## Chapter 1Introduction and Summary of results

1.1 Introduction

1.3 Conclusions

### 1.1 Introduction

This report shows the result of running Maple and Mathematica on my collection of diﬀerential equations. These were collected over time and stored in sqlite3 database. These were collected from a number of textbooks and other references such as Kamke and Murphy collections. All books used are listed here.

The current number of diﬀerential equations is [15472].

Both Maple and Mathematica are given a CPU time limit of 3 minutes to solve each ode else the problem is considered not solved and marked as failed.

When Mathematica returns DifferentialRoot as a solution to an ode then this is considered as not solved. Similarly, when Maple returns DESol or ODSESolStruc, then this is also considered as not solved.

If CAS solves the ODE within the timelimit, then it is counted as solved. No veriﬁcation is done to check that the solution is correct or not.

To reduce the size of latex output, in Maple the command simplify is called on the solution with timeout of 3 minutes. If this times out, then the unsimpliﬁed original ode solution is used otherwise the simpliﬁed one is used.

Similarly for Mathematica, Simplify is next called. If this timesout, then the unsimpliﬁed solution is used else the simpliﬁed one is used. The time used for simpliﬁcation is not counted in the CPU time used. The CPU time used only records the time used to solve the ode.

Tests are run under windows 10 with 128 GB RAM running on intel i9-12900K 3.20 GHz

### 1.2 Summary of results

#### 1.2.1 Percentage solved and CPU performance

The following table summarizes perentage solved for each CAS

 System % solved Number solved Number failed Maple 2024 95.489 14774 698 Mathematica 14 94.577 14633 839

The following table summarizes the run-time performance of each CAS system.

 System mean time (sec) mean leaf size total time (min) total leaf size Maple 2024 0.202 147.54 52.009 2282723 Mathematica 14 3.232 221.31 833.444 3424131

The problem which Mathematica produced largest leaf size of $$413606$$ is 9721.

The problem which Maple produced largest leaf size of $$949416$$ is 12388.

The problem which Mathematica used most CPU time of $$175.525$$ seconds is 6197.

The problem which Maple used most CPU time of $$140.984$$ seconds is 6839.

#### 1.2.2 Performance based on ODE type

The following gives the performance of each CAS based on the type of the ODE. Three diﬀerent classiﬁcations of ODE’s are used. The ﬁrst uses Maple’s own ode advisor classiﬁcation. The second uses own ODE classiﬁcation used in my ode solver. The third classiﬁcation uses a simpliﬁed classiﬁcation of ODE’s which is based on generic type of the ODE.

##### Performance using Maple’s ODE types classiﬁcation

This uses ODE classiﬁcations based on Maple’s ode advisor The following table gives count of the number of ODE’s for each ODE type, where the ODE type here is as classiﬁed by Maple’s odeadvisor, and the percentage of solved ODE’s of that type for each CAS. It also gives a direct link to the ODE’s that failed if any.

 Type of ODE Count Mathematica Maple [_quadrature] 873 98.05%[885, 4266, 4275, 12127, 12129, 12910, 12911, 12914, 12935, 12936, 12962, 12965, 12966, 12967, 14201, 15125, 15126] 99.77%[7303, 11994] [[_linear, ‘class A‘]] 304 100.00% 99.01%[7300, 7301, 11518] [_separable] 1196 99.16%[3022, 6264, 8667, 11415, 14980, 14999, 15000, 15001, 15002, 15006] 99.50%[408, 409, 6264, 6418, 11415, 15001] [_Riccati] 322 [[_homogeneous, ‘class G‘]] 70 94.29%[3232, 3236, 12148, 15095] 94.29%[3995, 4040, 8704, 8719] [_linear] 688 99.56%[6169, 11995, 15046] 99.56%[5502, 6169, 11995] [[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] 31 100.00% 100.00% [[_homogeneous, ‘class A‘], _rational, _Bernoulli] 102 99.02%[14378] 100.00% [[_homogeneous, ‘class A‘], _dAlembert] 150 99.33%[11212] 100.00% [[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] 96 100.00% 100.00% [[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] 60 100.00% 100.00% [[_homogeneous, ‘class A‘], _rational, _dAlembert] 219 98.63%[2085, 5761, 14384] 100.00% [[_homogeneous, ‘class C‘], _dAlembert] 81 91.36%[3000, 4260, 4278, 7102, 11240, 14439, 15129] 100.00% [[_homogeneous, ‘class C‘], _Riccati] 24 100.00% 100.00% [[_homogeneous, ‘class G‘], _rational, _Bernoulli] 75 100.00% 100.00% [_Bernoulli] 117 100.00% 100.00% [[_1st_order, _with_linear_symmetries], _Bernoulli] 10 100.00% 100.00% [[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] 48 100.00% 100.00% [‘y=_G(x,y’)‘] 145 62.76%[133, 485, 959, 961, 962, 964, 966, 968, 1703, 1706, 1707, 2026, 2316, 2319, 3363, 3368, 3384, 3463, 4011, 4216, 4261, 4287, 4299, 4951, 4995, 6549, 7063, 7253, 8411, 8416, 8419, 8457, 8706, 8731, 8795, 8796, 8841, 8845, 8866, 11219, 11224, 11404, 12214, 12220, 12239, 12636, 13289, 13348, 14046, 14133, 14296, 14313, 14441, 14941] 57.24%[133, 485, 959, 961, 962, 964, 966, 968, 1703, 1706, 1707, 2026, 2063, 2316, 2319, 3090, 3363, 3368, 3382, 3384, 3395, 3463, 3872, 4011, 4216, 4287, 4298, 4914, 4951, 4995, 6549, 7063, 7253, 8411, 8416, 8419, 8457, 8706, 8731, 8787, 8795, 8796, 8841, 8845, 8848, 8866, 8878, 11224, 11404, 12214, 12218, 12220, 12239, 12636, 13289, 13348, 14133, 14296, 14313, 14441, 14941, 15059] [[_1st_order, _with_linear_symmetries]] 104 91.35%[3229, 3231, 4290, 4294, 5346, 6797, 6807, 11215, 15124] 99.04%[8872] [[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] 38 97.37%[15074] 100.00% [_exact, _rational] 43 97.67%[119] 100.00% [_exact] 98 93.88%[3137, 14323, 14328, 15066, 15067, 15073] 97.96%[14323, 14328] [[_1st_order, _with_linear_symmetries], _exact, _rational] 4 100.00% 100.00% [[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] 4 100.00% 100.00% [[_homogeneous, ‘class G‘], _exact, _rational] 11 81.82%[146, 11610] 100.00% [[_2nd_order, _missing_x]] 833 96.52%[2307, 7411, 9934, 9935, 9936, 9938, 9939, 9941, 9959, 9960, 9962, 9967, 9985, 10031, 10033, 10156, 10159, 11589, 11590, 12570, 12571, 14516, 14517, 15204, 15444, 15446, 15449, 15453, 15459] 97.24%[7411, 9934, 9935, 9938, 9939, 9941, 9959, 9960, 9962, 9967, 9985, 10031, 10032, 10033, 10159, 11589, 11590, 12570, 12571, 15444, 15446, 15449, 15453] [[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] 121 97.52%[12614, 13569, 13570] 97.52%[12614, 13569, 13570] [[_Emden, _Fowler]] 348 100.00% 97.41%[2541, 4718, 5217, 5556, 5588, 5589, 6584, 6617, 12406] [[_2nd_order, _exact, _linear, _homogeneous]] 235 99.57%[12258] 98.30%[5590, 6460, 6618, 12407] [[_2nd_order, _missing_y]] 201 97.51%[6856, 6858, 7212, 10150, 11331] 99.00%[6443, 7305] [[_2nd_order, _with_linear_symmetries]] 2852 [[_2nd_order, _linear, _nonhomogeneous]] 1111 98.47%[1162, 1186, 7462, 9408, 12248, 12251, 12281, 12352, 12354, 12748, 12749, 14633, 14870, 15384, 15385, 15386, 15436] 97.66%[1162, 1186, 4722, 4723, 5500, 5501, 5833, 6513, 7224, 7225, 7226, 7230, 7231, 7233, 7241, 7306, 7307, 9408, 12248, 12251, 12281, 12352, 12354, 12749, 14633, 14870] [[_2nd_order, _exact, _linear, _nonhomogeneous]] 73 100.00% 100.00% system of linear ODEs 828 96.50%[6104, 6542, 6543, 10213, 10228, 10238, 10241, 10242, 10243, 10244, 10245, 10250, 10251, 10254, 10255, 10256, 10257, 10258, 10259, 10261, 12827, 12828, 12829, 12830, 12842, 14043, 15506, 15517, 15524] 96.74%[6104, 6542, 6543, 6716, 6719, 10213, 10228, 10238, 10241, 10242, 10243, 10244, 10245, 10250, 10251, 10254, 10256, 10257, 10259, 10261, 12827, 12828, 12829, 12830, 12842, 14043, 15524] [_Gegenbauer] 77 100.00% 100.00% [[_high_order, _missing_x]] 216 100.00% 100.00% [[_3rd_order, _missing_x]] 195 100.00% 100.00% [[_3rd_order, _missing_y]] 97 100.00% 100.00% [[_3rd_order, _exact, _linear, _homogeneous]] 15 100.00% 100.00% [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] 76 94.74%[10840, 10933, 11030, 11074] 98.68%[6459] [_Lienard] 59 100.00% 100.00% [[_homogeneous, ‘class A‘], _rational, _Riccati] 31 100.00% 100.00% [‘x=_G(y,y’)‘] 13 61.54%[550, 2713, 6183, 8907, 13034] 61.54%[550, 2713, 6183, 8907, 13034] [[_Abel, ‘2nd type‘, ‘class B‘]] 15 26.67%[553, 1046, 8586, 10667, 10670, 10690, 10691, 10692, 10712, 10725, 10730] 40.00%[553, 1046, 8586, 10670, 10690, 10691, 10692, 10712, 10725] [_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] 12 100.00% 100.00% [[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] 29 96.55%[2031] 100.00% [[_homogeneous, ‘class D‘], _rational] 3 100.00% 100.00% [[_1st_order, _with_exponential_symmetries]] 9 100.00% 100.00% [_rational] 111 82.88%[1039, 1075, 1953, 3118, 3192, 3193, 4146, 4315, 6111, 8815, 8817, 8824, 8838, 9219, 9228, 11198, 11604, 14101, 14126] 76.58%[1039, 1075, 1953, 3118, 3192, 3193, 3926, 4146, 4198, 4199, 4315, 6111, 8815, 8817, 8820, 8838, 9219, 9228, 9246, 9254, 11198, 11230, 11604, 12421, 14101, 14126] [_rational, [_Abel, ‘2nd type‘, ‘class B‘]] 136 [_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] 4 100.00% 100.00% [NONE] 86 37.21%[710, 1041, 7110, 7214, 8393, 8424, 8538, 8703, 8910, 8911, 9170, 9172, 9918, 9921, 9922, 9926, 9929, 9931, 9932, 9940, 9942, 9946, 9947, 9948, 9951, 9957, 9965, 9966, 9968, 9972, 9998, 10008, 10016, 10025, 10027, 10052, 10055, 10057, 10058, 10061, 10062, 10074, 10080, 10112, 10124, 10125, 10138, 10174, 12238, 12241, 12243, 13529, 14051, 14626] 33.72%[710, 6238, 7110, 7214, 8393, 8424, 8538, 8703, 8910, 8911, 9170, 9172, 9918, 9921, 9922, 9929, 9931, 9932, 9940, 9942, 9946, 9947, 9948, 9951, 9957, 9965, 9966, 9968, 9972, 9998, 10008, 10016, 10021, 10025, 10027, 10028, 10029, 10044, 10052, 10055, 10057, 10058, 10061, 10062, 10074, 10080, 10112, 10124, 10125, 10138, 10174, 12238, 12241, 12243, 13529, 14051, 14626] [_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] 29 100.00% 96.55%[1984] [[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] 58 98.28%[2083] 100.00% [_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] 21 100.00% 100.00% [[_high_order, _with_linear_symmetries]] 57 87.72%[813, 9865, 9866, 9867, 9868, 9895, 9913] 87.72%[813, 9865, 9866, 9867, 9868, 9905, 9913] [[_3rd_order, _with_linear_symmetries]] 158 88.61%[5817, 9784, 9785, 9786, 9787, 9788, 9789, 9790, 9800, 9801, 9803, 9811, 9816, 9827, 9840, 9841, 9856, 13559] 89.24%[5817, 9784, 9785, 9786, 9787, 9788, 9789, 9790, 9800, 9801, 9803, 9811, 9816, 9835, 9840, 9856, 13559] [[_high_order, _linear, _nonhomogeneous]] 89 97.75%[9875, 9904] 98.88%[9904] [[_1st_order, _with_linear_symmetries], _Clairaut] 76 100.00% 100.00% [[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] 52 100.00% 100.00% [[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] 79 98.73%[1985] 98.73%[1985] [[_homogeneous, ‘class C‘], _rational, _Riccati] 5 100.00% 100.00% [[_Abel, ‘2nd type‘, ‘class A‘]] 34 14.71%[3676, 3728, 4954, 8542, 8555, 10657, 10658, 10722, 10723, 10724, 10733, 10734, 10735, 10736, 10737, 10751, 10797, 10804, 10805, 10807, 10808, 10810, 10811, 10812, 10813, 10814, 10815, 10816, 10817] 35.29%[3676, 3728, 4954, 8542, 8555, 10722, 10723, 10724, 10733, 10734, 10735, 10736, 10737, 10751, 10797, 10805, 10808, 10812, 10813, 10815, 10816, 10817] [_rational, _Bernoulli] 46 100.00% 100.00% [[_homogeneous, ‘class A‘]] 7 100.00% 100.00% [[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] 162 98.15%[1941, 4451, 10819] 98.77%[1935, 1938] [[_homogeneous, ‘class G‘], _rational, _Riccati] 21 100.00% 100.00% [[_1st_order, _with_linear_symmetries], _Riccati] 10 100.00% 100.00% [[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘], _Riccati] 1 100.00% 100.00% [[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] 2 100.00% 100.00% [_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] 17 100.00% 100.00% [_exact, [_Abel, ‘2nd type‘, ‘class B‘]] 6 100.00% 100.00% [_exact, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] 10 100.00% 100.00% [_exact, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] 4 100.00% 100.00% [_exact, _Bernoulli] 7 100.00% 100.00% [[_homogeneous, ‘class A‘], _exact, _rational, _Bernoulli] 10 100.00% 100.00% [_rational, [_Abel, ‘2nd type‘, ‘class C‘]] 12 83.33%[4917, 4962] 83.33%[4917, 4962] [[_homogeneous, ‘class G‘], _rational] 98 98.98%[1986] 97.96%[4163, 6820] [[_homogeneous, ‘class D‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] 2 100.00% 100.00% [_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] 14 100.00% 100.00% [_rational, _Riccati] 102 95.10%[10349, 10392, 10401, 10405, 10406] 98.04%[10401, 10406] [[_3rd_order, _linear, _nonhomogeneous]] 93 97.85%[12223, 12227] 100.00% [[_high_order, _missing_y]] 57 98.25%[9909] 98.25%[9909] [[_3rd_order, _exact, _linear, _nonhomogeneous]] 6 100.00% 100.00% [[_high_order, _exact, _linear, _nonhomogeneous]] 7 100.00% 100.00% [[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] 34 100.00% 100.00% [_exact, [_Abel, ‘2nd type‘, ‘class A‘]] 2 100.00% 100.00% [[_1st_order, _with_linear_symmetries], [_Abel, ‘2nd type‘, ‘class A‘]] 2 100.00% 100.00% [[_Riccati, _special]] 26 100.00% 100.00% [_Abel] 30 66.67%[1704, 3352, 8384, 8385, 8386, 8387, 12134, 12631, 12938, 13057] 66.67%[1704, 3352, 8384, 8385, 8386, 8387, 12134, 12631, 12938, 13057] [_Laguerre] 39 100.00% 100.00% [_Laguerre, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] 4 100.00% 100.00% [_Bessel] 20 100.00% 100.00% [_rational, _Abel] 21 95.24%[1897] 100.00% [[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert] 3 100.00% 100.00% [[_homogeneous, ‘class D‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] 5 100.00% 100.00% [[_homogeneous, ‘class G‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] 8 100.00% 100.00% [_exact, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] 11 100.00% 100.00% [[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] 6 100.00% 100.00% [[_homogeneous, ‘class D‘], _rational, _Bernoulli] 36 100.00% 100.00% [[_homogeneous, ‘class D‘], _Bernoulli] 6 100.00% 100.00% [_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] 11 100.00% 100.00% [[_homogeneous, ‘class A‘], _exact, _dAlembert] 7 100.00% 100.00% [[_2nd_order, _quadrature]] 61 98.36%[12198] 98.36%[7304] [[_high_order, _quadrature]] 11 100.00% 100.00% [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] 78 89.74%[2308, 2376, 4658, 6100, 6839, 6840, 9914, 15211] 97.44%[2376, 15217] [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] 24 100.00% 100.00% [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]] 10 100.00% 100.00% [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] 17 58.82%[4668, 4839, 4840, 4841, 13523, 13524, 15210] 100.00% [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] 29 93.10%[2304, 13520] 93.10%[2304, 2309] [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] 11 100.00% 100.00% [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] 38 100.00% 97.37%[10131] [[_2nd_order, _missing_x], [_2nd_order, _with_potential_symmetries], [_2nd_order, _reducible, _mu_xy]] 5 100.00% 100.00% [[_homogeneous, ‘class C‘], _rational, _dAlembert] 13 100.00% 100.00% [_dAlembert] 25 100.00% 100.00% [[_1st_order, _with_linear_symmetries], _dAlembert] 64 82.81%[2350, 4251, 4252, 4253, 4274, 4305, 6811, 6813, 6874, 6878, 7254] 100.00% [[_homogeneous, ‘class G‘], _rational, _Clairaut] 10 100.00% 100.00% [[_homogeneous, ‘class G‘], _Clairaut] 3 100.00% 100.00% [[_1st_order, _with_linear_symmetries], _rational, _Clairaut] 17 100.00% 100.00% [[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class B‘]] 1 100.00% 100.00% [[_homogeneous, ‘class C‘], [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert] 6 100.00% 100.00% [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] 11 100.00% 100.00% [[_3rd_order, _exact, _nonlinear]] 3 66.67%[10164] 66.67%[10164] [_Jacobi] 37 100.00% 100.00% [_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class B‘]] 6 100.00% 100.00% [[_3rd_order, _quadrature]] 8 100.00% 100.00% [[_homogeneous, ‘class G‘], _exact] 3 100.00% 100.00% [[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli] 12 100.00% 100.00% [_exact, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] 1 100.00% 100.00% [[_homogeneous, ‘class A‘], _exact, _rational, _Riccati] 1 100.00% 100.00% [_erf] 4 100.00% 100.00% [[_homogeneous, ‘class D‘]] 13 100.00% 100.00% [_exact, _rational, _Riccati] 3 100.00% 100.00% [_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class B‘]] 7 100.00% 100.00% [[_1st_order, _with_linear_symmetries], _rational] 26 100.00% 100.00% [[_homogeneous, ‘class D‘], _rational, _Riccati] 20 100.00% 100.00% [[_1st_order, _with_linear_symmetries], _exact] 3 100.00% 100.00% [[_homogeneous, ‘class C‘], _exact, _dAlembert] 5 100.00% 100.00% [_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class A‘]] 2 100.00% 100.00% [[_homogeneous, ‘class D‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] 2 100.00% 100.00% [_rational, [_Abel, ‘2nd type‘, ‘class A‘]] 40 27.50%[3673, 5247, 8539, 8541, 10655, 10659, 10686, 10702, 10720, 10721, 10738, 10740, 10741, 10745, 10747, 10750, 10763, 10794, 10795, 10796, 10798, 10799, 10800, 10801, 10802, 10803, 10820, 10822, 11599] 45.00%[3673, 5247, 8539, 8541, 10655, 10659, 10720, 10721, 10741, 10747, 10750, 10763, 10794, 10795, 10798, 10799, 10800, 10801, 10802, 10820, 10822, 11599] [[_homogeneous, ‘class G‘], _dAlembert] 7 100.00% 100.00% [[_1st_order, _with_linear_symmetries], _rational, _Bernoulli] 5 100.00% 100.00% [[_homogeneous, ‘class G‘], _Abel] 4 100.00% 100.00% [[_homogeneous, ‘class G‘], _Chini] 4 100.00% 100.00% [_Chini] 4 0.00%[3355, 3642, 8392, 14440] 0.00%[3355, 3642, 8392, 14440] [_rational, [_Riccati, _special]] 9 100.00% 100.00% [[_1st_order, _with_linear_symmetries], _rational, _Riccati] 3 100.00% 100.00% [[_homogeneous, ‘class D‘], _Riccati] 20 100.00% 100.00% [[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]] 4 100.00% 100.00% [[_homogeneous, ‘class G‘], _Riccati] 4 100.00% 100.00% [_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class C‘]] 5 100.00% 100.00% [[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert] 6 100.00% 100.00% [_exact, _rational, _Bernoulli] 4 75.00%[14327] 75.00%[14327] [[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] 5 100.00% 100.00% [[_Abel, ‘2nd type‘, ‘class C‘]] 7 71.43%[3843, 8605] 71.43%[3843, 8605] [[_homogeneous, ‘class C‘], _rational] 8 100.00% 100.00% [_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] 2 100.00% 100.00% [_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] 17 100.00% 100.00% unknown 8 75.00%[8676, 10129] 62.50%[8676, 10129, 10158] [_rational, _dAlembert] 12 91.67%[8766] 100.00% [[_1st_order, _with_linear_symmetries], _rational, _dAlembert] 9 100.00% 100.00% [[_homogeneous, ‘class G‘], _rational, _dAlembert] 6 100.00% 100.00% [_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] 15 100.00% 100.00% [_Clairaut] 7 100.00% 85.71%[4343] [[_homogeneous, ‘class D‘], _exact, _rational, _Bernoulli] 1 100.00% 100.00% [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] 9 100.00% 100.00% [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] 10 90.00%[12495] 100.00% [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]] 3 66.67%[6246] 100.00% [_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] 9 100.00% 100.00% [[_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] 3 100.00% 100.00% [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] 3 100.00% 100.00% [[_homogeneous, ‘class G‘], _rational, _Abel] 2 100.00% 100.00% [[_elliptic, _class_I]] 2 100.00% 100.00% [[_elliptic, _class_II]] 2 100.00% 100.00% [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear]] 1 100.00% 100.00% [_Hermite] 16 100.00% 100.00% [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_y_y1]] 3 100.00% 100.00% [[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] 4 100.00% 100.00% [[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]] 3 100.00% 100.00% [[_1st_order, _with_linear_symmetries], _Chini] 2 100.00% 100.00% [_exact, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] 2 100.00% 100.00% [[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_exponential_symmetries], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2], [_3rd_order, _reducible, _mu_poly_yn]] 1 100.00% 100.00% [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] 3 100.00% 100.00% [[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] 39 100.00% 92.31%[9068, 9124, 9125] [[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]] 6 100.00% 83.33%[15197] [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_poly_yn]] 3 100.00% 100.00% [[_2nd_order, _missing_y], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_poly_yn]] 1 100.00% 100.00% [[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class C‘]] 3 100.00% 100.00% [[_2nd_order, _missing_y], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_poly_yn]] 1 100.00% 100.00% [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] 7 100.00% 100.00% [[_2nd_order, _missing_y], _Liouville, [_2nd_order, _reducible, _mu_xy]] 2 100.00% 100.00% [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_xy]] 2 100.00% 100.00% [[_Bessel, _modiﬁed]] 2 100.00% 100.00% [[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] 12 8.33%[7107, 7108, 9924, 9987, 10009, 10013, 10015, 10018, 10019, 12256, 13250] 25.00%[7107, 9924, 9987, 10009, 10013, 10015, 10018, 10019, 13250] [[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] 3 33.33%[9949, 12269] 33.33%[9949, 12269] [_Liouville, [_2nd_order, _reducible, _mu_xy]] 3 100.00% 100.00% [_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] 8 100.00% 100.00% [_Chini, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] 2 100.00% 100.00% [[_1st_order, _with_exponential_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] 1 100.00% 100.00% [[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] 1 100.00% 100.00% [[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] 1 100.00% 100.00% [[_homogeneous, ‘class G‘], [_Abel, ‘2nd type‘, ‘class C‘]] 1 100.00% 100.00% [[_1st_order, _with_linear_symmetries], [_Abel, ‘2nd type‘, ‘class C‘]] 7 100.00% 100.00% [[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] 8 100.00% 100.00% [[_Abel, ‘2nd type‘, ‘class C‘], [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] 4 100.00% 100.00% [[_1st_order, _with_linear_symmetries], _Abel] 13 100.00% 100.00% [[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] 7 100.00% 100.00% [[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘], [_Abel, ‘2nd type‘, ‘class C‘]] 2 100.00% 100.00% [[_homogeneous, ‘class D‘], _rational, _Abel] 3 100.00% 100.00% [[_homogeneous, ‘class C‘], _rational, _Abel] 3 100.00% 100.00% [_rational, [_Abel, ‘2nd type‘, ‘class C‘], [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] 3 100.00% 100.00% [[_homogeneous, ‘class D‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] 1 100.00% 100.00% [[_homogeneous, ‘class C‘], _Abel] 3 100.00% 100.00% [_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]] 6 100.00% 100.00% [[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] 5 100.00% 100.00% [_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] 10 100.00% 100.00% [[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]] 2 100.00% 100.00% [[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘], _Abel] 2 100.00% 100.00% [[_1st_order, _with_linear_symmetries], _rational, _Abel] 1 100.00% 100.00% [_Titchmarsh] 2 50.00%[9349] 50.00%[9349] [_ellipsoidal] 2 100.00% 100.00% [_Jacobi, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] 1 100.00% 100.00% [_Halm] 4 100.00% 100.00% [[_3rd_order, _fully, _exact, _linear]] 7 100.00% 100.00% [[_high_order, _fully, _exact, _linear]] 1 100.00% 100.00% [[_Painleve, ‘1st‘]] 1 0.00%[9916] 0.00%[9916] [[_Painleve, ‘2nd‘]] 1 0.00%[9919] 0.00%[9919] [[_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] 1 0.00%[9950] 0.00%[9950] [[_2nd_order, _with_potential_symmetries]] 2 100.00% 100.00% [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] 6 100.00% 100.00% [[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] 2 100.00% 100.00% [[_2nd_order, _reducible, _mu_xy]] 3 66.67%[10111] 66.67%[10111] [[_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] 1 0.00%[10036] 0.00%[10036] [[_Painleve, ‘4th‘]] 1 0.00%[10060] 0.00%[10060] [[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]] 3 100.00% 100.00% [[_Painleve, ‘3rd‘]] 1 0.00%[10084] 0.00%[10084] [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1]] 1 100.00% 100.00% [[_Painleve, ‘5th‘]] 1 0.00%[10120] 0.00%[10120] [[_Painleve, ‘6th‘]] 1 0.00%[10130] 0.00%[10130] [[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_poly_yn]] 1 0.00%[10139] 0.00%[10139] [[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_poly_yn]] 1 0.00%[10144] 0.00%[10144] [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1]] 1 0.00%[10148] 0.00%[10148] [[_3rd_order, _missing_x], [_3rd_order, _with_linear_symmetries]] 7 28.57%[10161, 10162, 10163, 10178, 13535] 28.57%[10161, 10162, 10163, 10178, 13535] [[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]] 1 100.00% 100.00% [[_3rd_order, _missing_x], [_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]] 1 100.00% 100.00% [[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2], [_3rd_order, _reducible, _mu_poly_yn]] 5 100.00% 100.00% [[_high_order, _missing_x], [_high_order, _missing_y], [_high_order, _with_linear_symmetries]] 2 50.00%[10173] 50.00%[10173] [[_high_order, _missing_x], [_high_order, _missing_y], [_high_order, _with_linear_symmetries], [_high_order, _reducible, _mu_poly_yn]] 2 100.00% 100.00% 96 100.00% 100.00% [[_1st_order, _with_linear_symmetries], [_Abel, ‘2nd type‘, ‘class B‘]] 1 100.00% 100.00% [[_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries]] 1 100.00% 100.00% [_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class B‘]] 1 100.00% 100.00% [[_high_order, _missing_x], [_high_order, _with_linear_symmetries]] 2 0.00%[12226, 12240] 0.00%[12226, 12240] [[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] 2 100.00% 100.00% [[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert] 1 100.00% 100.00% [[_2nd_order, _missing_x], _Duﬃng, [_2nd_order, _reducible, _mu_x_y1]] 2 100.00% 100.00% [[_1st_order, _with_exponential_symmetries], _exact] 1 100.00% 100.00% [[_homogeneous, ‘class C‘], _exact, _rational, _dAlembert] 1 100.00% 100.00% [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_poly_yn]] 2 100.00% 100.00% [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_poly_yn]] 1 100.00% 100.00% [[_high_order, _exact, _linear, _homogeneous]] 3 100.00% 100.00% [_exact, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] 1 100.00% 100.00% [_exact, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class B‘]] 1 100.00% 100.00% [[_2nd_order, _missing_x], _Van_der_Pol] 1 100.00% 100.00% [[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] 1 100.00% 100.00% [[_homogeneous, ‘class D‘], _exact, _rational] 1 100.00% 100.00% [_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class A‘]] 1 100.00% 100.00% [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_poly_yn]] 1 100.00% 100.00% [[_3rd_order, _missing_x], [_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]] 1 0.00%[15221] 100.00%
##### Performance using own ODE types classiﬁcation

The types of the ODE’s are described in my ode solver page at ode types The following table gives count of the number of ODE’s for each ODE type, where the ODE type here is as classiﬁed by my own ode solver, and the percentage of solved ODE’s of that type for each CAS. It also gives a direct link to the ODE’s that failed if any.

 Type of ODE Count Mathematica Maple quadrature 790 97.85%[885, 4266, 4275, 12127, 12129, 12910, 12911, 12914, 12935, 12936, 12962, 12965, 12966, 12967, 14201, 15125, 15126] 99.87%[11994] linear 69 98.55%[6169] 98.55%[6169] separable 126 100.00% 100.00% homogeneous 70 98.57%[5761] 100.00% homogeneousTypeD2 5 100.00% 100.00% exact 308 97.40%[119, 146, 3137, 14323, 14328, 15066, 15067, 15073] 99.35%[14323, 14328] exactWithIntegrationFactor 135 99.26%[8676] 96.30%[1984, 2063, 3090, 8676, 15059] exactByInspection 19 100.00% 94.74%[3926] bernoulli 25 100.00% 100.00% riccati 478 clairaut 115 100.00% 99.13%[4343] dAlembert 253 92.89%[2350, 3000, 4251, 4252, 4253, 4260, 4274, 4278, 4305, 6811, 6813, 6874, 6878, 7254, 8766, 11212, 11240, 15129] 100.00% isobaric 13 100.00% 100.00% polynomial 16 100.00% 100.00% abelFirstKind 58 82.76%[1704, 1897, 3352, 8384, 8385, 8387, 12134, 12631, 12938, 13057] 84.48%[1704, 3352, 8384, 8385, 8387, 12134, 12631, 12938, 13057] ﬁrst order ode series method. Taylor series method 12 100.00% 100.00% ﬁrst order ode series method. Regular singular point 8 100.00% 100.00% ﬁrst order ode series method. Irregular singular point 3 100.00% 0.00%[408, 409, 6418] ﬁrst_order_laplace 77 100.00% 100.00% ﬁrst_order_ode_lie_symmetry_calculated 498 95.98%[1941, 1986, 2083, 3229, 3231, 3232, 3236, 4290, 4294, 4451, 5346, 6797, 6807, 7102, 10819, 11215, 12148, 14439, 15095, 15124] 97.59%[3995, 4040, 4163, 4199, 6820, 8704, 8719, 8820, 8872, 9246, 11230, 12421] system of linear ODEs 802 96.88%[6104, 6542, 6543, 10228, 10238, 10241, 10242, 10243, 10244, 10245, 10254, 10255, 10256, 10257, 10258, 10259, 10261, 12827, 12828, 12829, 12830, 12842, 14043, 15506, 15524] 97.01%[6104, 6542, 6543, 6716, 6719, 10228, 10238, 10241, 10242, 10243, 10244, 10245, 10254, 10256, 10257, 10259, 10261, 12827, 12828, 12829, 12830, 12842, 14043, 15524] second_order_laplace 330 100.00% 99.70%[6513] reduction_of_order 160 98.12%[12050, 14472, 14473] 98.12%[12050, 14472, 14473] second_order_linear_constant_coeﬀ 2 100.00% 0.00%[7306, 7307] second_order_airy 15 100.00% 100.00% second_order_change_of_variable_on_x_method_1 1 100.00% 100.00% second_order_change_of_variable_on_x_method_2 5 100.00% 100.00% second_order_change_of_variable_on_y_method_2 18 83.33%[9747, 10927, 15436] 94.44%[10927] second_order_change_of_variable_on_y_method_1 4 100.00% 100.00% second_order_integrable_as_is 12 83.33%[10144, 12269] 83.33%[10144, 12269] second_order_ode_lagrange_adjoint_equation_method 9 88.89%[10881] 100.00% second_order_nonlinear_solved_by_mainardi_lioville_method 14 100.00% 100.00% second_order_bessel_ode 136 91.18%[7288, 9349, 9634, 9690, 10833, 10892, 10943, 10945, 11036, 11079, 12412, 12748] 97.79%[7288, 9349, 12412] second_order_bessel_ode_form_A 7 100.00% 100.00% second_order_ode_missing_x 168 86.90%[2307, 2308, 9934, 9935, 9936, 9939, 9941, 9959, 9960, 9962, 9985, 10031, 10033, 10156, 10159, 12495, 12570, 12571, 15211, 15444, 15446, 15449] 88.69%[9934, 9935, 9939, 9941, 9959, 9960, 9962, 9985, 10031, 10032, 10033, 10131, 10159, 12570, 12571, 15217, 15444, 15446, 15449] second_order_ode_missing_y 60 88.33%[2304, 6856, 6858, 7212, 10150, 11331, 13520] 96.67%[2304, 2309] second order series method. Taylor series method 8 87.50%[2376] 87.50%[2376] second order series method. Regular singular point. Diﬀerence not integer 266 100.00% 97.74%[7224, 7225, 7226, 7230, 7231, 7233] second order series method. Regular singular point. Repeated root 208 100.00% 99.52%[7241] second order series method. Regular singular point. Diﬀerence is integer 322 100.00% 99.69%[5501] second order series method. Irregular singular point 38 94.74%[5010, 6581] 0.00%[1794, 1797, 1805, 2400, 2541, 2920, 4701, 4714, 4718, 4722, 4723, 5003, 5010, 5217, 5500, 5521, 5526, 5556, 5564, 5588, 5589, 5590, 6042, 6441, 6443, 6449, 6459, 6460, 6581, 6584, 6592, 6617, 6618, 11904, 11905, 12406, 12407, 14803] second order series method. Regular singular point. Complex roots 30 100.00% 100.00% second_order_ode_high_degree 1 100.00% 100.00% higher_order_linear_constant_coeﬃcients_ODE 728 100.00% 100.00% higher_order_ODE_non_constant_coeﬃcients_of_type_Euler 96 100.00% 100.00% higher_order_laplace 29 100.00% 100.00%
##### Performance using simpliﬁed ODE types classiﬁcation

This chapter shows how each CAS performed based on the following basic diﬀerential equations types. A diﬀerential equation is classiﬁed as one of the following types.

1. First order ode.
2. Second and higher order ode.

For ﬁrst order ode, the following are the main classiﬁcations used.

1. First order ode $$f(x,y,y')=0$$ which is linear in $$y'(x)$$.
2. First order ode not linear in $$y'(x)$$ (such as d’Alembert, Clairaut). But it is important to note that in this case the ode is nonlinear in $$y'$$ when written in the form $$y=g(x,y')$$. For an example, lets look at this ode $y' = -\frac {x}{2}-1+\frac {\sqrt {x^{2}+4 x +4 y}}{2}$ Which is linear in $$y'$$ as it stands. But in d’Alembert, Clairaut we always look at the ode in the form $$y=g(x,y')$$. Hence, if we solve for $$y$$ ﬁrst, the above ode now becomes \begin {align*} y &= x y' + \left ( (y')^{2}+ 2 y' + 1 \right )\\ &= g(x,y') \end {align*}

Now we see that $$g(x,y')$$ is nonlinear in $$y'$$. The above ode happens to be of type Clairaut.

For second order and higher order ode’s, further classiﬁcation is

1. Linear ode.
2. non-linear ode.

Another classiﬁcation for second order and higher order ode’s is

1. Constant coeﬃcients ode.
2. Varying coeﬃcients ode

Another classiﬁcation for second order and higher order ode’s is

1. Homogeneous ode. (the right side is zero).
2. Non-homogeneous ode. (the right side is not zero).

All of the above can be combined to give this classiﬁcation

1. First order ode.

1. First order ode linear in $$y'(x)$$.
2. First order ode not linear in $$y'(x)$$ (such as d’Alembert, Clairaut).
2. Second and higher order ode

1. Linear second order ode.

1. Linear homogeneous ode. (the right side is zero).
2. Linear homogeneous and constant coeﬃcients ode.
3. Linear homogeneous and non-constant coeﬃcients ode.
4. Linear non-homogeneous ode. (the right side is not zero).
5. Linear non-homogeneous and constant coeﬃcients ode.
6. Linear non-homogeneous and non-constant coeﬃcients ode.
2. Nonlinear second order ode.

1. Nonlinear homogeneous ode.
2. Nonlinear non-homogeneous ode.

For system of diﬀerential equation the following classiﬁcation is used.

1. System of ﬁrst order odes.

1. Linear system of odes.
2. non-linear system of odes.
2. System of second order odes.

1. Linear system of odes.
2. non-linear system of odes.

The following gives count of the number of ODE’s for each ODE type as speciﬁed above, and the percentage of solved ODE’s of that type for each CAS. It also gives a direct link to the ODE’s that failed if any.

1. First order ode.

Number of problems 6876.

Solved by Mathematica: 93.22%

Solved by Maple: 94.90%

Links to problems not solved by Mathematica:

Links to problems not solved by Maple:

2. Second order linear ODE.

Number of problems 4610.

Solved by Mathematica: 96.79%

Solved by Maple: 97.83%

Links to problems not solved by Mathematica:

Links to problems not solved by Maple:

3. Second order ode.

Number of problems 5160.

Solved by Mathematica: 94.28%

Solved by Maple: 95.72%

Links to problems not solved by Mathematica:

Links to problems not solved by Maple:

4. Second ODE homogeneous ODE.

Number of problems 3169.

Solved by Mathematica: 92.84%

Solved by Maple: 94.79%

Links to problems not solved by Mathematica:

Links to problems not solved by Maple:

5. Second ODE non-homogeneous ODE.

Number of problems 1991.

Solved by Mathematica: 96.58%

Solved by Maple: 97.19%

Links to problems not solved by Mathematica:

Links to problems not solved by Maple:

6. Second order non-linear ODE.

Number of problems 550.

Solved by Mathematica: 73.27%

Solved by Maple: 78.00%

Links to problems not solved by Mathematica:

Links to problems not solved by Maple:

7. Solved using series method.

Number of problems 1555.

Solved by Mathematica: 99.74%

Solved by Maple: 96.27%

Links to problems not solved by Mathematica:

[2376, 5010, 6581, 15474]

Links to problems not solved by Maple:

[408, 409, 1794, 1797, 1805, 2376, 2400, 2541, 2920, 4701, 4714, 4718, 4722, 4723, 5003, 5010, 5217, 5500, 5501, 5502, 5521, 5526, 5556, 5564, 5588, 5589, 5590, 6042, 6418, 6441, 6443, 6449, 6459, 6460, 6581, 6584, 6592, 6617, 6618, 7224, 7225, 7226, 7230, 7231, 7233, 7241, 7300, 7301, 7303, 7304, 7305, 7306, 7307, 11904, 11905, 12406, 12407, 14803]

8. Third and higher order ode.

Number of problems 1054.

Solved by Mathematica: 95.73%

Solved by Maple: 96.11%

Links to problems not solved by Mathematica:

Links to problems not solved by Maple:

9. First order ode linear in derivative.

Number of problems 5933.

Solved by Mathematica: 93.22%

Solved by Maple: 94.77%

Links to problems not solved by Mathematica:

Links to problems not solved by Maple:

10. System of diﬀerential equations.

Number of problems 827.

Solved by Mathematica: 96.49%

Solved by Maple: 96.74%

Links to problems not solved by Mathematica:

Links to problems not solved by Maple:

11. Third and higher order homogeneous ODE.

Number of problems 599.

Solved by Mathematica: 94.32%

Solved by Maple: 94.49%

Links to problems not solved by Mathematica:

Links to problems not solved by Maple:

12. Third and higher order linear ODE.

Number of problems 1016.

Solved by Mathematica: 96.85%

Solved by Maple: 97.24%

Links to problems not solved by Mathematica:

[813, 5817, 9362, 9413, 9784, 9785, 9786, 9787, 9788, 9789, 9790, 9800, 9801, 9803, 9811, 9816, 9827, 9840, 9841, 9856, 9865, 9866, 9867, 9868, 9875, 9895, 9904, 9909, 9913, 12223, 12227, 13559]

Links to problems not solved by Maple:

[813, 5817, 9362, 9413, 9784, 9785, 9786, 9787, 9788, 9789, 9790, 9800, 9801, 9803, 9811, 9816, 9835, 9840, 9856, 9865, 9866, 9867, 9868, 9904, 9905, 9909, 9913, 13559]

13. Third and higher order non-linear ODE.

Number of problems 38.

Solved by Mathematica: 65.79%

Solved by Maple: 65.79%

Links to problems not solved by Mathematica:

Links to problems not solved by Maple:

14. First order ode non-linear in derivative.

Number of problems 943.

Solved by Mathematica: 93.21%

Solved by Maple: 95.65%

Links to problems not solved by Mathematica:

Links to problems not solved by Maple:

15. Higher order, non-linear and homogeneous.

Number of problems 26.

Solved by Mathematica: 73.08%

Solved by Maple: 73.08%

Links to problems not solved by Mathematica:

Links to problems not solved by Maple:

16. Higher order, non-linear and non-homogeneous.

Number of problems 12.

Solved by Mathematica: 50.00%

Solved by Maple: 50.00%

Links to problems not solved by Mathematica:

Links to problems not solved by Maple:

17. Second order, non-linear and homogeneous.

Number of problems 422.

Solved by Mathematica: 74.64%

Solved by Maple: 80.09%

Links to problems not solved by Mathematica:

Links to problems not solved by Maple:

[2304, 2309, 6238, 7214, 7411, 9924, 9928, 9929, 9934, 9935, 9938, 9939, 9940, 9942, 9946, 9948, 9949, 9957, 9959, 9960, 9962, 9965, 9966, 9967, 9968, 9971, 9972, 9981, 9982, 9983, 9985, 9986,