Problems 12201 to 12300

Table 2.247: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	time (sec)

12201	$y^{'} = y^{2} f (x) - a x^{n} g (x) y + a n x^{n - 1} + a^{2} x^{2 n} (g (x) - f (x))$	[_Riccati]	✗	187.173

12202	$y^{'} = a e^{λ x} y^{2} + a e^{λ x} f (x) y + λ f (x)$	[_Riccati]	✓	3.511

12203	$y^{'} = y^{2} f (x) - a e^{λ x} f (x) y + a λ e^{λ x}$	[_Riccati]	✓	4.787

12204	$y^{'} = y^{2} f (x) + a λ e^{λ x} - a^{2} e^{2 λ x} f (x)$	[_Riccati]	✗	8.958

12205	$y^{'} = y^{2} f (x) + λ y + a^{2} e^{2 λ x} f (x)$	[_Riccati]	✓	2.881

12206	$y^{'} = y^{2} f (x) - f (x) (a e^{λ x} + b) y + a λ e^{λ x}$	[_Riccati]	✓	5.712

12207	$y^{'} = e^{λ x} f (x) y^{2} + (a f (x) - λ) y + b e^{- λ x} f (x)$	[_Riccati]	✓	3.685

12208	$y^{'} = y^{2} f (x) + g (x) y + a λ e^{λ x} - a e^{λ x} g (x) - a^{2} e^{2 λ x} f (x)$	[_Riccati]	✗	77.107

12209	$y^{'} = y^{2} f (x) - a e^{λ x} g (x) y + a λ e^{λ x} + a^{2} e^{2 λ x} (g (x) - f (x))$	[_Riccati]	✗	87.641

12210	$y^{'} = y^{2} f (x) + 2 a λ x e^{λ x^{2}} - a^{2} f (x) e^{2 λ x^{2}}$	[_Riccati]	✗	9.576

12211	$y^{'} = y^{2} f (x) + λ x y + a f (x) e^{λ x}$	[_Riccati]	✗	4.778

12212	$y^{'} = y^{2} f (x) - a \tanh {(λ x)}^{2} (a f (x) + λ) + a λ$	[_Riccati]	✗	17.292

12213	$y^{'} = y^{2} f (x) - a \coth {(λ x)}^{2} (a f (x) + λ) + a λ$	[_Riccati]	✗	17.062

12214	$y^{'} = y^{2} f (x) - a^{2} f (x) + a λ \sinh (λ x) - a^{2} f (x) \sinh {(λ x)}^{2}$	[_Riccati]	✗	21.545

12215	$y^{'} x = y^{2} f (x) + a - a^{2} f (x) \ln {(x)}^{2}$	[_Riccati]	✗	4.071

12216	$y^{'} x = f (x) {(y + a \ln (x))}^{2} - a$	[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati]	✗	5.170

12217	$y^{'} = y^{2} f (x) - a x \ln (x) f (x) y + a \ln (x) + a$	[_Riccati]	✗	3.643

12218	$y^{'} = - a \ln (x) y^{2} + a f (x) (x \ln (x) - x) y - f (x)$	[_Riccati]	✓	3.497

12219	$y^{'} = λ \sin (λ x) y^{2} + f (x) \cos (λ x) y - f (x)$	[_Riccati]	✓	12.721

12220	$y^{'} = y^{2} f (x) - a^{2} f (x) + a λ \sin (λ x) + a^{2} f (x) \sin {(λ x)}^{2}$	[_Riccati]	✗	22.032

12221	$y^{'} = y^{2} f (x) - a^{2} f (x) + a λ \cos (λ x) + a^{2} f (x) \cos {(λ x)}^{2}$	[_Riccati]	✗	22.539

12222	$y^{'} = y^{2} f (x) - a \tan {(λ x)}^{2} (a f (x) - λ) + a λ$	[_Riccati]	✗	18.540

12223	$y^{'} = y^{2} f (x) - a \cot {(λ x)}^{2} (a f (x) - λ) + a λ$	[_Riccati]	✗	18.068

12224	$y^{'} = y^{2} - f {(x)}^{2} + f^{'} (x)$	[_Riccati]	✓	1.304

12225	$y^{'} = y^{2} f (x) - f (x) g (x) y + g^{'} (x)$	[_Riccati]	✓	1.908

12226	$y^{'} = - f^{'} (x) y^{2} + f (x) g (x) y - g (x)$	[_Riccati]	✓	2.092

12227	$y^{'} = g (x) {(y - f (x))}^{2} + f^{'} (x)$	[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati]	✓	2.031

12228	$y^{'} = \frac{y^{2} f^{'} (x)}{g (x)} - \frac{g^{'} (x)}{f (x)}$	[_Riccati]	✗	3.363

12229	$f {(x)}^{2} y^{'} - f^{'} (x) y^{2} + g (x) (y - f (x)) = 0$	[_Riccati]	✗	3.470

12230	$y^{'} = f^{'} (x) y^{2} + a e^{λ x} f (x) y + a e^{λ x}$	[_Riccati]	✓	2.602

12231	$y^{'} = y^{2} f (x) + g^{'} (x) y + a f (x) e^{2 g (x)}$	[_Riccati]	✓	2.383

12232	$y^{'} = y^{2} - \frac{f^{''} (x)}{f (x)}$	[_Riccati]	✓	1.273

12233	$y^{'} = y^{2} + a^{2} f (a x + b)$	[_Riccati]	✗	1.957

12234	$y^{'} = y^{2} + \frac{f (\frac{1}{x})}{x^{4}}$	[_Riccati]	✗	2.011

12235	$y^{'} = y^{2} + \frac{f (\frac{a x + b}{c x + d})}{{(c x + d)}^{4}}$	[_Riccati]	✗	4.811

12236	$x^{2} y^{'} = x^{4} f (x) y^{2} + 1$	[_Riccati]	✗	2.735

12237	$x^{2} y^{'} = x^{4} y^{2} + x^{2 n} f (a x^{n} + b) - \frac{n^{2}}{4} + \frac{1}{4}$	[_Riccati]	✗	6.942

12238	$y^{'} = y^{2} f (x) + g (x) y + h (x)$	[_Riccati]	✗	2.791

12239	$y^{'} = y^{2} + e^{2 λ x} f (e^{λ x}) - \frac{λ^{2}}{4}$	[_Riccati]	✗	4.098

12240	$y^{'} = y^{2} - \frac{λ^{2}}{4} + \frac{e^{2 λ x} f (\frac{a e^{λ x} + b}{c e^{λ x} + d})}{{(c e^{λ x} + d)}^{4}}$	[_Riccati]	✗	53.039

12241	$y^{'} = y^{2} - λ^{2} + \frac{f (\coth (λ x))}{\sinh {(λ x)}^{4}}$	[_Riccati]	✗	60.268

12242	$y^{'} = y^{2} - λ^{2} + \frac{f (\tanh (λ x))}{\cosh {(λ x)}^{4}}$	[_Riccati]	✗	24.517

12243	$x^{2} y^{'} = x^{2} y^{2} + f (a \ln (x) + b) + \frac{1}{4}$	[_Riccati]	✗	4.141

12244	$y^{'} = y^{2} + λ^{2} + \frac{f (\cot (λ x))}{\sin {(λ x)}^{4}}$	[_Riccati]	✗	118.751

12245	$y^{'} = y^{2} + λ^{2} + \frac{f (\tan (λ x))}{\cos {(λ x)}^{4}}$	[_Riccati]	✗	32.673

12246	$y^{'} = y^{2} + λ^{2} + \frac{f (\frac{\sin (λ x + a)}{\sin (λ x + b)})}{\sin {(λ x + b)}^{4}}$	[_Riccati]	✗	556.418

12247	$y y^{'} - y = A$	[_quadrature]	✓	1.022

12248	$y y^{'} - y = A x + B$	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	12.057

12249	$y y^{'} - y = - \frac{2 x}{9} + A + \frac{B}{\sqrt{x}}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	7.022

12250	$y y^{'} - y = 2 A (\sqrt{x} + 4 A + \frac{3 A^{2}}{\sqrt{x}})$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	7.163

12251	$y y^{'} - y = A x + \frac{B}{x} - \frac{B^{2}}{x^{3}}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	1.653

12252	$y y^{'} - y = A x^{k - 1} - k B x^{k} + k B^{2} x^{2 k - 1}$	[_rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✗	2.750

12253	$y y^{'} - y = \frac{A}{x} - \frac{A^{2}}{x^{3}}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	1.608

12254	$y y^{'} - y = A + B e^{- \frac{2 x}{A}}$	[[_Abel, ‘2nd type‘, ‘class A‘]]	✗	1.880

12255	$y y^{'} - y = A (e^{\frac{2 x}{A}} - 1)$	[[_Abel, ‘2nd type‘, ‘class A‘]]	✗	1.865

12256	$y y^{'} - y = - \frac{2 (m + 1)}{{(m + 3)}^{2}} + A x^{m}$	[_rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✗	2.841

12257	$y y^{'} - y = - \frac{2 x}{9} + 6 A^{2} (1 + \frac{2 A}{\sqrt{x}})$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	10.658

12258	$y y^{'} - y = \frac{2 m - 2}{{(m - 3)}^{2}} + \frac{2 A (m (m + 3) \sqrt{x} + (4 m^{2} + 3 m + 9) A + \frac{3 m (m + 3) A^{2}}{\sqrt{x}})}{{(m - 3)}^{2}}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	46.325

12259	$y y^{'} - y = \frac{(2 m + 1) x}{4 m^{2}} + \frac{A}{x} - \frac{A^{2}}{x^{3}}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	2.451

12260	$y y^{'} - y = \frac{4}{9} x + 2 A x^{2} + 2 A^{2} x^{3}$	[_rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✗	2.019

12261	$y y^{'} - y = - \frac{3 x}{16} + \frac{5 A}{x^{1 / 3}} - \frac{12 A^{2}}{x^{5 / 3}}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	5.600

12262	$y y^{'} - y = \frac{A}{x}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	1.003

12263	$y y^{'} - y = - \frac{x}{4} + \frac{A (\sqrt{x} + 5 A + \frac{3 A^{2}}{\sqrt{x}})}{4}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	11.920

12264	$y y^{'} - y = \frac{2 a^{2}}{\sqrt{8 a^{2} + x^{2}}}$	[[_Abel, ‘2nd type‘, ‘class B‘]]	✗	3.075

12265	$y y^{'} - y = 2 x + \frac{A}{x^{2}}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	1.218

12266	$y y^{'} - y = - \frac{6 X}{25} + \frac{2 A (2 \sqrt{x} + 19 A + \frac{6 A^{2}}{\sqrt{x}})}{25}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	30.978

12267	$y y^{'} - y = \frac{3 x}{8} + \frac{3 \sqrt{a^{2} + x^{2}}}{8} - \frac{a^{2}}{16 \sqrt{a^{2} + x^{2}}}$	[[_Abel, ‘2nd type‘, ‘class B‘]]	✗	6.825

12268	$y y^{'} - y = - \frac{4 x}{25} + \frac{A}{\sqrt{x}}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	7.985

12269	$y y^{'} - y = - \frac{9 x}{100} + \frac{A}{x^{5 / 3}}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	44.260

12270	$y y^{'} - y = - \frac{12 x}{49} + \frac{2 A (5 \sqrt{x} + 34 A + \frac{15 A^{2}}{\sqrt{x}})}{49}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	8.289

12271	$y y^{'} - y = - \frac{12 x}{49} + \frac{A (25 \sqrt{x} + 41 A + \frac{10 A^{2}}{\sqrt{x}})}{98}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	8.234

12272	$y y^{'} - y = - \frac{2 x}{9} + \frac{A}{\sqrt{x}}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	6.443

12273	$y y^{'} - y = - \frac{5 x}{36} + \frac{A}{x^{7 / 5}}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	54.498

12274	$y y^{'} - y = - \frac{12 x}{49} + \frac{6 A (- 3 \sqrt{x} + 23 A + \frac{12 A^{2}}{\sqrt{x}})}{49}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	8.176

12275	$y y^{'} - y = - \frac{30 x}{121} + \frac{3 A (21 \sqrt{x} + 35 A + \frac{6 A^{2}}{\sqrt{x}})}{242}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	7.822

12276	$y y^{'} - y = - \frac{3 x}{16} + \frac{A}{x^{5 / 3}}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	43.159

12277	$y y^{'} - y = - \frac{12 x}{49} + \frac{4 A (- 10 \sqrt{x} + 27 A + \frac{10 A^{2}}{\sqrt{x}})}{49}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	6.773

12278	$y y^{'} - y = \frac{A}{\sqrt{x}}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	2.164

12279	$y y^{'} - y = \frac{A}{x^{2}}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	0.952

12280	$y y^{'} - y = A (n + 2) (\sqrt{x} + 2 (n + 2) A + \frac{(n + 1) (n + 3) A^{2}}{\sqrt{x}})$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	21.089

12281	$y y^{'} - y = A (n + 2) (\sqrt{x} + 2 (n + 2) A + \frac{(2 n + 3) A^{2}}{\sqrt{x}})$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	22.614

12282	$y y^{'} - y = A \sqrt{x} + 2 A^{2} + \frac{B}{\sqrt{x}}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	13.714

12283	$y y^{'} - y = 2 A^{2} - A \sqrt{x}$	[_rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✗	2.963

12284	$y y^{'} - y = - \frac{x}{4} + \frac{6 A (\sqrt{x} + 8 A + \frac{5 A^{2}}{\sqrt{x}})}{49}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	9.032

12285	$y y^{'} - y = - \frac{6 x}{25} + \frac{6 A (2 \sqrt{x} + 7 A + \frac{4 A^{2}}{\sqrt{x}})}{25}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	8.159

12286	$y y^{'} - y = - \frac{3 x}{16} + \frac{3 A}{x^{1 / 3}} - \frac{12 A^{2}}{x^{5 / 3}}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	6.797

12287	$y y^{'} - y = \frac{3 x}{8} + \frac{3 \sqrt{b^{2} + x^{2}}}{8} + \frac{3 b^{2}}{16 \sqrt{b^{2} + x^{2}}}$	[[_Abel, ‘2nd type‘, ‘class B‘]]	✗	7.140

12288	$y y^{'} - y = \frac{9 x}{32} + \frac{15 \sqrt{b^{2} + x^{2}}}{32} + \frac{3 b^{2}}{64 \sqrt{b^{2} + x^{2}}}$	[[_Abel, ‘2nd type‘, ‘class B‘]]	✗	8.497

12289	$y y^{'} - y = - \frac{3 x}{32} - \frac{3 \sqrt{a^{2} + x^{2}}}{32} + \frac{15 a^{2}}{64 \sqrt{a^{2} + x^{2}}}$	[[_Abel, ‘2nd type‘, ‘class B‘]]	✗	7.502

12290	$y y^{'} - y = A x^{2} - \frac{9}{625 A}$	[_rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✗	1.463

12291	$y y^{'} - y = - \frac{6}{25} x - A x^{2}$	[_rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✗	1.304

12292	$y y^{'} - y = \frac{6}{25} x - A x^{2}$	[_rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✗	1.327

12293	$y y^{'} - y = 12 x + \frac{A}{x^{5 / 2}}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	7.416

12294	$y y^{'} - y = \frac{63 x}{4} + \frac{A}{x^{5 / 3}}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	51.626

12295	$y y^{'} - y = 2 x + 2 A (10 \sqrt{x} + 31 A + \frac{30 A^{2}}{\sqrt{x}})$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	8.671

12296	$y y^{'} - y = 2 x + 2 A (- 10 \sqrt{x} + 19 A + \frac{30 A^{2}}{\sqrt{x}})$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	7.582

12297	$y y^{'} - y = - \frac{28 x}{121} + \frac{2 A (5 \sqrt{x} + 106 A + \frac{65 A^{2}}{\sqrt{x}})}{121}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	8.278

12298	$y y^{'} - y = - \frac{12 x}{49} + \frac{A (5 \sqrt{x} + 262 A + \frac{65 A^{2}}{\sqrt{x}})}{49}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	9.017

12299	$y y^{'} - y = - \frac{12 x}{49} + A \sqrt{x}$	[_rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✗	5.856

12300	$y y^{'} - y = 6 x + \frac{A}{x^{4}}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	1.268

2.2.123 Problems 12201 to 12300