diff --git a/changelog b/changelog index 06f1da0..1a10e31 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,4 @@ +20080317 tpd src/input/kamke2.input check results using Maple 20080316 tpd src/input/kamke2.input check results using Mathematica. 20080316 acr src/algebra/mathml.spad invisibletimes == 20080314 tpd Makefile --enable-maxpage=512*1024 due to kamke2 diff --git a/src/input/kamke2.input.pamphlet b/src/input/kamke2.input.pamphlet index 7d31b41..c7bf2ba 100644 --- a/src/input/kamke2.input.pamphlet +++ b/src/input/kamke2.input.pamphlet @@ -50,8 +50,12 @@ ode101 := x*D(y(x),x) + x*y(x)**2 - y(x) --E 4 @ +Maple gives +$$\frac{2x}{x^2+2\_C1}$$ +which can be substituted and simplifies to 0. + Mathematica gives -$$y(x)=\frac{2*x}{x^2+2}$$ +$$y(x)=\frac{2x}{x^2+2}$$ which can be substituted and simplifies to 0. <<*>>= --S 5 of 131 @@ -86,6 +90,10 @@ ode102 := x*D(y(x),x) + x*y(x)**2 - y(x) - a*x**3 --E 7 @ +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 $$\sqrt{a}~x~ \tanh\left(\frac{1}{2}\left(\sqrt{a}~x^2+2\sqrt{a}~C[1]\right)\right)$$ @@ -214,6 +222,11 @@ ode103 := x*D(y(x),x) + x*y(x)**2 - (2*x**2+1)*y(x) - x**3 --E 10 @ +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 $$\frac{\left(e^{\sqrt{x}~x^2}+\sqrt{2}~e^{\sqrt{2}~x^2}+ e^{2\sqrt{2}~C[1]}-\sqrt{2}~e^{2\sqrt{2}~C[1]}\right)x} @@ -295,6 +308,14 @@ ode104 := x*D(y(x),x) + a*x*y(x)**2 + 2*y(x) + b*x --E 13 @ +Maple gets: +$$-\frac{\sqrt{x(a+b)} +\left(\_C1~{\rm BesselY}\left(3,2\sqrt{x(a+b)}\right)+ +{\rm BesselJ}\left(2,2\sqrt{x(a+b)}\right)\right)} +{\_C1~{\rm BesselY}\left(2,2\sqrt{x(a+b)}\right)+ +{\rm BesselJ}\left(2,2\sqrt{x(a+b)}\right)}$$ +which simplifies to 0 on substitution. + Mathematica gets: $$-\frac{1}{ax}-\sqrt{\frac{b}{a}}~\tan\left(a\sqrt{\frac{b}{a}}~x-C[1]\right)$$ but cannot simplify the substitution to 0. @@ -419,8 +440,19 @@ ode106 := x*D(y(x),x) + x**a*y(x)**2 + (a-b)*y(x)/2 + x**b --E 18 @ +Maple gets +$$-\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)$$ +$$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)$$ +which does not simplify to 0 on substitution. <<*>>= --S 19 of 131 yx:=solve(ode106,y,x) @@ -454,6 +486,10 @@ ode108 := x*D(y(x),x) - y(x)**2*log(x) + y(x) --R Type: Expression Integer --E 22 @ +Maple gets: +$$\frac{1}{1+\log(x)+x\_C1}$$ +which, on substitution, simplifies to 0. + Mathematica gets: $$\frac{1}{1+xC[1]+\log(x)}$$ which, on substitution, simplifies to 0. @@ -493,6 +529,11 @@ ode109 := x*D(y(x),x) - y(x)*(2*y(x)*log(x)-1) --E 25 @ +Maple gets: +$$\frac{1}{2+2\log(x)+x~\_C1}$$ +which simplifies to 0 on substitition. + + Mathematica gets $$\frac{1}{2+xC[1]+2\log(x)}$$ which simplifies to 0 on substitution. @@ -547,6 +588,10 @@ ode111 := x*D(y(x),x) + y(x)**3 + 3*x*y(x)**2 --R Type: Expression Integer --E 30 +@ +Maple gets 0 which simplifies to 0 on substitution. +<<*>>= + --S 31 of 131 yx:=solve(ode111,y,x) --R @@ -554,6 +599,10 @@ yx:=solve(ode111,y,x) --R Type: Union("failed",...) --E 31 +@ +Maple gets 0 but simplification gives the result $csgn(x)x$. +<<*>>= + --S 32 of 131 ode112 := x*D(y(x),x) - sqrt(y(x)**2 + x**2) - y(x) --R @@ -582,6 +631,8 @@ ode113 := x*D(y(x),x) + a*sqrt(y(x)**2 + x**2) - y(x) --E 34 @ +Maple gets 0 but on substitition this simplifies to $a~csgn(x)~x$ + Mathematica gets $$x*\sinh(C[1]+\log(x))$$ If we choose $C[1]=0$ this simplifies to @@ -606,6 +657,8 @@ ode114 := x*D(y(x),x) - x*sqrt(y(x)**2 + x**2) - y(x) --E 36 @ +Maple gets 0 but, on substitition, simplifies to $-x^2csqn(x)$. + Mathematica gets $$x\sinh(x+C[1])$$ but cannot simplify the substituted expression to 0. @@ -628,6 +681,9 @@ ode115 := x*D(y(x),x) - x*(y(x)-x)*sqrt(y(x)**2 + x**2) - y(x) --E 38 @ +Maple claims the result is 0 but simplifies it, on substitution, to +$x^3 csgn(x)$. + Mathematica claims that the equations appear to involve the variables to be solved for in an essentially non-algebraic way. <<*>>= @@ -649,6 +705,9 @@ ode116 := x*D(y(x),x) - x*sqrt((y(x)**2 - x**2)*(y(x)**2-4*x**2)) - y(x) --E 40 @ +Maple claims the answer is 0 but simplifies, on substitution, to +$-2x^3 csgn(x^2)$. + Mathematica says that a potential solution of ComplexInfinity was possibly discarded by the verifier and should be checked by hand, possibly using limits. And the equations appear to involve the variables to be solved @@ -673,6 +732,10 @@ ode117 := x*D(y(x),x) - x*exp(y(x)/x) - y(x) - x --E 42 @ +Maple gets: +$$\left(\log\left(-\frac{x}{-1+x~e^{\_C1}}\right)+\_C1\right)x$$ +which simplifies to 0 on substitution. + Mathematica says that inverse functions are being used by Solve, so some solutions may not be found and to use Reduce for complete solution information. It gets the answer: @@ -696,6 +759,10 @@ ode118 := x*D(y(x),x) - y(x)*log(y(x)) --E 44 @ +Maple gets +$$e^{(x~\_C1)}$$ +which, on substitution, does not simplify to 0. + Mathematics gets $$e^{e^{C[1]}x}$$ which, on substitution simplifies to @@ -733,6 +800,10 @@ ode119 := x*D(y(x),x) - y(x)*(log(x*y(x))-1) --E 47 @ +Maple get +$$\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)))}$$ which does not simplify to 0 on substitution. @@ -755,8 +826,13 @@ ode120 := x*D(y(x),x) - y(x)*(x*log(x**2/y(x))+2) --E 49 @ +Maple gets +$$\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$$ +which does not simplify to 0 on substitution. <<*>>= --S 50 of 131 yx:=solve(ode120,y,x) @@ -777,6 +853,7 @@ ode121 := x*D(y(x),x) + sin(y(x)-x) @ Mathematics gets $$\frac{\sin(x)}{1+\sin(x)}+x^{-sin(x)}C[1]$$ +which, on substitution, does not simplify to 0. <<*>>= --S 52 of 131 yx:=solve(ode121,y,x) @@ -795,6 +872,10 @@ ode122 := x*D(y(x),x) + (sin(y(x))-3*x**2*cos(y(x)))*cos(y(x)) --E 53 @ +Maple gets: +$$\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)$$ which, on substitution, simplifies to 0. @@ -816,6 +897,11 @@ ode123 := x*D(y(x),x) - x*sin(y(x)/x) - y(x) --E 55 @ +Maple gets: +$$\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]$$ which does not simplfy to 0 on substitution. @@ -837,6 +923,10 @@ ode124 := x*D(y(x),x) + x*cos(y(x)/x) - y(x) + x --E 57 @ +Maple gets +$$-2\arctan(\log(x)+~\_C1)x$$ +which, on substitution, does not simplify to 0. + Mathematics gets $$2x\arctan(C[1]-\log(x))$$ which does not simplify to 0 on substitution. @@ -858,6 +948,10 @@ ode125 := x*D(y(x),x) + x*tan(y(x)/x) - y(x) --E 59 @ +Maple gets +$$\arcsin\left(\frac{1}{x~\_C1}\right)x$$ +which, on substitution, simplifies to 0. + Mathematica gets $$\arcsin\left(\frac{e^{C[1]}}{x}\right)$$ which does not simplify to 0 on substitution. @@ -879,6 +973,11 @@ ode126 := x*D(y(x),x) - y(x)*f(x*y(x)) --E 61 @ +Maple gets +$$\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]}$$ which does not simplify to 0 on substitution. @@ -899,8 +998,11 @@ ode127 := x*D(y(x),x) - y(x)*f(x**a*y(x)**b) --R Type: Expression Integer --E 63 @ +Maple gives 0 which, on substitution simplifies to 0. + Mathematica gives: $$b\left(-\frac{f(x^a)}{a}-C[1]\right)^{-1/b}$$ +which, on substitution, does not simplify to 0. <<*>>= --S 64 of 131 yx:=solve(ode127,y,x) @@ -918,8 +1020,14 @@ ode128 := x*D(y(x),x) + a*y(x) - f(x)*g(x**a*y(x)) --R Type: Expression Integer --E 65 @ +Maple gives +$$\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]$$ +which, on substitution, does not simplify to 0. <<*>>= --S 66 of 131 yx:=solve(ode128,y,x) @@ -937,6 +1045,11 @@ ode129 := (x+1)*D(y(x),x) + y(x)*(y(x)-x) --R Type: Expression Integer --E 67 @ +Maple gives +$$\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)}$$ @@ -964,6 +1077,10 @@ ode130 := 2*x*D(y(x),x) - y(x) -2*x**3 --R Type: Expression Integer --E 69 @ +Maple gives +$$\frac{2x^3}{5}+\sqrt{x}~\_C1$$ +which, on substitution, simplifies to 0. + Mathematica gives $$\frac{2x^3}{5}+\sqrt{x}C[1]$$ which simplifies to 0 on substitution. @@ -1004,6 +1121,10 @@ ode131 := (2*x+1)*D(y(x),x) - 4*exp(-y(x)) + 2 --R Type: Expression Integer --E 73 @ +Maple gives +$$-\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)$$ which simplifies to 0 when substituted. @@ -1039,6 +1160,22 @@ ode132 := 3*x*D(y(x),x) - 3*x*log(x)*y(x)**4 - y(x) --R Type: Expression Integer --E 76 @ +Maple gives 3 solutions. +$$\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}}$$ @@ -1098,6 +1235,10 @@ ode133 := x**2*D(y(x),x) + y(x) - x --R Type: Expression Integer --E 79 @ +Maple gives +$$\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)$$ which simplifies to 0 on substitution. @@ -1128,6 +1269,10 @@ ode134 := x**2*D(y(x),x) - y(x) + x**2*exp(x-1/x) --R Type: Expression Integer --E 81 @ +Maple gets +$$(-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]$$ which does not simplify to 0 on substitution. @@ -1174,6 +1319,10 @@ ode135 := x**2*D(y(x),x) - (x-1)*y(x) --R Type: Expression Integer --E 85 @ +Maple gets +$$\_C1xe^{\left(\frac{1}{x}\right)}$$ +which simplifies to 0 when substituted. + Mathematica gets $$e^{1/x}xC[1]$$ which simplifies to 0 when substituted. @@ -1211,6 +1360,10 @@ ode136 := x**2*D(y(x),x) + y(x)**2 + x*y(x) + x**2 --R Type: Expression Integer --E 89 @ +Maple gets +$$-\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)}$$ which simplifies to 0 on substition. @@ -1252,6 +1405,10 @@ ode137 := x**2*D(y(x),x) - y(x)**2 - x*y(x) --R Type: Expression Integer --E 92 @ +Maple gets: +$$\frac{x}{-\log(x)+~\_C1}$$ +which simplifies to 0 on substitution. + Mathematica gets: $$\frac{x}{C[1]-\log(x)}$$ which simplifies to 0 on substitution. @@ -1276,11 +1433,7 @@ ode137expr := x**2*D(yx,x) - yx**2 - x*yx --R y(x) --R Type: Expression Integer --E 94 -@ -Mathematica get: -$$x\tan(C[2]+\log(x))$$ -which simplifies to 0 when substituted. -<<*>>= + --S 95 of 131 ode138 := x**2*D(y(x),x) - y(x)**2 - x*y(x) - x**2 --R @@ -1289,6 +1442,15 @@ ode138 := x**2*D(y(x),x) - y(x)**2 - x*y(x) - x**2 --R --R Type: Expression Integer --E 95 +@ +Maple gets +$$\tan(\log(x)+~\_C1)x$$ +which simplifies to 0 on substitution. + +Mathematica get: +$$x\tan(C[2]+\log(x))$$ +which simplifies to 0 when substituted. +<<*>>= --S 96 of 131 yx:=solve(ode138,y,x) @@ -1367,11 +1529,7 @@ yx:=solve(ode139,y,x) --R (99) "failed" --R Type: Union("failed",...) --E 99 -@ -Mathematica gets: -$$-\frac{2}{x}+\frac{1}{x+C[1]}$$ -which does not simplify. -<<*>>= + --S 100 of 131 ode140 := x**2*(D(y(x),x)+y(x)**2) + 4*x*y(x) + 2 --R @@ -1380,7 +1538,15 @@ ode140 := x**2*(D(y(x),x)+y(x)**2) + 4*x*y(x) + 2 --R --R Type: Expression Integer --E 100 +@ +Maple gets +$$-\frac{-2~\_C1+x}{x(-~\_C1+x)}$$ +which simplifies to 0 when substituted. +Mathematica gets: +$$-\frac{2}{x}+\frac{1}{x+C[1]}$$ +which does not simplify. +<<*>>= --S 101 of 131 yx:=solve(ode140,y,x) --R @@ -1736,6 +1902,9 @@ ode145 := x**2*D(y(x),x) + a*y(x)**3 - a*x**2*y(x)**2 --R Type: Expression Integer --E 114 +@ +Maple claims the result is 0, which when substituted, simplifies to 0 +<<*>>= --S 115 of 131 yx:=solve(ode145,y,x) --R @@ -1752,6 +1921,9 @@ ode146 := x**2*D(y(x),x) + x*y(x)**3 + a*y(x)**2 --R Type: Expression Integer --E 116 +@ +Maple gets 0 which, when substituted, simplifies to 0. +<<*>>= --S 117 of 131 yx:=solve(ode146,y,x) --R @@ -1767,18 +1939,16 @@ ode147 := x**2*D(y(x),x) + a*x**2*y(x)**3 + b*y(x)**2 --R --R Type: Expression Integer --E 118 - +@ +Maple gets 0 which, when substituted, results in 0. +<<*>>= --S 119 of 131 yx:=solve(ode147,y,x) --R --R (119) "failed" --R Type: Union("failed",...) --E 119 -@ -Mathematica gets -$$\frac{{\rm arcsinh}(x)}{\sqrt{1+x^2}}+\frac{C[1]}{\sqrt{1+x^2}}$$ -gives 0 when substituted. -<<*>>= + --S 120 of 131 ode148 := (x**2+1)*D(y(x),x) + x*y(x) - 1 --R @@ -1787,7 +1957,15 @@ ode148 := (x**2+1)*D(y(x),x) + x*y(x) - 1 --R --R Type: Expression Integer --E 120 +@ +Maple gets +$$\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}}$$ +gives 0 when substituted. +<<*>>= --S 121 of 131 ode148a:=solve(ode148,y,x) --R @@ -1820,11 +1998,7 @@ ode148expr := (x**2+1)*D(yx,x) + x*yx - 1 --R (123) 0 --R Type: Expression Integer --E 123 -@ -Mathematica gets -$$\frac{1}{3}(1+x^2)+\frac{C[1]}{\sqrt{1+x^2}}$$ -which simplifes to 0 when substituted. -<<*>>= + --S 124 of 131 ode149 := (x**2+1)*D(y(x),x) + x*y(x) - x*(x**2+1) --R @@ -1833,7 +2007,15 @@ ode149 := (x**2+1)*D(y(x),x) + x*y(x) - x*(x**2+1) --R --R Type: Expression Integer --E 124 +@ +Maple gets +$$\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}}$$ +which simplifes to 0 when substituted. +<<*>>= --S 125 of 131 ode149a:=solve(ode149,y,x) --R @@ -1863,11 +2045,6 @@ ode149expr := (x**2+1)*D(yx,x) + x*yx - x*(x**2+1) --R Type: Expression Integer --E 127 -@ -Mathematica gets: -$$\frac{2x^3}{3(1+x^2)}+\frac{C[1]}{1+x^2}$$ -which simplifies to 0 on substitution. -<<*>>= --S 128 of 131 ode150 := (x**2+1)*D(y(x),x) + 2*x*y(x) - 2*x**2 --R @@ -1876,6 +2053,15 @@ ode150 := (x**2+1)*D(y(x),x) + 2*x*y(x) - 2*x**2 --R --R Type: Expression Integer --E 128 +@ +Maple gets +$$\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 129 of 131 ode150a:=solve(ode150,y,x) @@ -1913,5 +2099,6 @@ ode150expr := (x**2+1)*D(yx,x) + 2*x*yx - 2*x**2 \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 \end{thebibliography} \end{document}