Problems 12401 to 12500

Table 2.251: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	time (sec)

12401	$y y^{'} = (a e^{x} + b) y + c e^{2 x} - a b e^{x} - b^{2}$	[[_Abel, ‘2nd type‘, ‘class A‘]]	✗	3.293

12402	$y y^{'} = (a (2 μ + λ) e^{λ x} + b) e^{μ x} y + (- a^{2} μ e^{2 λ x} - a b e^{λ x} + c) e^{2 μ x}$	[[_Abel, ‘2nd type‘, ‘class A‘]]	✗	7.637

12403	$y y^{'} = (a e^{λ x} + b) y + c (a^{2} e^{2 λ x} + a b (λ x + 1) e^{λ x} + b^{2} λ x)$	[[_Abel, ‘2nd type‘, ‘class A‘]]	✗	5.225

12404	$y y^{'} = e^{λ x} (2 a λ x + a + b) y - e^{2 λ x} (a^{2} λ x^{2} + a b x + c)$	[[_Abel, ‘2nd type‘, ‘class A‘]]	✗	70.757

12405	$y y^{'} = e^{a x} (2 a x^{2} + b + 2 x) y + e^{2 a x} (- a x^{4} - b x^{2} + c)$	[[_Abel, ‘2nd type‘, ‘class A‘]]	✗	6.033

12406	$y y^{'} + a (2 b x + 1) e^{b x} y = - a^{2} b x^{2} e^{2 b x}$	[[_Abel, ‘2nd type‘, ‘class A‘]]	✗	3.083

12407	$y y^{'} - a (1 + 2 n + 2 n (n + 1) x) e^{(n + 1) x} y = - a^{2} n (n + 1) (n x + 1) x e^{2 (n + 1) x}$	[[_Abel, ‘2nd type‘, ‘class A‘]]	✗	6.960

12408	$y y^{'} + a (1 + 2 b \sqrt{x}) e^{2 b \sqrt{x}} y = - a^{2} b x^{3 / 2} e^{4 b \sqrt{x}}$	[[_Abel, ‘2nd type‘, ‘class A‘]]	✗	7.936

12409	$y y^{'} = (a \cosh (x) + b) y - a b \sinh (x) + c$	[[_Abel, ‘2nd type‘, ‘class A‘]]	✗	9.683

12410	$y y^{'} = (a \sinh (x) + b) y - a b \cosh (x) + c$	[[_Abel, ‘2nd type‘, ‘class A‘]]	✗	9.710

12411	$y y^{'} = (2 \ln (x) + a + 1) y + x (- \ln {(x)}^{2} - a \ln (x) + b)$	[[_Abel, ‘2nd type‘, ‘class A‘]]	✗	2.210

12412	$y y^{'} = (2 \ln {(x)}^{2} + 2 \ln (x) + a) y + x (- \ln {(x)}^{4} - a \ln {(x)}^{2} + b)$	[[_Abel, ‘2nd type‘, ‘class A‘]]	✗	2.600

12413	$y y^{'} = a x \cos (λ x^{2}) y + x$	[[_Abel, ‘2nd type‘, ‘class A‘]]	✗	5.472

12414	$y y^{'} = a x \sin (λ x^{2}) y + x$	[[_Abel, ‘2nd type‘, ‘class A‘]]	✗	5.543

12415	$(A y + B x + a) y^{'} + B y + k x + b = 0$	[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	4.046

12416	$(y + a x + b) y^{'} = α y + β x + γ$	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	8.300

12417	$(y + a k x^{2} + b x + c) y^{'} = - a y^{2} + 2 a k x y + m y + k (k + b - m) x + s$	[_rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✗	7.608

12418	$(y + A x^{n} + a) y^{'} + n A x^{n - 1} y + k x^{m} + b = 0$	[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]]	✓	37.058

12419	$(y + a x^{n + 1} + b x^{n}) y^{'} = (a n x^{n} + c x^{n - 1}) y$	[_rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✗	4.321

12420	$x y y^{'} = a y^{2} + b y + c x^{n} + s$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	2.622

12421	$x y y^{'} = - n y^{2} + a (2 n + 1) x y + b y - a^{2} n x^{2} - a b x + c$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	2.281

12422	$y^{''} + a y = 0$	[[_2nd_order, _missing_x]]	✓	3.180

12423	$y^{''} - (a x + b) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.752

12424	$y^{''} - (a^{2} x^{2} + a) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.776

12425	$y^{''} - (a x^{2} + b) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.574

12426	$y^{''} + a^{3} x (- a x + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.450

12427	$y^{''} - (a x^{2} + b x c) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.519

12428	$y^{''} - a x^{n} y = 0$	[[_Emden, _Fowler]]	✓	0.691

12429	$y^{''} - a (a x^{2 n} + n x^{n - 1}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.340

12430	$y^{''} - a x^{n - 2} (a x^{n} + n + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.339

12431	$y^{''} + (a x^{2 n} + b x^{n - 1}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.329

12432	$y^{''} + a y^{'} + b y = 0$	[[_2nd_order, _missing_x]]	✓	3.134

12433	$y^{''} + a y^{'} + (b x + c) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.510

12434	$y^{''} + a y^{'} - (b x^{2} + c) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.704

12435	$y^{''} + a y^{'} + b (- b x^{2} + a x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	1.319

12436	$y^{''} + a y^{'} + b x (- b x^{3} + a x + 2) y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	1.020

12437	$y^{''} + a y^{'} + b (- b x^{2 n} + a x^{n} + n x^{n - 1}) y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✗	0.606

12438	$y^{''} + a y^{'} + b (- b x^{2 n} - a x^{n} + n x^{n - 1}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.595

12439	$y^{''} + x y^{'} + (n - 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.618

12440	$y^{''} - 2 x y^{'} + 2 n y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.595

12441	$y^{''} + a x y^{'} + b y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.691

12442	$y^{''} + a x y^{'} + b x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.664

12443	$y^{''} + a x y^{'} + (b x + c) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.747

12444	$y^{''} + 2 a x y^{'} + (b x^{4} + a^{2} x^{2} + c x + a) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.755

12445	$y^{''} + (a x + b) y^{'} - a y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.501

12446	$y^{''} + (a x + b) y^{'} + a y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	1.320

12447	$y^{''} + (a x + b) y^{'} + c (a x + b - c) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.405

12448	$y^{''} + (a x + 2 b) y^{'} + (a b x + b^{2} - a) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.135

12449	$y^{''} + (a x + b) y^{'} + (c x + d) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.841

12450	$y^{''} + (a x + b) y^{'} + c ((a - c) x^{2} + b x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.523

12451	$y^{''} + 2 (a x + b) y^{'} + (a^{2} x^{2} + 2 a b x + c) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	38.786

12452	$y^{''} + (a x + b) y^{'} + (α x^{2} + β x + γ) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	2.605

12453	$y^{''} + (a x + b) y^{'} + c (- c x^{2 n} + a x^{n + 1} + b x^{n} + n x^{n - 1}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.833

12454	$y^{''} + a (- b^{2} + x^{2}) y^{'} - a (x + b) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.304

12455	$y^{''} + (a x^{2} + b) y^{'} + c (a x^{2} + b - c) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.295

12456	$y^{''} + (a x^{2} + 2 b) y^{'} + (a b x^{2} - a x + b^{2}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.350

12457	$y^{''} + (2 x^{2} + a) y^{'} + (x^{4} + a x^{2} + b + 2 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	38.116

12458	$y^{''} + (a x^{2} + b x) y^{'} + (α x^{2} + β x + γ) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.868

12459	$y^{''} + (a b x^{2} + b x + 2 a) y^{'} + a^{2} (b x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.467

12460	$y^{''} + (a x^{2} + b x + c) y^{'} + x (a b x^{2} + c b + 2 a) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.361

12461	$y^{''} + (a x^{2} + b x + c) y^{'} + (a b x^{3} + a c x^{2} + b) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.358

12462	$y^{''} + (a x^{3} + 2 b) y^{'} + (a b x^{3} - a x^{2} + b^{2}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.372

12463	$y^{''} + (a x^{3} + b x) y^{'} + 2 (2 a x^{2} + b) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.063

12464	$y^{''} + (a b x^{3} + b x^{2} + 2 a) y^{'} + a^{2} (b x^{3} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.525

12465	$y^{''} + a x^{n} y^{'} = 0$	[[_2nd_order, _missing_y]]	✓	0.865

12466	$y^{''} + a x^{n} y^{'} + b x^{n - 1} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.815

12467	$y^{''} + 2 a x^{n} y^{'} + a (a x^{2 n} + n x^{n - 1}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.976

12468	$y^{''} + a x^{n} y^{'} + (b x^{2 n} + c x^{n - 1}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.957

12469	$y^{''} + a x^{n} y^{'} - b (a x^{m + n} + b x^{2 m} + m x^{m - 1}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.157

12470	$y^{''} + 2 a x^{n} y^{'} + (a^{2} x^{2 n} + b x^{2 m} + a n x^{n - 1} + c x^{m - 1}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.197

12471	$y^{''} + (a x^{n} + b) y^{'} + c (a x^{n} + b - c) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.803

12472	$y^{''} + (a x^{n} + 2 b) y^{'} + (a b x^{n} - a x^{n - 1} + b^{2}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.836

12473	$y^{''} + (a b x^{n} + b x^{n - 1} + 2 a) y^{'} + a^{2} (b x^{n} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.183

12474	$y^{''} + (a b x^{n} + 2 b x^{n - 1} - a^{2} x) y^{'} + a (a b x^{n} + b x^{n - 1} - a^{2} x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.609

12475	$y^{''} + x^{n} (a x^{2} + (a c + b) x + c b) y^{'} - x^{n} (a x + b) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.223

12476	$y^{''} + (a x^{n} + b x^{m}) y^{'} - (a x^{n - 1} + b x^{m - 1}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.516

12477	$y^{''} + (a x^{n} + b x^{m}) y^{'} + (a n x^{n - 1} + b m x^{m - 1}) y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	1.881

12478	$y^{''} + (a x^{n} + b x^{m}) y^{'} + (a (n + 1) x^{n - 1} + b (m + 1) x^{m - 1}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	3.104

12479	$y^{''} + (a x^{n} + b x^{m}) y^{'} + c (a x^{n} + b x^{m} - c) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.045

12480	$y^{''} + (a x^{n} + b x^{m}) y^{'} + (a b x^{m + n} + b (m + 1) x^{m - 1} - a x^{n - 1}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.138

12481	$y^{''} + (a x^{n} + b x^{m} + c) y^{'} + (a b x^{m + n} + b c x^{m} + a n x^{n - 1}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.184

12482	$x y^{''} + \frac{y^{'}}{2} + a y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	1.471

12483	$x y^{''} + a y^{'} + b y = 0$	[[_Emden, _Fowler]]	✓	0.785

12484	$x y^{''} + a y^{'} + b x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.926

12485	$x y^{''} + a y^{'} + (b x + c) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.938

12486	$x y^{''} + n y^{'} + b x^{1 - 2 n} y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	1.043

12487	$x y^{''} + (1 - 3 n) y^{'} - a^{2} n^{2} x^{2 n - 1} y = 0$	[[_Emden, _Fowler]]	✗	0.552

12488	$x y^{''} + a y^{'} + b x^{n} y = 0$	[[_Emden, _Fowler]]	✓	0.928

12489	$x y^{''} + a y^{'} + b x^{n} (- b x^{n + 1} + a + n) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.683

12490	$x y^{''} + a x y^{'} + a y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	1.221

12491	$x y^{''} + (b - x) y^{'} - a y = 0$	[_Laguerre]	✗	0.938

12492	$x y^{''} + (a x + b) y^{'} + c ((a - c) x + b) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.369

12493	$x y^{''} + (2 a x + b) y^{'} + a (a x + b) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.990

12494	$x y^{''} + ((a + b) x + n + m) y^{'} + (a b x + a n + b m) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.628

12495	$x y^{''} + (a x + b) y^{'} + (c x + d) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.951

12496	$x y^{''} - (a x + 1) y^{'} - b x^{2} (b x + a) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.434

12497	$x y^{''} - (2 a x + 1) y^{'} + (b x^{3} + a^{2} x + a) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.378

12498	$x y^{''} + (a x + b) y^{'} + c x (- c x^{2} + a x + b + 1) = 0$	[[_2nd_order, _missing_y]]	✓	1.145

12499	$x y^{''} - (2 a x + 1) y^{'} + b x^{3} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.749

12500	$x y^{''} + (a b x^{2} + b - 5) y^{'} + 2 a^{2} (b - 2) x^{3} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.678

2.2.125 Problems 12401 to 12500