diff --git a/changelog b/changelog index 1a10e31..4921b44 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,4 @@ +20080318 tpd src/input/kamke2.input check results using Maxima 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 == diff --git a/src/input/kamke2.input.pamphlet b/src/input/kamke2.input.pamphlet index c7bf2ba..0419904 100644 --- a/src/input/kamke2.input.pamphlet +++ b/src/input/kamke2.input.pamphlet @@ -40,6 +40,7 @@ g:=operator 'g --R Type: BasicOperator --E 3 +------------------------------------------------------------------- --S 4 of 131 ode101 := x*D(y(x),x) + x*y(x)**2 - y(x) --R @@ -50,6 +51,9 @@ ode101 := x*D(y(x),x) + x*y(x)**2 - y(x) --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. @@ -80,6 +84,7 @@ ode101expr := x*D(yx,x) + x*yx**2 - yx --R Type: Expression Integer --E 6 +------------------------------------------------------------------- --S 7 of 131 ode102 := x*D(y(x),x) + x*y(x)**2 - y(x) - a*x**3 --R @@ -90,6 +95,8 @@ ode102 := x*D(y(x),x) + x*y(x)**2 - y(x) - a*x**3 --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. @@ -212,6 +219,7 @@ ode102expr := x*D(yx,x) + x*yx**2 - yx - a*x**3 --R Type: Expression Integer --E 9 +------------------------------------------------------------------- --S 10 of 131 ode103 := x*D(y(x),x) + x*y(x)**2 - (2*x**2+1)*y(x) - x**3 --R @@ -222,6 +230,8 @@ ode103 := x*D(y(x),x) + x*y(x)**2 - (2*x**2+1)*y(x) - x**3 --E 10 @ +Maxima fails. + Maple gives $$\frac{1}{2}x\left(\sqrt{2}+ 2\tanh\left(\frac{(x^2+x\_C1)\sqrt{2}}{2}\right)\right)\sqrt{2}$$ @@ -298,6 +308,7 @@ ode103expr := x*D(yx,x) + x*yx**2 - (2*x**2+1)*yx - x**3 --R Type: Expression Integer --E 12 +------------------------------------------------------------------- --S 13 of 131 ode104 := x*D(y(x),x) + a*x*y(x)**2 + 2*y(x) + b*x --R @@ -308,6 +319,8 @@ ode104 := x*D(y(x),x) + a*x*y(x)**2 + 2*y(x) + b*x --E 13 @ +Maxima fails. + Maple gets: $$-\frac{\sqrt{x(a+b)} \left(\_C1~{\rm BesselY}\left(3,2\sqrt{x(a+b)}\right)+ @@ -403,6 +416,7 @@ ode104expr := x*D(yx,x) + a*x*yx**2 + 2*yx + b*x --R Type: Expression Integer --E 15 +------------------------------------------------------------------- --S 16 of 131 ode105 := x*D(y(x),x) + a*x*y(x)**2 + b*y(x) + c*x + d --R @@ -413,6 +427,8 @@ ode105 := x*D(y(x),x) + a*x*y(x)**2 + b*y(x) + c*x + d --E 16 @ +Maxima fails. + Note that this complains about being unable to factor $$x^3-3x^2+(-b^2+2b+2)x+b^2-2b$$ but MMA factors this instantly to be: @@ -428,6 +444,7 @@ yx:=solve(ode105,y,x) --R Type: Union("failed",...) --E 17 +------------------------------------------------------------------- --S 18 of 131 ode106 := x*D(y(x),x) + x**a*y(x)**2 + (a-b)*y(x)/2 + x**b --R @@ -440,6 +457,8 @@ ode106 := x*D(y(x),x) + x**a*y(x)**2 + (a-b)*y(x)/2 + x**b --E 18 @ +Maxima fails. + Maple gets $$-\frac{\tan\left( \frac{\displaystyle 2x^{\left(\displaystyle @@ -461,6 +480,7 @@ yx:=solve(ode106,y,x) --R Type: Union("failed",...) --E 19 +------------------------------------------------------------------- --S 20 of 131 ode107 := x*D(y(x),x) + a*x**alpha*y(x)**2 + b*y(x) - c*x**beta --R @@ -470,6 +490,9 @@ ode107 := x*D(y(x),x) + a*x**alpha*y(x)**2 + b*y(x) - c*x**beta --R Type: Expression Integer --E 20 +@ +Maxima fails. +<<*>>= --S 21 of 131 yx:=solve(ode107,y,x) --R @@ -477,6 +500,7 @@ yx:=solve(ode107,y,x) --R Type: Union("failed",...) --E 21 +------------------------------------------------------------------- --S 22 of 131 ode108 := x*D(y(x),x) - y(x)**2*log(x) + y(x) --R @@ -486,6 +510,10 @@ ode108 := x*D(y(x),x) - y(x)**2*log(x) + y(x) --R Type: Expression Integer --E 22 @ +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. @@ -519,6 +547,7 @@ ode108expr := x*D(yx,x) - yx**2*log(x) + yx --R Type: Expression Integer --E 24 +------------------------------------------------------------------- --S 25 of 131 ode109 := x*D(y(x),x) - y(x)*(2*y(x)*log(x)-1) --R @@ -529,11 +558,14 @@ ode109 := x*D(y(x),x) - y(x)*(2*y(x)*log(x)-1) --E 25 @ +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. @@ -563,6 +595,7 @@ ode109expr := x*D(yx,x) - yx*(2*yx*log(x)-1) --R Type: Expression Integer --E 27 +------------------------------------------------------------------- --S 28 of 131 ode110 := x*D(y(x),x) + f(x)*(y(x)**2-x**2) --R @@ -572,6 +605,9 @@ ode110 := x*D(y(x),x) + f(x)*(y(x)**2-x**2) --R Type: Expression Integer --E 28 +@ +Maxima failed. +<<*>>= --S 29 of 131 yx:=solve(ode110,y,x) --R @@ -579,6 +615,7 @@ yx:=solve(ode110,y,x) --R Type: Union("failed",...) --E 29 +------------------------------------------------------------------- --S 30 of 131 ode111 := x*D(y(x),x) + y(x)**3 + 3*x*y(x)**2 --R @@ -589,6 +626,8 @@ ode111 := x*D(y(x),x) + y(x)**3 + 3*x*y(x)**2 --E 30 @ +Maxima fails. + Maple gets 0 which simplifies to 0 on substitution. <<*>>= @@ -599,10 +638,7 @@ 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 @@ -613,6 +649,15 @@ ode112 := x*D(y(x),x) - sqrt(y(x)**2 + x**2) - y(x) --R Type: Expression Integer --E 32 +@ +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 33 of 131 yx:=solve(ode112,y,x) --R @@ -620,6 +665,7 @@ yx:=solve(ode112,y,x) --R Type: Union("failed",...) --E 33 +------------------------------------------------------------------- --S 34 of 131 ode113 := x*D(y(x),x) + a*sqrt(y(x)**2 + x**2) - y(x) --R @@ -631,6 +677,11 @@ ode113 := x*D(y(x),x) + a*sqrt(y(x)**2 + x**2) - y(x) --E 34 @ +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 @@ -646,6 +697,7 @@ yx:=solve(ode113,y,x) --R Type: Union("failed",...) --E 35 +------------------------------------------------------------------- --S 36 of 131 ode114 := x*D(y(x),x) - x*sqrt(y(x)**2 + x**2) - y(x) --R @@ -657,6 +709,8 @@ ode114 := x*D(y(x),x) - x*sqrt(y(x)**2 + x**2) - y(x) --E 36 @ +Maxima fails. + Maple gets 0 but, on substitition, simplifies to $-x^2csqn(x)$. Mathematica gets @@ -670,6 +724,7 @@ yx:=solve(ode114,y,x) --R Type: Union("failed",...) --E 37 +------------------------------------------------------------------- --S 38 of 131 ode115 := x*D(y(x),x) - x*(y(x)-x)*sqrt(y(x)**2 + x**2) - y(x) --R @@ -681,6 +736,8 @@ ode115 := x*D(y(x),x) - x*(y(x)-x)*sqrt(y(x)**2 + x**2) - y(x) --E 38 @ +Maxima failed. + Maple claims the result is 0 but simplifies it, on substitution, to $x^3 csgn(x)$. @@ -694,6 +751,7 @@ yx:=solve(ode115,y,x) --R Type: Union("failed",...) --E 39 +------------------------------------------------------------------- --S 40 of 131 ode116 := x*D(y(x),x) - x*sqrt((y(x)**2 - x**2)*(y(x)**2-4*x**2)) - y(x) --R @@ -705,6 +763,8 @@ ode116 := x*D(y(x),x) - x*sqrt((y(x)**2 - x**2)*(y(x)**2-4*x**2)) - y(x) --E 40 @ +Maxima failed. + Maple claims the answer is 0 but simplifies, on substitution, to $-2x^3 csgn(x^2)$. @@ -720,6 +780,7 @@ yx:=solve(ode116,y,x) --R Type: Union("failed",...) --E 41 +------------------------------------------------------------------- --S 42 of 131 ode117 := x*D(y(x),x) - x*exp(y(x)/x) - y(x) - x --R @@ -732,6 +793,10 @@ ode117 := x*D(y(x),x) - x*exp(y(x)/x) - y(x) - x --E 42 @ +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. @@ -749,6 +814,7 @@ yx:=solve(ode117,y,x) --R Type: Union("failed",...) --E 43 +------------------------------------------------------------------- --S 44 of 131 ode118 := x*D(y(x),x) - y(x)*log(y(x)) --R @@ -759,6 +825,10 @@ ode118 := x*D(y(x),x) - y(x)*log(y(x)) --E 44 @ +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. @@ -790,6 +860,7 @@ ode118expr := x*D(yx,x) - yx*log(yx) --R Type: Expression Integer --E 46 +------------------------------------------------------------------- --S 47 of 131 ode119 := x*D(y(x),x) - y(x)*(log(x*y(x))-1) --R @@ -800,6 +871,12 @@ ode119 := x*D(y(x),x) - y(x)*(log(x*y(x))-1) --E 47 @ +$$\frac{1}{x}$$ 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. @@ -815,6 +892,7 @@ yx:=solve(ode119,y,x) --R Type: Union("failed",...) --E 48 +------------------------------------------------------------------- --S 49 of 131 ode120 := x*D(y(x),x) - y(x)*(x*log(x**2/y(x))+2) --R @@ -826,6 +904,8 @@ ode120 := x*D(y(x),x) - y(x)*(x*log(x**2/y(x))+2) --E 49 @ +Maxima fails. + Maple gets $$\frac{x^2}{e^{\left(\frac{\_C1}{e^x}\right)}}$$ which, on substitution, does not simplify to 0. @@ -841,6 +921,7 @@ yx:=solve(ode120,y,x) --R Type: Union("failed",...) --E 50 +------------------------------------------------------------------- --S 51 of 131 ode121 := x*D(y(x),x) + sin(y(x)-x) --R @@ -851,6 +932,8 @@ ode121 := x*D(y(x),x) + sin(y(x)-x) --E 51 @ +Maxima fails. + Mathematics gets $$\frac{\sin(x)}{1+\sin(x)}+x^{-sin(x)}C[1]$$ which, on substitution, does not simplify to 0. @@ -862,6 +945,7 @@ yx:=solve(ode121,y,x) --R Type: Union("failed",...) --E 52 +------------------------------------------------------------------- --S 53 of 131 ode122 := x*D(y(x),x) + (sin(y(x))-3*x**2*cos(y(x)))*cos(y(x)) --R @@ -872,6 +956,8 @@ ode122 := x*D(y(x),x) + (sin(y(x))-3*x**2*cos(y(x)))*cos(y(x)) --E 53 @ +Maxima fails. + Maple gets: $$\arctan\left(\frac{x^3+2~\_C1}{x}\right)$$ which, on substitution, simplifies to 0. @@ -887,6 +973,7 @@ yx:=solve(ode122,y,x) --R Type: Union("failed",...) --E 54 +------------------------------------------------------------------- --S 55 of 131 ode123 := x*D(y(x),x) - x*sin(y(x)/x) - y(x) --R @@ -897,6 +984,12 @@ ode123 := x*D(y(x),x) - x*sin(y(x)/x) - y(x) --E 55 @ +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: $$\arctan\left(\frac{2x~\_C1}{1+x^2~\_C1^2}\quad,\quad -\frac{-1+x^2~\_C1^2}{1+x^2~\_C1^2}\right)x$$ @@ -913,6 +1006,7 @@ yx:=solve(ode123,y,x) --R Type: Union("failed",...) --E 56 +------------------------------------------------------------------- --S 57 of 131 ode124 := x*D(y(x),x) + x*cos(y(x)/x) - y(x) + x --R @@ -923,6 +1017,11 @@ ode124 := x*D(y(x),x) + x*cos(y(x)/x) - y(x) + x --E 57 @ +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. @@ -938,6 +1037,7 @@ yx:=solve(ode124,y,x) --R Type: Union("failed",...) --E 58 +------------------------------------------------------------------- --S 59 of 131 ode125 := x*D(y(x),x) + x*tan(y(x)/x) - y(x) --R @@ -948,6 +1048,10 @@ ode125 := x*D(y(x),x) + x*tan(y(x)/x) - y(x) --E 59 @ +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. @@ -963,6 +1067,7 @@ yx:=solve(ode125,y,x) --R Type: Union("failed",...) --E 60 +------------------------------------------------------------------- --S 61 of 131 ode126 := x*D(y(x),x) - y(x)*f(x*y(x)) --R @@ -973,6 +1078,8 @@ ode126 := x*D(y(x),x) - y(x)*f(x*y(x)) --E 61 @ +Maxima fails. + Maple gets $$\frac{{\rm RootOf}\left(-\log(x)+~\_C1+ \displaystyle\int^{\_Z}{\frac{1}{\displaystyle\_a(1+g(\_a))}}~d\_a\right)}{x}$$ @@ -989,6 +1096,7 @@ yx:=solve(ode126,y,x) --R Type: Union("failed",...) --E 62 +------------------------------------------------------------------- --S 63 of 131 ode127 := x*D(y(x),x) - y(x)*f(x**a*y(x)**b) --R @@ -998,6 +1106,8 @@ ode127 := x*D(y(x),x) - y(x)*f(x**a*y(x)**b) --R Type: Expression Integer --E 63 @ +Maxima fails. + Maple gives 0 which, on substitution simplifies to 0. Mathematica gives: @@ -1011,6 +1121,7 @@ yx:=solve(ode127,y,x) --R Type: Union("failed",...) --E 64 +------------------------------------------------------------------- --S 65 of 131 ode128 := x*D(y(x),x) + a*y(x) - f(x)*g(x**a*y(x)) --R @@ -1020,6 +1131,8 @@ ode128 := x*D(y(x),x) + a*y(x) - f(x)*g(x**a*y(x)) --R Type: Expression Integer --E 65 @ +Maxima fails. + Maple gives $$\frac{{\rm RootOf}\left( -\int{f(x)x^{(-1+a)}}~dx+\int^{\_Z}{\frac{1}{g(\_a)}~d\_a+\_C1}\right)}{x^a}$$ @@ -1036,6 +1149,7 @@ yx:=solve(ode128,y,x) --R Type: Union("failed",...) --E 66 +------------------------------------------------------------------- --S 67 of 131 ode129 := (x+1)*D(y(x),x) + y(x)*(y(x)-x) --R @@ -1045,6 +1159,10 @@ ode129 := (x+1)*D(y(x),x) + y(x)*(y(x)-x) --R Type: Expression Integer --E 67 @ +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 $$\frac{e^x} {-e^x-e^{(-1)}{\rm Ei}(1,-x-1)x-e^{(-1)}{\rm Ei}(1,-x-1)+x~\_C1+~\_C1}$$ @@ -1068,6 +1186,7 @@ yx:=solve(ode129,y,x) --R Type: Union(Expression Integer,...) --E 68 +------------------------------------------------------------------- --S 69 of 131 ode130 := 2*x*D(y(x),x) - y(x) -2*x**3 --R @@ -1077,6 +1196,11 @@ ode130 := 2*x*D(y(x),x) - y(x) -2*x**3 --R Type: Expression Integer --E 69 @ +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. @@ -1112,6 +1236,7 @@ ode130expr := 2*x*D(yx,x) - yx -2*x**3 --R Type: Expression Integer --E 72 +------------------------------------------------------------------- --S 73 of 131 ode131 := (2*x+1)*D(y(x),x) - 4*exp(-y(x)) + 2 --R @@ -1121,6 +1246,11 @@ ode131 := (2*x+1)*D(y(x),x) - 4*exp(-y(x)) + 2 --R Type: Expression Integer --E 73 @ +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. @@ -1151,6 +1281,7 @@ ode131expr := (2*x+1)*D(yx,x) - 4*exp(-yx) + 2 --R Type: Expression Integer --E 75 +------------------------------------------------------------------- --S 76 of 131 ode132 := 3*x*D(y(x),x) - 3*x*log(x)*y(x)**4 - y(x) --R @@ -1160,6 +1291,15 @@ ode132 := 3*x*D(y(x),x) - 3*x*log(x)*y(x)**4 - y(x) --R Type: Expression Integer --E 76 @ +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. $$\frac{\left(-4x(6x^2\log(x)-3x^2-4~\_C1)^2\right)^{(1/3)}} {6x^2\log(x)-3*x^2-4~\_C1}$$ @@ -1226,6 +1366,7 @@ ode132expr := 3*x*D(yx,x) - 3*x*log(x)*yx**4 - yx --R Type: Expression Integer --E 78 +------------------------------------------------------------------- --S 79 of 131 ode133 := x**2*D(y(x),x) + y(x) - x --R @@ -1235,6 +1376,11 @@ ode133 := x**2*D(y(x),x) + y(x) - x --R Type: Expression Integer --E 79 @ +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. @@ -1257,6 +1403,7 @@ yx:=solve(ode133,y,x) --RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) --E 80 +------------------------------------------------------------------- --S 81 of 131 ode134 := x**2*D(y(x),x) - y(x) + x**2*exp(x-1/x) --R @@ -1269,6 +1416,10 @@ ode134 := x**2*D(y(x),x) - y(x) + x**2*exp(x-1/x) --R Type: Expression Integer --E 81 @ +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. @@ -1310,6 +1461,7 @@ ode134expr := x**2*D(yx,x) - yx + x**2*exp(x-1/x) --R Type: Expression Integer --E 84 +------------------------------------------------------------------- --S 85 of 131 ode135 := x**2*D(y(x),x) - (x-1)*y(x) --R @@ -1319,6 +1471,10 @@ ode135 := x**2*D(y(x),x) - (x-1)*y(x) --R Type: Expression Integer --E 85 @ +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. @@ -1351,6 +1507,7 @@ ode135expr := x**2*D(yx,x) - (x-1)*yx --R Type: Expression Integer --E 88 +------------------------------------------------------------------- --S 89 of 131 ode136 := x**2*D(y(x),x) + y(x)**2 + x*y(x) + x**2 --R @@ -1360,6 +1517,10 @@ ode136 := x**2*D(y(x),x) + y(x)**2 + x*y(x) + x**2 --R Type: Expression Integer --E 89 @ +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. @@ -1396,6 +1557,7 @@ ode136expr := x**2*D(yx,x) + yx**2 + x*yx + x**2 --R Type: Expression Integer --E 91 +------------------------------------------------------------------- --S 92 of 131 ode137 := x**2*D(y(x),x) - y(x)**2 - x*y(x) --R @@ -1405,6 +1567,10 @@ ode137 := x**2*D(y(x),x) - y(x)**2 - x*y(x) --R Type: Expression Integer --E 92 @ +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. @@ -1434,6 +1600,7 @@ ode137expr := x**2*D(yx,x) - yx**2 - x*yx --R Type: Expression Integer --E 94 +------------------------------------------------------------------- --S 95 of 131 ode138 := x**2*D(y(x),x) - y(x)**2 - x*y(x) - x**2 --R @@ -1443,6 +1610,10 @@ ode138 := x**2*D(y(x),x) - y(x)**2 - x*y(x) - x**2 --R Type: Expression Integer --E 95 @ +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. @@ -1514,6 +1685,7 @@ ode138expr := x**2*D(yx,x) - yx**2 - x*yx - x**2 --R Type: Expression Integer --E 97 +------------------------------------------------------------------- --S 98 of 131 ode139 := x**2*(D(y(x),x)+y(x)**2) + a*x**k - b*(b-1) --R @@ -1523,6 +1695,14 @@ ode139 := x**2*(D(y(x),x)+y(x)**2) + a*x**k - b*(b-1) --R Type: Expression Integer --E 98 +@ +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 99 of 131 yx:=solve(ode139,y,x) --R @@ -1530,6 +1710,7 @@ yx:=solve(ode139,y,x) --R Type: Union("failed",...) --E 99 +------------------------------------------------------------------- --S 100 of 131 ode140 := x**2*(D(y(x),x)+y(x)**2) + 4*x*y(x) + 2 --R @@ -1539,6 +1720,10 @@ ode140 := x**2*(D(y(x),x)+y(x)**2) + 4*x*y(x) + 2 --R Type: Expression Integer --E 100 @ +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. @@ -1570,6 +1755,7 @@ ode140expr := x**2*(D(yx,x)+yx**2) + 4*x*yx + 2 --R Type: Expression Integer --E 102 +------------------------------------------------------------------- --S 103 of 131 ode141 := x**2*(D(y(x),x)+y(x)**2) + a*x*y(x) + b --R @@ -1579,6 +1765,14 @@ ode141 := x**2*(D(y(x),x)+y(x)**2) + a*x*y(x) + b --R Type: Expression Integer --E 103 +@ +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 104 of 131 yx:=solve(ode141,y,x) --R 2 @@ -1761,6 +1955,7 @@ ode141expr := x**2*(D(yx,x)+yx**2) + a*x*yx + b --R Type: Expression Integer --E 105 +------------------------------------------------------------------- --S 106 of 131 ode142 := x**2*(D(y(x),x)-y(x)**2) - a*x**2*y(x) + a*x + 2 --R @@ -1770,6 +1965,10 @@ ode142 := x**2*(D(y(x),x)-y(x)**2) - a*x**2*y(x) + a*x + 2 --R Type: Expression Integer --E 106 +@ +Maxima failed. +<<*>>= + --S 107 of 131 yx:=solve(ode142,y,x) --R @@ -1815,6 +2014,7 @@ ode142expr := x**2*(D(yx,x)-yx**2) - a*x**2*yx + a*x + 2 --R Type: Expression Integer --E 108 +------------------------------------------------------------------- --S 109 of 131 ode143 := x**2*(D(y(x),x)+a*y(x)**2) - b --R @@ -1824,6 +2024,23 @@ ode143 := x**2*(D(y(x),x)+a*y(x)**2) - b --R Type: Expression Integer --E 109 +@ +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 110 of 131 yx:=solve(ode143,y,x) --R 2 @@ -1877,6 +2094,7 @@ ode143expr := x**2*(D(yx,x)+a*yx**2) - b --R Type: Expression Integer --E 111 +------------------------------------------------------------------- --S 112 of 131 ode144 := x**2*(D(y(x),x)+a*y(x)**2) + b*x**alpha + c --R @@ -1886,6 +2104,9 @@ ode144 := x**2*(D(y(x),x)+a*y(x)**2) + b*x**alpha + c --R Type: Expression Integer --E 112 +@ +Maxima failed. +<<*>>= --S 113 of 131 yx:=solve(ode144,y,x) --R @@ -1893,6 +2114,7 @@ yx:=solve(ode144,y,x) --R Type: Union("failed",...) --E 113 +------------------------------------------------------------------- --S 114 of 131 ode145 := x**2*D(y(x),x) + a*y(x)**3 - a*x**2*y(x)**2 --R @@ -1903,6 +2125,8 @@ ode145 := x**2*D(y(x),x) + a*y(x)**3 - a*x**2*y(x)**2 --E 114 @ +Maxima failed. + Maple claims the result is 0, which when substituted, simplifies to 0 <<*>>= --S 115 of 131 @@ -1912,6 +2136,7 @@ yx:=solve(ode145,y,x) --R Type: Union("failed",...) --E 115 +------------------------------------------------------------------- --S 116 of 131 ode146 := x**2*D(y(x),x) + x*y(x)**3 + a*y(x)**2 --R @@ -1922,6 +2147,8 @@ ode146 := x**2*D(y(x),x) + x*y(x)**3 + a*y(x)**2 --E 116 @ +Maxima failed. + Maple gets 0 which, when substituted, simplifies to 0. <<*>>= --S 117 of 131 @@ -1931,6 +2158,7 @@ yx:=solve(ode146,y,x) --R Type: Union("failed",...) --E 117 +------------------------------------------------------------------- --S 118 of 131 ode147 := x**2*D(y(x),x) + a*x**2*y(x)**3 + b*y(x)**2 --R @@ -1940,6 +2168,8 @@ ode147 := x**2*D(y(x),x) + a*x**2*y(x)**3 + b*y(x)**2 --R Type: Expression Integer --E 118 @ +Maxima failed. + Maple gets 0 which, when substituted, results in 0. <<*>>= --S 119 of 131 @@ -1949,6 +2179,7 @@ yx:=solve(ode147,y,x) --R Type: Union("failed",...) --E 119 +------------------------------------------------------------------- --S 120 of 131 ode148 := (x**2+1)*D(y(x),x) + x*y(x) - 1 --R @@ -1958,6 +2189,10 @@ ode148 := (x**2+1)*D(y(x),x) + x*y(x) - 1 --R Type: Expression Integer --E 120 @ +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. @@ -1999,6 +2234,7 @@ ode148expr := (x**2+1)*D(yx,x) + x*yx - 1 --R Type: Expression Integer --E 123 +------------------------------------------------------------------- --S 124 of 131 ode149 := (x**2+1)*D(y(x),x) + x*y(x) - x*(x**2+1) --R @@ -2008,6 +2244,11 @@ ode149 := (x**2+1)*D(y(x),x) + x*y(x) - x*(x**2+1) --R Type: Expression Integer --E 124 @ +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. @@ -2045,6 +2286,7 @@ ode149expr := (x**2+1)*D(yx,x) + x*yx - x*(x**2+1) --R Type: Expression Integer --E 127 +------------------------------------------------------------------- --S 128 of 131 ode150 := (x**2+1)*D(y(x),x) + 2*x*y(x) - 2*x**2 --R @@ -2054,6 +2296,10 @@ ode150 := (x**2+1)*D(y(x),x) + 2*x*y(x) - 2*x**2 --R Type: Expression Integer --E 128 @ +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. @@ -2100,5 +2346,6 @@ ode150expr := (x**2+1)*D(yx,x) + 2*x*yx - 2*x**2 \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}