Problems 12701 to 12800

Table 2.257: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	time (sec)

12701	$y^{''} + 2 k e^{μ x} y^{'} + (a e^{2 λ x} + b e^{λ x} + k^{2} e^{2 μ x} + k μ e^{μ x} + c) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.153

12702	$y^{''} - (a + 2 b e^{a x}) y^{'} + b^{2} e^{2 a x} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.810

12703	$y^{''} + (a e^{2 λ x} + λ) y^{'} - a λ e^{2 λ x} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.893

12704	$y^{''} + (a e^{λ x} - λ) y^{'} + b e^{2 λ x} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.883

12705	$y^{''} + (a e^{λ x} + b) y^{'} + c (a e^{λ x} + b - c) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.024

12706	$y^{''} + (a + b e^{2 λ x}) y^{'} + λ (a - λ - b e^{2 λ x}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.040

12707	$y^{''} + (a + b e^{λ x} + b - 3 λ) y^{'} + a^{2} λ (b - λ) e^{2 λ x} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.345

12708	$y^{''} + (2 a e^{λ x} - λ) y^{'} + (a^{2} e^{2 λ x} + c e^{μ x}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.034

12709	$y^{''} + (2 a e^{λ x} + b) y^{'} + (a^{2} e^{2 λ x} + a (b + λ) e^{λ x} + c) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	7.145

12710	$y^{''} + (a e^{λ x} + 2 b - λ) y^{'} + (c e^{2 λ x} + a b e^{λ x} + b^{2} - b λ) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.300

12711	$y^{''} + (a e^{x} + b) y^{'} + (c (a - c) e^{2 x} + (a k + b c - 2 c k + c) e^{x} + k (b - k)) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.185

12712	$y^{''} + (a e^{λ x} + b) y^{'} + (α e^{2 λ x} + β e^{λ x} + γ) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.168

12713	$y^{''} + (2 a e^{λ x} - λ) y^{'} + (a^{2} e^{2 λ x} + b e^{2 μ x} + c e^{μ x} + k) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.171

12714	$y^{''} + (2 a e^{λ x} + b - λ) y^{'} + (a^{2} e^{2 λ x} + a b e^{λ x} + c e^{2 μ x} + d e^{μ x} + k) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.352

12715	$y^{''} + (a e^{λ x} + b e^{μ x}) y^{'} + a e^{λ x} (b e^{μ x} + λ) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.264

12716	$y^{''} + e^{λ x} (a e^{2 μ x} + b) y^{'} + μ (e^{λ x} (b - a e^{2 μ x}) - μ) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.549

12717	$y^{''} + (a e^{λ x} + b e^{μ x} + c) y^{'} + (a λ e^{λ x} + b μ e^{μ x}) y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	1.000

12718	$y^{''} + (a e^{λ x} + b e^{μ x} + c) y^{'} + (a b e^{(λ + μ) x} + a c e^{λ x} + b μ e^{μ x}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.375

12719	$y^{''} + (a e^{λ x} + 2 b e^{μ x} - λ) y^{'} + (a b e^{(λ + μ) x} + c e^{2 λ x} + b^{2} e^{2 μ x} + b (μ - λ) e^{μ x}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.562

12720	$y^{''} + (a e^{(λ + μ) x} + a λ e^{λ x} + b e^{μ x} - 2 λ) y^{'} + a^{2} b λ e^{(μ + 2 λ) x} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.967

12721	$y^{''} + a e^{b x^{n}} y^{'} + c (a e^{b x^{n}} - c) y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✗	1.085

12722	$(a e^{λ x} + b) y^{''} - a λ^{2} e^{λ x} y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.676

12723	$(a^{2} e^{2 λ x} + b) y^{''} - b λ y^{'} - a^{2} λ^{2} k^{2} e^{2 λ x} y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	5.169

12724	$2 (a e^{λ x} + b) y^{''} + a λ e^{λ x} y^{'} + c y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	41.131

12725	$(a e^{λ x} + b) y^{''} + (c e^{λ x} + d) y^{'} + k ((- a k + c) e^{λ x} + d - b k) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	2.153

12726	$(a e^{λ x} + b) y^{''} + (c e^{λ x} + d) y^{'} + (n e^{λ x} + m) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.665

12727	$\frac{1 + 2 x y}{y} + \frac{(y - x) y^{'}}{y^{2}} = 0$	[[_homogeneous, ‘class D‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	1.819

12728	$\frac{y^{2} - 2 x^{2}}{x y^{2} - x^{3}} + \frac{(2 y^{2} - x^{2}) y^{'}}{y^{3} - x^{2} y} = 0$	[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]	✓	159.192

12729	$\frac{1}{\sqrt{x^{2} + y^{2}}} + (\frac{1}{y} - \frac{x}{y \sqrt{x^{2} + y^{2}}}) y^{'} = 0$	[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]	✓	8.717

12730	$y^{'} x + x + y = 0$	[_linear]	✓	2.318

12731	$6 x - 2 y + 1 + (2 y - 2 x - 3) y^{'} = 0$	[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	3.435

12732	$\sec (x) \cos {(y)}^{2} - \cos (x) \sin (y) y^{'} = 0$	[_separable]	✓	7.801

12733	$(x + 1) y^{2} - x^{3} y^{'} = 0$	[_separable]	✓	1.503

12734	$(x^{2} + 1) (1 + y^{2}) y^{'} + 2 x y (1 - y^{2}) = 0$	[_separable]	✓	52.699

12735	$\sin (x) \cos {(y)}^{2} + \cos {(x)}^{2} y^{'} = 0$	[_separable]	✓	3.684

12736	$e^{\frac{y}{x}} x + y - y^{'} x = 0$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	5.275

12737	$2 x^{2} y + 3 y^{3} - (x^{3} + 2 x y^{2}) y^{'} = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	46.898

12738	$x^{2} y^{'} + y^{2} - x y = 0$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	1.785

12739	$2 x^{2} y + y^{3} - x^{3} y^{'} = 0$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	153.920

12740	$y^{3} + x^{3} y^{'} = 0$	[_separable]	✓	4.982

12741	$x + y \cos (\frac{y}{x}) - x \cos (\frac{y}{x}) y^{'} = 0$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	7.414

12742	$(x + y + 1) y^{'} + 1 + 4 x + 3 y = 0$	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	2.593

12743	$4 x - y + 2 + (x + y + 3) y^{'} = 0$	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	2.631

12744	$2 x + y - (4 x + 2 y - 1) y^{'} = 0$	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	2.028

12745	$y + 2 x y^{2} - x^{2} y^{3} + 2 x^{2} y y^{'} = 0$	[[_homogeneous, ‘class G‘], _rational, _Riccati]	✓	1.861

12746	$2 y + 3 x y^{2} + (x + 2 x^{2} y) y^{'} = 0$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	10.693

12747	$y + x y^{2} + (x - x^{2} y) y^{'} = 0$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	2.058

12748	$y^{'} + \cot (x) y = \sec (x)$	[_linear]	✓	1.838

12749	$y^{'} x + (x + 1) y = e^{x}$	[_linear]	✓	1.353

12750	$y^{'} - \frac{2 y}{x + 1} = {(x + 1)}^{3}$	[_linear]	✓	1.442

12751	$(x^{3} + x) y^{'} + 4 x^{2} y = 2$	[_linear]	✓	1.324

12752	$x^{2} y^{'} + (- 2 x + 1) y = x^{2}$	[_linear]	✓	1.642

12753	$(- x^{2} + 1) y^{'} - 2 (x + 1) y = y^{5 / 2}$	[_rational, _Bernoulli]	✓	1.888

12754	$y y^{'} + x y^{2} = x$	[_separable]	✓	2.033

12755	$y^{'} \sin (y) + \sin (x) \cos (y) = \sin (x)$	[_separable]	✓	38.071

12756	$4 y^{'} x + 3 y + e^{x} x^{4} y^{5} = 0$	[_Bernoulli]	✓	2.495

12757	$y^{'} - \frac{1 + y}{x + 1} = \sqrt{1 + y}$	[[_1st_order, _with_linear_symmetries]]	✓	2.498

12758	$x^{4} y (3 y + 2 y^{'} x) + x^{2} (4 y + 3 y^{'} x) = 0$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	3.731

12759	$y^{2} (3 y - 6 y^{'} x) - x (y - 2 y^{'} x) = 0$	[_separable]	✓	1.875

12760	$2 x^{3} y - y^{2} - (2 x^{4} + x y) y^{'} = 0$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	7.520

12761	$x^{2} y^{'} + y^{2} - x y = 0$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	1.920

12762	$\frac{y^{'} x - y}{\sqrt{x^{2} - y^{2}}} = y^{'} x$	[‘y=_G(x,y’)‘]	✓	2.287

12763	$x + y - (x - y) y^{'} = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	3.688

12764	$x^{2} + y^{2} - 2 x y^{'} y = 0$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	7.725

12765	$x - y^{2} + 2 x y^{'} y = 0$	[[_homogeneous, ‘class G‘], _rational, _Bernoulli]	✓	1.963

12766	$y^{'} x - y = x^{2} + y^{2}$	[[_homogeneous, ‘class D‘], _rational, _Riccati]	✓	1.541

12767	$3 x^{2} + 6 x y + 3 y^{2} + (2 x^{2} + 3 x y) y^{'} = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	6.026

12768	$(x^{2} + 2 y + y^{2}) y^{'} + 2 x = 0$	[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]]	✓	1.289

12769	$y^{4} + 2 y + (x y^{3} + 2 y^{4} - 4 x) y^{'} = 0$	[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]]	✓	2.365

12770	$x^{3} y - y^{4} + (x y^{3} - x^{4}) y^{'} = 0$	[_separable]	✓	1.537

12771	$y^{2} - x^{2} + 2 m x y + (m y^{2} - m x^{2} - 2 x y) y^{'} = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	25.455

12772	$y^{'} x - y + 2 x^{2} y - x^{3} = 0$	[_linear]	✓	1.402

12773	$(x + y) y^{'} - 1 = 0$	[[_homogeneous, ‘class C‘], [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert]	✓	1.816

12774	$x + y y^{'} + y - y^{'} x = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	3.515

12775	$y^{'} x - a y + b y^{2} = c x^{2 a}$	[_rational, _Riccati]	✓	3.456

12776	$x \sqrt{1 - y^{2}} + y \sqrt{- x^{2} + 1} y^{'} = 0$	[_separable]	✓	2.856

12777	$y^{'} \sqrt{- x^{2} + 1} + \sqrt{1 - y^{2}} = 0$	[_separable]	✓	19.938

12778	$y^{'} - x^{2} y = x^{5}$	[_linear]	✓	1.919

12779	${(y - x)}^{2} y^{'} = 1$	[[_homogeneous, ‘class C‘], _dAlembert]	✓	4.074

12780	$y^{'} x + y + x^{4} y^{4} e^{x} = 0$	[_Bernoulli]	✓	3.490

12781	$x (1 - y) y^{'} + (1 - x) y = 0$	[_separable]	✓	1.663

12782	$(y - x) y^{'} + y = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	2.906

12783	$y^{'} x - y = \sqrt{x^{2} + y^{2}}$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	5.293

12784	$y^{'} x - y = \sqrt{x^{2} - y^{2}}$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	116.650

12785	$x \sin (\frac{y}{x}) - y \cos (\frac{y}{x}) + x \cos (\frac{y}{x}) y^{'} = 0$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	4.039

12786	$(4 + 2 x - y) y^{'} + 5 + x - 2 y = 0$	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	3.477

12787	$y^{'} + \frac{y}{{(- x^{2} + 1)}^{3 / 2}} = \frac{x + \sqrt{- x^{2} + 1}}{{(- x^{2} + 1)}^{2}}$	[_linear]	✓	3.542

12788	$(- x^{2} + 1) y^{'} - x y = a x y^{2}$	[_separable]	✓	3.153

12789	$x y^{2} (y^{'} x + 3 y) - 2 y + y^{'} x = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	5.878

12790	$(x^{2} + 1) y^{'} + y = \arctan (x)$	[_linear]	✓	1.786

12791	$5 x y - 3 y^{3} + (3 x^{2} - 7 x y^{2}) y^{'} = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	2.360

12792	$y^{'} + \cos (x) y = \frac{\sin (2 x)}{2}$	[_linear]	✓	2.192

12793	$y + x y^{2} - y^{'} x = 0$	[[_homogeneous, ‘class D‘], _rational, _Bernoulli]	✓	1.884

12794	$(1 - x) y - x (1 + y) y^{'} = 0$	[_separable]	✓	1.579

12795	$3 x^{2} y + (x^{3} + x^{3} y^{2}) y^{'} = 0$	[_separable]	✓	3.052

12796	$(x^{2} + y^{2}) (y y^{'} + x) = (x^{2} + y^{2} + x) (y^{'} x - y)$	[_rational]	✗	3.223

12797	$2 x + 3 y - 1 + (2 x + 3 y - 5) y^{'} = 0$	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	1.955

12798	$y^{3} - 2 x^{2} y + (2 x y^{2} - x^{3}) y^{'} = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	48.436

12799	$2 x^{3} y^{2} - y + (2 x^{2} y^{3} - x) y^{'} = 0$	[_rational]	✓	1.685

12800	$(x^{2} + y^{2}) (y y^{'} + x) + \sqrt{x^{2} + y^{2} + 1} (y - y^{'} x) = 0$	[[_1st_order, _with_linear_symmetries]]	✓	2.579

2.2.128 Problems 12701 to 12800