +ode74 := D(y(x),x) - f(x)*(y(x)-g(x))*sqrt((y(x)-a)*(y(x)-b)) +--R +--R +---------------------------+ +--R , | 2 +--R (50) y (x) + (- f(x)y(x) + f(x)g(x))\|y(x) + (- b - a)y(x) + a b +--R +--R Type: Expression Integer +--E 53 + +--S 54 of 123 +ode74a:=solve(ode74,y,x) +--R +--R (51) "failed" +--R Type: Union("failed",...) +--E 54 + +--S 55 of 123 +ode75 := D(y(x),x) - exp(x-y(x)) + exp(x) +--R +--R , - y(x) + x x +--R (52) y (x) - %e + %e +--R +--R Type: Expression Integer +--E 55 + +--S 56 of 123 +ode75a:=solve(ode75,y,x) +--R +--R (53) "failed" +--R Type: Union("failed",...) +--E 56 + +--S 57 of 123 +ode76 := D(y(x),x) - a*cos(y(x)) + b +--R +--R , +--R (54) y (x) - a cos(y(x)) + b +--R +--R Type: Expression Integer +--E 57 + +--S 58 of 123 +yx:=solve(ode76,y,x) +--R +--R (55) +--R +---------+ +---------+ +--R 2 2 | 2 2 | 2 2 +--R (- b + a )sin(y(x)) + b\|- b + a cos(y(x)) - a\|- b + a +--R log(-------------------------------------------------------------) +--R a cos(y(x)) - b +--R + +--R +---------+ +--R | 2 2 +--R x\|- b + a +--R / +--R +---------+ +--R | 2 2 +--R \|- b + a +--R Type: Union(Expression Integer,...) +--E 58 + +--S 59 of 123 +ode76expr := D(yx,x) - a*cos(yx) + b +--R +--R (56) +--R 2 2 4 3 3 +--R ((- a b + a )cos(y(x)) + a b - a b)sin(y(x)) +--R + +--R +---------+ +---------+ +--R 2 | 2 2 2 2 3 | 2 2 +--R a b\|- b + a cos(y(x)) + (- a b - a )\|- b + a cos(y(x)) +--R + +--R +---------+ +--R 2 | 2 2 +--R a b\|- b + a +--R * +--R cos +--R log +--R +---------+ +---------+ +--R 2 2 | 2 2 | 2 2 +--R (- b + a )sin(y(x)) + b\|- b + a cos(y(x)) - a\|- b + a +--R ------------------------------------------------------------- +--R a cos(y(x)) - b +--R + +--R +---------+ +--R | 2 2 +--R x\|- b + a +--R / +--R +---------+ +--R | 2 2 +--R \|- b + a +--R + +--R +---------+ +--R | 2 2 2 2 2 +--R - a\|- b + a sin(y(x)) + (- b + a )sin(y(x)) +--R + +--R +---------+ +---------+ +--R | 2 2 2 | 2 2 +--R - a\|- b + a cos(y(x)) + b\|- b + a cos(y(x)) +--R * +--R , +--R y (x) +--R +--R + +--R 3 2 3 3 4 3 2 2 2 +--R ((a b + a b - a b - a )cos(y(x)) - b - b + a b + a b)sin(y(x)) +--R + +--R +---------+ +--R 2 | 2 2 2 +--R (- a b - a b)\|- b + a cos(y(x)) +--R + +--R +---------+ +---------+ +--R 3 2 2 2 | 2 2 2 | 2 2 +--R (b + b + a b + a )\|- b + a cos(y(x)) + (- a b - a b)\|- b + a +--R / +--R +---------+ +--R 2 3 3 2 | 2 2 2 +--R ((a b - a )cos(y(x)) - b + a b)sin(y(x)) - a b\|- b + a cos(y(x)) +--R + +--R +---------+ +---------+ +--R 2 2 | 2 2 | 2 2 +--R (b + a )\|- b + a cos(y(x)) - a b\|- b + a +--R Type: Expression Integer +--E 59 + +--S 60 of 123 +ode77 := D(y(x),x) - cos(a*y(x)+b*x) +--R +--R , +--R (57) y (x) - cos(a y(x) + b x) +--R +--R Type: Expression Integer +--E 60 + +--S 61 of 123 +ode77a:=solve(ode77,y,x) +--R +--R (58) "failed" +--R Type: Union("failed",...) +--E 61 + +--S 62 of 123 +ode78 := D(y(x),x) + a*sin(alpha*y(x)+beta*x) + b +--R +--R , +--R (59) y (x) + a sin(alpha y(x) + beta x) + b +--R +--R Type: Expression Integer +--E 62 + +--S 63 of 123 +ode78a:=solve(ode78,y,x) +--R +--R (60) "failed" +--R Type: Union("failed",...) +--E 63 + +--S 64 of 123 +ode79 := D(y(x),x) + f(x)*cos(a*y(x)) + g(x)*sin(a*y(x)) + h(x) +--R +--R , +--R (61) y (x) + g(x)sin(a y(x)) + f(x)cos(a y(x)) + h(x) +--R +--R Type: Expression Integer +--E 64 + +--S 65 of 123 +ode79a:=solve(ode79,y,x) +--R +--R (62) "failed" +--R Type: Union("failed",...) +--E 65 + +--S 66 of 123 +ode80 := D(y(x),x) + f(x)*sin(y(x)) + (1-D(f(x),x))*cos(y(x)) - D(f(x),x) - 1 +--R +--R , , +--R (63) y (x) + (- cos(y(x)) - 1)f (x) + f(x)sin(y(x)) + cos(y(x)) - 1 +--R +--R Type: Expression Integer +--E 66 + +--S 67 of 123 +ode80a:=solve(ode80,y,x) +--R +--R (64) "failed" +--R Type: Union("failed",...) +--E 67 + +--S 68 of 123 +ode81 := D(y(x),x) + 2*tan(y(x))*tan(x) - 1 +--R +--R , +--R (65) y (x) + 2tan(x)tan(y(x)) - 1 +--R +--R Type: Expression Integer +--E 68 + +--S 69 of 123 +ode81a:=solve(ode81,y,x) +--R +--R (66) "failed" +--R Type: Union("failed",...) +--E 69 + +--S 70 of 123 +ode82 := D(y(x),x) - a*(1+tan(y(x))**2) + tan(y(x))*tan(x) +--R +--R , 2 +--R (67) y (x) - a tan(y(x)) + tan(x)tan(y(x)) - a +--R +--R Type: Expression Integer +--E 70 + +--S 71 of 123 +ode82a:=solve(ode82,y,x) +--R +--R (68) "failed" +--R Type: Union("failed",...) +--E 71 + +--S 72 of 123 +ode83 := D(y(x),x) - tan(x*y(x)) +--R +--R , +--R (69) y (x) - tan(x y(x)) +--R +--R Type: Expression Integer +--E 72 + +--S 73 of 123 +ode83a:=solve(ode83,y,x) +--R +--R (70) "failed" +--R Type: Union("failed",...) +--E 73 + +--S 74 of 123 +ode84 := D(y(x),x) - f(a*x + b*y(x)) +--R +--R , +--R (71) y (x) - f(b y(x) + a x) +--R +--R Type: Expression Integer +--E 74 + +--S 75 of 123 +ode84a:=solve(ode84,y,x) +--R +--R (72) "failed" +--R Type: Union("failed",...) +--E 75 + +--S 76 of 123 +ode85 := D(y(x),x) - x**(a-1)*y(x)**(1-b)*f(x**a/a + y(x)**b/b) +--R +--R b a +--R a - 1 - b + 1 a y(x) + b x , +--R (73) - x y(x) f(--------------) + y (x) +--R a b +--R Type: Expression Integer +--E 76 + +--S 77 of 123 +ode85a:=solve(ode85,y,x) +--R +--R (74) "failed" +--R Type: Union("failed",...) +--E 77 + +--S 78 of 123 +ode86 := D(y(x),x) - (y(x)-x*f(x**2+a*y(x)**2))/(x+a*y(x)*f(x**2+a*y(x)**2)) +--R +--R 2 2 , 2 2 +--R (a y(x)f(a y(x) + x ) + x)y (x) + x f(a y(x) + x ) - y(x) +--R +--R (75) ----------------------------------------------------------- +--R 2 2 +--R a y(x)f(a y(x) + x ) + x +--R Type: Expression Integer +--E 78 + +--S 79 of 123 +ode86a:=solve(ode86,y,x) +--R +--R (76) "failed" +--R Type: Union("failed",...) +--E 79 + +--S 80 of 123 +ode87 := D(y(x),x) - (y(x)*a*f(x**c*y(x))+c*x**a*y(x)**b)/_ + (x*b*f(x**c*y(x))-x**a*y(x)**b) +--R +--R a b c , a b c +--R (x y(x) - b x f(y(x)x ))y (x) + c x y(x) + a y(x)f(y(x)x ) +--R +--R (77) ------------------------------------------------------------ +--R a b c +--R x y(x) - b x f(y(x)x ) +--R Type: Expression Integer +--E 80 + +--S 81 of 123 +ode87a:=solve(ode87,y,x) +--R +--R (78) "failed" +--R Type: Union("failed",...) +--E 81 + +--S 82 of 123 +ode88 := 2*D(y(x),x) - 3*y(x)**2 - 4*a*y(x) - b - c*exp(-2*a*x) +--R +--R , - 2a x 2 +--R (79) 2y (x) - c %e - 3y(x) - 4a y(x) - b +--R +--R Type: Expression Integer +--E 82 + +--S 83 of 123 +ode88a:=solve(ode88,y,x) +--R +--R (80) "failed" +--R Type: Union("failed",...) +--E 83 + +--S 84 of 123 +ode89 := x*D(y(x),x) - sqrt(a**2 - x**2) +--R +--R +---------+ +--R , | 2 2 +--R (81) xy (x) - \|- x + a +--R +--R Type: Expression Integer +--E 84 + +--S 85 of 123 +ode89a:=solve(ode89,y,x) +--R +--R (82) +--R +---------+ +--R +---------+ | 2 2 +--R | 2 2 2 \|- x + a - a 2 +--R (a\|- x + a - a )log(----------------) - x +--R x +--R [particular= ----------------------------------------------,basis= [1]] +--R +---------+ +--R | 2 2 +--R \|- x + a - a +--RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) +--E 85 + +--S 86 of 123 +yx:=ode89a.particular +--R +--R +---------+ +--R +---------+ | 2 2 +--R | 2 2 2 \|- x + a - a 2 +--R (a\|- x + a - a )log(----------------) - x +--R x +--R (83) ---------------------------------------------- +--R +---------+ +--R | 2 2 +--R \|- x + a - a +--R Type: Expression Integer +--E 86 + +--S 87 of 123 +ode89expr := x*D(yx,x) - sqrt(a**2 - x**2) +--R +--R (84) 0 +--R Type: Expression Integer +--E 87 + +--S 88 of 123 +ode90 := x*D(y(x),x) + y(x) - x*sin(x) +--R +--R , +--R (85) xy (x) - x sin(x) + y(x) +--R +--R Type: Expression Integer +--E 88 + +--S 89 of 123 +ode90a:=solve(ode90,y,x) +--R +--R sin(x) - x cos(x) 1 +--R (86) [particular= -----------------,basis= [-]] +--R x x +--RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) +--E 89 + +--S 90 of 123 +yx:=ode90a.particular +--R +--R sin(x) - x cos(x) +--R (87) ----------------- +--R x +--R Type: Expression Integer +--E 90 + +--S 91 of 123 +ode90expr := x*D(yx,x) + yx - x*sin(x) +--R +--R (88) 0 +--R Type: Expression Integer +--E 91 + +--S 92 of 123 +ode91 := x*D(y(x),x) - y(x) - x/log(x) +--R +--R , +--R x log(x)y (x) - y(x)log(x) - x +--R +--R (89) ------------------------------ +--R log(x) +--R Type: Expression Integer +--E 92 + +--S 93 of 123 +ode91a:=solve(ode91,y,x) +--R +--R (90) [particular= x log(log(x)),basis= [x]] +--RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) +--E 93 + +--S 94 of 123 +yx:=ode91a.particular +--R +--R (91) x log(log(x)) +--R Type: Expression Integer +--E 94 + +--S 95 of 123 +ode91expr := x*D(yx,x) - yx - x/log(x) +--R +--R (92) 0 +--R Type: Expression Integer +--E 95 + +--S 96 of 123 +ode92 := x*D(y(x),x) - y(x) - x**2*sin(x) +--R +--R , 2 +--R (93) xy (x) - x sin(x) - y(x) +--R +--R Type: Expression Integer +--E 96 + +--S 97 of 123 +ode92a:=solve(ode92,y,x) +--R +--R (94) [particular= - x cos(x),basis= [x]] +--RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) +--E 97 + +--S 98 of 123 +yx:=ode92a.particular +--R +--R (95) - x cos(x) +--R Type: Expression Integer +--E 98 + +--S 99 of 123 +ode92expr := x*D(yx,x) - yx - x**2*sin(x) +--R +--R (96) 0 +--R Type: Expression Integer +--E 99 + + +--S 100 of 123 +ode93 := x*D(y(x),x) - y(x) - x*cos(log(log(x)))/log(x) +--R +--R , +--R - x cos(log(log(x))) + x log(x)y (x) - y(x)log(x) +--R +--R (97) ------------------------------------------------- +--R log(x) +--R Type: Expression Integer +--E 100 + +--S 101 of 123 +ode93a:=solve(ode93,y,x) +--R +--R (98) [particular= x sin(log(log(x))),basis= [x]] +--RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) +--E 101 + +--S 102 of 123 +yx:=ode93a.particular +--R +--R (99) x sin(log(log(x))) +--R Type: Expression Integer +--E 102 + +--S 103 of 123 +ode93 := x*D(yx,x) - yx - x*cos(log(log(x)))/log(x) +--R +--R (100) 0 +--R Type: Expression Integer +--E 103 + +--S 104 of 123 +ode94 := x*D(y(x),x) +a*y(x) + b*x**n +--R +--R , n +--R (101) xy (x) + b x + a y(x) +--R +--R Type: Expression Integer +--E 104 + +--S 105 of 123 +ode94a:=solve(ode94,y,x) +--R +--R n log(x) +--R b %e - a log(x) +--R (102) [particular= - ------------,basis= [%e ]] +--R n + a +--RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) +--E 105 + +--S 106 of 123 +yx:=ode94a.particular +--R +--R n log(x) +--R b %e +--R (103) - ------------ +--R n + a +--R Type: Expression Integer +--E 106 + +--S 107 of 123 +ode94expr := x*D(yx,x) +a*yx + b*x**n +--R +--R n log(x) n +--R (104) - b %e + b x +--R Type: Expression Integer +--E 107 + +--S 108 of 123 +exprule := rule x^n == %e^(n*log(x)) +--R +--R n n log(x) +--R (105) x == %e +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 108 + +--S 109 of 123 +exprule ode94expr +--R +--R (106) 0 +--R Type: Expression Integer +--E 109 + +--S 110 of 123 +ode95 := x*D(y(x),x) + y(x)**2 + x**2 +--R +--R , 2 2 +--R (107) xy (x) + y(x) + x +--R +--R Type: Expression Integer +--E 110 + +--S 111 of 123 +ode95a:=solve(ode95,y,x) +--R +--R +--R (108) "failed" +--R Type: Union("failed",...) +--E 111 + +--S 112 of 123 +ode96 := x*D(y(x),x) - y(x)**2 + 1 +--R +--R +--R , 2 +--R (109) xy (x) - y(x) + 1 +--R +--R Type: Expression Integer +--E 112 + +--S 113 of 123 +yx:=solve(ode96,y,x) +--R +--R - x y(x) - x +--R (110) ---------------------- +--R +--------+ +--------+ +--R \|y(x) - 1 \|y(x) + 1 +--R Type: Union(Expression Integer,...) +--E 113 + +--S 114 of 123 +ode96expr := x*D(yx,x) - yx**2 + 1 +--R +--R (111) +--R 2 , 2 2 +--------+ +--------+ 2 +--R x y (x) + ((- x + 1)y(x) - x - 1)\|y(x) - 1 \|y(x) + 1 - x y(x) + x +--R +--R ----------------------------------------------------------------------- +--R +--------+ +--------+ +--R (y(x) - 1)\|y(x) - 1 \|y(x) + 1 +--R Type: Expression Integer +--E 114 + +--S 115 of 123 +ode97 := x*D(y(x),x) + a*y(x)**2 - y(x) + b*x**2 +--R +--R +--R , 2 2 +--R (112) xy (x) + a y(x) - y(x) + b x +--R +--R Type: Expression Integer +--E 115 + +--S 116 of 123 +yx:=solve(ode97,y,x) +--R +--R +--R +------+ +--R a y(x)\|- 4a b - 2a b x +--R (113) --------------------------------------- +--R +-----+ +--R +------+ 2x\|- a b +--R (2b x\|- 4a b - 4a b y(x))%e +--R Type: Union(Expression Integer,...) +--E 116 + +--S 117 of 123 +ode97expr := x*D(yx,x) + a*yx**2 - yx + b*x**2 +--R +--R (114) +--R +-----+ +--R 3 +------+ 2 2 2x\|- a b , +--R (- 4a b x \|- 4a b + 8a b x y(x))%e y (x) +--R +--R + +--R 2 3 2 3 5 +------+ 2 2 2 3 3 4 +--R ((12a b x y(x) - 4b x )\|- 4a b - 8a b x y(x) + 24a b x y(x)) +--R * +--R +-----+ 2 +--R 2x\|- a b +--R (%e ) +--R + +--R 2 3 3 +------+ 2 2 2 2 4 +--R ((- 4a x y(x) - 4a b x y(x))\|- 4a b - 8a b x y(x) - 8a b x ) +--R * +--R +-----+ +--R \|- a b +--R + +--R 2 3 2 +------+ 2 2 2 3 +--R (- 2a y(x) + 2a b x y(x))\|- 4a b - 12a b x y(x) - 4a b x +--R * +--R +-----+ +--R 2x\|- a b +--R %e +--R + +--R 3 2 2 3 +------+ 4 3 3 2 +--R (a x y(x) + a b x )\|- 4a b + 2a y(x) + 2a b x y(x) +--R / +--R 2 2 3 +------+ 2 3 2 2 +--R ((12a b x y(x) - 4b x )\|- 4a b - 8a b y(x) + 24a b x y(x)) +--R * +--R +-----+ 2 +--R 2x\|- a b +--R (%e ) +--R Type: Expression Integer +--E 117 + +--S 118 of 123 +ode98 := x*D(y(x),x) + a*y(x)**2 - b*y(x) + c*x**(2*b) +--R +--R +--R , 2b 2 +--R (115) xy (x) + c x + a y(x) - b y(x) +--R +--R Type: Expression Integer +--E 118 + +--S 119 of 123 +ode98a:=solve(ode98,y,x) +--R +--R (116) "failed" +--R Type: Union("failed",...) +--E 119 + +--S 120 of 123 +ode99 := x*D(y(x),x) + a*y(x)**2 - b*y(x) - c*x**beta +--R +--R +--R , beta 2 +--R (117) xy (x) - c x + a y(x) - b y(x) +--R +--R Type: Expression Integer +--E 120 + +--S 121 of 123 +ode99a:=solve(ode99,y,x) +--R +--R +--R (118) "failed" +--R Type: Union("failed",...) +--E 121 + +--S 122 of 123 +ode100 := x*D(y(x),x) + x*y(x)**2 + a +--R +--R +--R , 2 +--R (119) xy (x) + x y(x) + a +--R +--R Type: Expression Integer +--E 122 + +--S 123 of 123 +ode100a:=solve(ode100,y,x) +--R +--R +--R (120) "failed" +--R Type: Union("failed",...) +--E 123 +)spool +)lisp (bye) + +@ +\eject +\begin{thebibliography}{99} +\bibitem{1} {\bf http://www.cs.uwaterloo.ca/$\tilde{}$ecterrab/odetools.html} +\end{thebibliography} +\end{document}