$SPAD/src/input kamke2.input

Timothy Daly

December 3, 2016

Abstract

This is the 50 ODEs of the Kamke test suite as published by E. S. Cheb-Terrab[?]. They have been rewritten using Axiom syntax. Where possible we show that the particular solution actually satisfies the original ordinary differential equation.

Contents

2

— * —

)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((\frac{x^{2}\sqrt{a}}{2} + C1\sqrt{a}) x\sqrt{a}

which, upon substitution, simplifies to 0.

Mathematica gives

\sqrt{a} x tanh (\frac{1}{2} (\sqrt{a} x^{2} + 2\sqrt{a} C[1]))

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 (\sqrt{2} + 2 tanh (\frac{(x^{2} + x C1)\sqrt{2}}{2})) \sqrt{2}

which simplifies to 0 on substitution.

Mathematica gives

\frac{(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]}) 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 (\frac{2x(\frac{a}{2} + \frac{b}{2}) + C1 a + C1 b}{a + b})}{x^{(\frac{a}{2} − \frac{b}{2})}}

which simplifies to 0 on substitution.

Mathematica gets

e^{− \frac{1}{2} a log(x)+ \frac{1}{2} b log(x)} tan (\frac{2x^{\frac{a+b}{2}}}{a + b} − C[1])

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 (\frac{log(x)}{x} + \frac{1}{x} + %c)}

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 (%c − 2 (−\frac{log(x)}{x} − \frac{1}{x}))}

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\frac{xasinh (\frac{y}{x})}{|x|}

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−\frac{xasinh (\frac{y}{x})}{a|x|}

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^{2}csqn(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−\frac{x log(%e^{y/x} + 1) − y}{x}

which does not simplify to 0 on substitution.

Maple gets:

(log (−\frac{x}{−1 + x e ^{C1}} ) + C1) 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 (−1 + \frac{e^{−C[1]}}{x} )

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^{( ^{\frac{x}{C1}})}}{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^{( ^{\frac{C1}{e^{x}}})}}

which, on substitution, does not simplify to 0.

Mathematics get:

2e^{−e^{−x}C[1]+e^{−x}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 (\frac{x^{3} + 2 C1}{x})

which, on substitution, simplifies to 0.

Mathematica gets:

arctan (\frac{2x^{3} + C[1]}{2x})

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−\frac{log (cos (\frac{y}{x}) + 1) − log (cos (\frac{y}{x}) − 1)}{2}

which, on substitution, does not simplify to 0.

Maple gets:

arctan (\frac{2x C1}{1 + x^{2} C1^{2}} , −\frac{−1 + x^{2} C1^{2}}{1 + x^{2} C1^{2}} ) 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−\frac{sin (\frac{y}{x})}{cos (\frac{y}{x}) + 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 (\frac{1}{%c x}) x

which, on substitition, does simplifes to 0.

Maple gets

arcsin (\frac{1}{x C1}) x

which, on substitution, simplifies to 0.

Mathematica gets

arcsin (\frac{e^{C[1]}}{x} )

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{RootOf (− log(x) + C1 + ∫ ^{Z} \frac{1}{a(1 + g( a))} d a)}{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 (−\frac{f(x^{a})}{a} − C[1])^{−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{RootOf (− ∫ f(x)x^{(−1+a)} dx + ∫ ^{Z} \frac{1}{g( a)} d a + C1)}{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) (∫ \frac{%e^{x}}{(x+1)^{2}} dx + %c)}

which, on substitution, does not simplify to 0.

Maple gives

\frac{e^{x}}{−e^{x} − e^{(−1)}Ei(1, −x − 1)x − e^{(−1)}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] − ExpIntegralEi(1 + x) − xExpIntegralEi(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\frac{log(x)}{2} (\frac{2%e\frac{5 log(x)}{2}}{5} + %c)

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 (\frac{4%e^{2%c}x + 2%e^{2%c} + 1}{2%e^{2%c}x + %e^{2%c}})

which simplifies to 0 when substituted.

Maple gives

− log (\frac{2x + 1}{−1 + 4xe^{(2 C1)} + 2e^{(2 C1)}} ) − 2 C1

which simplifies to 0 when substituted.

Mathematica gives

log (2 + \frac{1}{1 + 2x})

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{(\sqrt{3} 4^{1/3}%i − 4^{1/3}) x^{1/3}}{2 (6x^{2} log(x) − 3x^{2} + 4%c)^{1/3}}

\frac{(\sqrt{3} 4^{1/3}%i + 4^{1/3}) x^{1/3}}{2 (6x^{2} log(x) − 3x^{2} + 4%c)^{1/3}}

−\frac{4^{1/3}x^{1/3}}{(6x^{2} log(x) − 3x^{2} + 4%c)^{1/3}}

which, on substitution, simplifies to 0.

Maple gives 3 solutions.

\frac{(−4x(6x^{2} log(x) − 3x^{2} − 4 C1)^{2})^{(1/3)}}{6x^{2} log(x) − 3 ∗ x^{2} − 4 C1}

−\frac{1}{2} \frac{(−4x(6x^{2} log(x) − 3x^{2} − 4 C1)^{2})^{(1/3)}}{6x^{2} log(x) − 3 ∗ x^{2} − 4 C1} + \frac{1}{2}I\sqrt{3}\frac{(−4x(6x^{2} log(x) − 3x^{2} − 4 C1)^{2})^{(1/3)}}{6x^{2} log(x) − 3 ∗ x^{2} − 4 C1}

−\frac{1}{2} \frac{(−4x(6x^{2} log(x) − 3x^{2} − 4 C1)^{2})^{(1/3)}}{6x^{2} log(x) − 3 ∗ x^{2} − 4 C1} − \frac{1}{2}I\sqrt{3}\frac{(−4x(6x^{2} log(x) − 3x^{2} − 4 C1)^{2})^{(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} (∫ \frac{%e^{− \frac{1}{x}}}{x} dx + %c)

which, on substitution, simplifies to 0.

Maple gives

(Ei (1, \frac{1}{x}) + C1) e^{( \frac{1}{x})}

which simplifies to 0 on substitution.

Mathematica gets:

e^{1/x}C[1] − e^{1/x}ExpIntegralEi (−\frac{1}{x})

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−\frac{1}{x} (%c − %e^{x})

which simplifies to 0 on substitution.

Maple gets

(−e^{x} + C1)e^{(− \frac{1}{x})}

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^{(^{\frac{1}{x}})}

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 (\frac{1}{%c x})}

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}^{(^{\frac{y}{x}})}

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{− (3^{5/6}%i (ax^{k} + %ckx − %cx + b^{2}k − bk − b^{2} + b)^{1/3} − 3^{1/3} (ax^{k} + %ckx − %cx + b^{2}k − bk − b^{2} + b)^{1/3})}{(2(k − 1)^{1/3}x^{1/3})}

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−a log(x) − 2x (%c − b ∫ \frac{%ea log(x) + 2x}{x^{2}} dx)

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^{− \begin{matrix}log & (−\frac{−2axy + \sqrt{4ab + 1} + 1}{2axy + \sqrt{4ab + 1} − 1}) \\ √ & \overline{4ab + 1} \end{matrix}}

and if 4ab + 1 < 0 gets:

x = %c%e^{− \frac{2 arctan (\frac{2axy − 1}{\sqrt{−4ab − 1}})}{\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

(asinh(x) + %c)%e^{− \frac{log(x^{2} + 1)}{2}}

which when substituted, does not simplify to 0.

Maple gets

\frac{arcsinh(x) + C1}{\sqrt{x^{2} + 1}}

which when substituted, simplifies to 0.

Mathematica gets

\frac{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

(\frac{(x^{2} + 1)^{3/2}}{3} + %c) %e−\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

\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)

———-

References

[1] http://www.cs.uwaterloo.ca/ẽcterrab/odetools.html [2] Mathematica 6.0.1.0

[3] Maple 11.01 Build ID 296069

[4] Maxima 5.13.0