\documentclass{article} \usepackage{axiom} \begin{document} \title{\$SPAD/src/input kamke2.input} \author{Timothy Daly} \maketitle \begin{abstract} This is the 50 ODEs of the Kamke test suite as published by E. S. Cheb-Terrab\cite{1}. They have been rewritten using Axiom syntax. Where possible we show that the particular solution actually satisfies the original ordinary differential equation. \end{abstract} \eject \tableofcontents \eject <<*>>= )spool kamke2.output )set break resume )set mes auto off )clear all --S 1 of 126 y:=operator 'y --R --R (1) y --R Type: BasicOperator --E 1 --S 2 of 126 f:=operator 'f --R --R (2) f --R Type: BasicOperator --E 2 --S 3 of 126 g:=operator 'g --R --R (3) g --R Type: BasicOperator --E 3 ------------------------------------------------------------------- --S 4 of 126 ode101 := x*D(y(x),x) + x*y(x)**2 - y(x) --R --R , 2 --R (4) xy (x) + x y(x) - y(x) --R --R Type: Expression Integer --E 4 @ Maxima gives $$\frac{2x}{x^2-2\%c}$$ which can be substituted and simplifies to 0. Maple gives $$\frac{2x}{x^2+2\_C1}$$ which can be substituted and simplifies to 0. Mathematica gives $$y(x)=\frac{2x}{x^2+2}$$ which can be substituted and simplifies to 0. <<*>>= --S 5 of 126 yx:=solve(ode101,y,x) --R --R 2 --R x y(x) - 2x --R (5) ----------- --R 2y(x) --R Type: Union(Expression Integer,...) --E 5 --S 6 of 126 ode101expr := x*D(yx,x) + x*yx**2 - yx --R --R 2 , 5 2 2 4 3 --R 4x y (x) + (x + 2x )y(x) - 4x y(x) + 4x --R --R (6) ------------------------------------------ --R 2 --R 4y(x) --R Type: Expression Integer --E 6 ------------------------------------------------------------------- --S 7 of 126 ode102 := x*D(y(x),x) + x*y(x)**2 - y(x) - a*x**3 --R --R , 2 3 --R (7) xy (x) + x y(x) - y(x) - a x --R --R Type: Expression Integer --E 7 @ Maxima fails. Maple gives $$\tanh(\left(\frac{x^2\sqrt{a}}{2}+\_C1\sqrt{a}\right)x\sqrt{a}$$ which, upon substitution, simplifies to 0. Mathematica gives $$\sqrt{a}~x~ \tanh\left(\frac{1}{2}\left(\sqrt{a}~x^2+2\sqrt{a}~C[1]\right)\right)$$ which, upon substitution, cannot be simplified to 0. <<*>>= --S 8 of 126 yx:=solve(ode102,y,x) --R --R +-+ --R (2y(x) + 3x)\|a + 3y(x) + 2a x --R (8) --------------------------------------------- --R 2 +-+ --R +-+ x \|a --R ((6y(x) - 4a x)\|a + 4a y(x) - 6a x)%e --R Type: Union(Expression Integer,...) --E 8 --S 9 of 126 ode102expr := x*D(yx,x) + x*yx**2 - yx - a*x**3 --R --R (9) --R 2 2 3 2 3 +-+ --R ((- 144a - 108a)x y(x) + (32a + 216a )x )\|a --R + --R 3 2 2 3 2 3 --R (- 32a - 216a )x y(x) + (144a + 108a )x --R * --R 2 +-+ --R x \|a , --R %e y (x) --R --R + --R 3 2 3 3 4 3 4 2 --R (- 144a - 108a )x y(x) + (96a + 648a )x y(x) --R + --R 4 3 5 5 4 6 --R (- 432a - 324a )x y(x) + (32a + 216a )x --R * --R +-+ --R \|a --R + --R 4 3 3 3 4 3 4 2 --R (- 32a - 216a )x y(x) + (432a + 324a )x y(x) --R + --R 5 4 5 5 4 6 --R (- 96a - 648a )x y(x) + (144a + 108a )x --R * --R 2 +-+ 2 --R x \|a --R (%e ) --R + --R 2 2 2 3 --R ((- 144a - 108a)x - 16a - 108a)y(x) --R + --R 3 2 3 2 2 --R ((32a + 216a )x + (216a + 162a)x)y(x) --R + --R 3 2 4 3 2 2 4 3 5 --R ((144a + 108a )x + (- 16a - 108a )x )y(x) + (- 32a - 216a )x --R + --R 3 2 3 --R (- 72a - 54a )x --R * --R +-+ --R \|a --R + --R 3 2 2 2 3 --R ((- 32a - 216a )x - 72a - 54a)y(x) --R + --R 3 2 3 3 2 2 --R ((144a + 108a )x + (48a + 324a )x)y(x) --R + --R 4 3 4 3 2 2 4 3 5 --R ((32a + 216a )x + (- 72a - 54a )x )y(x) + (- 144a - 108a )x --R + --R 4 3 3 --R (- 16a - 108a )x --R * --R 2 +-+ --R x \|a --R %e --R + --R 3 2 2 2 2 3 --R (36a + 27)x y(x) + (8a + 54a)x y(x) + (- 36a - 27a)x y(x) --R + --R 3 2 4 --R (- 8a - 54a )x --R * --R +-+ --R \|a --R + --R 2 3 2 2 2 3 2 3 --R (8a + 54a)x y(x) + (36a + 27a)x y(x) + (- 8a - 54a )x y(x) --R + --R 3 2 4 --R (- 36a - 27a )x --R / --R 2 3 3 2 2 --R (144a + 108a)y(x) + (- 96a - 648a )x y(x) --R + --R 3 2 2 4 3 3 --R (432a + 324a )x y(x) + (- 32a - 216a )x --R * --R +-+ --R \|a --R + --R 3 2 3 3 2 2 4 3 2 --R (32a + 216a )y(x) + (- 432a - 324a )x y(x) + (96a + 648a )x y(x) --R + --R 4 3 3 --R (- 144a - 108a )x --R * --R 2 +-+ 2 --R x \|a --R (%e ) --R Type: Expression Integer --E 9 ------------------------------------------------------------------- --S 10 of 126 ode103 := x*D(y(x),x) + x*y(x)**2 - (2*x**2+1)*y(x) - x**3 --R --R , 2 2 3 --R (10) xy (x) + x y(x) + (- 2x - 1)y(x) - x --R --R Type: Expression Integer --E 10 @ Maxima fails. Maple gives $$\frac{1}{2}x\left(\sqrt{2}+ 2\tanh\left(\frac{(x^2+x\_C1)\sqrt{2}}{2}\right)\right)\sqrt{2}$$ which simplifies to 0 on substitution. Mathematica gives $$\frac{\left(e^{\sqrt{x}~x^2}+\sqrt{2}~e^{\sqrt{2}~x^2}+ e^{2\sqrt{2}~C[1]}-\sqrt{2}~e^{2\sqrt{2}~C[1]}\right)x} {e^{\sqrt{2}~x^2}+e^{2*\sqrt{2}~C[1]}}$$ which does not simplify to 0 on substitution. <<*>>= --S 11 of 126 yx:=solve(ode103,y,x) --R --R +-+ +-+ --R (2\|2 + 3)y(x) + x\|2 + x --R (11) ----------------------------------------- --R 2 +-+ --R +-+ +-+ x \|2 --R ((6\|2 + 8)y(x) - 14x\|2 - 20x)%e --R Type: Union(Expression Integer,...) --E 11 --S 12 of 126 ode103expr := x*D(yx,x) + x*yx**2 - (2*x**2+1)*yx - x**3 --R --R (12) --R 2 +-+ --R 2 +-+ 2 3 +-+ 3 x \|2 , --R ((- 792x \|2 - 1120x )y(x) + 1912x \|2 + 2704x )%e y (x) --R --R + --R 3 +-+ 3 3 4 +-+ 4 2 --R (- 792x \|2 - 1120x )y(x) + (5736x \|2 + 8112x )y(x) --R + --R 5 +-+ 5 6 +-+ 6 --R (- 13848x \|2 - 19584x )y(x) + 11144x \|2 + 15760x --R * --R 2 +-+ 2 --R x \|2 --R (%e ) --R + --R 2 +-+ 2 3 --R ((- 1352x - 280)\|2 - 1912x - 396)y(x) --R + --R 3 +-+ 3 2 --R ((5968x + 2028x)\|2 + 8440x + 2868x)y(x) --R + --R 4 2 +-+ 4 2 --R ((- 5176x - 2984x )\|2 - 7320x - 4220x )y(x) --R + --R 5 3 +-+ 5 3 --R (- 3264x - 676x )\|2 - 4616x - 956x --R * --R 2 +-+ --R x \|2 --R %e --R + --R +-+ 3 2 +-+ 2 2 --R (99x\|2 + 140x)y(x) + (- 157x \|2 - 222x )y(x) --R + --R 3 +-+ 3 4 +-+ 4 --R (- 181x \|2 - 256x )y(x) - 41x \|2 - 58x --R / --R +-+ 3 +-+ 2 --R (792\|2 + 1120)y(x) + (- 5736x\|2 - 8112x)y(x) --R + --R 2 +-+ 2 3 +-+ 3 --R (13848x \|2 + 19584x )y(x) - 11144x \|2 - 15760x --R * --R 2 +-+ 2 --R x \|2 --R (%e ) --R Type: Expression Integer --E 12 ------------------------------------------------------------------- --S 13 of 126 ode106 := x*D(y(x),x) + x**a*y(x)**2 + (a-b)*y(x)/2 + x**b --R --R , b 2 a --R 2xy (x) + 2x + 2y(x) x + (- b + a)y(x) --R --R (13) ---------------------------------------- --R 2 --R Type: Expression Integer --E 13 @ Maxima fails. Maple gets $$-\frac{\tan\left( \frac{\displaystyle 2x^{\left(\displaystyle \frac{a}{2}+\frac{b}{2}\right)}+\displaystyle\_C1~a+\_C1~b} {\displaystyle a+b}\right)} {x^{\left(\displaystyle{\frac{a}{2}-\displaystyle\frac{b}{2}}\right)}}$$ which simplifies to 0 on substitution. Mathematica gets $$e^{-\frac{1}{2}a\log(x)+\frac{1}{2}b\log(x)} \tan\left(\frac{2x^{\frac{a+b}{2}}}{a+b}-C[1]\right)$$ which does not simplify to 0 on substitution. <<*>>= --S 14 of 126 yx:=solve(ode106,y,x) --R --R (14) "failed" --R Type: Union("failed",...) --E 14 ------------------------------------------------------------------- --S 15 of 126 ode107 := x*D(y(x),x) + a*x**alpha*y(x)**2 + b*y(x) - c*x**beta --R --R , beta 2 alpha --R (15) xy (x) - c x + a y(x) x + b y(x) --R --R Type: Expression Integer --E 15 @ Maxima fails. <<*>>= --S 16 of 126 yx:=solve(ode107,y,x) --R --R (16) "failed" --R Type: Union("failed",...) --E 16 ------------------------------------------------------------------- --S 17 of 126 ode108 := x*D(y(x),x) - y(x)**2*log(x) + y(x) --R --R , 2 --R (17) xy (x) - y(x) log(x) + y(x) --R --R Type: Expression Integer --E 17 @ Maxima gets: $$\frac{1}{x\left(\frac{\log(x)}{x}+\frac{1}{x}+\%c\right)}$$ which does not simplify on substitution. Maple gets: $$\frac{1}{1+\log(x)+x\_C1}$$ which, on substitution, simplifies to 0. Mathematica gets: $$\frac{1}{1+xC[1]+\log(x)}$$ which, on substitution, simplifies to 0. <<*>>= --S 18 of 126 yx:=solve(ode108,y,x) --R --R - y(x)log(x) - y(x) + 1 --R (18) ----------------------- --R x y(x) --R Type: Union(Expression Integer,...) --E 18 --S 19 of 126 ode108expr := x*D(yx,x) - yx**2*log(x) + yx --R --R (19) --R 2 , 2 3 2 2 --R - x y (x) - y(x) log(x) + (- 2y(x) + 2y(x))log(x) --R --R + --R 2 2 --R (- y(x) + 2y(x) - 1)log(x) - x y(x) --R / --R 2 2 --R x y(x) --R Type: Expression Integer --E 19 ------------------------------------------------------------------- --S 20 of 126 ode109 := x*D(y(x),x) - y(x)*(2*y(x)*log(x)-1) --R --R , 2 --R (20) xy (x) - 2y(x) log(x) + y(x) --R --R Type: Expression Integer --E 20 @ Maxima gets: $$\frac{1}{x\left(\%c-2\left(-\frac{\log(x)}{x}-\frac{1}{x}\right)\right)}$$ which does not simplify to 0 on substitution. Maple gets: $$\frac{1}{2+2\log(x)+x~\_C1}$$ which simplifies to 0 on substitition. Mathematica gets $$\frac{1}{2+xC[1]+2\log(x)}$$ which simplifies to 0 on substitution. <<*>>= --S 21 of 126 yx:=solve(ode109,y,x) --R --R - 2y(x)log(x) - 2y(x) + 1 --R (21) ------------------------- --R x y(x) --R Type: Union(Expression Integer,...) --E 21 --S 22 of 126 ode109expr := x*D(yx,x) - yx*(2*yx*log(x)-1) --R --R (22) --R 2 , 2 3 2 2 --R - x y (x) - 8y(x) log(x) + (- 16y(x) + 8y(x))log(x) --R --R + --R 2 2 --R (- 8y(x) + 8y(x) - 2)log(x) - 2x y(x) --R / --R 2 2 --R x y(x) --R Type: Expression Integer --E 22 ------------------------------------------------------------------- --S 23 of 126 ode110 := x*D(y(x),x) + f(x)*(y(x)**2-x**2) --R --R , 2 2 --R (23) xy (x) + f(x)y(x) - x f(x) --R --R Type: Expression Integer --E 23 @ Maxima failed. <<*>>= --S 24 of 126 yx:=solve(ode110,y,x) --R --R (24) "failed" --R Type: Union("failed",...) --E 24 ------------------------------------------------------------------- --S 25 of 126 ode111 := x*D(y(x),x) + y(x)**3 + 3*x*y(x)**2 --R --R , 3 2 --R (25) xy (x) + y(x) + 3x y(x) --R --R Type: Expression Integer --E 25 @ Maxima fails. Maple gets 0 which simplifies to 0 on substitution. <<*>>= --S 26 of 126 yx:=solve(ode111,y,x) --R --R (26) "failed" --R Type: Union("failed",...) --E 26 ------------------------------------------------------------------- --S 27 of 126 ode112 := x*D(y(x),x) - sqrt(y(x)**2 + x**2) - y(x) --R --R +----------+ --R , | 2 2 --R (27) xy (x) - \|y(x) + x - y(x) --R --R Type: Expression Integer --E 27 @ Maxima gets $$x=\%c \%e^{\displaystyle \frac{x {\rm asinh}\left(\frac{y}{x}\right)}{\vert x\vert}}$$ which does not simplify to 0 on substitution. Maple gets 0 but simplification gives the result$csgn(x)x$. <<*>>= --S 28 of 126 yx:=solve(ode112,y,x) --R --R (28) "failed" --R Type: Union("failed",...) --E 28 ------------------------------------------------------------------- --S 29 of 126 ode113 := x*D(y(x),x) + a*sqrt(y(x)**2 + x**2) - y(x) --R --R +----------+ --R , | 2 2 --R (29) xy (x) + a\|y(x) + x - y(x) --R --R Type: Expression Integer --E 29 @ Maxima gets $$x=\%c \%e^{\displaystyle -\frac{x {\rm asinh}\left(\frac{y}{x}\right)}{a\vert x\vert}}$$ which does not simplify to 0 on substitution. Maple gets 0 but on substitition this simplifies to$a~csgn(x)~x$Mathematica gets $$x*\sinh(C[1]+\log(x))$$ If we choose$C[1]=0$this simplifies to $$\frac{1}{2}(-1+x^2)$$ However, Mathematica cannot simplify either substition to 0. <<*>>= --S 30 of 126 yx:=solve(ode113,y,x) --R --R (30) "failed" --R Type: Union("failed",...) --E 30 ------------------------------------------------------------------- --S 31 of 126 ode114 := x*D(y(x),x) - x*sqrt(y(x)**2 + x**2) - y(x) --R --R +----------+ --R , | 2 2 --R (31) xy (x) - x\|y(x) + x - y(x) --R --R Type: Expression Integer --E 31 @ Maxima fails. Maple gets 0 but, on substitition, simplifies to$-x^2csqn(x)$. Mathematica gets $$x\sinh(x+C[1])$$ but cannot simplify the substituted expression to 0. <<*>>= --S 32 of 126 yx:=solve(ode114,y,x) --R --R (32) "failed" --R Type: Union("failed",...) --E 32 ------------------------------------------------------------------- --S 33 of 126 ode115 := x*D(y(x),x) - x*(y(x)-x)*sqrt(y(x)**2 + x**2) - y(x) --R --R +----------+ --R , 2 | 2 2 --R (33) xy (x) + (- x y(x) + x )\|y(x) + x - y(x) --R --R Type: Expression Integer --E 33 @ Maxima failed. Maple claims the result is 0 but simplifies it, on substitution, to$x^3 csgn(x)$. Mathematica claims that the equations appear to involve the variables to be solved for in an essentially non-algebraic way. <<*>>= --S 34 of 126 yx:=solve(ode115,y,x) --R --R (34) "failed" --R Type: Union("failed",...) --E 34 ------------------------------------------------------------------- --S 35 of 126 ode116 := x*D(y(x),x) - x*sqrt((y(x)**2 - x**2)*(y(x)**2-4*x**2)) - y(x) --R --R +----------------------+ --R , | 4 2 2 4 --R (35) xy (x) - x\|y(x) - 5x y(x) + 4x - y(x) --R --R Type: Expression Integer --E 35 @ Maxima failed. Maple claims the answer is 0 but simplifies, on substitution, to$-2x^3 csgn(x^2)$. Mathematica says that a potential solution of ComplexInfinity was possibly discarded by the verifier and should be checked by hand, possibly using limits. And the equations appear to involve the variables to be solved for in an essentially non-algebraic way. <<*>>= --S 36 of 126 yx:=solve(ode116,y,x) --R --R (36) "failed" --R Type: Union("failed",...) --E 36 ------------------------------------------------------------------- --S 37 of 126 ode117 := x*D(y(x),x) - x*exp(y(x)/x) - y(x) - x --R --R y(x) --R ---- --R , x --R (37) xy (x) - x %e - y(x) - x --R --R Type: Expression Integer --E 37 @ Maxima gets: $$\%c~x=\%e^{\displaystyle -\frac{x\log(\%e^{y/x}+1)-y}{x}}$$ which does not simplify to 0 on substitution. Maple gets: $$\left(\log\left(-\frac{x}{-1+x~e^{\_C1}}\right)+\_C1\right)x$$ which simplifies to 0 on substitution. Mathematica says that inverse functions are being used by Solve, so some solutions may not be found and to use Reduce for complete solution information. It gets the answer: $$-x\log\left(-1+\frac{e^{-C[1]}}{x}\right)$$ which simplifies to 0. <<*>>= --S 38 of 126 yx:=solve(ode117,y,x) --R --R (38) "failed" --R Type: Union("failed",...) --E 38 ------------------------------------------------------------------- --S 39 of 126 ode118 := x*D(y(x),x) - y(x)*log(y(x)) --R --R , --R (39) xy (x) - y(x)log(y(x)) --R --R Type: Expression Integer --E 39 @ Maxima gets $$\%e^{\%e^{\%c}x}$$ which, on substitution, simplifies to 0. Maple gets $$e^{(x~\_C1)}$$ which, on substitution, does not simplify to 0. Mathematics gets $$e^{e^{C[1]}x}$$ which, on substitution simplifies to $$e^x(x-\log(e^x))$$ which, if$log(e^x)$could simplify to$x$then the result would be 0. <<*>>= --S 40 of 126 yx:=solve(ode118,y,x) --R --R x --R (40) - --------- --R log(y(x)) --R Type: Union(Expression Integer,...) --E 40 --S 41 of 126 ode118expr := x*D(yx,x) - yx*log(yx) --R --R x 2 , --R x y(x)log(y(x))log(- ---------) + x y (x) - x y(x)log(y(x)) --R log(y(x)) --R (41) ----------------------------------------------------------- --R 2 --R y(x)log(y(x)) --R Type: Expression Integer --E 41 ------------------------------------------------------------------- --S 42 of 126 ode119 := x*D(y(x),x) - y(x)*(log(x*y(x))-1) --R --R , --R (42) xy (x) - y(x)log(x y(x)) + y(x) --R --R Type: Expression Integer --E 42 @ $$\frac{1}{x}$$ simplifies to 0. Maxima gets $$\frac{\%e^{x/\%c}}{x}$$ which, on substitution, does not simplify to 0. Maple get $$\frac{e^{\left(\frac{x}{\_C1}\right)}}{x}$$ which, on substitution, does not simplify to 0. Mathematica gets $$\frac{1}{x(C[1]-log(log(x)))}$$ which does not simplify to 0 on substitution. <<*>>= --S 43 of 126 yx:=solve(ode119,y,x) --R --R (43) "failed" --R Type: Union("failed",...) --E 43 ------------------------------------------------------------------- --S 44 of 126 ode120 := x*D(y(x),x) - y(x)*(x*log(x**2/y(x))+2) --R --R 2 --R , x --R (44) xy (x) - x y(x)log(----) - 2y(x) --R y(x) --R Type: Expression Integer --E 44 @ Maxima fails. Maple gets $$\frac{x^2}{e^{\left(\frac{\_C1}{e^x}\right)}}$$ which, on substitution, does not simplify to 0. Mathematics get: $$2e^{-e^{-x} C[1]+e^{-x}{\rm ExpIntegralEi}[x]}x$$ which does not simplify to 0 on substitution. <<*>>= --S 45 of 126 yx:=solve(ode120,y,x) --R --R (45) "failed" --R Type: Union("failed",...) --E 45 ------------------------------------------------------------------- --S 46 of 126 ode121 := x*D(y(x),x) + sin(y(x)-x) --R --R , --R (46) xy (x) + sin(y(x) - x) --R --R Type: Expression Integer --E 46 @ Maxima fails. Mathematics gets $$\frac{\sin(x)}{1+\sin(x)}+x^{-sin(x)}C[1]$$ which, on substitution, does not simplify to 0. <<*>>= --S 47 of 126 yx:=solve(ode121,y,x) --R --R (47) "failed" --R Type: Union("failed",...) --E 47 ------------------------------------------------------------------- --S 48 of 126 ode122 := x*D(y(x),x) + (sin(y(x))-3*x**2*cos(y(x)))*cos(y(x)) --R --R , 2 2 --R (48) xy (x) + cos(y(x))sin(y(x)) - 3x cos(y(x)) --R --R Type: Expression Integer --E 48 @ Maxima fails. Maple gets: $$\arctan\left(\frac{x^3+2~\_C1}{x}\right)$$ which, on substitution, simplifies to 0. Mathematica gets: $$\arctan\left(\frac{2x^3+C[1]}{2x}\right)$$ which, on substitution, simplifies to 0. <<*>>= --S 49 of 126 yx:=solve(ode122,y,x) --R --R (49) "failed" --R Type: Union("failed",...) --E 49 ------------------------------------------------------------------- --S 50 of 126 ode123 := x*D(y(x),x) - x*sin(y(x)/x) - y(x) --R --R , y(x) --R (50) xy (x) - x sin(----) - y(x) --R x --R Type: Expression Integer --E 50 @ Maxima gets: $$\%c~x=\%e^{\displaystyle -\frac{ \log\left(\cos\left(\frac{y}{x}\right)+1\right)- \log\left(\cos\left(\frac{y}{x}\right)-1\right)}{2}}$$ which, on substitution, does not simplify to 0. Maple gets: $$\arctan\left(\frac{2x~\_C1}{1+x^2~\_C1^2}\quad,\quad -\frac{-1+x^2~\_C1^2}{1+x^2~\_C1^2}\right)x$$ which, on substitution, simplifies to 0. Mathematica get: $$x^{1+sin(x)}C[1]$$ which does not simplfy to 0 on substitution. <<*>>= --S 51 of 126 yx:=solve(ode123,y,x) --R --R (51) "failed" --R Type: Union("failed",...) --E 51 ------------------------------------------------------------------- --S 52 of 126 ode124 := x*D(y(x),x) + x*cos(y(x)/x) - y(x) + x --R --R , y(x) --R (52) xy (x) + x cos(----) - y(x) + x --R x --R Type: Expression Integer --E 52 @ Maxima gets: $$\%c~x=\%e^{\displaystyle -\frac{\sin\left(\frac{y}{x}\right)} {\cos\left(\frac{y}{x}\right)+1}}$$ which, on substitution, does not simplify to 0. Maple gets $$-2\arctan(\log(x)+~\_C1)x$$ which, on substitution, does not simplify to 0. Mathematics gets $$2x\arctan(C[1]-\log(x))$$ which does not simplify to 0 on substitution. <<*>>= --S 53 of 126 yx:=solve(ode124,y,x) --R --R (53) "failed" --R Type: Union("failed",...) --E 53 ------------------------------------------------------------------- --S 54 of 126 ode125 := x*D(y(x),x) + x*tan(y(x)/x) - y(x) --R --R , y(x) --R (54) xy (x) + x tan(----) - y(x) --R x --R Type: Expression Integer --E 54 @ Maxima gets: $$\arcsin\left(\frac{1}{\%c~x}\right)x$$ which, on substitition, does simplifes to 0. Maple gets $$\arcsin\left(\frac{1}{x~\_C1}\right)x$$ which, on substitution, simplifies to 0. Mathematica gets $$\arcsin\left(\frac{e^{C[1]}}{x}\right)$$ which does not simplify to 0 on substitution. <<*>>= --S 55 of 126 yx:=solve(ode125,y,x) --R --R (55) "failed" --R Type: Union("failed",...) --E 55 ------------------------------------------------------------------- --S 56 of 126 ode126 := x*D(y(x),x) - y(x)*f(x*y(x)) --R --R , --R (56) xy (x) - y(x)f(x y(x)) --R --R Type: Expression Integer --E 56 @ Maxima fails. Maple gets $$\frac{{\rm RootOf}\left(-\log(x)+~\_C1+ \displaystyle\int^{\_Z}{\frac{1}{\displaystyle\_a(1+g(\_a))}}~d\_a\right)}{x}$$ which, on substitution, simplifies to 0. Mathematica gets $$\frac{1}{-f(x)-C[1]}$$ which does not simplify to 0 on substitution. <<*>>= --S 57 of 126 yx:=solve(ode126,y,x) --R --R (57) "failed" --R Type: Union("failed",...) --E 57 ------------------------------------------------------------------- --S 58 of 126 ode127 := x*D(y(x),x) - y(x)*f(x**a*y(x)**b) --R --R a b , --R (58) - y(x)f(x y(x) ) + xy (x) --R --R Type: Expression Integer --E 58 @ Maxima fails. Maple gives 0 which, on substitution simplifies to 0. Mathematica gives: $$b\left(-\frac{f(x^a)}{a}-C[1]\right)^{-1/b}$$ which, on substitution, does not simplify to 0. <<*>>= --S 59 of 126 yx:=solve(ode127,y,x) --R --R (59) "failed" --R Type: Union("failed",...) --E 59 ------------------------------------------------------------------- --S 60 of 126 ode128 := x*D(y(x),x) + a*y(x) - f(x)*g(x**a*y(x)) --R --R , a --R (60) xy (x) - f(x)g(y(x)x ) + a y(x) --R --R Type: Expression Integer --E 60 @ Maxima fails. Maple gives $$\frac{{\rm RootOf}\left( -\int{f(x)x^{(-1+a)}}~dx+\int^{\_Z}{\frac{1}{g(\_a)}~d\_a+\_C1}\right)}{x^a}$$ which, on substitution, gives 0. Mathematica gives $$e^{\frac{f(x)g(x^{1+a})}{1+a}-a\log(x)}C[1]$$ which, on substitution, does not simplify to 0. <<*>>= --S 61 of 126 yx:=solve(ode128,y,x) --R --R (61) "failed" --R Type: Union("failed",...) --E 61 ------------------------------------------------------------------- --S 62 of 126 ode129 := (x+1)*D(y(x),x) + y(x)*(y(x)-x) --R --R , 2 --R (62) (x + 1)y (x) + y(x) - x y(x) --R --R Type: Expression Integer --E 62 @ Maxima gets: $$\frac{\%e^x}{(x+1)\left(\int{\frac{\%e^x}{(x+1)^2}}~dx+\%c\right)}$$ which, on substitution, does not simplify to 0. Maple gives $$\frac{e^x} {-e^x-e^{(-1)}{\rm Ei}(1,-x-1)x-e^{(-1)}{\rm Ei}(1,-x-1)+x~\_C1+~\_C1}$$ which, on substitution, simplifies to 0. Mathematica gives $$-\frac{e^{1+x}}{e^{1+x}-eC[1]-exC[1]-{\rm ExpIntegralEi}(1+x)- x{\rm ExpIntegralEi}(1+x)}$$ <<*>>= --S 63 of 126 yx:=solve(ode129,y,x) --R --R --R x --R - x ++ 1 --I (- x - 1)y(x)%e | --------------------- d%U + 1 --I ++ 2 - %U --I (%U + 2%U + 1)%e --R (63) ----------------------------------------------------- --R - x --R (x + 1)y(x)%e --R Type: Union(Expression Integer,...) --E 63 ------------------------------------------------------------------- --S 64 of 126 ode130 := 2*x*D(y(x),x) - y(x) -2*x**3 --R --R , 3 --R (64) 2xy (x) - y(x) - 2x --R --R Type: Expression Integer --E 64 @ Maxima gets: $$\%e^{\displaystyle\frac{\log(x)}{2}}\displaystyle \left(\frac{2\%e^{\displaystyle\frac{5\log(x)}{2}}}{5}+\%c\right)$$ which, on substitution, does not give 0. Maple gives $$\frac{2x^3}{5}+\sqrt{x}~\_C1$$ which, on substitution, simplifies to 0. Mathematica gives $$\frac{2x^3}{5}+\sqrt{x}C[1]$$ which simplifies to 0 on substitution. <<*>>= --S 65 of 126 ode130a:=solve(ode130,y,x) --R --R 3 --R 2x +-+ --R (65) [particular= ---,basis= [\|x ]] --R 5 --RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) --E 65 --S 66 of 126 yx:=ode130a.particular --R --R 3 --R 2x --R (66) --- --R 5 --R Type: Expression Integer --E 66 --S 67 of 126 ode130expr := 2*x*D(yx,x) - yx -2*x**3 --R --R (67) 0 --R Type: Expression Integer --E 67 ------------------------------------------------------------------- --S 68 of 126 ode131 := (2*x+1)*D(y(x),x) - 4*exp(-y(x)) + 2 --R --R , - y(x) --R (68) (2x + 1)y (x) - 4%e + 2 --R --R Type: Expression Integer --E 68 @ Maxima gets: $$\log\left(\frac{4\%e^{2\%c}x+2\%e^{2\%c}+1} {2\%e^{2\%c}x+\%e^{2\%c}}\right)$$ which simplifies to 0 when substituted. Maple gives $$-\log\left(\frac{2x+1}{-1+4xe^{(2~\_C1)}+2e^{(2~\_C1)}}\right)-2~\_C1$$ which simplifies to 0 when substituted. Mathematica gives $$\log\left(2+\frac{1}{1+2x}\right)$$ which simplifies to 0 when substituted. <<*>>= --S 69 of 126 yx:=solve(ode131,y,x) --R --R - y(x) y(x) --R (69) (- 4x %e + 2x + 1)%e --R Type: Union(Expression Integer,...) --E 69 --S 70 of 126 ode131expr := (2*x+1)*D(yx,x) - 4*exp(-yx) + 2 --R --R (70) --R - y(x) y(x) --R (4x %e - 2x - 1)%e 2 y(x) , --R - 4%e + (4x + 4x + 1)%e y (x) --R --R + --R - y(x) y(x) --R ((- 8x - 4)%e + 4x + 2)%e + 2 --R Type: Expression Integer --E 70 ------------------------------------------------------------------- --S 71 of 126 ode132 := 3*x*D(y(x),x) - 3*x*log(x)*y(x)**4 - y(x) --R --R , 4 --R (71) 3xy (x) - 3x y(x) log(x) - y(x) --R --R Type: Expression Integer --E 71 @ Maxima gives 3 solutions. $$-\frac{\left(\sqrt{3}~4^{1/3}\%i-4^{1/3}\right)x^{1/3}} {2\left(6x^2\log(x)-3x^2+4\%c\right)^{1/3}}$$ $$\frac{\left(\sqrt{3}~4^{1/3}\%i+4^{1/3}\right)x^{1/3}} {2\left(6x^2\log(x)-3x^2+4\%c\right)^{1/3}}$$ $$-\frac{4^{1/3}x^{1/3}}{\left(6x^2\log(x)-3x^2+4\%c\right)^{1/3}}$$ which, on substitution, simplifies to 0. Maple gives 3 solutions. $$\frac{\left(-4x(6x^2\log(x)-3x^2-4~\_C1)^2\right)^{(1/3)}} {6x^2\log(x)-3*x^2-4~\_C1}$$ $$-\frac{1}{2}\frac{\left(-4x(6x^2\log(x)-3x^2-4~\_C1)^2\right)^{(1/3)}} {6x^2\log(x)-3*x^2-4~\_C1} +\frac{1}{2}I\sqrt{3} \frac{\left(-4x(6x^2\log(x)-3x^2-4~\_C1)^2\right)^{(1/3)}} {6x^2\log(x)-3*x^2-4~\_C1}$$ $$-\frac{1}{2}\frac{\left(-4x(6x^2\log(x)-3x^2-4~\_C1)^2\right)^{(1/3)}} {6x^2\log(x)-3*x^2-4~\_C1} -\frac{1}{2}I\sqrt{3} \frac{\left(-4x(6x^2\log(x)-3x^2-4~\_C1)^2\right)^{(1/3)}} {6x^2\log(x)-3*x^2-4~\_C1}$$ which, on substitution, simplifies to 0. Mathematica gives 3 solutions, $$\frac{(-2)^{2/3}x^{1/3}}{(3x^2+4C[1]-6x^2\log(x))^{1/3}}$$ $$\frac{( 2)^{2/3}x^{1/3}}{(3x^2+4C[1]-6x^2\log(x))^{1/3}}$$ $$\frac{(-1)^{1/3}2^{2/3}x^{1/3}}{(3x^2+4C[1]-6x^2\log(x))^{1/3}}$$ which do not simplify to 0 on substitution. <<*>>= --S 72 of 126 yx:=solve(ode132,y,x) --R --R 2 3 2 3 --R - 6x y(x) log(x) + 3x y(x) - 4x --R (72) -------------------------------- --R 3 --R 4y(x) --R Type: Union(Expression Integer,...) --E 72 --S 73 of 126 ode132expr := 3*x*D(yx,x) - 3*x*log(x)*yx**4 - yx --R --R (73) --R 2 8 , 9 12 5 --R 2304x y(x) y (x) - 3888x y(x) log(x) --R --R + --R 9 12 8 9 4 --R (7776x y(x) - 10368x y(x) )log(x) --R + --R 9 12 8 9 7 6 3 --R (- 5832x y(x) + 15552x y(x) - 10368x y(x) )log(x) --R + --R 9 12 8 9 7 6 6 3 2 --R (1944x y(x) - 7776x y(x) + 10368x y(x) - 4608x y(x) )log(x) --R + --R 9 2 12 8 9 7 6 6 3 --R (- 243x - 1920x )y(x) + 1296x y(x) - 2592x y(x) + 2304x y(x) --R + --R 5 --R - 768x --R * --R log(x) --R + --R 2 12 9 --R - 192x y(x) - 512x y(x) --R / --R 12 --R 256y(x) --R Type: Expression Integer --E 73 ------------------------------------------------------------------- --S 74 of 126 ode133 := x**2*D(y(x),x) + y(x) - x --R --R 2 , --R (74) x y (x) + y(x) - x --R --R Type: Expression Integer --E 74 @ Maxima gets $$\%e^{1/x} \left(\int{\displaystyle\frac{\%e^{-\frac{1}{x}}}{x}}~dx+\%c\right)$$ which, on substitution, simplifies to 0. Maple gives $$\left({\rm Ei}\left(1,\frac{1}{x}\right)+~\_C1\right)e^{(\frac{1}{x})}$$ which simplifies to 0 on substitution. Mathematica gets: $$e^{1/x}C[1]-e^{1/x}{\rm ExpIntegralEi}\left(-\frac{1}{x}\right)$$ which simplifies to 0 on substitution. <<*>>= --S 75 of 126 yx:=solve(ode133,y,x) --R --R --R 1 1 --R - x - --R x ++ 1 x --I (75) [particular= %e | ------- d%U ,basis= [%e ]] --R ++ 1 --R -- --I %U --I %U %e --RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) --E 75 ------------------------------------------------------------------- --S 76 of 126 ode134 := x**2*D(y(x),x) - y(x) + x**2*exp(x-1/x) --R --R 2 --R x - 1 --R ------ --R 2 , 2 x --R (76) x y (x) + x %e - y(x) --R --R Type: Expression Integer --E 76 @ Maxima gets $$\%e^{\displaystyle -\frac{1}{x}}\left(\%c-\%e^x\right)$$ which simplifies to 0 on substitution. Maple gets $$(-e^x+~\_C1)e^{\left(-\frac{1}{x}\right)}$$ which simplifies to 0 on substitution. Mathematics get $$-e^{-\frac{1}{x}+x}+e^{-1/x}C[1]$$ which does not simplify to 0 on substitution. This is curious because the basis element is the same one computed by Axiom, which Axiom cannot simplify either. However, Axiom can simplify the particular element to 0 and Mathematica cannot. <<*>>= --S 77 of 126 ode134a:=solve(ode134,y,x) --R --R 2 --R x - 1 1 --R ------ - - --R x x --R (77) [particular= - %e ,basis= [%e ]] --RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) --E 77 --S 78 of 126 yx:=ode134a.particular --R --R 2 --R x - 1 --R ------ --R x --R (78) - %e --R Type: Expression Integer --E 78 --S 79 of 126 ode134expr := x**2*D(yx,x) - yx + x**2*exp(x-1/x) --R --R (79) 0 --R Type: Expression Integer --E 79 ------------------------------------------------------------------- --S 80 of 126 ode135 := x**2*D(y(x),x) - (x-1)*y(x) --R --R 2 , --R (80) x y (x) + (- x + 1)y(x) --R --R Type: Expression Integer --E 80 @ Maxima gets $$\%c~x\%e^{1/x}$$ which simplifies to 0 when substituted. Maple gets $$\_C1xe^{\left(\frac{1}{x}\right)}$$ which simplifies to 0 when substituted. Mathematica gets $$e^{1/x}xC[1]$$ which simplifies to 0 when substituted. <<*>>= --S 81 of 126 ode135a:=solve(ode135,y,x) --R --R 1 --R - --R x --R (81) [particular= 0,basis= [x %e ]] --RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) --E 81 --S 82 of 126 yx:=ode135a.particular --R --R (82) 0 --R Type: Expression Integer --E 82 --S 83 of 126 ode135expr := x**2*D(yx,x) - (x-1)*yx --R --R (83) 0 --R Type: Expression Integer --E 83 ------------------------------------------------------------------- --S 84 of 126 ode136 := x**2*D(y(x),x) + y(x)**2 + x*y(x) + x**2 --R --R 2 , 2 2 --R (84) x y (x) + y(x) + x y(x) + x --R --R Type: Expression Integer --E 84 @ Maxima gets $$-\frac{x\log(\%c~x)-x}{log(\%c~x)}$$ which simplifies to 0 on substitution. Maple gets $$-\frac{x(-1+\log(x)+~\_C1)}{\log(x)+~\_C1}$$ which simplifies to 0 on substitution. Mathematica gets $$\frac{-x-xC[1]+x\log(x)}{C[1]-\log(x)}$$ which simplifies to 0 on substition. <<*>>= --S 85 of 126 yx:=solve(ode136,y,x) --R --R (- y(x) - x)log(x) + x --R (85) ---------------------- --R y(x) + x --R Type: Union(Expression Integer,...) --E 85 --S 86 of 126 ode136expr := x**2*D(yx,x) + yx**2 + x*yx + x**2 --R --R (86) --R 3 , 2 2 2 --R - x y (x) + (y(x) + 2x y(x) + x )log(x) --R --R + --R 2 2 3 2 2 2 3 --R (- x y(x) + (- 2x - 2x)y(x) - x - 2x )log(x) + (x - x)y(x) + 2x y(x) --R + --R 4 2 --R x + x --R / --R 2 2 --R y(x) + 2x y(x) + x --R Type: Expression Integer --E 86 ------------------------------------------------------------------- --S 87 of 126 ode137 := x**2*D(y(x),x) - y(x)**2 - x*y(x) --R --R 2 , 2 --R (87) x y (x) - y(x) - x y(x) --R --R Type: Expression Integer --E 87 @ Maxima gets $$\frac{x}{\log\left(\displaystyle \frac{1}{\%c~x}\right)}$$ which simplifies to 0 on substitution. Maple gets: $$\frac{x}{-\log(x)+~\_C1}$$ which simplifies to 0 on substitution. Mathematica gets: $$\frac{x}{C[1]-\log(x)}$$ which simplifies to 0 on substitution. <<*>>= --S 88 of 126 yx:=solve(ode137,y,x) --R --R y(x)log(x) + x --R (88) -------------- --R y(x) --R Type: Union(Expression Integer,...) --E 88 --S 89 of 126 ode137expr := x**2*D(yx,x) - yx**2 - x*yx --R --R 3 , 2 2 2 2 2 --R - x y (x) - y(x) log(x) + (- x y(x) - 2x y(x))log(x) + x y(x) - x --R --R (89) --------------------------------------------------------------------- --R 2 --R y(x) --R Type: Expression Integer --E 89 ------------------------------------------------------------------- --S 90 of 126 ode138 := x**2*D(y(x),x) - y(x)**2 - x*y(x) - x**2 --R --R 2 , 2 2 --R (90) x y (x) - y(x) - x y(x) - x --R --R Type: Expression Integer --E 90 @ Maxima gets $$\%c~x=\%e^{\arctan\left(\frac{y}{x}\right)}$$ which does not simplify to 0 when substituted. Maple gets $$\tan(\log(x)+~\_C1)x$$ which simplifies to 0 on substitution. Mathematica get: $$x\tan(C[2]+\log(x))$$ which simplifies to 0 when substituted. <<*>>= --S 91 of 126 yx:=solve(ode138,y,x) --R --R +---+ +---+ --R (- 7\|- 1 + 9)y(x) + 9x\|- 1 + 7x --R (91) -------------------------------------------------------- --R +---+ --R +---+ +---+ - 2\|- 1 log(x) --R ((18\|- 1 + 14)y(x) - 14x\|- 1 + 18x)%e --R Type: Union(Expression Integer,...) --E 91 --S 92 of 126 ode138expr := x**2*D(yx,x) - yx**2 - x*yx - x**2 --R --R (92) --R 3 +---+ 3 4 +---+ 4 --R ((- 1188x \|- 1 + 2716x )y(x) - 2716x \|- 1 - 1188x ) --R * --R +---+ --R - 2\|- 1 log(x) , --R %e y (x) --R --R + --R 2 +---+ 2 3 3 +---+ 3 2 --R (- 1188x \|- 1 + 2716x )y(x) + (- 8148x \|- 1 - 3564x )y(x) --R + --R 4 +---+ 4 5 +---+ 5 --R (3564x \|- 1 - 8148x )y(x) + 2716x \|- 1 + 1188x --R * --R +---+ 2 --R - 2\|- 1 log(x) --R (%e ) --R + --R +---+ 3 2 +---+ 2 2 --R (- 170x\|- 1 - 3310x)y(x) + (4498x \|- 1 - 2886x )y(x) --R + --R 3 +---+ 3 4 +---+ 4 --R (2546x \|- 1 - 2122x )y(x) + 3310x \|- 1 - 170x --R * --R +---+ --R - 2\|- 1 log(x) --R %e --R + --R +---+ 3 +---+ 2 --R (297\|- 1 - 679)y(x) + (- 679x\|- 1 - 297x)y(x) --R + --R 2 +---+ 2 3 +---+ 3 --R (297x \|- 1 - 679x )y(x) - 679x \|- 1 - 297x --R / --R +---+ 3 +---+ 2 --R (1188\|- 1 - 2716)y(x) + (8148x\|- 1 + 3564x)y(x) --R + --R 2 +---+ 2 3 +---+ 3 --R (- 3564x \|- 1 + 8148x )y(x) - 2716x \|- 1 - 1188x --R * --R +---+ 2 --R - 2\|- 1 log(x) --R (%e ) --R Type: Expression Integer --E 92 ------------------------------------------------------------------- --S 93 of 126 ode139 := x**2*(D(y(x),x)+y(x)**2) + a*x**k - b*(b-1) --R --R 2 , k 2 2 2 --R (93) x y (x) + a x + x y(x) - b + b --R --R Type: Expression Integer --E 93 @ Maxima gets 6 answers, one of which is: $$\frac{-\left(3^{5/6}\%i\left(ax^k+\%ckx-\%cx+b^2k-bk-b^2+b\right)^{1/3}- 3^{1/3}\left(ax^k+\%ckx-\%cx+b^2k-bk-b^2+b\right)^{1/3}\right)} {\left(2(k-1)^{1/3}x^{1/3}\right)}$$ which simplifies to 0 on substitution. <<*>>= --S 94 of 126 yx:=solve(ode139,y,x) --R --R (94) "failed" --R Type: Union("failed",...) --E 94 ------------------------------------------------------------------- --S 95 of 126 ode140 := x**2*(D(y(x),x)+y(x)**2) + 4*x*y(x) + 2 --R --R 2 , 2 2 --R (95) x y (x) + x y(x) + 4x y(x) + 2 --R --R Type: Expression Integer --E 95 @ Maxima gets $$-\frac{x-2\%c}{x^2-\%c~x}$$ which simplifies to 0 when substituted. Maple gets $$-\frac{-2~\_C1+x}{x(-~\_C1+x)}$$ which simplifies to 0 when substituted. Mathematica gets: $$-\frac{2}{x}+\frac{1}{x+C[1]}$$ which does not simplify. <<*>>= --S 96 of 126 yx:=solve(ode140,y,x) --R --R x y(x) + 2 --R (96) -------------------- --R 2 --R (x - x)y(x) + x - 2 --R Type: Union(Expression Integer,...) --E 96 --S 97 of 126 ode140expr := x**2*(D(yx,x)+yx**2) + 4*x*yx + 2 --R --R (97) --R 4 , 4 3 2 2 3 2 2 --R - x y (x) + (6x - 8x + 2x )y(x) + (16x - 28x + 8x)y(x) + 12x - 24x + 8 --R --R ---------------------------------------------------------------------------- --R 4 3 2 2 3 2 2 --R (x - 2x + x )y(x) + (2x - 6x + 4x)y(x) + x - 4x + 4 --R Type: Expression Integer --E 97 ------------------------------------------------------------------- --S 98 of 126 ode141 := x**2*(D(y(x),x)+y(x)**2) + a*x*y(x) + b --R --R 2 , 2 2 --R (98) x y (x) + x y(x) + a x y(x) + b --R --R Type: Expression Integer --E 98 @ Maxima gets: $$\%e^{\displaystyle -a\log(x)-2x} \left(\%c-b \int{\displaystyle \frac{\%e^{\displaystyle a\log(x)+2x}}{x^2}}~dx\right)$$ which, when substituted, simplifies to 0. <<*>>= --S 99 of 126 yx:=solve(ode141,y,x) --R 2 --R WARNING (genufact): No known algorithm to factor ? + (a - 1)? + b --R , trying square-free. --R --R (99) --R +------------------+ --R | 2 --R \|- 4b + a - 2a + 1 - 2x y(x) - a + 1 --R / --R +------------------+ --R | 2 2 --R ((2x y(x) + a - 1)\|- 4b + a - 2a + 1 - 4b + a - 2a + 1) --R * --R +------------------+ --R | 2 --R - log(x)\|- 4b + a - 2a + 1 --R %e --R Type: Union(Expression Integer,...) --E 99 --S 100 of 126 ode141expr := x**2*(D(yx,x)+yx**2) + a*x*yx + b --R --R (100) --R 2 4 3 2 3 --R ((- 8b + 2a - 4a + 2)x y(x) + ((- 4a + 4)b + a - 3a + 3a - 1)x ) --R * --R +------------------+ --R | 2 --R \|- 4b + a - 2a + 1 --R + --R 2 2 4 3 2 3 --R (16b + (- 8a + 16a - 8)b + a - 4a + 6a - 4a + 1)x --R * --R +------------------+ --R | 2 --R - log(x)\|- 4b + a - 2a + 1 , --R %e y (x) --R --R + --R 2 2 3 3 --R (8b + (- 2a + 4a - 2)b)x y(x) --R + --R 2 3 2 2 2 --R ((12a - 12)b + (- 3a + 9a - 9a + 3)b)x y(x) --R + --R 3 2 2 --R - 24b + (18a - 36a + 18)b --R + --R 4 3 2 --R (- 3a + 12a - 18a + 12a - 3)b --R * --R x y(x) --R + --R 3 3 2 2 --R (- 12a + 12)b + (7a - 21a + 21a - 7)b --R + --R 5 4 3 2 --R (- a + 5a - 10a + 10a - 5a + 1)b --R * --R +------------------+ --R | 2 --R \|- 4b + a - 2a + 1 --R + --R 3 2 2 4 3 2 2 --R (- 48b + (24a - 48a + 24)b + (- 3a + 12a - 18a + 12a - 3)b)x --R * --R 2 --R y(x) --R + --R 3 3 2 2 --R (- 48a + 48)b + (24a - 72a + 72a - 24)b --R + --R 5 4 3 2 --R (- 3a + 15a - 30a + 30a - 15a + 3)b --R * --R x y(x) --R + --R 4 2 3 4 3 2 2 --R 16b + (- 24a + 48a - 24)b + (9a - 36a + 54a - 36a + 9)b --R + --R 6 5 4 3 2 --R (- a + 6a - 15a + 20a - 15a + 6a - 1)b --R * --R +------------------+ 2 --R | 2 --R - log(x)\|- 4b + a - 2a + 1 --R (%e ) --R + --R 2 4 3 --R (- 8b + 2a - 4a + 2)x y(x) --R + --R 3 2 3 2 --R ((- 16a + 4)b + 4a - 9a + 6a - 1)x y(x) --R + --R 2 2 4 3 2 2 --R (- 8b + (- 6a + 4a + 2)b + 2a - 6a + 6a - 2a)x y(x) --R + --R 2 3 2 --R ((- 8a + 4)b + (2a - 5a + 4a - 1)b)x --R * --R +------------------+ --R | 2 --R \|- 4b + a - 2a + 1 --R + --R 3 2 4 3 --R (- 8a b + 2a - 4a + 2a)x y(x) --R + --R 2 2 4 3 2 3 2 --R (16b + (- 20a + 28a - 8)b + 4a - 13a + 15a - 7a + 1)x y(x) --R + --R 2 3 2 5 4 3 2 2 --R (8a b + (- 10a + 20a - 10a)b + 2a - 8a + 12a - 8a + 2a)x y(x) --R + --R 3 2 2 4 3 2 --R (16b + (- 12a + 20a - 8)b + (2a - 7a + 9a - 5a + 1)b)x --R * --R +------------------+ --R | 2 --R - log(x)\|- 4b + a - 2a + 1 --R %e --R + --R 5 3 4 2 2 3 --R - 2x y(x) + (- 3a + 3)x y(x) + (- 2b - a + 2a - 1)x y(x) --R + --R 2 --R (- a + 1)b x --R * --R +------------------+ --R | 2 --R \|- 4b + a - 2a + 1 --R + --R 2 4 2 3 2 3 --R (- 4b + a - 2a + 1)x y(x) + ((- 4a + 4)b + a - 3a + 3a - 1)x y(x) --R + --R 2 2 2 --R (- 4b + (a - 2a + 1)b)x --R / --R 2 3 3 --R (8b - 2a + 4a - 2)x y(x) --R + --R 3 2 2 2 --R ((12a - 12)b - 3a + 9a - 9a + 3)x y(x) --R + --R 2 2 4 3 2 --R (- 24b + (18a - 36a + 18)b - 3a + 12a - 18a + 12a - 3)x y(x) --R + --R 2 3 2 5 4 3 2 --R (- 12a + 12)b + (7a - 21a + 21a - 7)b - a + 5a - 10a + 10a --R + --R - 5a + 1 --R * --R +------------------+ --R | 2 --R \|- 4b + a - 2a + 1 --R + --R 2 2 4 3 2 2 2 --R (- 48b + (24a - 48a + 24)b - 3a + 12a - 18a + 12a - 3)x y(x) --R + --R 2 3 2 5 4 3 --R (- 48a + 48)b + (24a - 72a + 72a - 24)b - 3a + 15a - 30a --R + --R 2 --R 30a - 15a + 3 --R * --R x y(x) --R + --R 3 2 2 4 3 2 6 --R 16b + (- 24a + 48a - 24)b + (9a - 36a + 54a - 36a + 9)b - a --R + --R 5 4 3 2 --R 6a - 15a + 20a - 15a + 6a - 1 --R * --R +------------------+ 2 --R | 2 --R - log(x)\|- 4b + a - 2a + 1 --R (%e ) --R Type: Expression Integer --E 100 ------------------------------------------------------------------- --S 101 of 126 ode142 := x**2*(D(y(x),x)-y(x)**2) - a*x**2*y(x) + a*x + 2 --R --R 2 , 2 2 2 --R (101) x y (x) - x y(x) - a x y(x) + a x + 2 --R --R Type: Expression Integer --E 101 @ Maxima failed. <<*>>= --S 102 of 126 yx:=solve(ode142,y,x) --R --R 2 3 2 3 3 2 2 --R (a x - 2a x + 2x)y(x) + a x - a x + 2a x - 2 --R (102) ------------------------------------------------ --R 3 3 - a x --R (a x y(x) - a )%e --R Type: Union(Expression Integer,...) --E 102 --S 103 of 126 ode142expr := x**2*(D(yx,x)-yx**2) - a*x**2*yx + a*x + 2 --R --R (103) --R 6 6 - a x , --R - a x %e y (x) --R --R + --R 7 3 6 2 2 7 2 6 7 6 - a x 2 --R ((a x + 2a x )y(x) + (- 2a x - 4a x)y(x) + a x + 2a )(%e ) --R + --R 5 5 4 4 2 6 5 5 4 4 3 6 4 5 3 --R (2a x - 2a x )y(x) + (2a x - 4a x + 4a x )y(x) - 3a x + 2a x --R + --R 4 2 --R - 2a x --R * --R - a x --R %e --R + --R 4 8 3 7 2 6 5 4 2 --R (- a x + 4a x - 8a x + 8a x - 4x )y(x) --R + --R 5 8 4 7 3 6 2 5 4 3 6 8 5 7 --R (- 2a x + 6a x - 12a x + 16a x - 16a x + 8x )y(x) - a x + 2a x --R + --R 4 6 3 5 2 4 3 2 --R - 5a x + 8a x - 8a x + 8a x - 4x --R / --R 6 2 2 6 6 - a x 2 --R (a x y(x) - 2a x y(x) + a )(%e ) --R Type: Expression Integer --E 103 ------------------------------------------------------------------- --S 104 of 126 ode143 := x**2*(D(y(x),x)+a*y(x)**2) - b --R --R 2 , 2 2 --R (104) x y (x) + a x y(x) - b --R --R Type: Expression Integer --E 104 @ Maxima, if$4ab+1 >= 0$gets: $$x=\%c\%e^{ -\frac{\displaystyle\log\left( -\frac{\displaystyle -2axy+\sqrt{4ab+1}+1} {\displaystyle 2axy+\sqrt{4ab+1}-1}\right)} {\displaystyle\sqrt{4ab+1}}}$$ and if$4ab+1 < 0$gets: $$x=\%c\%e^{ -\frac{\displaystyle 2\arctan\left( \frac{\displaystyle 2axy-1}{\displaystyle\sqrt{-4ab-1}}\right)} {\displaystyle\sqrt{-4ab-1}}}$$ neither of which simplify to 0 on substitution. <<*>>= --S 105 of 126 yx:=solve(ode143,y,x) --R 2 --R WARNING (genufact): No known algorithm to factor ? - ? - a b --R , trying square-free. --R --R +--------+ 2 --R a\|4a b + 1 - 2a x y(x) + a --R (105) ------------------------------------------------------------ --R +--------+ --R +--------+ - log(x)\|4a b + 1 --R ((2a x y(x) - 1)\|4a b + 1 + 4a b + 1)%e --R Type: Union(Expression Integer,...) --E 105 --S 106 of 126 ode143expr := x**2*(D(yx,x)+a*yx**2) - b --R --R (106) --R +--------+ --R 3 2 3 - log(x)\|4a b + 1 , --R (- 8a b - 2a )x %e y (x) --R --R + --R 2 2 2 +--------+ --R ((- 8a b - 2a b)x y(x) + 4a b + b)\|4a b + 1 --R + --R 3 2 2 2 2 2 2 2 3 2 --R (- 8a b - 2a b)x y(x) + (8a b + 2a b)x y(x) - 8a b - 6a b - b --R * --R +--------+ 2 --R - log(x)\|4a b + 1 --R (%e ) --R + --R +--------+ --R 4 3 3 2 3 2 2 - log(x)\|4a b + 1 --R ((- 8a b - 2a )x y(x) + (8a b + 2a b)x)%e --R + --R 4 3 3 2 +--------+ 5 4 2 4 3 4 3 2 --R (- 2a x y(x) + a x )\|4a b + 1 + 2a x y(x) - 2a x y(x) + (2a b + a )x --R / --R 2 +--------+ 3 2 2 2 --R ((8a b + 2a)x y(x) - 4a b - 1)\|4a b + 1 + (8a b + 2a )x y(x) --R + --R 2 2 2 --R (- 8a b - 2a)x y(x) + 8a b + 6a b + 1 --R * --R +--------+ 2 --R - log(x)\|4a b + 1 --R (%e ) --R Type: Expression Integer --E 106 ------------------------------------------------------------------- --S 107 of 126 ode144 := x**2*(D(y(x),x)+a*y(x)**2) + b*x**alpha + c --R --R 2 , alpha 2 2 --R (107) x y (x) + b x + a x y(x) + c --R --R Type: Expression Integer --E 107 @ Maxima failed. <<*>>= --S 108 of 126 yx:=solve(ode144,y,x) --R --R (108) "failed" --R Type: Union("failed",...) --E 108 ------------------------------------------------------------------- --S 109 of 126 ode145 := x**2*D(y(x),x) + a*y(x)**3 - a*x**2*y(x)**2 --R --R 2 , 3 2 2 --R (109) x y (x) + a y(x) - a x y(x) --R --R Type: Expression Integer --E 109 @ Maxima failed. Maple claims the result is 0, which when substituted, simplifies to 0 <<*>>= --S 110 of 126 yx:=solve(ode145,y,x) --R --R (110) "failed" --R Type: Union("failed",...) --E 110 ------------------------------------------------------------------- --S 111 of 126 ode146 := x**2*D(y(x),x) + x*y(x)**3 + a*y(x)**2 --R --R 2 , 3 2 --R (111) x y (x) + x y(x) + a y(x) --R --R Type: Expression Integer --E 111 @ Maxima failed. Maple gets 0 which, when substituted, simplifies to 0. <<*>>= --S 112 of 126 yx:=solve(ode146,y,x) --R --R (112) "failed" --R Type: Union("failed",...) --E 112 ------------------------------------------------------------------- --S 113 of 126 ode147 := x**2*D(y(x),x) + a*x**2*y(x)**3 + b*y(x)**2 --R --R 2 , 2 3 2 --R (113) x y (x) + a x y(x) + b y(x) --R --R Type: Expression Integer --E 113 @ Maxima failed. Maple gets 0 which, when substituted, results in 0. <<*>>= --S 114 of 126 yx:=solve(ode147,y,x) --R --R (114) "failed" --R Type: Union("failed",...) --E 114 ------------------------------------------------------------------- --S 115 of 126 ode148 := (x**2+1)*D(y(x),x) + x*y(x) - 1 --R --R 2 , --R (115) (x + 1)y (x) + x y(x) - 1 --R --R Type: Expression Integer --E 115 @ Maxima gets $$({\rm asinh}(x)+\%c)\%e^{-\frac{\displaystyle\log(x^2+1)}{\displaystyle 2}}$$ which when substituted, does not simplify to 0. Maple gets $$\frac{{\rm arcsinh}(x)+~\_C1}{\sqrt{x^2+1}}$$ which when substituted, simplifies to 0. Mathematica gets $$\frac{{\rm arcsinh}(x)}{\sqrt{1+x^2}}+\frac{C[1]}{\sqrt{1+x^2}}$$ gives 0 when substituted. <<*>>= --S 116 of 126 ode148a:=solve(ode148,y,x) --R --R +------+ --R | 2 --R log(\|x + 1 - x) 1 --R (116) [particular= - ------------------,basis= [---------]] --R +------+ +------+ --R | 2 | 2 --R \|x + 1 \|x + 1 --RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) --E 116 --S 117 of 126 yx:=ode148a.particular --R --R +------+ --R | 2 --R log(\|x + 1 - x) --R (117) - ------------------ --R +------+ --R | 2 --R \|x + 1 --R Type: Expression Integer --E 117 --S 118 of 126 ode148expr := (x**2+1)*D(yx,x) + x*yx - 1 --R --R (118) 0 --R Type: Expression Integer --E 118 ------------------------------------------------------------------- --S 119 of 126 ode149 := (x**2+1)*D(y(x),x) + x*y(x) - x*(x**2+1) --R --R 2 , 3 --R (119) (x + 1)y (x) + x y(x) - x - x --R --R Type: Expression Integer --E 119 @ Maxima gets $$\left(\displaystyle\frac{(x^2+1)^{3/2}}{3}+\%c\right) \%e^{\displaystyle -\frac{log(x^2+1)}{2}}$$ which simplifies to 0 when substituted. Maple gets $$\frac{x^2}{3}+\frac{1}{3}+\frac{\_C1}{\sqrt{x^2+1}}$$ which simplifies to 0 when substituted. Mathematica gets $$\frac{1}{3}(1+x^2)+\frac{C[1]}{\sqrt{1+x^2}}$$ which simplifes to 0 when substituted. <<*>>= --S 120 of 126 ode149a:=solve(ode149,y,x) --R --R 2 --R x + 1 1 --R (120) [particular= ------,basis= [---------]] --R 3 +------+ --R | 2 --R \|x + 1 --RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) --E 120 --S 121 of 126 yx:=ode149a.particular --R --R 2 --R x + 1 --R (121) ------ --R 3 --R Type: Expression Integer --E 121 --S 122 of 126 ode149expr := (x**2+1)*D(yx,x) + x*yx - x*(x**2+1) --R --R (122) 0 --R Type: Expression Integer --E 122 ------------------------------------------------------------------- --S 123 of 126 ode150 := (x**2+1)*D(y(x),x) + 2*x*y(x) - 2*x**2 --R --R 2 , 2 --R (123) (x + 1)y (x) + 2x y(x) - 2x --R --R Type: Expression Integer --E 123 @ Maxima gets $$\displaystyle\frac{\frac{2x^3}{3}+\%c}{x^2+1}$$ which simplifies to 0 on substitution. Maple gets $$\frac{\frac{2x^3}{3}+~\_C1}{x^2+1}$$ which simplifies to 0 on substitution. Mathematica gets: $$\frac{2x^3}{3(1+x^2)}+\frac{C[1]}{1+x^2}$$ which simplifies to 0 on substitution. <<*>>= --S 124 of 126 ode150a:=solve(ode150,y,x) --R --R 3 --R 2x + 3 1 --R (124) [particular= -------,basis= [------]] --R 2 2 --R 3x + 3 x + 1 --RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) --E 124 --S 125 of 126 yx:=ode150a.particular --R --R 3 --R 2x + 3 --R (125) ------- --R 2 --R 3x + 3 --R Type: Expression Integer --E 125 --S 126 of 126 ode150expr := (x**2+1)*D(yx,x) + 2*x*yx - 2*x**2 --R --R (126) 0 --R Type: Expression Integer --E 126 )spool )lisp (bye) @ \eject \begin{thebibliography}{99} \bibitem{1} {\bf http://www.cs.uwaterloo.ca/$\tilde{}\$ecterrab/odetools.html} \bibitem{2} Mathematica 6.0.1.0 \bibitem{3} Maple 11.01 Build ID 296069 \bibitem{4} Maxima 5.13.0 \end{thebibliography} \end{document}