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]]	✓	0.898

6702	$y^{'''} + y^{''} - 2 y^{'} = 0$	[[_3rd_order, _missing_x]]	✓	0.049

6703	$y^{''} + 6 y^{'} + 9 y = 0$	[[_2nd_order, _missing_x]]	✓	0.981

6704	$y^{''''} - 6 y^{'''} + 12 y^{''} - 8 y^{'} = 0$	[[_high_order, _missing_x]]	✓	0.059

6705	$y^{''} - 4 y^{'} + 13 y = 0$	[[_2nd_order, _missing_x]]	✓	2.765

6706	$y^{''} + 25 y = 0$	[[_2nd_order, _missing_x]]	✓	2.215

6707	$y^{'''} - y^{''} + 9 y^{'} - 9 y = 0$	[[_3rd_order, _missing_x]]	✓	0.056

6708	$y^{''''} + 4 y^{''} = 0$	[[_high_order, _missing_x]]	✓	0.055

6709	$y^{''''} - 6 y^{'''} + 13 y^{''} - 12 y^{'} + 4 y = 0$	[[_high_order, _missing_x]]	✓	0.059

6710	$y^{(6)} + 9 y^{''''} + 24 y^{''} + 16 y = 0$	[[_high_order, _missing_x]]	✓	0.080

6711	$y^{''} - 4 y^{'} + 3 y = 1$	[[_2nd_order, _missing_x]]	✓	1.005

6712	$y^{''} - 4 y^{'} = 5$	[[_2nd_order, _missing_x]]	✓	2.006

6713	$y^{'''} - 4 y^{''} = 5$	[[_3rd_order, _missing_x]]	✓	0.082

6714	$y^{(5)} - 4 y^{'''} = 5$	[[_high_order, _missing_x]]	✓	0.095

6715	$y^{'''} - 4 y^{'} = x$	[[_3rd_order, _missing_y]]	✓	0.088

6716	$y^{''} - 6 y^{'} + 9 y = e^{2 x}$	[[_2nd_order, _with_linear_symmetries]]	✓	1.115

6717	$y^{''} + y^{'} - 2 y = - 2 x^{2} + 2 x + 2$	[[_2nd_order, _with_linear_symmetries]]	✓	1.150

6718	$y^{''} - y = 4 x e^{x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.210

6719	$y^{''} - y = \sin {(x)}^{2}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.843

6720	$y^{''} - y = \frac{1}{{(1 + e^{- x})}^{2}}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.344

6721	$y^{''} + y = \csc (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	3.800

6722	$y^{''} - 3 y^{'} + 2 y = \sin (e^{- x})$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.459

6723	$y^{''} + y = \csc (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	3.764

6724	$y^{''} + 4 y = 4 \sec {(x)}^{2}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	4.637

6725	$y^{''} - 4 y^{'} + 3 y = \frac{1}{1 + e^{- x}}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.217

6726	$y^{''} - y = e^{- x} \sin (e^{- x}) + \cos (e^{- x})$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.990

6727	$y^{''} - y = \frac{1}{{(1 + e^{- x})}^{2}}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.352

6728	$y^{''} + 2 y = 2 + e^{x}$	[[_2nd_order, _with_linear_symmetries]]	✓	3.579

6729	$y^{''} - y = e^{x} \sin (2 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.825

6730	$y^{''} + 2 y^{'} + 2 y = x^{2} + \sin (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	15.195

6731	$y^{''} - 9 y = x + e^{2 x} - \sin (2 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	2.162

6732	$y^{'''} + 3 y^{''} + 2 y^{'} = x^{2} + 4 x + 8$	[[_3rd_order, _missing_y]]	✓	0.103

6733	$y^{''} + y = - 2 \sin (x) + 4 x \cos (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	3.427

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]]	✓	0.161

6735	$y^{''} + y^{'} + y = e^{3 x} + 6 e^{x} - 3 e^{- 2 x} + 5$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	64.818

6736	$y^{''} - y = e^{x}$	[[_2nd_order, _with_linear_symmetries]]	✓	1.165

6737	$y^{''} - 4 y^{'} + 4 y = e^{x} + x e^{2 x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.325

6738	$y^{''''} - y = \sin (2 x)$	[[_high_order, _linear, _nonhomogeneous]]	✓	0.130

6739	$y^{'''} + y = \cos (x)$	[[_3rd_order, _linear, _nonhomogeneous]]	✓	0.122

6740	$y^{''} + 4 y = \sin (2 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	3.565

6741	$y^{''} + 5 y = \cos (\sqrt{5} x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	4.017

6742	$y^{'''} + y^{''} + y^{'} + y = e^{x} + e^{- x} + \sin (x)$	[[_3rd_order, _linear, _nonhomogeneous]]	✓	0.739

6743	$y^{''} - y = x^{2}$	[[_2nd_order, _with_linear_symmetries]]	✓	1.124

6744	$y^{''} + 2 y = x^{3} + x^{2} + e^{- 2 x} + \cos (3 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	8.562

6745	$y^{''} - 2 y^{'} - y = e^{x} \cos (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.827

6746	$y^{''} - 4 y^{'} + 4 y = \frac{e^{2 x}}{x^{2}}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.310

6747	$y^{''} - y = x e^{3 x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.180

6748	$y^{''} + 5 y^{'} + 6 y = e^{- 2 x} \sec {(x)}^{2} (1 + 2 \tan (x))$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	2.102

6749	$x^{2} y^{''} - 3 x y^{'} + 4 y = x + x^{2} \ln (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.513

6750	$x^{2} y^{''} - 2 x y^{'} + 2 y = \ln {(x)}^{2} - \ln (x^{2})$	[[_2nd_order, _with_linear_symmetries]]	✓	2.750

6751	$x^{3} y^{'''} + 2 x^{2} y^{''} = x + \sin (\ln (x))$	[[_3rd_order, _missing_y]]	✓	0.416

6752	$x^{3} y^{'''} + x y^{'} - y = 3 x^{4}$	[[_3rd_order, _with_linear_symmetries]]	✓	0.293

6753	${(x + 1)}^{2} y^{''} + (x + 1) y^{'} - y = \ln {(x + 1)}^{2} + x - 1$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓	2.819

6754	${(2 x + 1)}^{2} y^{''} - 2 (2 x + 1) y^{'} - 12 y = 6 x$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓	1.416

6755	$x y^{''} - (x + 2) y^{'} + 2 y = 0$	[_Laguerre]	✓	0.811

6756	$(x^{2} + 1) y^{''} - 2 x y^{'} + 2 y = 2$	[[_2nd_order, _with_linear_symmetries]]	✓	1.475

6757	$(x^{2} + 4) y^{''} - 2 x y^{'} + 2 y = 8$	[[_2nd_order, _with_linear_symmetries]]	✓	1.434

6758	$(x + 1) y^{''} - (2 x + 3) y^{'} + (x + 2) y = (x^{2} + 2 x + 1) e^{2 x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.993

6759	$y^{''} - 2 \tan (x) y^{'} - 10 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	2.985

6760	$x^{2} y^{''} - x (2 x + 3) y^{'} + (x^{2} + 3 x + 3) y = (- x^{2} + 6) e^{x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.874

6761	$4 x^{2} y^{''} + 4 x^{3} y^{'} + {(x^{2} + 1)}^{2} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.797

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]]	✓	1.445

6763	$x y^{''} - y^{'} + 4 x^{3} y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	1.332

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]]	✓	3.719

6766	$(x \sin (x) + \cos (x)) y^{''} - x \cos (x) y^{'} + y \cos (x) = x$	[[_2nd_order, _with_linear_symmetries]]	✓	11.346

6767	$x y^{''} - 3 y^{'} + \frac{3 y}{x} = x + 2$	[[_2nd_order, _with_linear_symmetries]]	✓	1.499

6768	$(x + 1) y^{''} - (3 x + 4) y^{'} + 3 y = (3 x + 2) e^{3 x}$	[[_2nd_order, _with_linear_symmetries]]	✓	1.414

6769	$x^{2} y^{''} - 4 x y^{'} + (9 x^{2} + 6) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	2.879

6770	$x y^{''} + 2 y^{'} + 4 x y = 4$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	3.506

6771	$(x^{2} + 1) y^{''} - 2 x y^{'} + 2 y = \frac{- x^{2} + 1}{x}$	[[_2nd_order, _with_linear_symmetries]]	✓	1.604

6772	$y^{''} + {y^{'}}^{2} + 1 = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]	✓	1.622

6773	$(x^{2} + 1) y^{''} + 2 x y^{'} = \frac{2}{x^{3}}$	[[_2nd_order, _missing_y]]	✓	1.383

6774	$x y^{''} - y^{'} = - \frac{2}{x} - \ln (x)$	[[_2nd_order, _missing_y]]	✓	1.248

6775	$y^{'''} + y^{''} = x^{2}$	[[_3rd_order, _missing_y]]	✓	0.093

6776	$y y^{''} + {y^{'}}^{3} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]	✓	0.380

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]]	✓	1.090

6778	$y y^{''} = {y^{'}}^{2} (1 - y^{'} \cos (y) + y y^{'} \sin (y))$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_y_y1]]	✓	1.058

6779	$(2 x - 3) y^{'''} - (6 x - 7) y^{''} + 4 x y^{'} - 4 y = 8$	[[_3rd_order, _with_linear_symmetries]]	✗	0.040

6780	$(2 x^{3} - 1) y^{'''} - 6 x^{2} y^{''} + 6 x y^{'} = 0$	[[_3rd_order, _missing_y]]	✓	0.384

6781	$y y^{''} - {y^{'}}^{2} = y^{2} \ln (y)$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]	✓	2.020

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]]	✓	0.947

6783	$(1 + 2 y + 3 y^{2}) y^{'''} + 6 y^{'} (y^{''} + {y^{'}}^{2} + 3 y y^{''}) = x$	[[_3rd_order, _exact, _nonlinear]]	✗	0.043

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]]	✗	0.058

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]]	✗	0.049

6786	$2 (y + 1) y^{''} + 2 {y^{'}}^{2} + y^{2} + 2 y = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]	✓	4.609

6787	$[\begin{matrix} x^{'} - y^{'} + y = - e^{t} \\ x + y^{'} - y = e^{2 t} \end{matrix}]$	system_of_ODEs	✓	0.509

6788	$[\begin{matrix} x^{'} + 2 x + y^{'} + y = t \\ 5 x + y^{'} + 3 y = t^{2} \end{matrix}]$	system_of_ODEs	✓	0.813

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	✓	0.854

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	✓	0.214

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	✓	0.504

6792	$(1 - x) y^{'} = x^{2} - y$	[_linear]	✓	0.362

6793	$x y^{'} = 1 - x + 2 y$	[_linear]	✓	0.438

6794	$x y^{'} = 1 - x + 2 y$	[_linear]	✓	1.781

6795	$y^{'} = 2 x^{2} + 3 y$	[[_linear, ‘class A‘]]	✓	0.444

6796	$(x + 1) y^{'} = x^{2} - 2 x + y$	[_linear]	✓	0.356

6797	$y^{''} + x y = 0$	[[_Emden, _Fowler]]	✓	0.229

6798	$y^{''} + 2 x^{2} y = 0$	[[_Emden, _Fowler]]	✓	0.225

6799	$y^{''} - x y^{'} + x^{2} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.335

6800	$(- x^{2} + 1) y^{''} - 2 x y^{'} + p (p + 1) y = 0$	[_Gegenbauer]	✓	0.579

2.2.68 Problems 6701 to 6800