Problems 12501 to 12600

Table 2.253: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	time (sec)

12501	$x y^{''} + (a x^{2} + b x) y^{'} - (a c x^{2} + (c b + c^{2} + a) x + b + 2 c) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.466

12502	$x y^{''} + (a x^{2} + b x + 2) y^{'} + b y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.602

12503	$x y^{''} + (a x^{2} + b x + c) y^{'} + (2 a x + b) y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	1.754

12504	$x y^{''} + (a x^{2} + b x + c) y^{'} + (c - 1) (a x + b) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.349

12505	$x y^{''} + (a x^{2} + b x + c) y^{'} + (A x^{2} + B x + C 0) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.164

12506	$x y^{''} + (a x^{2} + b x + 2) y^{'} + (c x^{2} + d x + b) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.053

12507	$x y^{''} + (a x^{3} + b) y^{'} + a (b - 1) x^{2} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.833

12508	$x y^{''} + x (a x^{2} + b) y^{'} + (3 a x^{2} + b) y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	1.387

12509	$x y^{''} + (a x^{3} + b x^{2} + 2) y^{'} + b x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.663

12510	$x y^{''} + (a b x^{3} + b x^{2} + a x - 1) y^{'} + a^{2} b x^{3} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.671

12511	$x y^{''} + (a x^{3} + b x^{2} + c x + d) y^{'} + (d - 1) (a x^{2} + b x + c) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.550

12512	$x y^{''} + a x^{n} y^{'} + (a b x^{n} - a x^{n - 1} - b^{2} x + 2 b) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.903

12513	$x y^{''} + (a x^{n} + 2) y^{'} + a x^{n - 1} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.948

12514	$x y^{''} + (x^{n} + 1 - n) y^{'} + b x^{2 n - 1} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.710

12515	$x y^{''} + (a x^{n} + b) y^{'} + a n x^{n - 1} y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	1.704

12516	$x y^{''} + (a x^{n} + b) y^{'} + a (b - 1) x^{n - 1} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.128

12517	$x y^{''} + (a x^{n} + b) y^{'} + a (n + b - 1) x^{n - 1} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.204

12518	$x y^{''} + (a x^{n} + b) y^{'} + c (a x^{n} - c x + b) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.928

12519	$x y^{''} + (a b x^{n} + b - 3 n + 1) y^{'} + a^{2} n (b - n) x^{2 n - 1} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.977

12520	$x y^{''} + (a x^{n} + b) y^{'} + (c x^{2 n - 1} + d x^{n - 1}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.897

12521	$x y^{''} + (a x^{n} + b x^{n - 1} + 2) y^{'} + b x^{n - 2} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.140

12522	$x y^{''} + (a x^{n} + b x) y^{'} + (a b x^{n} + a n x^{n - 1} - b) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.814

12523	$x y^{''} + (a b x^{n} + b x^{n - 1} + a x - 1) y^{'} + a^{2} b x^{n} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.451

12524	$x y^{''} + (a x^{n} + b x^{m} + c) y^{'} + (c - 1) (a x^{n - 1} + b x^{m - 1}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.404

12525	$x y^{''} + (a b x^{m + n} + a n x^{n} + b x^{m} + 1 - 2 n) y^{'} + a^{2} b n x^{2 n + m - 1} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.515

12526	$(x + a) y^{''} + (b x + c) y^{'} + b y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	1.523

12527	$(a_{1} x + a_{0}) y^{''} + (b_{1} x + b_{0}) y^{'} - m b_{1} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.400

12528	$(a x + b) y^{''} + s (c x + d) y^{'} - s^{2} ((a + c) x + b + d) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.870

12529	$(a_{2} x + b_{2}) y^{''} + (a_{1} x + b_{1}) y^{'} + (a_{0} x + b_{0}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	7.578

12530	$(x + γ) y^{''} + (a x^{n} + b x^{m} + c) y^{'} + (a n x^{n - 1} + b m x^{m - 1}) y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	1.436

12531	$x^{2} y^{''} + a y = 0$	[[_Emden, _Fowler]]	✓	0.724

12532	$x^{2} y^{''} + (a x + b) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.576

12533	$x^{2} y^{''} + (a^{2} x^{2} - (n + 1) n) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.826

12534	$x^{2} y^{''} - (a^{2} x^{2} + (n + 1) n) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.882

12535	$x^{2} y^{''} - (a^{2} x^{2} + 2 a b x + b^{2} - b) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.711

12536	$x^{2} y^{''} + (a x^{2} + b x + c) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.727

12537	$x^{2} y^{''} - (a x^{3} + \frac{5}{16}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	4.342

12538	$x^{2} y^{''} - (a^{2} x^{4} + a (2 b - 1) x^{2} + b (b + 1)) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.771

12539	$x^{2} y^{''} + (a x^{n} + b) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.755

12540	$x^{2} y^{''} - (a^{2} x^{2 n} + a (2 b + n - 1) x^{n} + b (b - 1)) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.382

12541	$x^{2} y^{''} + (a x^{2 n} + b x^{n} + c) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.347

12542	$x^{2} y^{''} + (a x^{3 n} + b x^{2 n} + \frac{1}{4} - \frac{n^{2}}{4}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.370

12543	$x^{2} y^{''} + (a x^{2 n} {(b x^{n} + c)}^{m} + \frac{1}{4} - \frac{n^{2}}{4}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.562

12544	$x^{2} y^{''} + a x y^{'} + b y = 0$	[[_Emden, _Fowler]]	✓	1.645

12545	$x^{2} y^{''} + x y^{'} + (x^{2} - {(n + \frac{1}{2})}^{2}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.895

12546	$x^{2} y^{''} + x y^{'} - (x^{2} + {(n + \frac{1}{2})}^{2}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.922

12547	$x^{2} y^{''} + x y^{'} + (- ν^{2} + x^{2}) y = 0$	[_Bessel]	✓	0.823

12548	$x^{2} y^{''} + x y^{'} - (ν^{2} + x^{2}) y = 0$	[[_Bessel, _modiﬁed]]	✓	0.921

12549	$x^{2} y^{''} + 2 x y^{'} - (a^{2} x^{2} + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	3.951

12550	$x^{2} y^{''} - 2 a x y^{'} + (b^{2} x^{2} + a (a + 1)) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	2.388

12551	$x^{2} y^{''} - 2 a x y^{'} + (- b^{2} x^{2} + a (a + 1)) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.329

12552	$x^{2} y^{''} + λ x y^{'} + (a x^{2} + b x + c) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.995

12553	$x^{2} y^{''} + a x y^{'} + (b x^{n} + c) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.947

12554	$x^{2} y^{''} + a x y^{'} + x^{n} (b x^{n} + c) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.556

12555	$x^{2} y^{''} + (a x + b) y^{'} + c y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.000

12556	$x^{2} y^{''} + a x^{2} y^{'} + (b x^{2} + c x + d) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.146

12557	$x^{2} y^{''} + (a x^{2} + b) y^{'} + c ((a - c) x^{2} + b) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.586

12558	$x^{2} y^{''} + (a x^{2} + b x) y^{'} - b y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.112

12559	$x^{2} y^{''} + (a x^{2} + b x) y^{'} + (k (a - k) x^{2} + (a n + b k - 2 k n) x + n (- n + b - 1)) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.704

12560	$a_{2} x^{2} y^{''} + (a_{1} x^{2} + b_{1} x) y^{'} + (a_{0} x^{2} + b_{0} x + c_{0}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	2.691

12561	$x^{2} y^{''} + (a x^{2} + (a b - 1) x + b) y^{'} + a^{2} b x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	11.241

12562	$x^{2} y^{''} - 2 x (x^{2} - a) y^{'} + (2 n x^{2} + ({(- 1)}^{n} - 1) a) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.494

12563	$x^{2} y^{''} + x (a x^{2} + b x + c) y^{'} + (A x^{3} + B x^{2} + C x + d) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.259

12564	$x^{2} y^{''} + a x^{n} y^{'} - (a b x^{n} + a c x^{n - 1} + b^{2} x^{2} + 2 b x c + c^{2} - c) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.251

12565	$x^{2} y^{''} + a x^{n} y^{'} + (a b x^{n + 2 m} - b^{2} x^{4 m + 2} + a m x^{n - 1} - m^{2} - m) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.319

12566	$x^{2} y^{''} + x (a x^{n} + b) y^{'} + b (a x^{n} - 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.020

12567	$x^{2} y^{''} + x (a x^{n} + b) y^{'} + (α x^{2 n} + β x^{n} + γ) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.859

12568	$x^{2} y^{''} + x (2 a x^{n} + b) y^{'} + (a^{2} x^{2 n} + a (n + b - 1) x^{n} + α x^{2 m} + β x^{m} + γ) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.122

12569	$x^{2} y^{''} + (a x^{2 + n} + b x^{2} + c) y^{'} + (a n x^{n + 1} + a c x^{n} + c b) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.126

12570	$(- x^{2} + 1) y^{''} + n (n - 1) y = 0$	[_Gegenbauer]	✗	0.524

12571	$(- a^{2} + x^{2}) y^{''} + b y^{'} - 6 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.225

12572	$(x^{2} - 1) y^{''} + x y^{'} + a y = 0$	[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	2.997

12573	$(- x^{2} + 1) y^{''} - x y^{'} + n^{2} y = 0$	[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	1.516

12574	$(- x^{2} + 1) y^{''} - 2 x y^{'} + n (n + 1) y = 0$	[_Gegenbauer]	✗	0.945

12575	$(- x^{2} + 1) y^{''} - 2 x y^{'} + ν (ν + 1) y = 0$	[_Gegenbauer]	✗	0.939

12576	$(- x^{2} + 1) y^{''} - 3 x y^{'} + n (2 + n) y = 0$	[_Gegenbauer]	✓	1.882

12577	$(x^{2} - 1) y^{''} + 2 (n + 1) x y^{'} - (ν + n + 1) (ν - n) y = 0$	[_Gegenbauer]	✗	1.376

12578	$(x^{2} - 1) y^{''} - 2 (n - 1) x y^{'} - (ν - n + 1) (ν + n) y = 0$	[_Gegenbauer]	✗	1.383

12579	$(x^{2} - 1) y^{''} + (2 a + 1) y^{'} - b (2 a + b) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.214

12580	$(- x^{2} + 1) y^{''} + (2 a - 3) x y^{'} + (n + 1) (n + 2 a - 1) y = 0$	[_Gegenbauer]	✗	1.355

12581	$(- x^{2} + 1) y^{''} + (β - α - (α + β + 2) x) y^{'} + n (n + α + β + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.545

12582	$(- x^{2} + 1) y^{''} + (α - β + (α + β - 2) x) y^{'} + (n + 1) (n + α + β) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.530

12583	$(a x^{2} + b) y^{''} + a x y^{'} + c y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	323.755

12584	$(x^{2} + a) y^{''} + 2 b x y^{'} + 2 (b - 1) y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	1.648

12585	$(- a^{2} + x^{2}) y^{''} + 2 b x y^{'} + b (b - 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	63.664

12586	$(a^{2} + x^{2}) y^{''} + 2 b x y^{'} + b (b - 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	2.239

12587	$(a x^{2} + b) y^{''} + (2 n + 1) a x y^{'} + c y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	497.694

12588	$(- x^{2} + 1) y^{''} - x y^{'} + (2 a x^{2} + b) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.118

12589	$(- x^{2} + 1) y^{''} + (a x + b) y^{'} + c y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.331

12590	$(a x^{2} + b) y^{''} + (c x^{2} + d) y^{'} + λ ((- a λ + c) x^{2} + d - b λ) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	2.421

12591	$(a x^{2} + b) y^{''} + (λ (a + c) x^{2} + (c - a) x + 2 b λ) y^{'} + λ^{2} (c x^{2} + b) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	3.396

12592	$x (- 1 + x) y^{''} + ((α + β + 1) x - γ) y^{'} + α β y = 0$	[_Jacobi]	✗	1.557

12593	$x (x + a) y^{''} + (b x + c) y^{'} + d y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.388

12594	$2 x (- 1 + x) y^{''} + (2 x - 1) y^{'} + (a x + b) y = 0$	[_Jacobi]	✗	0.729

12595	$(2 a x + x^{2} + b) y^{''} + (x + a) y^{'} - m^{2} y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	2.067

12596	$(a x^{2} + b x + c) y^{''} + (d x + k) y^{'} + (d - 2 a) y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	37.506

12597	$(a x^{2} + b x + c) y^{''} + (k x + d) y^{'} - k y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	4.719

12598	$(a x^{2} + 2 b x + c) y^{''} + (a x + b) y^{'} + d y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	6.300

12599	$(a x^{2} + 2 b x + c) y^{''} + 3 (a x + b) y^{'} + d y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	4.589

12600	$(a_{2} x^{2} + b_{2} x + c_{2}) y^{''} + (b_{1} x + c_{1}) y^{'} + c_{0} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	3.923

2.2.126 Problems 12501 to 12600