Chapter 1
Introduction 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 in the links below.

The current number of differential equations is [10044]. 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

Table 1.1: Summary of final results

System	% solved	Number solved	Number failed
Maple 2022.2	94.454	9487	557
Mathematica 13.1	93.260	9367	677

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

Table 1.2: Summary of run time performance of each CAS system

System	mean time (sec)	mean leaf size	total time (min)	total leaf size
Maple 2022.2	0.285	273.73	47.695	2749388
Mathematica 13.1	2.448	845.31	409.857	8490335

The problem which Mathematica produced largest leaf size of \(2733033\) is 9606The problem which Maple produced largest leaf size of \(540884\) is 9648The problem which Mathematica used most CPU time of \(178.125\) seconds is 3759The problem which Maple used most CPU time of \(1997.156\) seconds is 8492

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.

Table 1.3: Percentage solved per Maple ODE type

Type of ODE	Count	Mathematica	Maple
[_quadrature]	459	99.13% [885, 3741, 3758, 3767]	99.78% [6550]
[[_linear, class A]]	148	100.00%	98.65% [6547, 6548]
[_separable]	752	99.47% [944, 2513, 5511, 7911]	99.47% [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]	454	99.78% [5416]	99.56% [4749, 5416]
[[_homogeneous, class A], _exact, _rational, [_Abel, 2nd type, class A]]	18	100.00%	100.00%
[[_homogeneous, class A], _rational, _Bernoulli]	59	100.00%	100.00%
[[_homogeneous, class A], _dAlembert]	111	100.00%	100.00%
[[_homogeneous, class A], _rational, [_Abel, 2nd type, class A]]	68	98.53% [5501]	100.00%
[[_homogeneous, class A], _rational, [_Abel, 2nd type, class B]]	45	100.00%	100.00%
[[_homogeneous, class A], _rational, _dAlembert]	161	98.14% [3703, 5008, 5509]	99.38% [5509]
[[_homogeneous, class C], _dAlembert]	57	92.98% [2491, 3752, 3770, 6349]	98.25% [3752]
[[_homogeneous, class C], _Riccati]	15	100.00%	100.00%
[[_homogeneous, class G], _rational, _Bernoulli]	47	100.00%	100.00%
[_Bernoulli]	84	97.62% [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’)]	109	65.14% [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, 8086, 8087, 8090, 8111, 8442]	59.63% [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, 8032, 8040, 8041, 8086, 8087, 8090, 8111, 8123, 8140]
[[_1st_order, _with_linear_symmetries]]	89	94.38% [2720, 2722, 3782, 3786, 6054]	98.88% [8117]
[[_homogeneous, class A], _exact, _rational, _dAlembert]	24	100.00%	100.00%
[_exact, _rational]	31	96.77% [119]	100.00%
[_exact]	60	98.33% [2628]	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]]	408	96.57% [6655, 9190, 9191, 9194, 9195, 9197, 9215, 9216, 9218, 9223, 9241, 9287, 9289, 9415]	96.32% [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]]	76	93.42% [6070, 6103, 6105, 6459, 9406]	97.37% [5690, 6552]
[[_2nd_order, _with_linear_symmetries]]	2113	96.07% [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]	96.83% [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]
[[_2nd_order, _linear, _nonhomogeneous]]	541	99.26% [1162, 1186, 6706, 8656]	96.67% [1162, 1186, 4214, 4215, 4747, 4748, 5080, 5760, 6471, 6472, 6473, 6477, 6478, 6480, 6488, 6553, 6554, 8656]
[[_2nd_order, _exact, _linear, _nonhomogeneous]]	33	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]	95.77% [5351, 5789, 5790, 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]]	94	96.81% [9123, 9126, 9155]	100.00%
[[_3rd_order, _missing_x]]	77	100.00%	100.00%
[[_3rd_order, _missing_y]]	30	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]	95	85.26% [1039, 1075, 2609, 2683, 2684, 3638, 3807, 5358, 8060, 8062, 8069, 8083, 8465, 8474]	81.05% [1039, 1075, 2609, 2683, 2684, 3418, 3638, 3690, 3691, 3807, 5358, 8060, 8062, 8083, 8288, 8465, 8474, 8500]
[_rational, [_Abel, 2nd type, class B]]	133	27.07% [1069, 2481, 2583, 3268, 3275, 5432, 6563, 7814, 7817, 7833, 7845, 9910, 9911, 9918, 9919, 9921, 9923, 9926, 9928, 9930, 9931, 9933, 9934, 9935, 9936, 9937, 9940, 9941, 9942, 9944, 9945, 9946, 9953, 9954, 9955, 9956, 9957, 9958, 9961, 9962, 9963, 9964, 9965, 9966, 9967, 9968, 9970, 9971, 9972, 9973, 9974, 9975, 9976, 9988, 10005, 10006, 10009, 10012, 10013, 10014, 10015, 10016, 10017, 10018, 10019, 10021, 10022, 10023, 10024, 10025, 10026, 10027, 10028, 10029, 10030, 10031, 10032, 10033, 10034, 10035, 10036, 10037, 10038, 10039, 10040, 10041, 10042, 10043, 10044, 10045, 10046, 10047, 10048, 10049, 10050, 10080, 10081]	51.88% [2481, 2583, 3268, 3275, 5432, 6563, 7814, 7817, 7833, 7845, 9918, 9921, 9926, 9933, 9934, 9935, 9937, 9944, 9945, 9954, 9956, 9957, 9961, 9962, 9965, 9966, 9967, 9968, 9970, 9972, 9973, 9974, 9975, 9976, 10005, 10006, 10012, 10014, 10015, 10016, 10017, 10018, 10021, 10023, 10024, 10026, 10027, 10028, 10029, 10031, 10032, 10034, 10035, 10036, 10038, 10039, 10040, 10041, 10042, 10043, 10044, 10048, 10049, 10080]
[_rational, [_1st_order, _with_symmetry_[F(x),G(x)]], [_Abel, 2nd type, class A]]	4	100.00%	100.00%
[NONE]	81	39.51% [710, 1041, 4481, 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]	35.80% [710, 4481, 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]]]	19	100.00%	100.00%
[[_homogeneous, class G], _rational, [_Abel, 2nd type, class B]]	36	100.00%	100.00%
[_Gegenbauer, [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]	11	100.00%	100.00%
[[_high_order, _with_linear_symmetries]]	37	81.08% [813, 9119, 9120, 9121, 9122, 9151, 9169]	81.08% [813, 9119, 9120, 9121, 9122, 9161, 9169]
[[_3rd_order, _with_linear_symmetries]]	96	82.29% [5064, 9036, 9037, 9038, 9039, 9040, 9041, 9042, 9052, 9053, 9055, 9063, 9068, 9079, 9094, 9095, 9110]	83.33% [5064, 9036, 9037, 9038, 9039, 9040, 9041, 9042, 9052, 9053, 9055, 9063, 9068, 9089, 9094, 9110]
[[_high_order, _linear, _nonhomogeneous]]	49	95.92% [9131, 9160]	97.96% [9160]
[[_1st_order, _with_linear_symmetries], _Clairaut]	45	100.00%	100.00%
[[_1st_order, _with_symmetry_[F(x),G(y)]]]	48	100.00%	100.00%
[[_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]	70	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]	38	100.00%	100.00%
[[_homogeneous, class A]]	7	100.00%	100.00%
[[_homogeneous, class C], _rational, [_Abel, 2nd type, class A]]	101	98.02% [3943, 10076]	100.00%
[[_homogeneous, class G], _rational, _Riccati]	18	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]	69	100.00%	97.10% [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]	100	94.00% [9610, 9641, 9649, 9658, 9662, 9663]	97.00% [9658, 9662, 9663]
[[_3rd_order, _linear, _nonhomogeneous]]	51	100.00%	100.00%
[[_high_order, _missing_y]]	18	94.44% [9165]	94.44% [9165]
[[_3rd_order, _exact, _linear, _nonhomogeneous]]	4	100.00%	100.00%
[[_high_order, _exact, _linear, _nonhomogeneous]]	5	100.00%	100.00%
[[_homogeneous, class C], _exact, _rational, [_Abel, 2nd type, class A]]	19	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]	33	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]	3	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]]	10	100.00%	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]]	32	100.00%	96.88% [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]	10	100.00%	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]	18	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]	48.65% [3165, 7783, 7785, 9912, 9916, 9977, 9978, 9998, 10004, 10007, 10020, 10051, 10052, 10055, 10056, 10057, 10058, 10059, 10079]
[[_homogeneous, class G], _dAlembert]	5	100.00%	100.00%
[[_1st_order, _with_linear_symmetries], _rational, _Bernoulli]	3	100.00%	100.00%
[[_homogeneous, class D], _rational, _Bernoulli]	23	100.00%	100.00%
[[_1st_order, _with_linear_symmetries], _dAlembert]	45	75.56% [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)]]]	16	100.00%	87.50% [3642, 8028]
unknown	7	71.43% [7920, 9385]	0.00% [3472, 4272, 7649, 7920, 7932, 9385, 9414]
[_dAlembert]	17	100.00%	100.00%
[_rational, _dAlembert]	11	90.91% [8010]	100.00%
[[_homogeneous, class G], _rational, _dAlembert]	7	100.00%	100.00%
[[_homogeneous, class G], _rational, _Clairaut]	4	100.00%	100.00%
[[_1st_order, _with_linear_symmetries], _rational, _dAlembert]	10	100.00%	100.00%
[[_homogeneous, class C], _rational, _dAlembert]	9	100.00%	100.00%
[_rational, [_1st_order, _with_symmetry_[F(x),G(y)]]]	14	100.00%	92.86% [3702]
[[_homogeneous, class G], _Clairaut]	1	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]]	2	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]]	53	94.34% [5347, 6086, 6087]	96.23% [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]]	15	93.33% [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]]	2	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]]	2	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]]	2	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]]	4	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]]	2	50.00% [9367]	50.00% [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%

Performance using own ODE types classification

The following gives the ODE types used and a short description of each type.

1. polynomial. First order Polynomial type such as \(y'=\frac {-6x+y-3}{2x-y-1}\).
2. quadrature. First order quadrature type ode such as \(y'=1\).
3. linear. First order linear ode such as \(y'+y=x\).
4. separable. First order separable such as \(y'=xy\).
5. riccati. First order Riccati such as \(y'=x^2-y^2\).
6. exact or exactWithIntegrationFactor. First order exact such as \((x^2+y) \mathrm {d}x+(e^y+x)\mathrm {d}y=0\).
7. homogeneous. First order homogeneous such as \((x+y)y'=x-y\).
8. bernoulli. First order Bernoulli such as \(2xyy'=x^2+y^2\).
9. dAlembert. First order dAlembert such as \(y'=\sqrt {x+y}\).
10. clairaut. First order Clairaut such as \(y=xy'+(y')^3\).
11. polynomial. First order Polynomial type such as \(y'=\frac {-6x+y-3}{2x-y-1}\).
12. isobaric. First order isobaric such as \(2 x^3 y'=1+\sqrt {1+ x^2 y}\).
13. abelFirstKind. First order Abel such as \(y'=e^{-5 x}+y^3\).
14. system of linear ODEs. These are system of first order odes.
15. first order ode series method. Ordinary point. First order ode solved using series method. Ordinary point.
16. first order ode series method. Regular singular point. First order ode solved using series method. Regular singular point.
17. second order constant coefficients. standard second order ode with constant coefficients.
18. second order Euler ode (type 7). standard second order Euler ode such as \(x^2 y''+x y+y=0\).
19. second order quadrature. Such as \(A(x) y'=F(x)\).
20. second order missing y (type 3). Such as \(x^2 y''+x y'=\sin x\).
21. second order Airy ode. Such as \(y''-xy=0\) or \(y''+y'-xy=f(x)\).
22. second order type 5. Such as \(3 y''-y^3=5\).
23. second order type 6. Such as \(y^2 y''+y^2=5\).
24. second order with basic integrating factor (type 13). Such as \(y''+4 x y'+(2+4 x^2)y=0\).
25. second order. Tranformation on independent variable. p=0 method. (type 15). Transformation on independent variable.
26. second order. Tranformation on independent variable. q=constant method. (type 8). Transformation on independent variable.
27. second order. Tranformation on dependent variable. special case method. (type 9). Transformation on dependent variable using \(y(x)=v(x)x^n\).
28. second order type 11. Such as \(x y y''+x (y')^2 - y y'=0\).
29. kovacic type. Any second order ODE which is solvable using Kovacic algorithm.
30. reduction of order. Second order ode where one solution is given.
31. second order series method. Ordinary point. Second order solved using series method. Ordinary point.
32. second order series method. Regular singular point. Complex roots. Second order solved using series method. Regular singular point. Complex roots.
33. second order series method. Regular singular point. Difference is integer. Second order solved using series method. Regular singular point. Difference between roots is integer.
34. second order series method. Regular singular point. Difference not integer. Second order solved using series method. Regular singular point. Difference between roots is not integer.
35. second order series method. Regular singular point. Repeated root. Second order solved using series method. Regular singular point. root is repeated.
36. second order series method. Irregular singular point. Second order solved using Asymptotic methods. Irregular singular point.
37. second order ode. Lagrange adjoint equation method (type 16). Second order solved using Lagrange adjoint method such as \(y''+x^2 y'+y=0\).
38. second order transformation, Reduction to Lower Order. (type 17). Such as \(y (y'')^2 + *y')^3=0\).
39. second order ode. Liouville ode (type 18). Such as \(y''+(3+x+\sin x)y'+(1+y) (y')^2 = 0\).
40. second order ode. Liouville transformation (type 19). Such as \(y''-\frac {1}{\sqrt {x}}+\frac {1}{4 x^2}(x+\sqrt {x}-8) y=0\).
41. Higher order linear constant coefficients ODE.
42. Higher order ODE, non constant coefficients of type Euler. Higher order but Euler type.
43. second order ode with degree not 1. Any second order of degree not one.
44. second order ode. Lagrange adjoint equation method (type 16). Uses transformation to adjoint form.
45. unknown or NONE Any unknown ode type.

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.

Table 1.4: Percentage solved per own ODE type

Type of ODE	Count	Mathematica	Maple
quadrature	160	100.00%	100.00%
linear	747	99.73% [5416, 5511]	99.73% [5416, 5511]
separable	730	99.04% [885, 944, 2513, 3741, 3758, 3767, 7911]	100.00%
homogeneous	455	99.12% [3703, 5008, 5501, 5509]	99.78% [5509]
homogeneousTypeC	22	100.00%	100.00%
exact	218	98.17% [119, 146, 2628, 4481]	98.17% [3472, 4272, 4481, 7932]
exactWithIntegrationFactor	186	99.46% [7920]	97.31% [2581, 3642, 7649, 7920, 8028]
bernoulli	261	99.23% [4607, 6377]	100.00%
riccati	548	73.72% [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, 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]	83.03% [958, 1697, 1700, 1701, 1702, 2198, 2815, 2817, 2830, 3878, 6592, 7691, 8313, 8369, 8370, 9596, 9603, 9616, 9618, 9620, 9658, 9662, 9663, 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]
clairaut	74	100.00%	98.65% [3845]
dAlembert	147	89.12% [2491, 3743, 3744, 3745, 3752, 3766, 3770, 3797, 6058, 6060, 6062, 6121, 6125, 6349, 6501, 8010]	99.32% [3752]
isobaric	199	93.97% [2720, 2722, 2723, 2727, 2888, 3532, 3782, 3786, 6054, 7963, 8465, 8474]	95.48% [3487, 3532, 3655, 6067, 7948, 7963, 8117, 8465, 8474]
first order special form ID 1	5	100.00%	100.00%
polynomial	92	97.83% [3943, 10076]	100.00%
abelFirstKind	78	92.31% [1704, 1897, 2843, 7628, 7629, 7631]	93.59% [1704, 2843, 7628, 7629, 7631]
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]	99.26% [5351, 5790, 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	20	100.00%	100.00%
second_order_linear_constant_coeff	685	100.00%	100.00%
second_order_airy	23	100.00%	100.00%
second_order_euler_ode	164	100.00%	100.00%
second_order_change_of_variable_on_y_general_n	51	96.08% [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	20	85.00% [5347, 6086, 6087]	90.00% [6086, 6087]
second_order_ode_solved_by_an_integrating_factor	17	100.00%	100.00%
second_order_change_of_variable_on_x_p1_zero_method	65	96.92% [8654, 9023]	100.00%
second_order_ode_lagrange_adjoint_equation_method	17	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	36	97.22% [8706]	100.00%
second_order_bessel_ode	112	100.00%	100.00%
second_order_ode_missing_x	128	89.06% [4158, 9190, 9191, 9195, 9197, 9215, 9216, 9218, 9223, 9241, 9287, 9289, 9404, 9415]	88.28% [9190, 9191, 9195, 9197, 9215, 9216, 9218, 9223, 9241, 9287, 9288, 9289, 9387, 9404, 9415]
second_order_ode_missing_y	69	91.30% [6070, 6103, 6105, 6459, 6464, 9406]	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	300	99.00% [9123, 9126, 9155]	100.00%
Higher order ODE, non constant coefficients of type Euler	41	100.00%	100.00%
higher_order_laplace	9	100.00%	100.00%

1.3 Links to problems based on solution result

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

Not solved by Mathematica

(677) [119, 133, 146, 485, 550, 553, 710, 813, 885, 944, 958, 959, 961, 962, 964, 966, 968, 1039, 1041, 1046, 1069, 1075, 1105, 1138, 1162, 1186, 1697, 1698, 1700, 1701, 1702, 1703, 1704, 1706, 1707, 1897, 2198, 2204, 2481, 2491, 2513, 2583, 2609, 2628, 2683, 2684, 2720, 2722, 2723, 2727, 2795, 2815, 2817, 2830, 2843, 2846, 2854, 2859, 2876, 2888, 2955, 3131, 3134, 3165, 3168, 3220, 3268, 3275, 3335, 3503, 3532, 3638, 3703, 3708, 3741, 3743, 3744, 3745, 3752, 3753, 3758, 3766, 3767, 3770, 3779, 3782, 3786, 3791, 3797, 3807, 3878, 3943, 4158, 4159, 4160, 4331, 4332, 4333, 4409, 4443, 4446, 4454, 4481, 4487, 4502, 4607, 4741, 4742, 4743, 5008, 5060, 5064, 5065, 5347, 5351, 5358, 5416, 5430, 5432, 5493, 5501, 5509, 5511, 5590, 5591, 5789, 5790, 5796, 5828, 6054, 6058, 6060, 6062, 6070, 6086, 6087, 6103, 6105, 6121, 6125, 6310, 6343, 6349, 6354, 6355, 6357, 6377, 6425, 6426, 6429, 6430, 6434, 6436, 6459, 6461, 6464, 6500, 6501, 6535, 6563, 6592, 6655, 6706, 6798, 6800, 7186, 7220, 7222, 7628, 7629, 7630, 7631, 7636, 7637, 7655, 7660, 7663, 7668, 7691, 7701, 7782, 7783, 7785, 7786, 7799, 7814, 7817, 7830, 7833, 7845, 7849, 7911, 7920, 7947, 7950, 7963, 7975, 8010, 8040, 8041, 8060, 8062, 8069, 8083, 8086, 8087, 8090, 8111, 8152, 8155, 8156, 8415, 8417, 8442, 8465, 8474, 8595, 8599, 8606, 8608, 8610, 8611, 8612, 8618, 8652, 8653, 8654, 8655, 8656, 8657, 8661, 8662, 8663, 8679, 8706, 8737, 8785, 8792, 8796, 8816, 8858, 8885, 8902, 8941, 8987, 8998, 9018, 9019, 9020, 9022, 9023, 9036, 9037, 9038, 9039, 9040, 9041, 9042, 9052, 9053, 9055, 9063, 9068, 9079, 9094, 9095, 9110, 9119, 9120, 9121, 9122, 9123, 9126, 9131, 9151, 9155, 9160, 9165, 9169, 9172, 9174, 9175, 9177, 9178, 9180, 9182, 9184, 9185, 9187, 9188, 9190, 9191, 9194, 9195, 9196, 9197, 9198, 9202, 9203, 9204, 9205, 9206, 9207, 9213, 9215, 9216, 9218, 9221, 9222, 9223, 9224, 9227, 9228, 9237, 9238, 9239, 9241, 9242, 9243, 9244, 9245, 9246, 9251, 9252, 9254, 9256, 9257, 9259, 9263, 9264, 9265, 9269, 9271, 9272, 9274, 9275, 9281, 9283, 9287, 9289, 9292, 9298, 9308, 9311, 9313, 9314, 9316, 9317, 9318, 9321, 9330, 9336, 9340, 9341, 9356, 9358, 9359, 9367, 9368, 9376, 9380, 9381, 9385, 9386, 9390, 9394, 9395, 9397, 9398, 9399, 9400, 9404, 9406, 9410, 9411, 9415, 9417, 9418, 9419, 9420, 9429, 9430, 9434, 9469, 9484, 9494, 9497, 9498, 9499, 9500, 9501, 9506, 9507, 9508, 9511, 9512, 9513, 9514, 9515, 9516, 9518, 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, 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, 9910, 9911, 9912, 9914, 9915, 9916, 9918, 9919, 9921, 9923, 9924, 9926, 9927, 9928, 9930, 9931, 9933, 9934, 9935, 9936, 9937, 9940, 9941, 9942, 9943, 9944, 9945, 9946, 9947, 9948, 9949, 9953, 9954, 9955, 9956, 9957, 9958, 9959, 9961, 9962, 9963, 9964, 9965, 9966, 9967, 9968, 9969, 9970, 9971, 9972, 9973, 9974, 9975, 9976, 9977, 9978, 9979, 9980, 9981, 9982, 9987, 9988, 9990, 9991, 9992, 9993, 9994, 9995, 9997, 9998, 10002, 10004, 10005, 10006, 10007, 10008, 10009, 10012, 10013, 10014, 10015, 10016, 10017, 10018, 10019, 10020, 10021, 10022, 10023, 10024, 10025, 10026, 10027, 10028, 10029, 10030, 10031, 10032, 10033, 10034, 10035, 10036, 10037, 10038, 10039, 10040, 10041, 10042, 10043, 10044, 10045, 10046, 10047, 10048, 10049, 10050, 10051, 10052, 10053, 10054, 10055, 10056, 10057, 10058, 10059, 10060, 10061, 10062, 10064, 10065, 10067, 10068, 10069, 10070, 10071, 10072, 10073, 10074, 10076, 10077, 10079, 10080, 10081, 10089, 10090, 10097, 10098]

Not solved by Maple

(557) [133, 408, 409, 485, 550, 553, 710, 813, 958, 959, 961, 962, 964, 966, 968, 1039, 1046, 1075, 1162, 1186, 1697, 1700, 1701, 1702, 1703, 1704, 1706, 1707, 1794, 1797, 1805, 2032, 2198, 2204, 2411, 2481, 2581, 2583, 2609, 2683, 2684, 2815, 2817, 2830, 2843, 2846, 2854, 2859, 2874, 2876, 2887, 2955, 3134, 3165, 3168, 3220, 3268, 3275, 3335, 3364, 3418, 3472, 3487, 3503, 3532, 3638, 3642, 3655, 3690, 3691, 3702, 3708, 3752, 3779, 3790, 3807, 3845, 3878, 4193, 4206, 4210, 4214, 4215, 4272, 4406, 4409, 4443, 4446, 4454, 4481, 4487, 4495, 4502, 4709, 4747, 4748, 4749, 4768, 4773, 4803, 4811, 4835, 4836, 4837, 5064, 5065, 5080, 5289, 5351, 5358, 5416, 5430, 5432, 5485, 5509, 5511, 5665, 5688, 5690, 5696, 5706, 5707, 5760, 5789, 5790, 5796, 5828, 5831, 5839, 5864, 5865, 6067, 6086, 6087, 6310, 6354, 6357, 6426, 6434, 6436, 6461, 6471, 6472, 6473, 6477, 6478, 6480, 6488, 6500, 6535, 6547, 6548, 6550, 6551, 6552, 6553, 6554, 6563, 6592, 6655, 7628, 7629, 7630, 7631, 7636, 7637, 7649, 7655, 7660, 7663, 7668, 7691, 7701, 7782, 7783, 7785, 7786, 7799, 7814, 7817, 7830, 7833, 7845, 7849, 7920, 7932, 7947, 7948, 7950, 7963, 7975, 8028, 8032, 8040, 8041, 8060, 8062, 8083, 8086, 8087, 8090, 8111, 8117, 8123, 8140, 8152, 8155, 8156, 8288, 8313, 8369, 8370, 8415, 8417, 8465, 8474, 8500, 8595, 8599, 8606, 8608, 8610, 8611, 8618, 8652, 8653, 8655, 8656, 8657, 8661, 8737, 8785, 8792, 8796, 8816, 8858, 8987, 9018, 9019, 9020, 9022, 9036, 9037, 9038, 9039, 9040, 9041, 9042, 9052, 9053, 9055, 9063, 9068, 9089, 9094, 9110, 9119, 9120, 9121, 9122, 9160, 9161, 9165, 9169, 9172, 9174, 9175, 9177, 9178, 9180, 9184, 9185, 9187, 9188, 9190, 9191, 9194, 9195, 9196, 9197, 9198, 9202, 9203, 9204, 9205, 9206, 9207, 9213, 9215, 9216, 9218, 9221, 9222, 9223, 9224, 9227, 9228, 9237, 9238, 9239, 9241, 9242, 9243, 9244, 9245, 9246, 9251, 9252, 9254, 9256, 9259, 9261, 9263, 9264, 9265, 9269, 9271, 9272, 9274, 9275, 9277, 9281, 9283, 9284, 9285, 9287, 9288, 9289, 9292, 9298, 9300, 9308, 9311, 9313, 9314, 9316, 9317, 9318, 9321, 9330, 9336, 9340, 9341, 9356, 9367, 9368, 9376, 9380, 9381, 9385, 9386, 9387, 9390, 9394, 9395, 9399, 9400, 9404, 9410, 9411, 9413, 9414, 9415, 9417, 9418, 9419, 9420, 9429, 9430, 9434, 9469, 9484, 9494, 9497, 9498, 9499, 9500, 9501, 9506, 9507, 9508, 9511, 9513, 9514, 9516, 9518, 9596, 9603, 9616, 9618, 9620, 9658, 9662, 9663, 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, 9912, 9916, 9918, 9921, 9926, 9927, 9933, 9934, 9935, 9937, 9944, 9945, 9947, 9948, 9949, 9954, 9956, 9957, 9961, 9962, 9965, 9966, 9967, 9968, 9969, 9970, 9972, 9973, 9974, 9975, 9976, 9977, 9978, 9979, 9980, 9981, 9982, 9990, 9991, 9992, 9993, 9994, 9998, 10004, 10005, 10006, 10007, 10008, 10012, 10014, 10015, 10016, 10017, 10018, 10020, 10021, 10023, 10024, 10026, 10027, 10028, 10029, 10031, 10032, 10034, 10035, 10036, 10038, 10039, 10040, 10041, 10042, 10043, 10044, 10048, 10049, 10051, 10052, 10054, 10055, 10056, 10057, 10058, 10059, 10062, 10065, 10069, 10070, 10072, 10073, 10074, 10079, 10080]

Solved by Maple but not by Mathematica

(216) [119, 146, 885, 944, 1041, 1069, 1105, 1138, 1698, 1897, 2491, 2513, 2628, 2720, 2722, 2723, 2727, 2795, 2888, 3131, 3703, 3741, 3743, 3744, 3745, 3753, 3758, 3766, 3767, 3770, 3782, 3786, 3791, 3797, 3943, 4158, 4159, 4160, 4331, 4332, 4333, 4607, 4741, 4742, 4743, 5008, 5060, 5347, 5493, 5501, 5590, 5591, 6054, 6058, 6060, 6062, 6070, 6103, 6105, 6121, 6125, 6343, 6349, 6355, 6377, 6425, 6429, 6430, 6459, 6464, 6501, 6706, 6798, 6800, 7186, 7220, 7222, 7911, 8010, 8069, 8442, 8612, 8654, 8662, 8663, 8679, 8706, 8885, 8902, 8941, 8998, 9023, 9079, 9095, 9123, 9126, 9131, 9151, 9155, 9182, 9257, 9358, 9359, 9397, 9398, 9406, 9512, 9515, 9592, 9597, 9598, 9610, 9619, 9641, 9649, 9672, 9696, 9709, 9710, 9716, 9717, 9719, 9720, 9734, 9735, 9739, 9741, 9745, 9753, 9757, 9761, 9762, 9763, 9764, 9770, 9774, 9775, 9776, 9777, 9782, 9783, 9788, 9793, 9794, 9795, 9798, 9803, 9805, 9806, 9812, 9813, 9814, 9817, 9826, 9835, 9844, 9856, 9878, 9879, 9884, 9886, 9890, 9910, 9911, 9914, 9915, 9919, 9923, 9924, 9928, 9930, 9931, 9936, 9940, 9941, 9942, 9943, 9946, 9953, 9955, 9958, 9959, 9963, 9964, 9971, 9987, 9988, 9995, 9997, 10002, 10009, 10013, 10019, 10022, 10025, 10030, 10033, 10037, 10045, 10046, 10047, 10050, 10053, 10060, 10061, 10064, 10067, 10068, 10071, 10076, 10077, 10081, 10089, 10090, 10097, 10098]

Solved by Mathematica but not by Maple

(96) [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, 4272, 4406, 4495, 4709, 4747, 4748, 4749, 4768, 4773, 4803, 4811, 4835, 4836, 4837, 5080, 5289, 5485, 5665, 5688, 5690, 5696, 5706, 5707, 5760, 5831, 5839, 5864, 5865, 6067, 6471, 6472, 6473, 6477, 6478, 6480, 6488, 6547, 6548, 6550, 6551, 6552, 6553, 6554, 7649, 7932, 7948, 8028, 8032, 8117, 8123, 8140, 8288, 8313, 8369, 8370, 8500, 9089, 9161, 9261, 9277, 9284, 9285, 9288, 9300, 9387, 9413, 9414, 9675, 9683, 9804]

Both systems unable to solve

(461) [133, 485, 550, 553, 710, 813, 958, 959, 961, 962, 964, 966, 968, 1039, 1046, 1075, 1162, 1186, 1697, 1700, 1701, 1702, 1703, 1704, 1706, 1707, 2198, 2204, 2481, 2583, 2609, 2683, 2684, 2815, 2817, 2830, 2843, 2846, 2854, 2859, 2876, 2955, 3134, 3165, 3168, 3220, 3268, 3275, 3335, 3503, 3532, 3638, 3708, 3752, 3779, 3807, 3878, 4409, 4443, 4446, 4454, 4481, 4487, 4502, 5064, 5065, 5351, 5358, 5416, 5430, 5432, 5509, 5511, 5789, 5790, 5796, 5828, 6086, 6087, 6310, 6354, 6357, 6426, 6434, 6436, 6461, 6500, 6535, 6563, 6592, 6655, 7628, 7629, 7630, 7631, 7636, 7637, 7655, 7660, 7663, 7668, 7691, 7701, 7782, 7783, 7785, 7786, 7799, 7814, 7817, 7830, 7833, 7845, 7849, 7920, 7947, 7950, 7963, 7975, 8040, 8041, 8060, 8062, 8083, 8086, 8087, 8090, 8111, 8152, 8155, 8156, 8415, 8417, 8465, 8474, 8595, 8599, 8606, 8608, 8610, 8611, 8618, 8652, 8653, 8655, 8656, 8657, 8661, 8737, 8785, 8792, 8796, 8816, 8858, 8987, 9018, 9019, 9020, 9022, 9036, 9037, 9038, 9039, 9040, 9041, 9042, 9052, 9053, 9055, 9063, 9068, 9094, 9110, 9119, 9120, 9121, 9122, 9160, 9165, 9169, 9172, 9174, 9175, 9177, 9178, 9180, 9184, 9185, 9187, 9188, 9190, 9191, 9194, 9195, 9196, 9197, 9198, 9202, 9203, 9204, 9205, 9206, 9207, 9213, 9215, 9216, 9218, 9221, 9222, 9223, 9224, 9227, 9228, 9237, 9238, 9239, 9241, 9242, 9243, 9244, 9245, 9246, 9251, 9252, 9254, 9256, 9259, 9263, 9264, 9265, 9269, 9271, 9272, 9274, 9275, 9281, 9283, 9287, 9289, 9292, 9298, 9308, 9311, 9313, 9314, 9316, 9317, 9318, 9321, 9330, 9336, 9340, 9341, 9356, 9367, 9368, 9376, 9380, 9381, 9385, 9386, 9390, 9394, 9395, 9399, 9400, 9404, 9410, 9411, 9415, 9417, 9418, 9419, 9420, 9429, 9430, 9434, 9469, 9484, 9494, 9497, 9498, 9499, 9500, 9501, 9506, 9507, 9508, 9511, 9513, 9514, 9516, 9518, 9596, 9603, 9616, 9618, 9620, 9658, 9662, 9663, 9689, 9693, 9695, 9697, 9702, 9718, 9733, 9736, 9737, 9738, 9740, 9744, 9758, 9760, 9771, 9773, 9789, 9802, 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, 9912, 9916, 9918, 9921, 9926, 9927, 9933, 9934, 9935, 9937, 9944, 9945, 9947, 9948, 9949, 9954, 9956, 9957, 9961, 9962, 9965, 9966, 9967, 9968, 9969, 9970, 9972, 9973, 9974, 9975, 9976, 9977, 9978, 9979, 9980, 9981, 9982, 9990, 9991, 9992, 9993, 9994, 9998, 10004, 10005, 10006, 10007, 10008, 10012, 10014, 10015, 10016, 10017, 10018, 10020, 10021, 10023, 10024, 10026, 10027, 10028, 10029, 10031, 10032, 10034, 10035, 10036, 10038, 10039, 10040, 10041, 10042, 10043, 10044, 10048, 10049, 10051, 10052, 10054, 10055, 10056, 10057, 10058, 10059, 10062, 10065, 10069, 10070, 10072, 10073, 10074, 10079, 10080]