Chapter 1Introduction and Summary of results

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 [10258]. 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, FullSimplify is called on the solution with timeout of 3 minutes. If this timesout, then Simplify is next called. If this also 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 2022.2 94.521 9696 562 Mathematica 13.2 93.264 9567 691

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 2022.2 0.180 271.14 30.743 2781365 Mathematica 13.2 4.046 829.60 691.715 8510026

The problem which Mathematica produced largest leaf size of $$2733033$$ is 9606.

The problem which Maple produced largest leaf size of $$545927$$ is 9648.

The problem which Mathematica used most CPU time of $$178.017$$ seconds is 3759.

The problem which Maple used most CPU time of $$118.453$$ seconds is 3752.

1.2.2 Performance based on ODE type

The following gives the performance of each CAS based on the type of the ODE. The ﬁrst subsection uses the types as classiﬁed by Maple ode advisor.The next subsection uses my own ode solver ODE classiﬁcaiton.

Performance using Maple’s ODE types classiﬁcation

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] 467 99.14%[885, 3741, 3758, 3767] 99.79%[6550] [[_linear, class A]] 148 100.00% 98.65%[6547, 6548] [_separable] 769 99.48%[944, 2513, 5511, 7911] 99.48%[408, 409, 5511, 5665] [_Riccati] 308 55.19%[958, 1697, 1698, 1700, 1701, 1702, 2198, 2795, 2815, 2817, 2830, 3131, 3878, 6592, 7691, 9592, 9596, 9597, 9598, 9603, 9616, 9618, 9619, 9620, 9672, 9689, 9693, 9695, 9696, 9697, 9702, 9709, 9710, 9716, 9717, 9718, 9719, 9720, 9733, 9734, 9735, 9736, 9737, 9738, 9739, 9740, 9741, 9744, 9745, 9753, 9757, 9758, 9760, 9761, 9762, 9763, 9764, 9770, 9771, 9773, 9774, 9775, 9776, 9777, 9782, 9783, 9788, 9789, 9793, 9794, 9795, 9798, 9802, 9803, 9805, 9806, 9811, 9812, 9813, 9814, 9817, 9819, 9820, 9823, 9826, 9828, 9829, 9832, 9835, 9837, 9838, 9841, 9844, 9846, 9847, 9850, 9854, 9855, 9856, 9860, 9861, 9863, 9864, 9866, 9868, 9869, 9870, 9871, 9872, 9873, 9874, 9875, 9877, 9878, 9879, 9880, 9881, 9882, 9883, 9884, 9885, 9886, 9889, 9890, 9893, 9894, 9895, 9896, 9897, 9898, 9899, 9900, 9901, 9902, 9903, 9904, 9905, 9906] 71.75%[958, 1697, 1700, 1701, 1702, 2198, 2815, 2817, 2830, 3878, 6592, 7691, 9596, 9603, 9616, 9618, 9620, 9675, 9683, 9689, 9693, 9695, 9697, 9702, 9718, 9733, 9736, 9737, 9738, 9740, 9744, 9758, 9760, 9771, 9773, 9789, 9802, 9804, 9811, 9819, 9820, 9823, 9828, 9829, 9832, 9837, 9838, 9841, 9846, 9847, 9850, 9854, 9855, 9860, 9861, 9863, 9864, 9866, 9868, 9869, 9870, 9871, 9872, 9873, 9874, 9875, 9877, 9880, 9881, 9882, 9883, 9885, 9889, 9893, 9894, 9895, 9896, 9897, 9898, 9899, 9900, 9901, 9902, 9903, 9904, 9905, 9906] [[_homogeneous, class G]] 62 91.94%[2723, 2727, 2888, 3532, 7963] 93.55%[3487, 3532, 7948, 7963] [_linear] 466 99.79%[5416] 99.57%[4749, 5416] [[_homogeneous, class A], _exact, _rational, [_Abel, 2nd type, class A]] 18 100.00% 100.00% [[_homogeneous, class A], _rational, _Bernoulli] 63 100.00% 100.00% [[_homogeneous, class A], _dAlembert] 118 99.15%[10198] 99.15%[6370] [[_homogeneous, class A], _rational, [_Abel, 2nd type, class A]] 71 98.59%[5501] 100.00% [[_homogeneous, class A], _rational, [_Abel, 2nd type, class B]] 46 100.00% 100.00% [[_homogeneous, class A], _rational, _dAlembert] 174 98.85%[5008, 5509] 99.43%[5509] [[_homogeneous, class C], _dAlembert] 60 86.67%[2491, 3561, 3570, 3752, 3770, 6349, 10204, 10226] 100.00% [[_homogeneous, class C], _Riccati] 16 100.00% 100.00% [[_homogeneous, class G], _rational, _Bernoulli] 48 100.00% 100.00% [_Bernoulli] 87 97.70%[4607, 6377] 100.00% [[_1st_order, _with_linear_symmetries], _Bernoulli] 3 100.00% 100.00% [[_1st_order, _with_symmetry_[F(x),G(x)]]] 45 100.00% 100.00% [y=_G(x,y’)] 112 63.39%[133, 485, 959, 961, 962, 964, 966, 968, 1703, 1706, 1707, 2854, 2859, 2876, 2955, 3503, 3708, 3753, 3779, 3791, 4443, 4487, 5796, 6310, 6500, 7655, 7660, 7663, 7701, 7950, 7975, 8040, 8041, 8083, 8086, 8087, 8090, 8111, 8442, 10205, 10210] 59.82%[133, 485, 959, 961, 962, 964, 966, 968, 1703, 1706, 1707, 2581, 2854, 2859, 2874, 2876, 2887, 2955, 3364, 3503, 3708, 3779, 3790, 4406, 4443, 4487, 5796, 6310, 6500, 7655, 7660, 7663, 7701, 7950, 7975, 8040, 8041, 8083, 8086, 8087, 8090, 8111, 8123, 8140, 10210] [[_1st_order, _with_linear_symmetries]] 94 93.62%[2720, 2722, 3782, 3786, 6054, 10201] 98.94%[8117] [[_homogeneous, class A], _exact, _rational, _dAlembert] 26 100.00% 100.00% [_exact, _rational] 31 96.77%[119] 100.00% [_exact] 62 96.77%[2628, 5444] 100.00% [[_1st_order, _with_linear_symmetries], _exact, _rational] 3 100.00% 100.00% [[_homogeneous, class A], _exact, _rational, [_Abel, 2nd type, class B]] 2 100.00% 100.00% [[_homogeneous, class G], _exact, _rational] 3 66.67%[146] 100.00% [[_2nd_order, _missing_x]] 410 96.59%[6655, 9190, 9191, 9194, 9195, 9197, 9215, 9216, 9218, 9223, 9241, 9287, 9289, 9415] 96.34%[6655, 9190, 9191, 9194, 9195, 9197, 9215, 9216, 9218, 9223, 9241, 9287, 9288, 9289, 9415] [[_Emden, _Fowler], [_2nd_order, _linear, _with_symmetry_[0,F(x)]]] 58 100.00% 100.00% [[_Emden, _Fowler]] 233 99.57%[5591] 96.57%[2032, 4210, 4709, 4803, 4835, 4836, 5831, 5864] [[_2nd_order, _exact, _linear, _homogeneous]] 176 100.00% 98.30%[4837, 5707, 5865] [[_2nd_order, _missing_y]] 80 92.50%[6070, 6103, 6105, 6459, 9406, 10317] 97.50%[5690, 6552] [[_2nd_order, _with_linear_symmetries]] 2134 95.97%[1105, 1138, 4502, 4741, 4742, 4743, 5060, 5065, 5590, 5828, 6343, 6425, 6426, 6429, 6430, 6434, 6436, 6535, 6798, 6800, 7186, 7220, 7222, 8599, 8606, 8608, 8610, 8611, 8612, 8618, 8652, 8653, 8655, 8657, 8661, 8662, 8663, 8679, 8706, 8737, 8785, 8792, 8796, 8816, 8858, 8885, 8941, 8987, 8998, 9018, 9019, 9020, 9022, 9184, 9227, 9237, 9238, 9239, 9242, 9244, 9245, 9246, 9251, 9252, 9256, 9257, 9259, 9263, 9298, 9321, 9341, 9356, 9358, 9359, 9390, 9397, 9398, 9399, 9410, 9411, 10089, 10090, 10098, 10296, 10297, 10315] 96.81%[1794, 1797, 1805, 2411, 4193, 4206, 4495, 4502, 4768, 4773, 4811, 5065, 5289, 5688, 5696, 5828, 5839, 6426, 6434, 6436, 6535, 8599, 8606, 8608, 8610, 8611, 8618, 8652, 8653, 8655, 8657, 8661, 8737, 8785, 8792, 8796, 8816, 8858, 8987, 9018, 9019, 9020, 9022, 9184, 9227, 9237, 9238, 9239, 9242, 9244, 9245, 9246, 9251, 9252, 9256, 9259, 9261, 9263, 9298, 9321, 9341, 9356, 9390, 9399, 9410, 9411, 9413, 10315] [[_2nd_order, _linear, _nonhomogeneous]] 560 99.29%[1162, 1186, 6706, 8656] 96.96%[1162, 1186, 4214, 4215, 4747, 4748, 5760, 6471, 6472, 6473, 6477, 6478, 6480, 6488, 6553, 6554, 8656] [[_2nd_order, _exact, _linear, _nonhomogeneous]] 36 100.00% 100.00% system of linear ODEs 449 95.32%[5351, 5789, 5790, 9469, 9484, 9494, 9497, 9498, 9499, 9500, 9501, 9506, 9507, 9508, 9511, 9512, 9513, 9514, 9515, 9516, 9518] 95.32%[5351, 5789, 5790, 5963, 5966, 9469, 9484, 9494, 9497, 9498, 9499, 9500, 9501, 9506, 9507, 9508, 9511, 9513, 9514, 9516, 9518] [_Gegenbauer] 63 100.00% 100.00% [[_high_order, _missing_x]] 96 96.88%[9123, 9126, 9155] 100.00% [[_3rd_order, _missing_x]] 83 100.00% 100.00% [[_3rd_order, _missing_y]] 35 100.00% 100.00% [[_3rd_order, _exact, _linear, _homogeneous]] 12 100.00% 100.00% [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, _with_symmetry_[0,F(x)]]] 54 92.59%[8654, 8902, 9023, 10097] 98.15%[5706] [_Lienard] 47 100.00% 100.00% [[_homogeneous, class A], _rational, _Riccati] 27 100.00% 100.00% [x=_G(y,y’)] 12 66.67%[550, 2204, 5430, 8152] 66.67%[550, 2204, 5430, 8152] [[_Abel, 2nd type, class B]] 15 26.67%[553, 1046, 7830, 9924, 9927, 9947, 9948, 9949, 9969, 9982, 9987] 40.00%[553, 1046, 7830, 9927, 9947, 9948, 9949, 9969, 9982] [_exact, _rational, [_1st_order, _with_symmetry_[F(x),G(x)]], [_Abel, 2nd type, class A]] 6 100.00% 100.00% [[_1st_order, _with_symmetry_[F(x)*G(y),0]]] 21 100.00% 100.00% [[_homogeneous, class D], _rational] 2 100.00% 100.00% [[_1st_order, _with_exponential_symmetries]] 5 100.00% 100.00% [_rational] 99 85.86%[1039, 1075, 2609, 2683, 2684, 3638, 3807, 5358, 8060, 8062, 8069, 8465, 8474, 10184] 77.78%[1039, 1075, 2609, 2683, 2684, 3418, 3638, 3690, 3691, 3807, 5358, 8032, 8060, 8062, 8065, 8288, 8465, 8474, 8492, 8500, 10184, 10216] [_rational, [_Abel, 2nd type, class B]] 133 [_rational, [_1st_order, _with_symmetry_[F(x),G(x)]], [_Abel, 2nd type, class A]] 4 100.00% 100.00% [NONE] 80 40.00%[710, 1041, 6357, 6461, 7637, 7668, 7782, 7947, 8155, 8156, 8415, 8417, 9174, 9177, 9178, 9182, 9185, 9187, 9188, 9196, 9198, 9202, 9203, 9204, 9207, 9213, 9221, 9222, 9224, 9228, 9254, 9264, 9272, 9281, 9283, 9308, 9311, 9313, 9314, 9317, 9318, 9330, 9336, 9368, 9380, 9381, 9394, 9430] 36.25%[710, 5485, 6357, 6461, 7637, 7668, 7782, 7947, 8155, 8156, 8415, 8417, 9174, 9177, 9178, 9185, 9187, 9188, 9196, 9198, 9202, 9203, 9204, 9207, 9213, 9221, 9222, 9224, 9228, 9254, 9264, 9272, 9277, 9281, 9283, 9284, 9285, 9300, 9308, 9311, 9313, 9314, 9317, 9318, 9330, 9336, 9368, 9380, 9381, 9394, 9430] [_rational, [_1st_order, _with_symmetry_[F(x)*G(y),0]]] 21 100.00% 100.00% [[_homogeneous, class G], _rational, [_Abel, 2nd type, class B]] 40 100.00% 100.00% [_Gegenbauer, [_2nd_order, _linear, _with_symmetry_[0,F(x)]]] 12 100.00% 100.00% [[_high_order, _with_linear_symmetries]] 39 82.05%[813, 9119, 9120, 9121, 9122, 9151, 9169] 82.05%[813, 9119, 9120, 9121, 9122, 9161, 9169] [[_3rd_order, _with_linear_symmetries]] 103 83.50%[5064, 9036, 9037, 9038, 9039, 9040, 9041, 9042, 9052, 9053, 9055, 9063, 9068, 9079, 9094, 9095, 9110] 84.47%[5064, 9036, 9037, 9038, 9039, 9040, 9041, 9042, 9052, 9053, 9055, 9063, 9068, 9089, 9094, 9110] [[_high_order, _linear, _nonhomogeneous]] 53 96.23%[9131, 9160] 98.11%[9160] [[_1st_order, _with_linear_symmetries], _Clairaut] 44 100.00% 100.00% [[_1st_order, _with_symmetry_[F(x),G(y)]]] 49 100.00% 100.00% [[_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]] 71 100.00% 100.00% [[_homogeneous, class C], _rational, _Riccati] 5 100.00% 100.00% [[_Abel, 2nd type, class A]] 34 14.71%[3168, 3220, 4446, 7786, 7799, 9914, 9915, 9979, 9980, 9981, 9990, 9991, 9992, 9993, 9994, 10008, 10054, 10061, 10062, 10064, 10065, 10067, 10068, 10069, 10070, 10071, 10072, 10073, 10074] 35.29%[3168, 3220, 4446, 7786, 7799, 9979, 9980, 9981, 9990, 9991, 9992, 9993, 9994, 10008, 10054, 10062, 10065, 10069, 10070, 10072, 10073, 10074] [_rational, _Bernoulli] 39 100.00% 100.00% [[_homogeneous, class A]] 7 100.00% 100.00% [[_homogeneous, class C], _rational, [_Abel, 2nd type, class A]] 106 98.11%[3943, 10076] 100.00% [[_homogeneous, class G], _rational, _Riccati] 19 100.00% 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]] 1 100.00% 100.00% [_exact, _rational, [_Abel, 2nd type, class B]] 14 100.00% 100.00% [_exact, [_Abel, 2nd type, class B]] 2 100.00% 100.00% [_exact, [_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]] 6 100.00% 100.00% [_exact, [_1st_order, _with_symmetry_[F(x),G(x)]], [_Abel, 2nd type, class A]] 2 100.00% 100.00% [_exact, _Bernoulli] 6 100.00% 100.00% [[_homogeneous, class A], _exact, _rational, _Bernoulli] 4 100.00% 100.00% [_rational, [_Abel, 2nd type, class C]] 12 83.33%[4409, 4454] 83.33%[4409, 4454] [[_homogeneous, class G], _rational] 74 100.00% 97.30%[3655, 6067] [[_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] 101 94.06%[9610, 9641, 9649, 9658, 9662, 9663] 97.03%[9658, 9662, 9663] [[_3rd_order, _linear, _nonhomogeneous]] 53 100.00% 100.00% [[_high_order, _missing_y]] 18 94.44%[9165] 94.44%[9165] [[_3rd_order, _exact, _linear, _nonhomogeneous]] 6 100.00% 100.00% [[_high_order, _exact, _linear, _nonhomogeneous]] 5 100.00% 100.00% [[_homogeneous, class C], _exact, _rational, [_Abel, 2nd type, class A]] 20 100.00% 100.00% [_exact, [_Abel, 2nd type, class A]] 1 100.00% 100.00% [[_1st_order, _with_linear_symmetries], [_Abel, 2nd type, class A]] 2 100.00% 100.00% [[_Riccati, _special]] 14 100.00% 100.00% [_Abel] 25 76.00%[1704, 2843, 7628, 7629, 7630, 7631] 76.00%[1704, 2843, 7628, 7629, 7630, 7631] [_Laguerre] 34 100.00% 100.00% [_Laguerre, [_2nd_order, _linear, _with_symmetry_[0,F(x)]]] 4 100.00% 100.00% [_Bessel] 15 100.00% 100.00% [_rational, _Abel] 21 95.24%[1897] 100.00% [_exact, _rational, [_1st_order, _with_symmetry_[F(x)*G(y),0]]] 9 100.00% 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] 4 100.00% 100.00% [[_homogeneous, class G], _exact, _rational, [_Abel, 2nd type, class B]] 6 100.00% 100.00% [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]] 3 100.00% 100.00% [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] 11 90.91%[9355] 100.00% [[_3rd_order, _exact, _nonlinear]] 2 50.00%[9420] 50.00%[9420] [_Jacobi] 30 100.00% 100.00% [_rational, [_1st_order, _with_symmetry_[F(x)*G(y),0]], [_Abel, 2nd type, class B]] 5 100.00% 100.00% [[_2nd_order, _quadrature]] 33 100.00% 96.97%[6551] [[_3rd_order, _quadrature]] 3 100.00% 100.00% [[_homogeneous, class D], _Bernoulli] 3 100.00% 100.00% [[_homogeneous, class G], _exact] 1 100.00% 100.00% [_exact, [_1st_order, _with_symmetry_[F(x)*G(y),0]]] 5 100.00% 100.00% [[_homogeneous, class G], _exact, _rational, _Bernoulli] 8 100.00% 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% [[_1st_order, _with_linear_symmetries], _rational, _Clairaut] 14 92.86%[10212] 100.00% [[_homogeneous, class D]] 8 100.00% 100.00% [_exact, _rational, _Riccati] 3 100.00% 100.00% [_rational, [_1st_order, _with_symmetry_[F(x),G(x)]], [_Abel, 2nd type, class B]] 5 100.00% 100.00% [[_1st_order, _with_linear_symmetries], _rational] 22 100.00% 100.00% [[_homogeneous, class D], _rational, _Riccati] 19 100.00% 100.00% [[_1st_order, _with_linear_symmetries], _exact] 2 100.00% 100.00% [[_homogeneous, class C], _exact, _dAlembert] 3 100.00% 100.00% [_exact, _rational, [_1st_order, _with_symmetry_[F(x)*G(y),0]], [_Abel, 2nd type, class A]] 1 100.00% 100.00% [[_homogeneous, class D], _rational, [_Abel, 2nd type, class A]] 2 100.00% 100.00% [_rational, [_Abel, 2nd type, class A]] 37 27.03%[3165, 7783, 7785, 9912, 9916, 9943, 9959, 9977, 9978, 9995, 9997, 9998, 10002, 10004, 10007, 10020, 10051, 10052, 10053, 10055, 10056, 10057, 10058, 10059, 10060, 10077, 10079] 45.95%[3165, 7783, 7785, 9912, 9916, 9977, 9978, 9998, 10004, 10007, 10020, 10051, 10052, 10055, 10056, 10057, 10058, 10059, 10077, 10079] [[_homogeneous, class G], _dAlembert] 6 100.00% 100.00% [[_1st_order, _with_linear_symmetries], _rational, _Bernoulli] 3 100.00% 100.00% [[_homogeneous, class D], _rational, _Bernoulli] 24 100.00% 100.00% [[_1st_order, _with_linear_symmetries], _dAlembert] 47 76.60%[3743, 3744, 3745, 3766, 3797, 6058, 6060, 6062, 6121, 6125, 6501] 100.00% [[_homogeneous, class G], _Abel] 4 100.00% 100.00% [[_homogeneous, class G], _Chini] 4 100.00% 100.00% [_Chini] 3 0.00%[2846, 3134, 7636] 0.00%[2846, 3134, 7636] [_rational, [_Riccati, _special]] 9 100.00% 100.00% [[_1st_order, _with_linear_symmetries], _rational, _Riccati] 2 100.00% 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 G], _rational, [_Abel, 2nd type, class A]] 3 100.00% 100.00% [[_homogeneous, class C], _rational, [_Abel, 2nd type, class C], _dAlembert] 1 100.00% 100.00% [[_homogeneous, class A], _rational, [_Abel, 2nd type, class C], _dAlembert] 4 100.00% 100.00% [_exact, _rational, _Bernoulli] 1 100.00% 100.00% [[_homogeneous, class G], _rational, [_Abel, 2nd type, class C]] 5 100.00% 100.00% [[_Abel, 2nd type, class C]] 7 71.43%[3335, 7849] 71.43%[3335, 7849] [[_homogeneous, class C], _rational] 7 100.00% 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% 88.24%[3642, 8028] unknown 5 60.00%[7920, 9385] 0.00%[3472, 7920, 7932, 9385, 9414] [_dAlembert] 17 100.00% 100.00% [_rational, _dAlembert] 11 90.91%[8010] 100.00% [[_homogeneous, class G], _rational, _Clairaut] 5 100.00% 100.00% [[_1st_order, _with_linear_symmetries], _rational, _dAlembert] 10 100.00% 100.00% [[_homogeneous, class C], _rational, _dAlembert] 10 100.00% 100.00% [[_homogeneous, class G], _rational, _dAlembert] 6 100.00% 100.00% [_rational, [_1st_order, _with_symmetry_[F(x),G(y)]]] 14 100.00% 92.86%[3702] [[_homogeneous, class G], _Clairaut] 2 100.00% 100.00% [_Clairaut] 7 100.00% 85.71%[3845] [[_homogeneous, class A], _exact, _dAlembert] 2 100.00% 100.00% [[_homogeneous, class D], _exact, _rational, _Bernoulli] 1 100.00% 100.00% [[_homogeneous, class D], _exact, _rational, [_Abel, 2nd type, class A]] 3 100.00% 100.00% [[_high_order, _quadrature]] 6 100.00% 100.00% [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] 8 50.00%[4160, 4331, 4332, 4333] 100.00% [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] 55 94.55%[5347, 6086, 6087] 96.36%[6086, 6087] [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] 24 95.83%[4158] 95.83%[9387] [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] 4 100.00% 100.00% [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] 6 100.00% 100.00% [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] 16 93.75%[6464] 100.00% [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]] 2 0.00%[4159, 5493] 100.00% [_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] 7 100.00% 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] 12 100.00% 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] 36 100.00% 91.67%[8313, 8369, 8370] [[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]] 3 100.00% 100.00% [[_2nd_order, _missing_x], [_2nd_order, _with_potential_symmetries], [_2nd_order, _reducible, _mu_xy]] 3 100.00% 100.00% [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] 11 100.00% 100.00% [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_poly_yn]] 1 100.00% 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_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] 2 100.00% 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]] 3 100.00% 100.00% [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_y_y1]] 2 100.00% 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]] 1 100.00% 100.00% [[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] 10 10.00%[6354, 6355, 9180, 9243, 9265, 9269, 9271, 9274, 9275] 20.00%[6354, 9180, 9243, 9265, 9269, 9271, 9274, 9275] [[_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]] 3 100.00% 100.00% [[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] 2 50.00%[9205] 50.00%[9205] [_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] 1 0.00%[8595] 0.00%[8595] [_ellipsoidal] 2 100.00% 100.00% [_Jacobi, [_2nd_order, _linear, _with_symmetry_[0,F(x)]]] 1 100.00% 100.00% [_Halm] 2 100.00% 100.00% [[_3rd_order, _fully, _exact, _linear]] 5 100.00% 100.00% [[_high_order, _fully, _exact, _linear]] 1 100.00% 100.00% [[_Painleve, 1st]] 1 0.00%[9172] 0.00%[9172] [[_Painleve, 2nd]] 1 0.00%[9175] 0.00%[9175] [[_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] 1 0.00%[9206] 0.00%[9206] [[_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]] 1 100.00% 100.00% [[_2nd_order, _reducible, _mu_xy]] 3 66.67%[9367] 66.67%[9367] [[_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] 1 0.00%[9292] 0.00%[9292] [[_Painleve, 4th]] 1 0.00%[9316] 0.00%[9316] [[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]] 1 100.00% 100.00% [[_Painleve, 3rd]] 1 0.00%[9340] 0.00%[9340] [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1]] 1 100.00% 100.00% [[_Painleve, 5th]] 1 0.00%[9376] 0.00%[9376] [[_Painleve, 6th]] 1 0.00%[9386] 0.00%[9386] [[_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%[9395] 0.00%[9395] [[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_poly_yn]] 1 0.00%[9400] 0.00%[9400] [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1]] 1 0.00%[9404] 0.00%[9404] [[_3rd_order, _missing_x], [_3rd_order, _with_linear_symmetries]] 6 33.33%[9417, 9418, 9419, 9434] 33.33%[9417, 9418, 9419, 9434] [[_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]] 2 100.00% 100.00% [[_high_order, _missing_x], [_high_order, _missing_y], [_high_order, _with_linear_symmetries]] 2 50.00%[9429] 50.00%[9429] [[_high_order, _missing_x], [_high_order, _missing_y], [_high_order, _with_linear_symmetries], [_high_order, _reducible, _mu_poly_yn]] 1 100.00% 100.00% 62 100.00% 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% [[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]] 1 100.00% 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 178 100.00% 100.00% linear 761 99.74%[5416, 5511] 99.74%[5416, 5511] separable 681 98.97%[885, 944, 2513, 3741, 3758, 3767, 7911] 100.00% homogeneous 481 99.17%[5008, 5501, 5509, 10198] 99.79%[5509] homogeneousTypeC 24 100.00% 100.00% exact 202 98.02%[119, 146, 2628, 5444] 99.01%[3472, 7932] exactWithIntegrationFactor 268 99.63%[7920] 98.51%[2581, 3642, 7920, 8028] exactByInspection 32 100.00% 96.88%[3418] bernoulli 339 99.41%[4607, 6377] 100.00% riccati 525 72.95%[958, 1697, 1698, 1700, 1701, 1702, 2198, 2795, 2815, 2817, 2830, 3131, 3878, 6592, 7691, 9592, 9596, 9597, 9598, 9603, 9610, 9616, 9618, 9619, 9620, 9641, 9649, 9658, 9662, 9663, 9672, 9689, 9693, 9695, 9696, 9697, 9702, 9709, 9710, 9716, 9717, 9718, 9719, 9720, 9733, 9734, 9735, 9736, 9737, 9738, 9739, 9740, 9741, 9744, 9745, 9753, 9757, 9760, 9761, 9762, 9763, 9764, 9770, 9773, 9774, 9775, 9776, 9777, 9782, 9783, 9788, 9789, 9793, 9794, 9795, 9798, 9802, 9803, 9805, 9806, 9811, 9812, 9813, 9814, 9817, 9819, 9820, 9823, 9826, 9828, 9829, 9832, 9835, 9837, 9838, 9841, 9844, 9846, 9847, 9850, 9854, 9855, 9856, 9860, 9861, 9863, 9864, 9866, 9868, 9869, 9870, 9871, 9872, 9873, 9874, 9875, 9877, 9878, 9879, 9880, 9881, 9882, 9883, 9884, 9885, 9886, 9889, 9890, 9893, 9894, 9895, 9896, 9897, 9898, 9899, 9900, 9901, 9902, 9903, 9904, 9905, 9906] 83.24%[958, 1697, 1700, 1701, 1702, 2198, 2815, 2817, 2830, 3878, 6592, 7691, 9596, 9603, 9616, 9618, 9620, 9658, 9662, 9663, 9675, 9683, 9689, 9693, 9695, 9697, 9702, 9718, 9733, 9736, 9737, 9738, 9740, 9744, 9760, 9773, 9789, 9802, 9804, 9811, 9819, 9820, 9823, 9828, 9829, 9832, 9837, 9838, 9841, 9846, 9847, 9850, 9854, 9855, 9860, 9861, 9863, 9864, 9866, 9868, 9869, 9870, 9871, 9872, 9873, 9874, 9875, 9877, 9880, 9881, 9882, 9883, 9885, 9889, 9893, 9894, 9895, 9896, 9897, 9898, 9899, 9900, 9901, 9902, 9903, 9904, 9905, 9906] clairaut 79 98.73%[10212] 98.73%[3845] dAlembert 151 86.75%[2491, 3561, 3570, 3743, 3744, 3745, 3752, 3766, 3770, 3797, 6058, 6060, 6062, 6121, 6125, 6349, 6501, 8010, 10204, 10226] 99.34%[6370] isobaric 166 93.37%[2720, 2722, 2723, 2727, 2888, 3532, 3782, 3786, 6054, 7963, 10201] 95.78%[3487, 3532, 3655, 6067, 7948, 7963, 8117] ﬁrst order special form ID 1 5 100.00% 100.00% polynomial 97 97.94%[3943, 10076] 100.00% abelFirstKind 6 100.00% 100.00% diﬀerentialType 67 100.00% 100.00% ﬁrst order ode series method. Ordinary point 42 100.00% 92.86%[6547, 6548, 6550] ﬁrst order ode series method. Regular singular point 9 100.00% 88.89%[4749] ﬁrst order ode series method. Irregular singular point 3 100.00% 0.00%[408, 409, 5665] ﬁrst_order_laplace 42 100.00% 100.00% system of linear ODEs 403 99.26%[5351, 5790, 9484] 98.76%[5351, 5790, 5963, 5966, 9484] second_order_laplace 159 100.00% 99.37%[5760] reduction_of_order 90 96.67%[1138, 5590, 5591] 100.00% second_order_ode_quadrature 21 100.00% 100.00% second_order_linear_constant_coeﬀ 703 100.00% 100.00% second_order_airy 23 100.00% 100.00% second_order_euler_ode 166 100.00% 100.00% second_order_change_of_variable_on_x_q1_constant_method 1 100.00% 100.00% second_order_change_of_variable_on_y_general_n 58 96.55%[8902, 8998] 100.00% second_order_integrable_as_is 69 86.96%[4159, 4160, 4331, 4332, 4333, 5493, 9205, 9395, 9400] 95.65%[9205, 9395, 9400] second_order_ode_can_be_made_integrable 21 85.71%[5347, 6086, 6087] 90.48%[6086, 6087] second_order_ode_solved_by_an_integrating_factor 18 100.00% 100.00% second_order_change_of_variable_on_x_p1_zero_method 70 97.14%[8654, 9023] 100.00% second_order_ode_lagrange_adjoint_equation_method 18 100.00% 100.00% second_order_nonlinear_solved_by_mainardi_lioville_method 10 100.00% 100.00% second_order_change_of_variable_on_y_n_one_case 40 97.50%[8706] 100.00% second_order_bessel_ode 116 100.00% 100.00% second_order_ode_missing_x 146 91.78%[4158, 9190, 9191, 9195, 9197, 9215, 9216, 9218, 9241, 9287, 9289, 9415] 91.10%[9190, 9191, 9195, 9197, 9215, 9216, 9218, 9241, 9287, 9288, 9289, 9387, 9415] second_order_ode_missing_y 49 85.71%[6070, 6103, 6105, 6459, 6464, 9406, 10317] 100.00% second order series method. Ordinary point 456 100.00% 100.00% second order series method. Regular singular point. Diﬀerence not integer 188 100.00% 100.00% second order series method. Regular singular point. Repeated root 167 100.00% 100.00% second order series method. Regular singular point. Diﬀerence is integer 260 100.00% 100.00% second order series method. Irregular singular point 29 93.10%[4502, 5828] 0.00%[1794, 1797, 1805, 2032, 2411, 4193, 4206, 4210, 4495, 4502, 4709, 4768, 4773, 4803, 4811, 4835, 4836, 4837, 5289, 5688, 5690, 5696, 5706, 5707, 5828, 5831, 5839, 5864, 5865] second order series method. Regular singular point. Complex roots 24 87.50%[4741, 4742, 4743] 100.00% second_order_ode_high_degree 1 100.00% 100.00% Higher order linear constant coeﬃcients ODE 322 99.07%[9123, 9126, 9155] 100.00% Higher order ODE, non constant coeﬃcients of type Euler 45 100.00% 100.00% higher_order_laplace 9 100.00% 100.00%

These are direct links to the ode problems based on status of solving.

Not solved by Mathematica

Not solved by Maple

Solved by Maple but not by Mathematica

Solved by Mathematica but not by Maple

(99) [408, 409, 1794, 1797, 1805, 2032, 2411, 2581, 2874, 2887, 3364, 3418, 3472, 3487, 3642, 3655, 3690, 3691, 3702, 3790, 3845, 4193, 4206, 4210, 4214, 4215, 4406, 4495, 4709, 4747, 4748, 4749, 4768, 4773, 4803, 4811, 4835, 4836, 4837, 5289, 5485, 5665, 5688, 5690, 5696, 5706, 5707, 5760, 5831, 5839, 5864, 5865, 5963, 5966, 6067, 6370, 6471, 6472, 6473, 6477, 6478, 6480, 6488, 6547, 6548, 6550, 6551, 6552, 6553, 6554, 7932, 7948, 8028, 8032, 8065, 8117, 8123, 8140, 8288, 8313, 8369, 8370, 8492, 8500, 9089, 9161, 9261, 9277, 9284, 9285, 9288, 9300, 9387, 9413, 9414, 9675, 9683, 9804, 10216]

Both systems unable to solve