diff --git a/changelog b/changelog index 0f1d325..e60b026 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,6 @@ +20091028 tpd src/axiom-website/patches.html 20091028.02.tpd.patch +20091028 tpd src/input/Makefile add nonlinhomodiffeq.input +20091028 tpd src/input/nonlinhomodiffeq.input added 20091028 tpd src/axiom-website/patches.html 20091028.01.rhx.patch 20091028 tpd src/input/Makefile add cachedf.input 20091028 rhx src/input/cachedf.input added diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html index 41bde76..23e026a 100644 --- a/src/axiom-website/patches.html +++ b/src/axiom-website/patches.html @@ -2181,5 +2181,7 @@ slam, buildom, interp-proclaims remove vmlisp
src/interp/vmlisp migrate functions to boot package
20091028.01.rhx.patch src/input/cachedf.input added
+20091028.02.tpd.patch +src/input/nonlinhomodiffeq.input added
diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet index 0d879e9..08d52e5 100644 --- a/src/input/Makefile.pamphlet +++ b/src/input/Makefile.pamphlet @@ -349,7 +349,8 @@ REGRES= algaggr.regress algbrbf.regress algfacob.regress alist.regress \ mfinfact.regress mkfunc.regress mpoly.regress mset2.regress \ mset.regress multfact.regress multiple.regress ndftip.regress \ negfloats.regress nepip.regress newlodo.regress newton.regress \ - nlode.regress none.regress noonburg.regress noptip.regress \ + nlode.regress nonlinhomodiffeq.regress \ + none.regress noonburg.regress noptip.regress \ nqip.regress nsfip.regress numbers.regress octonion.regress \ oct.regress ode.regress odpol.regress op1.regress \ opalg.regress operator.regress op.regress ovar.regress \ @@ -643,6 +644,7 @@ FILES= ${OUT}/algaggr.input ${OUT}/algbrbf.input ${OUT}/algfacob.input \ ${OUT}/ndftip.input ${OUT}/newlodo.input \ ${OUT}/negfloats.input \ ${OUT}/nepip.input ${OUT}/newton.input \ + ${OUT}/nonlinhomodiffeq.input \ ${OUT}/nlode.input ${OUT}/none.input ${OUT}/noonburg.input \ ${OUT}/noptip.input ${OUT}/nqip.input ${OUT}/nsfip.input \ ${OUT}/ntube.input ${OUT}/oct.input ${OUT}/ode.input \ @@ -964,7 +966,8 @@ DOCFILES= \ ${DOC}/ndftip.input.dvi ${DOC}/negfloats.input.dvi \ ${DOC}/nepip.input.dvi ${DOC}/newlodo.input.dvi \ ${DOC}/newton.input.dvi ${DOC}/nlode.input.dvi \ - ${DOC}/none.input.dvi ${DOC}/noonburg.input.dvi \ + ${DOC}/none.input.dvi ${DOC}/nonlinhomodiffeq.input.dvi \ + ${DOC}/noonburg.input.dvi \ ${DOC}/noptip.input.dvi ${DOC}/nqip.input.dvi \ ${DOC}/nsfip.input.dvi ${DOC}/ntube.input.dvi \ ${DOC}/numbers.input.dvi ${DOC}/oct.input.dvi \ diff --git a/src/input/nonlinhomodiffeq.input.pamphlet b/src/input/nonlinhomodiffeq.input.pamphlet new file mode 100644 index 0000000..eda7437 --- /dev/null +++ b/src/input/nonlinhomodiffeq.input.pamphlet @@ -0,0 +1,839 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/src/input nonlinhomodiffeq.input} +\author{jv1} +\maketitle +\begin{abstract} +\end{abstract} +\eject +\tableofcontents +\eject +\begin{chunk}{*} +)set break resume +)sys rm -f nonlinhomodiffeq.output +)spool nonlinhomodiffeq.output +)set message test on +)set message auto off +)clear all + +\end{chunk} +Test 1: solve a simple nonlinear homogeneous differential equation +\begin{chunk}{*} +--S 1 of 59 +y := operator y +--R +--R +--R (1) y +--R Type: BasicOperator +--E 1 + +--S 2 of 59 +deq1 := D(y(x),x) = 9-y(x)^2 +--R +--R +--R , 2 +--R (2) y (x)= - y(x) + 9 +--R +--R Type: Equation Expression Integer +--E 2 + +--S 3 of 59 +solve(deq1,y,x) +--R +--R +--R - log(y(x) + 3) + log(y(x) - 3) + 6x +--R (3) ------------------------------------ +--R 6 +--R Type: Union(Expression Integer,...) +--E 3 + +\end{chunk} +Test 2: solve a class of simple nonlinear homogeneous differential equations +\begin{chunk}{*} +--S 4 of 59 +deq2a := D(y(x),x) = c - p*y(x)^2 +--R +--R +--R , 2 +--R (4) y (x)= - p y(x) + c +--R +--R Type: Equation Expression Integer +--E 4 + +--S 5 of 59 +xpr2b := solve(deq2a,y,x) +--R +--R +--R 2 +---+ +--R (p y(x) + c)\|c p - 2c p y(x) +---+ +--R log(-------------------------------) + 2x\|c p +--R 2 +--R p y(x) - c +--R (5) ----------------------------------------------- +--R +---+ +--R 2\|c p +--R Type: Union(Expression Integer,...) +--E 5 + +--S 6 of 59 +simplify(x-xpr2b) +--R +--R +--R 2 +---+ +--R (p y(x) + c)\|c p - 2c p y(x) +--R log(-------------------------------) +--R 2 +--R p y(x) - c +--R (6) - ------------------------------------ +--R +---+ +--R 2\|c p +--R Type: Expression Integer +--E 6 + +\end{chunk} +Test 3; find general solutions for nonlinear homogeneous differential eqautions +\begin{chunk}{*} +--S 7 of 59 +f := operator f +--R +--R +--R (7) f +--R Type: BasicOperator +--E 7 + +--S 8 of 59 +deq3 := D(y(x),x) = f(y(x)) +--R +--R +--R , +--R (8) y (x)= f(y(x)) +--R +--R Type: Equation Expression Integer +--E 8 + +--S 9 of 59 +solve(deq3,y,x) +--R +--R +--R y(x) +--R ++ 1 +--R (9) | ----- d%P - x +--R ++ f(%P) +--R Type: Union(Expression Integer,...) +--E 9 + +\end{chunk} +Test 4: integration +\begin{chunk}{*} +--S 10 of 59 +integrate(1/(1-x^2),x) +--R +--R +--R log(x + 1) - log(x - 1) +--R (10) ----------------------- +--R 2 +--R Type: Union(Expression Integer,...) +--E 10 + +\end{chunk} +Test 5: check result +\begin{chunk}{*} +--S 11 of 59 +xpr5 := (log(x+1)-log(x-1))/2 +--R +--R +--R log(x + 1) - log(x - 1) +--R (11) ----------------------- +--R 2 +--R Type: Expression Integer +--E 11 + +--S 12 of 59 +D(xpr5,x) +--R +--R +--R 1 +--R (12) - ------ +--R 2 +--R x - 1 +--R Type: Expression Integer +--E 12 + +\end{chunk} +Test 6: check simplification +\begin{chunk}{*} +--S 13 of 59 +xpr6 := log(1+2/(x-1))/2 +--R +--R +--R x + 1 +--R log(-----) +--R x - 1 +--R (13) ---------- +--R 2 +--R Type: Expression Integer +--E 13 + +--S 14 of 59 +D(xpr6,x) +--R +--R +--R 1 +--R (14) - ------ +--R 2 +--R x - 1 +--R Type: Expression Integer +--E 14 + +\end{chunk} +Test 7: express y(x) as a function of x (replacing y(x) with z) +\begin{chunk}{*} +--S 15 of 59 +eq7a := x = (log(z+1)-log(z-1))/2 +--R +--R +--R log(z + 1) - log(z - 1) +--R (15) x= ----------------------- +--R 2 +--R Type: Equation Expression Integer +--E 15 + +--S 16 of 59 +solve(eq7a,z) +--R +--R +--R - 2x +--R - %e - 1 +--R (16) [z= ------------] +--R - 2x +--R %e - 1 +--R Type: List Equation Expression Integer +--E 16 + +--S 17 of 59 +xpr7b := (1+exp(-2*z))/(1-exp(-2*z)) +--R +--R +--R - 2z +--R - %e - 1 +--R (17) ------------ +--R - 2z +--R %e - 1 +--R Type: Expression Integer +--E 17 + +--S 18 of 59 +simplify(xpr7b) +--R +--R +--R - 2z +--R - %e - 1 +--R (18) ------------ +--R - 2z +--R %e - 1 +--R Type: Expression Integer +--E 18 + +\end{chunk} +Test 8 check simplified result +\begin{chunk}{*} +--S 19 of 59 +xpr8 := (1+exp(-2*x))/(1-exp(-2*x)) - (1+2/(exp(2*x)-1)) +--R +--R +--R - 2x 2x +--R - 2%e %e + 2 +--R (19) ----------------------------- +--R - 2x 2x - 2x +--R (%e - 1)%e - %e + 1 +--R Type: Expression Integer +--E 19 + +--S 20 of 59 +simplify(xpr8) +--R +--R +--R (20) 0 +--R Type: Expression Integer +--E 20 + +\end{chunk} +Test 9 check result by substitution in the DEQ +\begin{chunk}{*} +--S 21 of 59 +xpr9a := (1+2/(exp(2*x)-1)) +--R +--R +--R 2x +--R %e + 1 +--R (21) -------- +--R 2x +--R %e - 1 +--R Type: Expression Integer +--E 21 + +--S 22 of 59 +xpr9b := D(xpr9a,x) +--R +--R +--R 2x +--R 4%e +--R (22) - ------------------- +--R 2x 2 2x +--R (%e ) - 2%e + 1 +--R Type: Expression Integer +--E 22 + +--S 23 of 59 +xpr9c := 1-(xpr9a)^2 +--R +--R +--R 2x +--R 4%e +--R (23) - ------------------- +--R 2x 2 2x +--R (%e ) - 2%e + 1 +--R Type: Expression Integer +--E 23 + +\end{chunk} +Test 10 finding the explicit solution for the deq in test 2 +\begin{chunk}{*} +--S 24 of 59 +wcp := sqrt(c*p) +--R +--R +--R +---+ +--R (24) \|c p +--R Type: Expression Integer +--E 24 + +--S 25 of 59 +xpr10a := log(((p*z^2+c)*wcp-2*c*p*z)/(p*z^2-c))/(2*wcp) +--R +--R +--R 2 +---+ +--R (p z + c)\|c p - 2c p z +--R log(-------------------------) +--R 2 +--R p z - c +--R (25) ------------------------------ +--R +---+ +--R 2\|c p +--R Type: Expression Integer +--E 25 + +--S 26 of 59 +solve(x = xpr10a,z) +--R +--R +--R (26) +--R [ +--R z = +--R +---+ 2 +---+ +--R 2x\|c p +---+ 2x\|c p +--R (- (%e ) + \|c p %e ) +--R * +--R ROOT +--R +---+ +--R 2x\|c p +---+ +--R c %e - c\|c p +--R / +--R +---+ 3 +---+ 2 +---+ +--R 2x\|c p +---+ 2x\|c p 2 2x\|c p +--R p (%e ) - 3p\|c p (%e ) + 3c p %e +--R + +--R 2 +---+ +--R - c p \|c p +--R + +--R - c +--R / +--R +---+ +--R 2x\|c p +---+ +--R %e - \|c p +--R , +--R +--R z = +--R +---+ 2 +---+ +--R 2x\|c p +---+ 2x\|c p +--R ((%e ) - \|c p %e ) +--R * +--R ROOT +--R +---+ +--R 2x\|c p +---+ +--R c %e - c\|c p +--R / +--R +---+ 3 +---+ 2 +---+ +--R 2x\|c p +---+ 2x\|c p 2 2x\|c p +--R p (%e ) - 3p\|c p (%e ) + 3c p %e +--R + +--R 2 +---+ +--R - c p \|c p +--R + +--R - c +--R / +--R +---+ +--R 2x\|c p +---+ +--R %e - \|c p +--R ] +--R Type: List Equation Expression Integer +--E 26 + +--S 27 of 59 +eq10c := (p*z^2+c)*wcp - 2*c*p*z +--R +--R +--R 2 +---+ +--R (27) (p z + c)\|c p - 2c p z +--R Type: Expression Integer +--E 27 + +--S 28 of 59 +solve(eq10c=0,z) +--R +--R +--R c c +--R (28) [z= ------,z= ------] +--R +---+ +---+ +--R \|c p \|c p +--R Type: List Equation Expression Integer +--E 28 + +--S 29 of 59 +xpr10d := p*wcp*(z-wcp/p)^2 +--R +--R +--R 2 +---+ +--R (29) (p z + c)\|c p - 2c p z +--R Type: Expression Integer +--E 29 + +--S 30 of 59 +simplify(xpr10d/eq10c) +--R +--R +--R (30) 1 +--R Type: Expression Integer +--E 30 + +--S 31 of 59 +xpr10e := (wcp/p)*(2/(1-exp(-2*wcp*x)/wcp)-1) +--R +--R +--R +---+ +--R +---+ - 2x\|c p +--R - \|c p %e - c p +--R (31) -------------------------- +--R +---+ +--R - 2x\|c p +---+ +--R p %e - p\|c p +--R Type: Expression Integer +--E 31 + +--S 32 of 59 +xpr10f := D(xpr10e,x) +--R +--R +--R +---+ +--R +---+ - 2x\|c p +--R 4c\|c p %e +--R (32) - ------------------------------------------- +--R +---+ 2 +---+ +--R - 2x\|c p +---+ - 2x\|c p +--R (%e ) - 2\|c p %e + c p +--R Type: Expression Integer +--E 32 + +--S 33 of 59 +xpr10g := c-p*(xpr10e)^2 +--R +--R +--R +---+ +--R +---+ - 2x\|c p +--R 4c\|c p %e +--R (33) - ------------------------------------------- +--R +---+ 2 +---+ +--R - 2x\|c p +---+ - 2x\|c p +--R (%e ) - 2\|c p %e + c p +--R Type: Expression Integer +--E 33 + +--S 34 of 59 +simplify(xpr10f/xpr10g) +--R +--R +--R (34) 1 +--R Type: Expression Integer +--E 34 + +\end{chunk} +Test 11 non-homogeneous generalization: c as a linear function of x +\begin{chunk}{*} +--S 35 of 59 +deq11a := D(y(x),x) = a*x+b-p*y(x)^2 +--R +--R +--R , 2 +--R (35) y (x)= - p y(x) + a x + b +--R +--R Type: Equation Expression Integer +--E 35 + +--S 36 of 59 +solve(deq11a,y,x) +--R +--R +--R (36) "failed" +--R Type: Union("failed",...) +--E 36 + +\end{chunk} +Test 12 non-homogeneous generalization: c as arbitrary function of x +\begin{chunk}{*} +--S 37 of 59 +c := operator c +--R +--R +--R (37) c +--R Type: BasicOperator +--E 37 + +--S 38 of 59 +deq12a := D(y(x),x) = c(x) - p*y(x)^2 +--R +--R +--R , 2 +--R (38) y (x)= - p y(x) + c(x) +--R +--R Type: Equation Expression Integer +--E 38 + +--S 39 of 59 +solve(deq12a,y,x) +--R +--R +--R (39) "failed" +--R Type: Union("failed",...) +--E 39 + +\end{chunk} +Test 13 guessing solution for deq12a +\begin{chunk}{*} +--S 40 of 59 +wpcx := sqrt(c(x))*sqrt(p) +--R +--R +--R +-+ +----+ +--R (40) \|p \|c(x) +--R Type: Expression Integer +--E 40 + +--S 41 of 59 +xpr13a := log((p*wpcx*(z-wpcx/p)^2)/(p*z^2-c(x)))/(2*wpcx) +--R +--R +--R 2 +-+ +----+ +--R (- c(x) - p z )\|p \|c(x) + 2p z c(x) +--R log(--------------------------------------) +--R 2 +--R c(x) - p z +--R (41) ------------------------------------------- +--R +-+ +----+ +--R 2\|p \|c(x) +--R Type: Expression Integer +--E 41 + +--S 42 of 59 +xpr13b := (wpcx/p)*(2/(1-exp(-2*wpcx*x)/wpcx)-1) +--R +--R +--R +-+ +----+ +--R +-+ +----+ - 2x\|p \|c(x) +--R - \|p \|c(x) %e - p c(x) +--R (42) --------------------------------------- +--R +-+ +----+ +--R - 2x\|p \|c(x) +-+ +----+ +--R p %e - p\|p \|c(x) +--R Type: Expression Integer +--E 42 + +--S 43 of 59 +xpr13c := D(xpr13b,x) +--R +--R +--R (43) +--R +-+ +----+ 2 +--R +-+ , - 2x\|p \|c(x) +--R - \|p c (x)(%e ) +--R +--R + +--R +-+ +----+ +--R +----+ +-+ , 2 +-+ - 2x\|p \|c(x) +--R ((- 2p\|c(x) - 4p x c(x)\|p )c (x) - 8p c(x) \|p )%e +--R +--R + +--R +-+ , +--R p c(x)\|p c (x) +--R +--R / +--R +-+ +----+ 2 +-+ +----+ +--R +----+ - 2x\|p \|c(x) +-+ - 2x\|p \|c(x) +--R 2p\|c(x) (%e ) - 4p c(x)\|p %e +--R + +--R 2 +----+ +--R 2p c(x)\|c(x) +--R Type: Expression Integer +--E 43 + +--S 44 of 59 +xpr13d := c(x) - p*(xpr13b)^2 +--R +--R +--R +-+ +----+ +--R +-+ +----+ - 2x\|p \|c(x) +--R 4c(x)\|p \|c(x) %e +--R (44) - ------------------------------------------------------------- +--R +-+ +----+ 2 +-+ +----+ +--R - 2x\|p \|c(x) +-+ +----+ - 2x\|p \|c(x) +--R (%e ) - 2\|p \|c(x) %e + p c(x) +--R Type: Expression Integer +--E 44 + +--S 45 of 59 +simplify(xpr13d/xpr13c) +--R +--R +--R (45) +--R +-+ +----+ +-+ +----+ +--R 2 3 +-+ - 2x\|p \|c(x) 2 2 +----+ - 4x\|p \|c(x) +--R 8p c(x) \|p %e - 16p c(x) \|c(x) %e +--R + +--R +-+ +----+ +--R 2 +-+ - 6x\|p \|c(x) +--R 8p c(x) \|p %e +--R / +--R +-+ +----+ +--R 2 +----+ 2 2 +-+ , 2 3 +-+ - 2x\|p \|c(x) +--R ((4p c(x)\|c(x) + 4p x c(x) \|p )c (x) + 8p c(x) \|p )%e +--R +--R + +--R 2 +----+ +-+ , 2 2 +----+ +--R ((- 8p x c(x)\|c(x) - 4p c(x)\|p )c (x) - 16p c(x) \|c(x) ) +--R +--R * +--R +-+ +----+ +--R - 4x\|p \|c(x) +--R %e +--R + +--R +-+ +----+ +--R +-+ , 2 +-+ - 6x\|p \|c(x) +--R (4p x c(x)\|p c (x) + 8p c(x) \|p )%e +--R +--R + +--R +-+ +----+ +--R +-+ , - 8x\|p \|c(x) 2 2 +-+ , +--R \|p c (x)%e - p c(x) \|p c (x) +--R +--R Type: Expression Integer +--E 45 + +--S 46 of 59 +xpr13e := simplify(D((2/(1-exp(-2*wpcx*x)/wpcx)-1),x)) +--R +--R +--R +-+ +----+ +--R +----+ +-+ , +----+ - 2x\|p \|c(x) +--R ((2p x\|c(x) + \|p )c (x) + 4p c(x)\|c(x) )%e +--R +--R (46) --------------------------------------------------------------------- +--R +-+ +----+ +-+ +----+ +--R +-+ - 2x\|p \|c(x) +----+ - 4x\|p \|c(x) +----+ +--R 2c(x)\|p %e - \|c(x) %e - p c(x)\|c(x) +--R Type: Expression Integer +--E 46 + +--S 47 of 59 +xpr13f := (wpcx-exp(-2*wpcx*x))^2 +--R +--R +--R +-+ +----+ 2 +-+ +----+ +--R - 2x\|p \|c(x) +-+ +----+ - 2x\|p \|c(x) +--R (47) (%e ) - 2\|p \|c(x) %e + p c(x) +--R Type: Expression Integer +--E 47 + +--S 48 of 59 +xpr13g := simplify(((2*p*c(x)*x+wpcx)*D(c(x),x)+4*p*c(x)^2)/wpcx) +--R +--R +--R +-+ +----+ , 2 +--R (\|p \|c(x) + 2p x c(x))c (x) + 4p c(x) +--R +--R (48) ----------------------------------------- +--R +-+ +----+ +--R \|p \|c(x) +--R Type: Expression Integer +--E 48 + +--S 49 of 59 +xpr13h := simplify((1+2*wpcx*x)*D(c(x),x)+4*wpcx*c(x)) +--R +--R +--R +-+ +----+ , +-+ +----+ +--R (49) (2x\|p \|c(x) + 1)c (x) + 4c(x)\|p \|c(x) +--R +--R Type: Expression Integer +--E 49 + +--S 50 of 59 +simplify(xpr13h/xpr13g) +--R +--R +--R (50) 1 +--R Type: Expression Integer +--E 50 + +--S 51 of 59 +xpr13i:=simplify((1/(wpcx-z)-1/(2*wpcx))*D(c(x),x)+_ + ((1+2*wpcx*x)*D(c(x),x)+4*wpcx*c(x))*z/(wpcx-z)^2) +--R +--R +--R (51) +--R 2 +-+ +----+ 2 3 , +--R (((- 4p x z - p)c(x) + 3z )\|p \|c(x) + (4p x z - p z)c(x) - z )c (x) +--R +--R + +--R 2 +-+ +----+ 2 2 +--R - 8p z c(x) \|p \|c(x) + 8p z c(x) +--R / +--R 3 +-+ +----+ 2 2 2 +--R (6p z c(x) + 2z )\|p \|c(x) - 2p c(x) - 6p z c(x) +--R Type: Expression Integer +--E 51 + +--S 52 of 59 +w := operator w +--R +--R +--R (52) w +--R Type: BasicOperator +--E 52 + +--S 53 of 59 +z := operator z +--R +--R +--R (53) z +--R Type: BasicOperator +--E 53 + +--S 54 of 59 +simplify(D((2/(1-z(x)/w(x))-1)*c(x)/w(x),x)) +--R +--R +--R (54) +--R 2 , 2 2 , +--R 2c(x)w(x) z (x) + (c(x)z(x) - 2c(x)w(x)z(x) - c(x)w(x) )w (x) +--R +--R + +--R 2 3 , +--R (- w(x)z(x) + w(x) )c (x) +--R +--R / +--R 2 2 3 4 +--R w(x) z(x) - 2w(x) z(x) + w(x) +--R Type: Expression Integer +--E 54 + +--S 55 of 59 +simplify((2*c(x)*w(x)^2*D(z(x),x)+(z(x)^2-2*w(x)*z(x)-w(x)^2)*c(x)*D(w(x),x)+_ + (w(x)^3-w(x)*z(x)^2)*D(c(x),x))/(w(x)^2*(z(x)-w(x))^2)) +--R +--R +--R (55) +--R 2 , 2 2 , +--R 2c(x)w(x) z (x) + (c(x)z(x) - 2c(x)w(x)z(x) - c(x)w(x) )w (x) +--R +--R + +--R 2 3 , +--R (- w(x)z(x) + w(x) )c (x) +--R +--R / +--R 2 2 3 4 +--R w(x) z(x) - 2w(x) z(x) + w(x) +--R Type: Expression Integer +--E 55 + +--S 56 of 59 +xpr13j:=simplify((2*c(x)*w(x)^2*w(x)*z(x)*(x/c(x)-1/2)+(z(x)^2-2*w(x)*z(x)-_ + w(x)^2)*c(x)*(-w(x)/(2*c(x)))+(w(x)^2-w(x)*z(x)^2)*D(c(x),x))_ + /(w(x)^2*(z(x)-w(x))^2)) +--R +--R +--R (56) +--R 2 , 2 2 2 +--R (- 2z(x) + 2w(x))c (x) - z(x) + ((- 2c(x) + 4x)w(x) + 2w(x))z(x) + w(x) +--R +--R --------------------------------------------------------------------------- +--R 2 2 3 +--R 2w(x)z(x) - 4w(x) z(x) + 2w(x) +--R Type: Expression Integer +--E 56 + +--S 57 of 59 +xpr13k:=simplify((1/(w(x)-z(x))-1/(2*w(x)))*D(c(x),x)+_ + ((1+2*w(x)*x)*D(c(x),x)+4*w(x)*c(x))*z(x)/(w(x)-z(x))^2) +--R +--R +--R 2 2 2 , 2 +--R (- z(x) + (4x w(x) + 2w(x))z(x) + w(x) )c (x) + 8c(x)w(x) z(x) +--R +--R (57) ---------------------------------------------------------------- +--R 2 2 3 +--R 2w(x)z(x) - 4w(x) z(x) + 2w(x) +--R Type: Expression Integer +--E 57 + +--S 58 of 59 +simplify(xpr13k/xpr13j) +--R +--R +--R (58) +--R 2 2 2 , 2 +--R (z(x) + (- 4x w(x) - 2w(x))z(x) - w(x) )c (x) - 8c(x)w(x) z(x) +--R +--R ----------------------------------------------------------------------- +--R 2 , 2 2 2 +--R (2z(x) - 2w(x))c (x) + z(x) + ((2c(x) - 4x)w(x) - 2w(x))z(x) - w(x) +--R +--R Type: Expression Integer +--E 58 + +--S 59 of 59 +simplify(xpr13k-xpr13j) +--R +--R +--R (59) +--R 2 2 2 , 2 +--R (z(x) + (4x w(x) + 2w(x))z(x) + w(x) - 2w(x))c (x) + z(x) +--R +--R + +--R 2 2 +--R ((10c(x) - 4x)w(x) - 2w(x))z(x) - w(x) +--R / +--R 2 2 3 +--R 2w(x)z(x) - 4w(x) z(x) + 2w(x) +--R Type: Expression Integer +--E 59 + +)spool +)lisp (bye) + +\end{chunk} +\eject +\begin{thebibliography}{99} +\bibitem{1} http://axiom-wiki.newsynthesis.org/SandBox13 +\end{thebibliography} +\end{document} +