Problems 12001 to 12100

Table 2.243: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	time (sec)

12001	$x^{2} (a x^{n} - 1) (y^{'} + λ y^{2}) + (p x^{n} + q) x y + r x^{n} + s = 0$	[_rational, _Riccati]	✓	30.041

12002	$(a x^{n} + b x^{m} + c) y^{'} = c y^{2} - b x^{m - 1} y + a x^{n - 2}$	[_rational, _Riccati]	✗	94.688

12003	$(a x^{n} + b x^{m} + c) y^{'} = a x^{n - 2} y^{2} + b x^{m - 1} y + c$	[_rational, _Riccati]	✗	91.352

12004	$(a x^{n} + b x^{m} + c) y^{'} = α x^{k} y^{2} + β x^{s} y - α λ^{2} x^{k} + β λ x^{s}$	[_rational, _Riccati]	✗	144.528

12005	$(a x^{n} + b x^{m} + c) (y^{'} x - y) + s x^{k} (y^{2} - λ x^{2}) = 0$	[[_homogeneous, ‘class D‘], _rational, _Riccati]	✓	46.855

12006	$y^{'} = a y^{2} + b e^{λ x}$	[_Riccati]	✓	1.911

12007	$y^{'} = y^{2} + a λ e^{λ x} - a^{2} e^{2 λ x}$	[_Riccati]	✓	2.588

12008	$y^{'} = σ y^{2} + a + b e^{λ x} + c e^{2 λ x}$	[_Riccati]	✓	4.536

12009	$y^{'} = σ y^{2} + a y + b e^{x} + c$	[_Riccati]	✓	2.370

12010	$y^{'} = y^{2} + b y + a (λ - b) e^{λ x} - a^{2} e^{2 λ x}$	[_Riccati]	✓	2.799

12011	$y^{'} = y^{2} + a e^{λ x} y - a b e^{λ x} - b^{2}$	[_Riccati]	✗	6.482

12012	$y^{'} = y^{2} + a e^{2 λ x} {(e^{λ x} + b)}^{n} - \frac{λ^{2}}{4}$	[_Riccati]	✓	46.485

12013	$y^{'} = y^{2} + a e^{8 λ x} + b e^{6 λ x} + c e^{4 λ x} - λ^{2}$	[_Riccati]	✓	11.131

12014	$y^{'} = a e^{k x} y^{2} + b e^{s x}$	[_Riccati]	✓	3.184

12015	$y^{'} = b e^{μ x} y^{2} + a λ e^{λ x} - a^{2} b e^{(μ + 2 λ) x}$	[_Riccati]	✗	5.069

12016	$y^{'} = a e^{λ x} y^{2} + b y + c e^{- λ x}$	[[_1st_order, _with_linear_symmetries], _Riccati]	✓	6.738

12017	$y^{'} = a e^{μ x} y^{2} + λ y - a b^{2} e^{(μ + 2 λ) x}$	[_Riccati]	✓	2.754

12018	$y^{'} = e^{λ x} y^{2} + a e^{μ x} y + a λ e^{(μ - λ) x}$	[_Riccati]	✓	4.629

12019	$y^{'} = - λ e^{λ x} y^{2} + a e^{μ x} y - a e^{(μ - λ) x}$	[_Riccati]	✓	4.547

12020	$y^{'} = a e^{μ x} y^{2} + a b e^{(λ + μ) x} y - b λ e^{λ x}$	[_Riccati]	✗	82.087

12021	$y^{'} = a e^{k x} y^{2} + b y + c e^{s x} + d e^{- k x}$	[_Riccati]	✓	4.538

12022	$y^{'} = a e^{(μ + 2 λ) x} y^{2} + (b e^{(λ + μ) x} - λ) y + c e^{μ x}$	[_Riccati]	✓	3.846

12023	$y^{'} = a e^{k x} y^{2} + b y + c e^{k n x} + d e^{k (2 n + 1) x}$	[_Riccati]	✗	5.687

12024	$y^{'} = e^{μ x} {(y - b e^{λ x})}^{2} + b λ e^{λ x}$	[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati]	✓	2.716

12025	$(a e^{λ x} + b e^{μ x} + c) y^{'} = y^{2} + k e^{ν x} y - m^{2} + k m e^{ν x}$	[_Riccati]	✗	25.445

12026	$(a e^{λ x} + b e^{μ x} + c) (y^{'} - y^{2}) + a λ^{2} e^{λ x} + b μ^{2} e^{μ x} = 0$	[_Riccati]	✓	4.520

12027	$y^{'} = y^{2} + a x e^{λ x} y + a e^{λ x}$	[_Riccati]	✓	2.931

12028	$y^{'} = a e^{λ x} y^{2} + b e^{- λ x}$	[[_1st_order, _with_linear_symmetries], _Riccati]	✓	5.476

12029	$y^{'} = a e^{λ x} y^{2} + b n x^{n - 1} - a b^{2} e^{λ x} x^{2 n}$	[_Riccati]	✗	7.090

12030	$y^{'} = e^{λ x} y^{2} + a x^{n} y + a λ x^{n} e^{- λ x}$	[_Riccati]	✗	5.653

12031	$y^{'} = - λ e^{λ x} y^{2} + a x^{n} e^{λ x} y - a x^{n}$	[_Riccati]	✓	3.515

12032	$y^{'} = a e^{λ x} y^{2} - a b x^{n} e^{λ x} y + b n x^{n - 1}$	[_Riccati]	✓	5.317

12033	$y^{'} = a x^{n} y^{2} + b λ e^{λ x} - a b^{2} x^{n} e^{2 λ x}$	[_Riccati]	✗	6.362

12034	$y^{'} = a x^{n} y^{2} + λ y - a b^{2} x^{n} e^{2 λ x}$	[_Riccati]	✓	3.377

12035	$y^{'} = a x^{n} y^{2} - a b x^{n} e^{λ x} y + b λ e^{λ x}$	[_Riccati]	✗	5.931

12036	$y^{'} = - (k + 1) x^{k} y^{2} + a x^{k + 1} e^{λ x} y - a e^{λ x}$	[_Riccati]	✓	6.509

12037	$y^{'} = a x^{n} y^{2} - a x^{n} (b e^{λ x} + c) y + c x^{n}$	[_Riccati]	✗	7.525

12038	$y^{'} = a x^{n} e^{2 λ x} y^{2} + (b x^{n} e^{λ x} - λ) y + c x^{n}$	[_Riccati]	✓	6.068

12039	$y^{'} = a e^{λ x} {(y - b x^{n} - c)}^{2} + b n x^{n - 1}$	[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati]	✓	4.752

12040	$y^{'} x = a e^{λ x} y^{2} + k y + a b^{2} x^{2 k} e^{λ x}$	[_Riccati]	✓	4.004

12041	$y^{'} x = a x^{2 n} e^{λ x} y^{2} + (b x^{n} e^{λ x} - n) y + c e^{λ x}$	[_Riccati]	✗	8.963

12042	$y^{'} = y^{2} + 2 a λ x e^{λ x^{2}} - a^{2} e^{2 λ x^{2}}$	[_Riccati]	✗	3.709

12043	$y^{'} = a e^{- λ x^{2}} y^{2} + λ x y + a b^{2}$	[_Riccati]	✓	2.660

12044	$y^{'} = a x^{n} y^{2} + λ x y + a b^{2} x^{n} e^{λ x^{2}}$	[_Riccati]	✓	3.934

12045	$x^{4} (y^{'} - y^{2}) = a + b e^{\frac{k}{x}} + c e^{\frac{2 k}{x}}$	[_Riccati]	✓	4.799

12046	$y^{'} = y^{2} - a^{2} + a λ \sinh (λ x) - a^{2} \sinh {(λ x)}^{2}$	[_Riccati]	✓	9.519

12047	$y^{'} = y^{2} + a \sinh (β x) y + a b \sinh (β x) - b^{2}$	[_Riccati]	✓	4.474

12048	$y^{'} = y^{2} + a x \sinh {(b x)}^{m} y + a \sinh {(b x)}^{m}$	[_Riccati]	✓	11.534

12049	$y^{'} = λ \sinh (λ x) y^{2} - λ \sinh {(λ x)}^{3}$	[_Riccati]	✓	9.244

12050	$y^{'} = (a \sinh {(λ x)}^{2} - λ) y^{2} - a \sinh {(λ x)}^{2} + λ - a$	[_Riccati]	✓	29.765

12051	$(a \sinh (λ x) + b) y^{'} = y^{2} + c \sinh (μ x) y - d^{2} + c d \sinh (μ x)$	[_Riccati]	✗	116.438

12052	$(a \sinh (λ x) + b) (y^{'} - y^{2}) + a λ^{2} \sinh (λ x) = 0$	[_Riccati]	✓	5.821

12053	$y^{'} = α y^{2} + β + γ \cosh (x)$	[_Riccati]	✓	3.084

12054	$y^{'} = y^{2} + a \cosh (β x) y + a b \cosh (β x) - b^{2}$	[_Riccati]	✓	5.410

12055	$y^{'} = y^{2} + a x \cosh {(b x)}^{m} y + a \cosh {(b x)}^{m}$	[_Riccati]	✓	9.979

12056	$y^{'} = (a \cosh {(λ x)}^{2} - λ) y^{2} + a + λ - a \cosh {(λ x)}^{2}$	[_Riccati]	✓	27.898

12057	$2 y^{'} = (a - λ + a \cosh (λ x)) y^{2} + a + λ - a \cosh (λ x)$	[_Riccati]	✓	26.847

12058	$y^{'} = y^{2} - λ^{2} + a \cosh {(λ x)}^{n} \sinh {(λ x)}^{- n - 4}$	[_Riccati]	✗	31.109

12059	$y^{'} = a \sinh (λ x) y^{2} + b \sinh (λ x) \cosh {(λ x)}^{n}$	[_Riccati]	✓	11.448

12060	$y^{'} = a \cosh (λ x) y^{2} + b \cosh (λ x) \sinh {(λ x)}^{n}$	[_Riccati]	✗	86.373

12061	$(a \cosh (λ x) + b) y^{'} = y^{2} + c \cosh (μ x) y - d^{2} + c d \cosh (μ x)$	[_Riccati]	✗	91.412

12062	$(a \cosh (λ x) + b) (y^{'} - y^{2}) + a λ^{2} \cosh (λ x) = 0$	[_Riccati]	✓	5.178

12063	$y^{'} = y^{2} + a λ - a (a + λ) \tanh {(λ x)}^{2}$	[_Riccati]	✗	16.730

12064	$y^{'} = y^{2} + 3 a λ - λ^{2} - a (a + λ) \tanh {(λ x)}^{2}$	[_Riccati]	✗	17.942

12065	$y^{'} = y^{2} + a x \tanh {(b x)}^{m} y + a \tanh {(b x)}^{m}$	[_Riccati]	✓	8.923

12066	$(a \tanh (λ x) + b) y^{'} = y^{2} + c \tanh (μ x) y - d^{2} + c d \tanh (μ x)$	[_Riccati]	✗	179.373

12067	$y^{'} = y^{2} + a λ - a (a + λ) \coth {(λ x)}^{2}$	[_Riccati]	✗	16.258

12068	$y^{'} = y^{2} - λ^{2} + 3 a λ - a (a + λ) \coth {(λ x)}^{2}$	[_Riccati]	✗	17.524

12069	$y^{'} = y^{2} + a x \coth {(b x)}^{m} y + a \coth {(b x)}^{m}$	[_Riccati]	✓	10.077

12070	$(a \coth (λ x) + b) y^{'} = y^{2} + c \coth (μ x) y - d^{2} + c d \coth (μ x)$	[_Riccati]	✗	178.308

12071	$y^{'} = y^{2} - 2 λ^{2} \tanh {(λ x)}^{2} - 2 λ^{2} \coth {(λ x)}^{2}$	[_Riccati]	✓	17.736

12072	$y^{'} = y^{2} + a λ + b λ - 2 a b - a (a + λ) \tanh {(λ x)}^{2} - b (b + λ) \coth {(λ x)}^{2}$	[_Riccati]	✓	26.297

12073	$y^{'} = a \ln {(x)}^{n} y^{2} + b m x^{m - 1} - a b^{2} x^{2 m} \ln {(x)}^{n}$	[_Riccati]	✗	7.792

12074	$y^{'} x = a y^{2} + b \ln (x) + c$	[_Riccati]	✓	2.498

12075	$y^{'} x = a y^{2} + b \ln {(x)}^{k} + c \ln {(x)}^{2 k + 2}$	[_Riccati]	✓	39.670

12076	$y^{'} x = x y^{2} - a^{2} x \ln {(β x)}^{2} + a$	[_Riccati]	✗	2.766

12077	$y^{'} x = x y^{2} - a^{2} x \ln {(β x)}^{2 k} + a k \ln {(β x)}^{k - 1}$	[_Riccati]	✗	5.278

12078	$y^{'} x = a x^{n} y^{2} + b - a b^{2} x^{n} \ln {(x)}^{2}$	[_Riccati]	✗	4.061

12079	$x^{2} y^{'} = x^{2} y^{2} + a \ln {(x)}^{2} + b \ln (x) + c$	[_Riccati]	✓	8.409

12080	$x^{2} y^{'} = x^{2} y^{2} + a {(b \ln (x) + c)}^{n} + \frac{1}{4}$	[_Riccati]	✗	4.341

12081	$x^{2} \ln (a x) (y^{'} - y^{2}) = 1$	[_Riccati]	✓	2.003

12082	$y^{'} = y^{2} + a \ln (β x) y - a b \ln (β x) - b^{2}$	[_Riccati]	✓	2.320

12083	$y^{'} = y^{2} + a x \ln {(b x)}^{m} y + a \ln {(b x)}^{m}$	[_Riccati]	✓	2.820

12084	$y^{'} = a x^{n} y^{2} - a b x^{n + 1} \ln (x) y + b \ln (x) + b$	[_Riccati]	✗	5.565

12085	$y^{'} = - (n + 1) x^{n} y^{2} + a x^{n + 1} \ln {(x)}^{m} y - a \ln {(x)}^{m}$	[_Riccati]	✓	5.116

12086	$y^{'} = a \ln {(x)}^{n} y - a b x \ln {(x)}^{n + 1} y + b \ln (x) + b$	[_linear]	✓	1.690

12087	$y^{'} = a \ln {(x)}^{k} {(y - b x^{n} - c)}^{2} + b n x^{n - 1}$	[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati]	✓	6.058

12088	$y^{'} = a \ln {(x)}^{n} y^{2} + b \ln {(x)}^{m} y + b c \ln {(x)}^{m} - a c^{2} \ln {(x)}^{n}$	[_Riccati]	✗	6.296

12089	$y^{'} x = {(a y + b \ln (x))}^{2}$	[[_1st_order, _with_linear_symmetries], _Riccati]	✓	1.892

12090	$y^{'} x = a \ln {(λ x)}^{m} y^{2} + k y + a b^{2} x^{2 k} \ln {(λ x)}^{m}$	[_Riccati]	✓	41.752

12091	$y^{'} x = a x^{n} {(y + b \ln (x))}^{2} - b$	[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati]	✓	3.252

12092	$y^{'} x = a x^{2 n} \ln (x) y^{2} + (b x^{n} \ln (x) - n) y + c \ln (x)$	[_Riccati]	✓	5.061

12093	$x^{2} y^{'} = y^{2} a^{2} x^{2} - x y + b^{2} \ln {(x)}^{n}$	[_Riccati]	✓	4.267

12094	$(a \ln (x) + b) y^{'} = y^{2} + c \ln {(x)}^{n} y - λ^{2} + λ c \ln {(x)}^{n}$	[_Riccati]	✗	10.035

12095	$(a \ln (x) + b) y^{'} = \ln {(x)}^{n} y^{2} + c y - λ^{2} \ln {(x)}^{n} + λ c$	[_Riccati]	✗	15.577

12096	$y^{'} = α y^{2} + β + γ \sin (λ x)$	[_Riccati]	✓	3.779

12097	$y^{'} = y^{2} - a^{2} + a λ \sin (λ x) + a^{2} \sin {(λ x)}^{2}$	[_Riccati]	✓	8.582

12098	$y^{'} = y^{2} + λ^{2} + c \sin {(λ x + a)}^{n} \sin {(λ x + b)}^{- n - 4}$	[_Riccati]	✗	150.170

12099	$y^{'} = y^{2} + a \sin (β x) y + a b \sin (β x) - b^{2}$	[_Riccati]	✓	4.627

12100	$y^{'} = y^{2} + a \sin {(b x)}^{m} y + a \sin {(b x)}^{m}$	[_Riccati]	✗	16.519

2.2.121 Problems 12001 to 12100