Problems 6701 to 6800

Table 2.137: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	time (sec)

6701	$y^{''} + 2 y^{'} - 15 y = 0$	[[_2nd_order, _missing_x]]	✓	0.347

6702	$y^{'''} + y^{''} - 2 y^{'} = 0$	[[_3rd_order, _missing_x]]	✓	0.080

6703	$y^{''} + 6 y^{'} + 9 y = 0$	[[_2nd_order, _missing_x]]	✓	0.385

6704	$y^{''''} - 6 y^{'''} + 12 y^{''} - 8 y^{'} = 0$	[[_high_order, _missing_x]]	✓	0.077

6705	$y^{''} - 4 y^{'} + 13 y = 0$	[[_2nd_order, _missing_x]]	✓	0.498

6706	$y^{''} + 25 y = 0$	[[_2nd_order, _missing_x]]	✓	1.410

6707	$y^{'''} - y^{''} + 9 y^{'} - 9 y = 0$	[[_3rd_order, _missing_x]]	✓	0.085

6708	$y^{''''} + 4 y^{''} = 0$	[[_high_order, _missing_x]]	✓	0.079

6709	$y^{''''} - 6 y^{'''} + 13 y^{''} - 12 y^{'} + 4 y = 0$	[[_high_order, _missing_x]]	✓	0.084

6710	$y^{(6)} + 9 y^{''''} + 24 y^{''} + 16 y = 0$	[[_high_order, _missing_x]]	✓	0.100

6711	$y^{''} - 4 y^{'} + 3 y = 1$	[[_2nd_order, _missing_x]]	✓	0.423

6712	$y^{''} - 4 y^{'} = 5$	[[_2nd_order, _missing_x]]	✓	1.059

6713	$y^{'''} - 4 y^{''} = 5$	[[_3rd_order, _missing_x]]	✓	0.119

6714	$y^{(5)} - 4 y^{'''} = 5$	[[_high_order, _missing_x]]	✓	0.126

6715	$y^{'''} - 4 y^{'} = x$	[[_3rd_order, _missing_y]]	✓	0.128

6716	$y^{''} - 6 y^{'} + 9 y = e^{2 x}$	[[_2nd_order, _with_linear_symmetries]]	✓	0.550

6717	$y^{''} + y^{'} - 2 y = - 2 x^{2} + 2 x + 2$	[[_2nd_order, _with_linear_symmetries]]	✓	0.440

6718	$y^{''} - y = 4 x e^{x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.592

6719	$y^{''} - y = \sin {(x)}^{2}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.715

6720	$y^{''} - y = \frac{1}{{(1 + e^{- x})}^{2}}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.982

6721	$y^{''} + y = \csc (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.692

6722	$y^{''} - 3 y^{'} + 2 y = \sin (e^{- x})$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.724

6723	$y^{''} + y = \csc (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.701

6724	$y^{''} + 4 y = 4 \sec {(x)}^{2}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.004

6725	$y^{''} - 4 y^{'} + 3 y = \frac{1}{1 + e^{- x}}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.634

6726	$y^{''} - y = e^{- x} \sin (e^{- x}) + \cos (e^{- x})$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.543

6727	$y^{''} - y = \frac{1}{{(1 + e^{- x})}^{2}}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.950

6728	$y^{''} + 2 y = e^{x} + 2$	[[_2nd_order, _with_linear_symmetries]]	✓	0.695

6729	$y^{''} - y = e^{x} \sin (2 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.762

6730	$y^{''} + 2 y^{'} + 2 y = x^{2} + \sin (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.717

6731	$y^{''} - 9 y = x + e^{2 x} - \sin (2 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.993

6732	$y^{'''} + 3 y^{''} + 2 y^{'} = x^{2} + 4 x + 8$	[[_3rd_order, _missing_y]]	✓	0.134

6733	$y^{''} + y = - 2 \sin (x) + 4 x \cos (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.944

6734	$y^{'''} - y^{''} - 4 y^{'} + 4 y = 2 x^{2} - 4 x - 1 + 2 x^{2} e^{2 x} + 5 x e^{2 x} + e^{2 x}$	[[_3rd_order, _linear, _nonhomogeneous]]	✓	0.239

6735	$y^{''} + y^{'} + y = e^{3 x} + 6 e^{x} - 3 e^{- 2 x} + 5$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.846

6736	$y^{''} - y = e^{x}$	[[_2nd_order, _with_linear_symmetries]]	✓	0.506

6737	$y^{''} - 4 y^{'} + 4 y = e^{x} + x e^{2 x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.652

6738	$y^{''''} - y = \sin (2 x)$	[[_high_order, _linear, _nonhomogeneous]]	✓	0.196

6739	$y^{'''} + y = \cos (x)$	[[_3rd_order, _linear, _nonhomogeneous]]	✓	0.179

6740	$y^{''} + 4 y = \sin (2 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.736

6741	$y^{''} + 5 y = \cos (\sqrt{5} x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.832

6742	$y^{'''} + y^{''} + y^{'} + y = e^{x} + e^{- x} + \sin (x)$	[[_3rd_order, _linear, _nonhomogeneous]]	✓	1.183

6743	$y^{''} - y = x^{2}$	[[_2nd_order, _with_linear_symmetries]]	✓	0.456

6744	$y^{''} + 2 y = x^{3} + x^{2} + e^{- 2 x} + \cos (3 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.276

6745	$y^{''} - 2 y^{'} - y = e^{x} \cos (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.600

6746	$y^{''} - 4 y^{'} + 4 y = \frac{e^{2 x}}{x^{2}}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.667

6747	$y^{''} - y = x e^{3 x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.563

6748	$y^{''} + 5 y^{'} + 6 y = e^{- 2 x} \sec {(x)}^{2} (1 + 2 \tan (x))$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.938

6749	$x^{2} y^{''} - 3 y^{'} x + 4 y = x + x^{2} \ln (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.396

6750	$x^{2} y^{''} - 2 y^{'} x + 2 y = \ln {(x)}^{2} - \ln (x^{2})$	[[_2nd_order, _with_linear_symmetries]]	✓	2.647

6751	$x^{3} y^{'''} + 2 x^{2} y^{''} = x + \sin (\ln (x))$	[[_3rd_order, _missing_y]]	✓	0.462

6752	$x^{3} y^{'''} + y^{'} x - y = 3 x^{4}$	[[_3rd_order, _with_linear_symmetries]]	✓	0.265

6753	${(x + 1)}^{2} y^{''} + (x + 1) y^{'} - y = \ln {(x + 1)}^{2} + x - 1$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓	2.546

6754	${(2 x + 1)}^{2} y^{''} - 2 (2 x + 1) y^{'} - 12 y = 6 x$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓	1.011

6755	$x y^{''} - (x + 2) y^{'} + 2 y = 0$	[_Laguerre]	✓	0.504

6756	$(x^{2} + 1) y^{''} - 2 y^{'} x + 2 y = 2$	[[_2nd_order, _with_linear_symmetries]]	✓	1.134

6757	$(x^{2} + 4) y^{''} - 2 y^{'} x + 2 y = 8$	[[_2nd_order, _with_linear_symmetries]]	✓	1.182

6758	$(x + 1) y^{''} - (2 x + 3) y^{'} + (x + 2) y = (x^{2} + 2 x + 1) e^{2 x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.594

6759	$y^{''} - 2 y^{'} \tan (x) - 10 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.663

6760	$x^{2} y^{''} - x (2 x + 3) y^{'} + (x^{2} + 3 x + 3) y = (- x^{2} + 6) e^{x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.608

6761	$4 x^{2} y^{''} + 4 x^{3} y^{'} + {(x^{2} + 1)}^{2} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.515

6762	$x^{2} y^{''} + (- 4 x^{2} + x) y^{'} + (4 x^{2} - 2 x + 1) y = (x^{2} - x + 1) e^{x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.627

6763	$x y^{''} - y^{'} + 4 x^{3} y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	0.913

6764	$x^{4} y^{''} + 2 x^{3} y^{'} + y = \frac{x + 1}{x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	107.299

6765	$x^{8} y^{''} + 4 x^{7} y^{'} + y = \frac{1}{x^{3}}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.562

6766	$(x \sin (x) + \cos (x)) y^{''} - x \cos (x) y^{'} + \cos (x) y = x$	[[_2nd_order, _with_linear_symmetries]]	✓	2.069

6767	$x y^{''} - 3 y^{'} + \frac{3 y}{x} = x + 2$	[[_2nd_order, _with_linear_symmetries]]	✓	1.338

6768	$(x + 1) y^{''} - (3 x + 4) y^{'} + 3 y = (3 x + 2) e^{3 x}$	[[_2nd_order, _with_linear_symmetries]]	✓	0.764

6769	$x^{2} y^{''} - 4 y^{'} x + (9 x^{2} + 6) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.708

6770	$x y^{''} + 2 y^{'} + 4 x y = 4$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.921

6771	$(x^{2} + 1) y^{''} - 2 y^{'} x + 2 y = \frac{- x^{2} + 1}{x}$	[[_2nd_order, _with_linear_symmetries]]	✓	1.276

6772	$y^{''} + {y^{'}}^{2} + 1 = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]	✓	1.987

6773	$(x^{2} + 1) y^{''} + 2 y^{'} x = \frac{2}{x^{3}}$	[[_2nd_order, _missing_y]]	✓	0.997

6774	$x y^{''} - y^{'} = - \frac{2}{x} - \ln (x)$	[[_2nd_order, _missing_y]]	✓	1.059

6775	$y^{'''} + y^{''} = x^{2}$	[[_3rd_order, _missing_y]]	✓	0.142

6776	$y y^{''} + {y^{'}}^{3} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]	✓	0.545

6777	$y y^{''} + {y^{'}}^{2} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]	✓	1.164

6778	$y y^{''} = {y^{'}}^{2} (1 - y^{'} \cos (y) + y y^{'} \sin (y))$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_y_y1]]	✓	2.897

6779	$(2 x - 3) y^{'''} - (6 x - 7) y^{''} + 4 y^{'} x - 4 y = 8$	[[_3rd_order, _with_linear_symmetries]]	✗	0.075

6780	$(2 x^{3} - 1) y^{'''} - 6 x^{2} y^{''} + 6 y^{'} x = 0$	[[_3rd_order, _missing_y]]	✓	1.004

6781	$y y^{''} - {y^{'}}^{2} = y^{2} \ln (y)$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]	✓	4.250

6782	$(x + 2 y) y^{''} + 2 {y^{'}}^{2} + 2 y^{'} = 2$	[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]	✓	1.117

6783	$(1 + 2 y + 3 y^{2}) y^{'''} + 6 y^{'} (y^{''} + {y^{'}}^{2} + 3 y y^{''}) = x$	[[_3rd_order, _exact, _nonlinear]]	✗	0.088

6784	$3 x (y^{2} y^{'''} + 6 y y^{'} y^{''} + 2 {y^{'}}^{3}) - 3 y (y y^{''} + 2 {y^{'}}^{2}) = - \frac{2}{x}$	[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]]	✗	0.100

6785	$y y^{'''} + 3 y^{'} y^{''} - 2 y y^{''} - 2 {y^{'}}^{2} + y y^{'} = e^{2 x}$	[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]]	✗	0.101

6786	$2 (1 + y) y^{''} + 2 {y^{'}}^{2} + y^{2} + 2 y = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]	✓	7.848

6787	$[\begin{matrix} x^{'} - y^{'} + y = - e^{t} \\ x + y^{'} - y = e^{2 t} \end{matrix}]$	system_of_ODEs	✓	0.725

6788	$[\begin{matrix} x^{'} + 2 x + y^{'} + y = t \\ 5 x + y^{'} + 3 y = t^{2} \end{matrix}]$	system_of_ODEs	✓	0.902

6789	$[\begin{matrix} x^{'} + x + 2 y^{'} + 7 y = e^{t} + 2 \\ - 2 x + y^{'} + 3 y = e^{t} - 1 \end{matrix}]$	system_of_ODEs	✓	1.004

6790	$[\begin{matrix} x^{'} - x + y^{'} + 3 y = e^{- t} - 1 \\ x^{'} + 2 x + y^{'} + 3 y = 1 + e^{2 t} \end{matrix}]$	system_of_ODEs	✓	0.320

6791	$[\begin{matrix} x^{'} - x + y^{'} + 2 y = 1 + e^{t} \\ y^{'} + 2 y + z^{'} + z = e^{t} + 2 \\ x^{'} - x + z^{'} + z = 3 + e^{t} \end{matrix}]$	system_of_ODEs	✓	0.589

6792	$(1 - x) y^{'} = x^{2} - y$	[_linear]	✓	0.353

6793	$y^{'} x = 1 - x + 2 y$	[_linear]	✓	0.418

6794	$y^{'} x = 1 - x + 2 y$	[_linear]	✓	1.404

6795	$y^{'} = 2 x^{2} + 3 y$	[[_linear, ‘class A‘]]	✓	0.324

6796	$(x + 1) y^{'} = x^{2} - 2 x + y$	[_linear]	✓	0.317

6797	$y^{''} + x y = 0$	[[_Emden, _Fowler]]	✓	0.350

6798	$y^{''} + 2 x^{2} y = 0$	[[_Emden, _Fowler]]	✓	0.287

6799	$y^{''} - y^{'} x + x^{2} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.414

6800	$(- x^{2} + 1) y^{''} - 2 y^{'} x + p (p + 1) y = 0$	[_Gegenbauer]	✓	0.644

2.2.68 Problems 6701 to 6800