Problems 12901 to 13000

Table 2.261: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	time (sec)

12901	$x y^{''} + 2 y^{'} - x y = 2 e^{x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.138

12902	$y^{''} + (2 e^{x} - 1) y^{'} + e^{2 x} y = e^{4 x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.513

12903	$(- x^{2} + 1) y^{''} - y^{'} x + 4 y = 0$	[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	0.855

12904	$y^{''} + y^{'} \tan (x) + \cos {(x)}^{2} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.020

12905	$x^{6} y^{''} + 3 x^{5} y^{'} + y = \frac{1}{x^{2}}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.417

12906	$x y^{''} - (2 x^{2} + 1) y^{'} - 8 x^{3} y = 4 x^{3} e^{- x^{2}}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.999

12907	$x y^{''} - (3 + x) y^{'} + 3 y = 0$	[_Laguerre]	✓	0.518

12908	$(x - 3) y^{''} - (4 x - 9) y^{'} + (3 x - 6) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.607

12909	$x^{2} y^{''} + 4 y^{'} x + (- x^{2} + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.496

12910	$(x^{2} + 1) y^{''} - 2 y^{'} x + 2 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.978

12911	$x y^{''} - (2 x - 1) y^{'} + (x - 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.465

12912	$x^{2} y^{''} - 4 y^{'} x + (x^{2} + 6) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.645

12913	$(2 x^{3} - 1) y^{''} - 6 x^{2} y^{'} + 6 x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.963

12914	$x^{2} y^{''} - 2 x (x + 1) y^{'} + 2 (x + 1) y = x^{3}$	[[_2nd_order, _with_linear_symmetries]]	✓	0.864

12915	$x^{2} y^{''} - 2 n x (x + 1) y^{'} + (a^{2} x^{2} + n^{2} + n) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.619

12916	$x^{4} y^{''} + 2 x^{3} (x + 1) y^{'} + n^{2} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.203

12917	$(x^{2} + 1) y^{''} + 1 + {y^{'}}^{2} = 0$	[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]	✓	0.824

12918	${(x y^{'''} - y^{''})}^{2} = {y^{'''}}^{2} + 1$	[[_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries]]	✓	1.314

12919	$y^{''} + y^{'} x = x$	[[_2nd_order, _missing_y]]	✓	0.867

12920	$y^{''} = x e^{x}$	[[_2nd_order, _quadrature]]	✓	0.966

12921	${(y^{'} - x y^{''})}^{2} = 1 + {y^{''}}^{2}$	[[_2nd_order, _missing_y]]	✓	1.069

12922	$y y^{''} - y^{2} y^{'} - {y^{'}}^{2} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _with_potential_symmetries], [_2nd_order, _reducible, _mu_xy]]	✓	0.717

12923	$y y^{''} - {y^{'}}^{2} + 1 = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓	1.452

12924	$2 y^{''} = e^{y}$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓	1.647

12925	$y y^{''} + 2 y^{'} - {y^{'}}^{2} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]	✓	0.534

12926	$(x^{2} - 2 x + 2) y^{'''} - x^{2} y^{''} + 2 y^{'} x - 2 y = 0$	[[_3rd_order, _with_linear_symmetries]]	✗	0.077

12927	$x y^{'''} - y^{''} - y^{'} x + y = - x^{2} + 1$	[[_3rd_order, _with_linear_symmetries]]	✗	0.075

12928	${(x + 2)}^{2} y^{'''} + (x + 2) y^{''} + y^{'} = 1$	[[_3rd_order, _missing_y]]	✓	0.567

12929	$x^{2} y^{''} + 3 y^{'} x + y = x$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓	1.459

12930	${(x - 1)}^{2} y^{''} + 4 (x - 1) y^{'} + 2 y = \cos (x)$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓	1.724

12931	$(x^{3} - x) y^{'''} + (8 x^{2} - 3) y^{''} + 14 y^{'} x + 4 y = 0$	[[_3rd_order, _fully, _exact, _linear]]	✓	1.455

12932	$2 x^{3} y y^{'''} + 6 x^{3} y^{'} y^{''} + 18 x^{2} y y^{''} + 18 x^{2} {y^{'}}^{2} + 36 x y^{'} y + 6 y^{2} = 0$	[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]]	✗	0.091

12933	$x^{5} y^{''} + (2 x^{4} - x) y^{'} - (2 x^{3} - 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.890

12934	$x^{2} (- x^{3} + 1) y^{''} - x^{3} y^{'} - 2 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	143.601

12935	$x^{2} y^{'''} - 5 x y^{''} + (4 x^{4} + 5) y^{'} - 8 x^{3} y = 0$	[[_3rd_order, _with_linear_symmetries]]	✗	0.075

12936	$y^{''} + 2 \cot (x) y^{'} + 2 \tan (x) {y^{'}}^{2} = 0$	[[_2nd_order, _missing_y]]	✓	1.288

12937	$x^{2} y y^{''} + {(y^{'} x - y)}^{2} = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]	✗	0.293

12938	$x^{3} y^{''} - {(y^{'} x - y)}^{2} = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]	✗	0.299

12939	$y y^{''} - {y^{'}}^{2} = y^{2} \ln (y) - x^{2} y^{2}$	[[_2nd_order, _reducible, _mu_xy]]	✗	0.303

12940	$\sin {(x)}^{2} y^{''} - 2 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.094

12941	$y^{''} = 1 + {y^{'}}^{2}$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]	✓	1.621

12942	$(- x^{2} + 1) y^{''} - y^{'} x = 2$	[[_2nd_order, _missing_y]]	✓	0.984

12943	$y^{''} + y y^{'} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]	✓	1.150

12944	$(x^{3} + 1) y^{'''} + 9 x^{2} y^{''} + 18 y^{'} x + 6 y = 0$	[[_3rd_order, _fully, _exact, _linear]]	✓	1.166

12945	$(x^{2} - x) y^{''} + (2 + 4 x) y^{'} + 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.821

12946	$y (1 - \ln (y)) y^{''} + (1 + \ln (y)) {y^{'}}^{2} = 0$	[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]	✓	0.859

12947	$y^{''} + \frac{y^{'}}{x} = 0$	[[_2nd_order, _missing_y]]	✓	0.635

12948	$x (x + 2 y) y^{''} + 2 x {y^{'}}^{2} + 4 (x + y) y^{'} + 2 y + x^{2} = 0$	[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]]	✓	1.490

12949	$y^{''} + {y^{'}}^{2} + 1 = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]	✓	1.831

12950	$(- x^{2} + 1) y^{''} - \frac{y^{'}}{x} + x^{2} = 0$	[[_2nd_order, _missing_y]]	✓	1.301

12951	$4 x^{2} y^{'''} + 8 x y^{''} + y^{'} = 0$	[[_3rd_order, _missing_y]]	✓	0.286

12952	$\sin (x) y^{''} - \cos (x) y^{'} + 2 y \sin (x) = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.970

12953	$[\begin{matrix} 3 x^{'} + 3 x + 2 y = e^{t} \\ 4 x - 3 y^{'} + 3 y = 3 t \end{matrix}]$	system_of_ODEs	✓	0.955

12954	$x^{'} = \frac{2 x}{t}$	[_separable]	✓	1.543

12955	$x^{'} = - \frac{t}{x}$	[_separable]	✓	3.194

12956	$x^{'} = - x^{2}$	[_quadrature]	✓	1.386

12957	$x^{''} + 2 x^{'} + 2 x = 0$	[[_2nd_order, _missing_x]]	✓	0.440

12958	$x^{'} = e^{- x}$	[_quadrature]	✓	0.655

12959	$x^{'} + 2 x = t^{2} + 4 t + 7$	[[_linear, ‘class A‘]]	✓	1.109

12960	$2 t x^{'} = x$	[_separable]	✓	1.506

12961	$t^{2} x^{''} - 6 x = 0$	[[_Emden, _Fowler]]	✓	0.359

12962	$2 x^{''} - 5 x^{'} - 3 x = 0$	[[_2nd_order, _missing_x]]	✓	0.340

12963	$x^{'} = x (1 - \frac{x}{4})$	[_quadrature]	✓	1.013

12964	$x^{'} = x^{2} + t^{2}$	[[_Riccati, _special]]	✓	1.394

12965	$x^{'} = t \cos (t^{2})$ i.c.	[_quadrature]	✓	0.628

12966	$x^{'} = \frac{1 + t}{\sqrt{t}}$ i.c.	[_quadrature]	✓	0.640

12967	$x^{''} = - 3 \sqrt{t}$ i.c.	[[_2nd_order, _quadrature]]	✓	1.375

12968	$x^{'} = t e^{- 2 t}$	[_quadrature]	✓	0.429

12969	$x^{'} = \frac{1}{t \ln (t)}$	[_quadrature]	✓	0.326

12970	$\sqrt{t} x^{'} = \cos (\sqrt{t})$	[_quadrature]	✓	0.537

12971	$x^{'} = \frac{e^{- t}}{\sqrt{t}}$ i.c.	[_quadrature]	✓	0.683

12972	$x^{'} + t x^{''} = 1$ i.c.	[[_2nd_order, _missing_y]]	✓	1.386

12973	$x^{'} = \sqrt{x}$ i.c.	[_quadrature]	✓	1.172

12974	$x^{'} = e^{- 2 x}$ i.c.	[_quadrature]	✓	1.370

12975	$y^{'} = 1 + y^{2}$	[_quadrature]	✓	3.638

12976	$u^{'} = \frac{1}{5 - 2 u}$	[_quadrature]	✓	0.903

12977	$x^{'} = a x + b$	[_quadrature]	✓	0.669

12978	$Q^{'} = \frac{Q}{4 + Q^{2}}$	[_quadrature]	✓	0.997

12979	$x^{'} = e^{x^{2}}$	[_quadrature]	✓	1.408

12980	$y^{'} = r (a - y)$	[_quadrature]	✓	0.673

12981	$x^{'} = \frac{2 x}{1 + t}$	[_separable]	✓	1.356

12982	$θ^{'} = t \sqrt{t^{2} + 1} \sec (θ)$	[_separable]	✓	1.912

12983	$(2 u + 1) u^{'} - 1 - t = 0$	[_separable]	✓	2.311

12984	$R^{'} = (1 + t) (1 + R^{2})$	[_separable]	✓	2.212

12985	$y^{'} + y + \frac{1}{y} = 0$	[_quadrature]	✓	21.887

12986	$(1 + t) x^{'} + x^{2} = 0$	[_separable]	✓	1.093

12987	$y^{'} = \frac{1}{2 y + 1}$ i.c.	[_quadrature]	✓	0.794

12988	$x^{'} = {(4 t - x)}^{2}$ i.c.	[[_homogeneous, ‘class C‘], _Riccati]	✓	2.288

12989	$x^{'} = 2 t x^{2}$ i.c.	[_separable]	✓	1.856

12990	$x^{'} = t^{2} e^{- x}$ i.c.	[_separable]	✓	1.799

12991	$x^{'} = x (4 + x)$ i.c.	[_quadrature]	✓	1.318

12992	$x^{'} = e^{t + x}$ i.c.	[_separable]	✓	2.589

12993	$T^{'} = 2 a t (T^{2} - a^{2})$ i.c.	[_separable]	✓	2.854

12994	$y^{'} = t^{2} \tan (y)$ i.c.	[_separable]	✓	2.564

12995	$x^{'} = \frac{(4 + 2 t) x}{\ln (x)}$ i.c.	[_separable]	✓	2.297

12996	$y^{'} = \frac{2 t y^{2}}{t^{2} + 1}$ i.c.	[_separable]	✓	1.999

12997	$x^{'} = \frac{t^{2}}{1 - x^{2}}$ i.c.	[_separable]	✓	2.861

12998	$x^{'} = 6 t {(x - 1)}^{2 / 3}$	[_separable]	✓	4.490

12999	$x^{'} = \frac{4 t^{2} + 3 x^{2}}{2 t x}$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	4.497

13000	$x^{'} e^{2 t} + 2 x e^{2 t} = e^{- t}$ i.c.	[[_linear, ‘class A‘]]	✓	1.536

2.2.130 Problems 12901 to 13000