diff --git a/changelog b/changelog index 443b749..06f1da0 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,4 @@ +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 20080314 tpd src/input/Makefile add heugcd.input diff --git a/src/input/kamke2.input.pamphlet b/src/input/kamke2.input.pamphlet index 2c8be63..7d31b41 100644 --- a/src/input/kamke2.input.pamphlet +++ b/src/input/kamke2.input.pamphlet @@ -49,6 +49,11 @@ ode101 := x*D(y(x),x) + x*y(x)**2 - y(x) --R Type: Expression Integer --E 4 +@ +Mathematica gives +$$y(x)=\frac{2*x}{x^2+2}$$ +which can be substituted and simplifies to 0. +<<*>>= --S 5 of 131 yx:=solve(ode101,y,x) --R @@ -80,6 +85,12 @@ ode102 := x*D(y(x),x) + x*y(x)**2 - y(x) - a*x**3 --R Type: Expression Integer --E 7 +@ +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 131 yx:=solve(ode102,y,x) --R @@ -202,6 +213,13 @@ ode103 := x*D(y(x),x) + x*y(x)**2 - (2*x**2+1)*y(x) - x**3 --R Type: Expression Integer --E 10 +@ +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 131 yx:=solve(ode103,y,x) --R @@ -276,6 +294,11 @@ ode104 := x*D(y(x),x) + a*x*y(x)**2 + 2*y(x) + b*x --R Type: Expression Integer --E 13 +@ +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. +<<*>>= --S 14 of 131 yx:=solve(ode104,y,x) --R @@ -368,6 +391,12 @@ ode105 := x*D(y(x),x) + a*x*y(x)**2 + b*y(x) + c*x + d --R Type: Expression Integer --E 16 +@ +Note that this complains about being unable to factor +$$x^3-3x^2+(-b^2+2b+2)x+b^2-2b$$ +but MMA factors this instantly to be: +$$-((b-x) (-1+x) (-2+b+x))$$ +<<*>>= --S 17 of 131 yx:=solve(ode105,y,x) --R WARNING (genufact): No known algorithm to factor @@ -389,6 +418,10 @@ ode106 := x*D(y(x),x) + x**a*y(x)**2 + (a-b)*y(x)/2 + x**b --R Type: Expression Integer --E 18 +@ +Mathematica gets +$$e^{-\frac{1}{2}a\log(x)+\frac{1}{2}b\log(x)}\tan\left(\frac{2x^{\frac{a+b}{2}}}{a+b}-C[1]\right)$$ +<<*>>= --S 19 of 131 yx:=solve(ode106,y,x) --R @@ -420,7 +453,11 @@ ode108 := x*D(y(x),x) - y(x)**2*log(x) + y(x) --R --R Type: Expression Integer --E 22 - +@ +Mathematica gets: +$$\frac{1}{1+xC[1]+\log(x)}$$ +which, on substitution, simplifies to 0. +<<*>>= --S 23 of 131 yx:=solve(ode108,y,x) --R @@ -455,6 +492,11 @@ ode109 := x*D(y(x),x) - y(x)*(2*y(x)*log(x)-1) --R Type: Expression Integer --E 25 +@ +Mathematica gets +$$\frac{1}{2+xC[1]+2\log(x)}$$ +which simplifies to 0 on substitution. +<<*>>= --S 26 of 131 yx:=solve(ode109,y,x) --R @@ -539,6 +581,13 @@ ode113 := x*D(y(x),x) + a*sqrt(y(x)**2 + x**2) - y(x) --R Type: Expression Integer --E 34 +@ +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 35 of 131 yx:=solve(ode113,y,x) --R @@ -556,6 +605,11 @@ ode114 := x*D(y(x),x) - x*sqrt(y(x)**2 + x**2) - y(x) --R Type: Expression Integer --E 36 +@ +Mathematica gets +$$x\sinh(x+C[1])$$ +but cannot simplify the substituted expression to 0. +<<*>>= --S 37 of 131 yx:=solve(ode114,y,x) --R @@ -573,6 +627,10 @@ ode115 := x*D(y(x),x) - x*(y(x)-x)*sqrt(y(x)**2 + x**2) - y(x) --R Type: Expression Integer --E 38 +@ +Mathematica claims that the equations appear to involve the variables +to be solved for in an essentially non-algebraic way. +<<*>>= --S 39 of 131 yx:=solve(ode115,y,x) --R @@ -590,6 +648,12 @@ ode116 := x*D(y(x),x) - x*sqrt((y(x)**2 - x**2)*(y(x)**2-4*x**2)) - y(x) --R Type: Expression Integer --E 40 +@ +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 41 of 131 yx:=solve(ode116,y,x) --R @@ -608,6 +672,13 @@ ode117 := x*D(y(x),x) - x*exp(y(x)/x) - y(x) - x --R Type: Expression Integer --E 42 +@ +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 43 of 131 yx:=solve(ode117,y,x) --R @@ -624,6 +695,13 @@ ode118 := x*D(y(x),x) - y(x)*log(y(x)) --R Type: Expression Integer --E 44 +@ +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 45 of 131 yx:=solve(ode118,y,x) --R @@ -654,6 +732,11 @@ ode119 := x*D(y(x),x) - y(x)*(log(x*y(x))-1) --R Type: Expression Integer --E 47 +@ +Mathematica gets +$$\frac{1}{x(C[1]-log(log(x)))}$$ +which does not simplify to 0 on substitution. +<<*>>= --S 48 of 131 yx:=solve(ode119,y,x) --R @@ -671,6 +754,10 @@ ode120 := x*D(y(x),x) - y(x)*(x*log(x**2/y(x))+2) --R Type: Expression Integer --E 49 +@ +Mathematics get: +$$2e^{-e^{-x} C[1]+e^{-x}{\rm ExpIntegralEi}[x]}x$$ +<<*>>= --S 50 of 131 yx:=solve(ode120,y,x) --R @@ -687,6 +774,10 @@ ode121 := x*D(y(x),x) + sin(y(x)-x) --R Type: Expression Integer --E 51 +@ +Mathematics gets +$$\frac{\sin(x)}{1+\sin(x)}+x^{-sin(x)}C[1]$$ +<<*>>= --S 52 of 131 yx:=solve(ode121,y,x) --R @@ -703,6 +794,11 @@ ode122 := x*D(y(x),x) + (sin(y(x))-3*x**2*cos(y(x)))*cos(y(x)) --R Type: Expression Integer --E 53 +@ +Mathematica gets: +$$\arctan\left(\frac{2x^3+C[1]}{2x}\right)$$ +which, on substitution, simplifies to 0. +<<*>>= --S 54 of 131 yx:=solve(ode122,y,x) --R @@ -719,6 +815,11 @@ ode123 := x*D(y(x),x) - x*sin(y(x)/x) - y(x) --R Type: Expression Integer --E 55 +@ +Mathematica get: +$$x^{1+sin(x)}C[1]$$ +which does not simplfy to 0 on substitution. +<<*>>= --S 56 of 131 yx:=solve(ode123,y,x) --R @@ -735,6 +836,11 @@ ode124 := x*D(y(x),x) + x*cos(y(x)/x) - y(x) + x --R Type: Expression Integer --E 57 +@ +Mathematics gets +$$2x\arctan(C[1]-\log(x))$$ +which does not simplify to 0 on substitution. +<<*>>= --S 58 of 131 yx:=solve(ode124,y,x) --R @@ -751,6 +857,11 @@ ode125 := x*D(y(x),x) + x*tan(y(x)/x) - y(x) --R Type: Expression Integer --E 59 +@ +Mathematica gets +$$\arcsin\left(\frac{e^{C[1]}}{x}\right)$$ +which does not simplify to 0 on substitution. +<<*>>= --S 60 of 131 yx:=solve(ode125,y,x) --R @@ -767,6 +878,11 @@ ode126 := x*D(y(x),x) - y(x)*f(x*y(x)) --R Type: Expression Integer --E 61 +@ +Mathematica gets +$$\frac{1}{-f(x)-C[1]}$$ +which does not simplify to 0 on substitution. +<<*>>= --S 62 of 131 yx:=solve(ode126,y,x) --R @@ -782,7 +898,10 @@ ode127 := x*D(y(x),x) - y(x)*f(x**a*y(x)**b) --R --R Type: Expression Integer --E 63 - +@ +Mathematica gives: +$$b\left(-\frac{f(x^a)}{a}-C[1]\right)^{-1/b}$$ +<<*>>= --S 64 of 131 yx:=solve(ode127,y,x) --R @@ -798,7 +917,10 @@ ode128 := x*D(y(x),x) + a*y(x) - f(x)*g(x**a*y(x)) --R --R Type: Expression Integer --E 65 - +@ +Mathematica gives +$$e^{\frac{f(x)g(x^{1+a})}{1+a}-a\log(x)}C[1]$$ +<<*>>= --S 66 of 131 yx:=solve(ode128,y,x) --R @@ -814,7 +936,11 @@ ode129 := (x+1)*D(y(x),x) + y(x)*(y(x)-x) --R --R Type: Expression Integer --E 67 - +@ +Mathematica gives +$$-\frac{e^{1+x}}{e^{1+x}-eC[1]-exC[1]-{\rm ExpIntegralEi}(1+x)- +x{\rm ExpIntegralEi}(1+x)}$$ +<<*>>= --S 68 of 131 yx:=solve(ode129,y,x) --R @@ -837,7 +963,11 @@ ode130 := 2*x*D(y(x),x) - y(x) -2*x**3 --R --R Type: Expression Integer --E 69 - +@ +Mathematica gives +$$\frac{2x^3}{5}+\sqrt{x}C[1]$$ +which simplifies to 0 on substitution. +<<*>>= --S 70 of 131 ode130a:=solve(ode130,y,x) --R @@ -873,7 +1003,11 @@ ode131 := (2*x+1)*D(y(x),x) - 4*exp(-y(x)) + 2 --R --R Type: Expression Integer --E 73 - +@ +Mathematica gives +$$\log\left(2+\frac{1}{1+2x}\right)$$ +which simplifies to 0 when substituted. +<<*>>= --S 74 of 131 yx:=solve(ode131,y,x) --R @@ -904,7 +1038,13 @@ ode132 := 3*x*D(y(x),x) - 3*x*log(x)*y(x)**4 - y(x) --R --R Type: Expression Integer --E 76 - +@ +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 77 of 131 yx:=solve(ode132,y,x) --R @@ -957,7 +1097,11 @@ ode133 := x**2*D(y(x),x) + y(x) - x --R --R Type: Expression Integer --E 79 - +@ +Mathematica gets: +$$e^{1/x}C[1]-e^{1/x}{\rm ExpIntegralEi}\left(-\frac{1}{x}\right)$$ +which simplifies to 0 on substitution. +<<*>>= --S 80 of 131 yx:=solve(ode133,y,x) --R @@ -983,7 +1127,15 @@ ode134 := x**2*D(y(x),x) - y(x) + x**2*exp(x-1/x) --R --R Type: Expression Integer --E 81 - +@ +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 82 of 131 ode134a:=solve(ode134,y,x) --R @@ -1021,7 +1173,11 @@ ode135 := x**2*D(y(x),x) - (x-1)*y(x) --R --R Type: Expression Integer --E 85 - +@ +Mathematica gets +$$e^{1/x}xC[1]$$ +which simplifies to 0 when substituted. +<<*>>= --S 86 of 131 ode135a:=solve(ode135,y,x) --R @@ -1054,7 +1210,11 @@ ode136 := x**2*D(y(x),x) + y(x)**2 + x*y(x) + x**2 --R --R Type: Expression Integer --E 89 - +@ +Mathematica gets +$$\frac{-x-xC[1]+x\log(x)}{C[1]-\log(x)}$$ +which simplifies to 0 on substition. +<<*>>= --S 90 of 131 yx:=solve(ode136,y,x) --R @@ -1091,7 +1251,11 @@ ode137 := x**2*D(y(x),x) - y(x)**2 - x*y(x) --R --R Type: Expression Integer --E 92 - +@ +Mathematica gets: +$$\frac{x}{C[1]-\log(x)}$$ +which simplifies to 0 on substitution. +<<*>>= --S 93 of 131 yx:=solve(ode137,y,x) --R @@ -1112,7 +1276,11 @@ 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 @@ -1199,7 +1367,11 @@ 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 @@ -1602,7 +1774,11 @@ yx:=solve(ode147,y,x) --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 @@ -1644,7 +1820,11 @@ 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 @@ -1683,6 +1863,11 @@ 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 @@ -1727,5 +1912,6 @@ ode150expr := (x**2+1)*D(yx,x) + 2*x*yx - 2*x**2 \eject \begin{thebibliography}{99} \bibitem{1} {\bf http://www.cs.uwaterloo.ca/$\tilde{}$ecterrab/odetools.html} +\bibitem{2} Mathematica 6.0.1.0 \end{thebibliography} \end{document}