Problems 12601 to 12700

Table 2.255: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	time (sec)

12601	$(a x^{2} + b x + c) y^{''} - (- k^{2} + x^{2}) y^{'} + (x + k) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	5.665

12602	$(a x^{2} + b x + c) y^{''} + (k^{3} + x^{3}) y^{'} - (k^{2} - k x + x^{2}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	5.359

12603	$x^{3} y^{''} + (a x + b) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.380

12604	$x^{3} y^{''} + (a x^{2} + b) y^{'} + c x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.602

12605	$x^{3} y^{''} + (a x^{2} + b x) y^{'} + b y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.505

12606	$x^{3} y^{''} + (a x^{2} + b x) y^{'} + c y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.583

12607	$x^{3} y^{''} + (a x^{2} + b x) y^{'} + (c x + d) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.547

12608	$x^{3} y^{''} + (a x^{3} + a b x - x^{2} + b) y^{'} + a^{2} b x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.699

12609	$x^{3} y^{''} + (a x^{n} + b) y^{'} x - (a x^{n} - a b x^{n - 1} + b) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.393

12610	$x (a x^{2} + b) y^{''} + 2 (a x^{2} + b) y^{'} - 2 y a x = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	1.010

12611	$x (x^{2} + a) y^{''} + (b x^{2} + c) y^{'} + s x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	2.307

12612	$x^{2} (a x + b) y^{''} + (c x^{2} + (a λ + 2 b) x + b λ) y^{'} + λ (c - 2 a) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	2.557

12613	$x^{2} (a x + b) y^{''} - 2 x (a x + 2 b) y^{'} + 2 (a x + 3 b) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.903

12614	$x^{2} (a x + b) y^{''} + (a (2 - n - m) x^{2} - b (n + m) x) y^{'} + (a m (n - 1) x + b n (m + 1)) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.011

12615	$x^{2} (x + a_{2}) y^{''} + x (b_{1} x + a_{1}) y^{'} + (b_{0} x + a_{0}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	2.085

12616	$(a x^{3} + b x^{2} + c x) y^{''} + (α x^{2} + β x + 2 c) y^{'} + (β - 2 b) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	5.001

12617	$(a x^{3} + b x^{2} + c x) y^{''} + (α x^{2} + β x + 2 c) y^{'} - (α x + 2 b - β) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	4.952

12618	$(a x^{3} + b x^{2} + c x) y^{''} + (- 2 a x^{2} - (b + 1) x + k) y^{'} + 2 (a x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	4.048

12619	$(a x^{3} + b x^{2} + c x) y^{''} + (n x^{2} + m x + k) y^{'} + (k - 1) ((- a k + n) x + m - b k) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	5.293

12620	$(a x^{3} + b x^{2} + c x) y^{''} + ((m - a) x^{2} + (2 c m - 1) x - c) y^{'} + (- 2 m x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	656.792

12621	$(a x^{3} + b x^{2} + c x) y^{''} + (n x^{2} + m x + k) y^{'} + (- 2 (a + n) x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	60.101

12622	$(a x^{3} + x^{2} + b) y^{''} + a^{2} x (x^{2} - b) y^{'} - a^{3} b x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	21.261

12623	$2 x (a x^{2} + b x + c) y^{''} + (a x^{2} - c) y^{'} + λ x^{2} y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	153.933

12624	$x (a x^{2} + b x + 1) y^{''} + (α x^{2} + β x + γ) y^{'} + (n x + m) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	6.160

12625	$x (x - 1) (x - a) y^{''} + ((α + β + 1) x^{2} - (α + β + 1 + a (γ + d) - a) x + a γ) y^{'} + (α β x - q) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	3.621

12626	$(a x^{3} + b x^{2} + c x + d) y^{''} - (- λ^{2} + x^{2}) y^{'} + (x + λ) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	174.268

12627	$2 (a x^{3} + b x^{2} + c x + d) y^{''} + (3 a x^{2} + 2 b x + c) y^{'} + λ y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	189.795

12628	$2 (a x^{3} + b x^{2} + c x + d) y^{''} + 3 (3 a x^{2} + 2 b x + c) y^{'} + (6 a x + 2 b + λ) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	107.531

12629	$(a x^{3} + b x^{2} + c x + d) y^{''} + (α x^{2} + (α γ + β) x + β λ) y^{'} - (α x + β) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	442.490

12630	$(a x^{3} + b x^{2} + c x + d) y^{''} + (λ^{3} + x^{3}) y^{'} - (λ^{2} - λ x + x^{2}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1420.685

12631	$2 x (a x^{2} + b x + c) y^{''} + (a (2 - k) x^{2} + b (1 - k) x - c k) y^{'} + λ x^{k + 1} y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	2.301

12632	$x^{4} y^{''} + a y = 0$	[[_Emden, _Fowler]]	✓	0.611

12633	$x^{4} y^{''} + (a x^{2} + b x + c) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.892

12634	$x^{4} y^{''} - (a + b) x^{2} y^{'} + ((a + b) x + a b) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.853

12635	$x^{4} y^{''} + 2 x^{2} (x + a) y^{'} + b y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.163

12636	$x^{4} y^{''} + a x^{n} y^{'} - (a x^{n - 1} + a b x^{n - 2} + b^{2}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.562

12637	$x^{2} {(x - a)}^{2} y^{''} + b y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.849

12638	$x^{2} {(x - a)}^{2} y^{''} + b y = c x^{2} {(x - a)}^{2}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.754

12639	$a x^{2} {(x - 1)}^{2} y^{''} + (b x^{2} + c x + d) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.274

12640	$x^{2} (x^{2} + a) y^{''} + (b x^{2} + c) x y^{'} + d y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	2.282

12641	${(x^{2} + 1)}^{2} y^{''} + a y = 0$	[_Halm]	✓	0.637

12642	${(x^{2} - 1)}^{2} y^{''} + a y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.668

12643	${(a^{2} + x^{2})}^{2} y^{''} + b^{2} y = 0$	[[_Emden, _Fowler]]	✓	0.817

12644	${(- a^{2} + x^{2})}^{2} y^{''} + b^{2} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.847

12645	$4 {(x^{2} + 1)}^{2} y^{''} + (a x^{2} + a - 3) y = 0$	[_Halm]	✓	0.807

12646	${(a x^{2} + b)}^{2} y^{''} + 2 a x (a x^{2} + b) y^{'} + c y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	1.457

12647	${(x^{2} - 1)}^{2} y^{''} + 2 x (x^{2} - 1) y^{'} - (ν (ν + 1) (x^{2} - 1) + n^{2}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.557

12648	${(- x^{2} + 1)}^{2} y^{''} - 2 x (- x^{2} + 1) y^{'} + (ν (ν + 1) (- x^{2} + 1) - μ^{2}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.597

12649	$a {(x^{2} - 1)}^{2} y^{''} + b x (x^{2} - 1) y^{'} + (c x^{2} + d x + e) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.882

12650	${(a x^{2} + b)}^{2} y^{''} + (2 a x + c) (a x^{2} + b) y^{'} + k y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.421

12651	${(a x^{2} + b)}^{2} y^{''} + (a x^{2} + b) (c x^{2} + d) y^{'} + 2 (- a d + b c) x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	2.734

12652	${(x^{2} + a)}^{2} y^{''} + b x^{n} (x^{2} + a) y^{'} - (b x^{n + 1} + a) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.449

12653	${(x^{2} + a)}^{2} y^{''} + b x^{n} (x^{2} + a) y^{'} - m (b x^{n + 1} + (m - 1) x^{2} + a) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.550

12654	${(x - a)}^{2} {(x - b)}^{2} y^{''} - c y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.088

12655	${(x - a)}^{2} {(x - b)}^{2} y^{''} + (x - a) (x - b) (2 x + λ) y^{'} + μ y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	2.027

12656	${(a x^{2} + b x + c)}^{2} y^{''} + A y = 0$	[[_Emden, _Fowler]]	✓	1.188

12657	${(x^{2} - 1)}^{2} y^{''} + 2 x (x^{2} - 1) y^{'} + ((x^{2} - 1) (a^{2} x^{2} - λ) - m^{2}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.350

12658	${(x^{2} + 1)}^{2} y^{''} + 2 x (x^{2} + 1) y^{'} + ((x^{2} + 1) (a^{2} x^{2} - λ) + m^{2}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.451

12659	${(a x^{2} + b x + c)}^{2} y^{''} + (2 a x + k) (a x^{2} + b x + c) y^{'} + m y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	2.100

12660	$x^{6} y^{''} - x^{5} y^{'} + a y = 0$	[[_Emden, _Fowler]]	✓	0.523

12661	$x^{6} y^{''} + (3 x^{2} + a) x^{3} y^{'} + b y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.960

12662	$x^{n} y^{''} + c {(a x + b)}^{n - 4} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.584

12663	$x^{n} y^{''} + a x y^{'} - (b^{2} x^{n} + 2 b x^{n - 1} + a b x + a) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.497

12664	$x^{n} y^{''} + (a x + b) y^{'} - a y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.353

12665	$x^{n} y^{''} + (a x^{n - 1} + b x) y^{'} + (a - 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.503

12666	$x^{n} y^{''} + (2 x^{n - 1} + a x^{2} + b x) y^{'} + b y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.490

12667	$x^{n} y^{''} + (a x^{n} + b) y^{'} + c ((a - c) x^{n} + b) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.254

12668	$x^{n} y^{''} + (a x^{n} - x^{n - 1} + a b x + b) y^{'} + a^{2} b x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	3.922

12669	$x^{n} y^{''} + (a x^{n + m} + 1) y^{'} + a x^{m} (1 + m x^{n - 1}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.909

12670	$(a x^{n} + b) y^{''} + (c x^{n} + d) y^{'} + λ ((- a λ + c) x^{n} + d - b λ) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.948

12671	$(a x^{n} + b x + c) y^{''} = a n (n - 1) x^{n - 2} y$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	5.839

12672	$x (x^{n} + 1) y^{''} + ((a - b) x^{n} + a - n) y^{'} + b (1 - a) x^{n - 1} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	2.000

12673	$x (x^{2 n} + a) y^{''} + (x^{2 n} + a - a n) y^{'} - b^{2} x^{2 n - 1} y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	0.914

12674	$x^{2} (a^{2} x^{2 n} - 1) y^{''} + x (a^{2} (n + 1) x^{2 n} + n - 1) y^{'} - ν (ν + 1) a^{2} n^{2} x^{2 n} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.993

12675	$x^{2} (a^{2} x^{2 n} - 1) y^{''} + x (a p x^{n} + q) y^{'} + (a r x^{n} + s) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	11.917

12676	${(x^{n} + a)}^{2} y^{''} - b x^{n - 2} ((b - 1) x^{n} + a (n - 1)) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.755

12677	${(a x^{n} + b)}^{2} y^{''} + (a x^{n} + b) (c x^{n} + d) y^{'} + n (- a d + b c) x^{n - 1} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.696

12678	${(x^{n} + a)}^{2} y^{''} + b x^{m} (x^{n} + a) y^{'} - x^{n - 2} (b x^{m + 1} + a n - a) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	2.211

12679	${(a x^{n} + b)}^{2} y^{''} + c x^{m} (a x^{n} + b) y^{'} + (c x^{m} - a n x^{n - 1} - 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	2.562

12680	$x^{2} {(a x^{n} + b)}^{2} y^{''} + (n + 1) x (a^{2} x^{2 n} - b^{2}) y^{'} + c y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	4.580

12681	${(a x^{n + 1} + b x^{n} + c)}^{2} y^{''} + (α x^{n} + β x^{n - 1} + γ) y^{'} + (n (- a n - a + α) x^{n - 1} + (n - 1) (- b n + β) x^{n - 2}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	6.555

12682	$(a x^{n} + b x^{m} + c) y^{''} + (λ - x) y^{'} + y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.655

12683	$(a x^{n} + b x^{m} + c) y^{''} + (λ^{2} - x^{2}) y^{'} + (x + λ) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.719

12684	$2 (a x^{n} + b x^{m} + c) y^{''} + a n x^{n - 1} b m x^{m - 1} y^{'} + d y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	2.149

12685	${(a x^{n} + b)}^{m + 1} y^{''} + (a x^{n} + b) y^{'} - a n m x^{n - 1} y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	1.399

12686	$y^{''} + a e^{λ x} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.288

12687	$y^{''} + (a e^{x} - b) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.313

12688	$y^{''} + a (λ e^{λ x} - a e^{2 λ x}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.625

12689	$y^{''} - (a^{2} e^{2 x} + a (2 b + 1) e^{x} + b^{2}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.675

12690	$y^{''} - (a e^{2 λ x} + b e^{λ x} + c) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.616

12691	$y^{''} + (a e^{4 λ x} + b e^{3 λ x} + c e^{2 λ x} - \frac{λ^{2}}{4}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.750

12692	$y^{''} + (a e^{2 λ x} {(b e^{λ x} + c)}^{n} - \frac{λ^{2}}{4}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.788

12693	$y^{''} + a y^{'} + b e^{2 a x} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.335

12694	$y^{''} - a y^{'} + b e^{2 a x} y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	1.136

12695	$y^{''} + a y^{'} + (b e^{λ x} + c) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.456

12696	$y^{''} - y^{'} + (a e^{3 λ x} + b e^{2 λ x} + \frac{1}{4} - \frac{λ^{2}}{4}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.824

12697	$y^{''} - y^{'} + (a e^{2 λ x} {(b e^{λ x} + c)}^{n} + \frac{1}{4} - \frac{λ^{2}}{4}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.074

12698	$y^{''} + 2 a e^{λ x} y^{'} + a e^{λ x} (a e^{λ x} + λ) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.700

12699	$y^{''} + (a + b) e^{λ x} y^{'} + a e^{λ x} (b e^{λ x} + λ) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.957

12700	$y^{''} + a e^{λ x} y^{'} - b e^{μ x} (a e^{λ x} + b e^{μ x} + μ) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.196

2.2.127 Problems 12601 to 12700