Problems 4401 to 4500

Table 2.91: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	Maple	Mma	Sympy

4401	$2 \sqrt{x y} - y - y^{'} x = 0$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	✓	✓	✓

4402	$y^{'} = e^{\frac{x y^{'}}{y}}$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	✓	✓	✗

4403	$2 x^{3} y^{2} - y + (2 x^{2} y^{3} - x) y^{'} = 0$	[_rational]	✓	✓	✓	✗

4404	$y - 1 - x y + y^{'} x = 0$	[_linear]	✓	✓	✓	✓

4405	$y^{'} x - y = x \tan (\frac{y}{x})$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	✓	✓	✓

4406	$y^{'} + \frac{y}{x} = e^{x y}$	[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]	✓	✓	✓	✗

4407	$y y^{''} - y y^{'} = {y^{'}}^{2}$	[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]	✓	✓	✓	✗

4408	$2 y - x (\ln (x^{2} y) - 1) y^{'} = 0$	[[_homogeneous, ‘class G‘]]	✓	✓	✓	✓

4409	$y^{'} = \frac{1}{x y + x^{3} y^{3}}$	[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]]	✓	✓	✓	✓

4410	$y^{'} = \frac{2 {(y + 2)}^{2}}{{(x + y - 1)}^{2}}$	[[_homogeneous, ‘class C‘], _rational]	✓	✓	✓	✗

4411	$e^{x} + 3 y^{2} + 2 x y^{'} y = 0$	[_Bernoulli]	✓	✓	✓	✓

4412	$x y + 2 x^{3} y + x^{2} y^{'} = 0$	[_separable]	✓	✓	✓	✓

4413	$4 x - 2 y y^{'} + x {y^{'}}^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	✓	✓	✓

4414	$y^{'''} = 2 (y^{''} - 1) \cot (x)$	[[_3rd_order, _missing_y]]	✓	✓	✓	✓

4415	$y + 3 x^{4} y^{2} + (x + 2 x^{2} y^{3}) y^{'} = 0$	[_rational]	✓	✓	✓	✗

4416	$y^{'} x = y + \sqrt{x^{2} - y^{2}}$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	✓	✓	✓

4417	$2 y (x e^{x^{2}} + \sin (x) \cos (x) y) + (2 e^{x^{2}} + 3 y \sin {(x)}^{2}) y^{'} = 0$	[[_Abel, ‘2nd type‘, ‘class B‘]]	✓	✓	✓	✗

4418	$\cos (y) + \sin (y) (x - \sin (y) \cos (y)) y^{'} = 0$	[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]]	✓	✓	✓	✗

4419	$y^{3} + (3 x^{2} - 2 x y^{2}) y^{'} = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	✓	✓	✓

4420	$(y^{'} + 1) \ln (\frac{x + y}{3 + x}) = \frac{x + y}{3 + x}$	[[_homogeneous, ‘class C‘], _exact, _dAlembert]	✓	✓	✓	✓

4421	$2 x^{3} y y^{'} + 3 x^{2} y^{2} + 7 = 0$	[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli]	✓	✓	✓	✓

4422	$x - y \cos (\frac{y}{x}) + x \cos (\frac{y}{x}) y^{'} = 0$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	✓	✓	✓

4423	$x^{2} (y^{'} x - y) = y (x + y)$	[[_homogeneous, ‘class D‘], _rational, _Bernoulli]	✓	✓	✓	✓

4424	$y^{4} + x y + (x y^{3} - x^{2}) y^{'} = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	✓	✓	✗

4425	$x^{2} + 3 \ln (y) - \frac{x y^{'}}{y} = 0$	[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]	✓	✓	✓	✓

4426	$x y^{''} = x + y^{'}$	[[_2nd_order, _missing_y]]	✓	✓	✓	✓

4427	$y + (x y - x - y^{3}) y^{'} = 0$	[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]]	✓	✓	✓	✓

4428	$y + 2 y^{3} y^{'} = (x + 4 y \ln (y)) y^{'}$	[[_1st_order, _with_linear_symmetries]]	✓	✓	✓	✓

4429	$y \ln (x) \ln (y) + y^{'} = 0$	[_separable]	✓	✓	✓	✓

4430	$2 x^{3 / 2} + x^{2} + y^{2} + 2 y \sqrt{x} y^{'} = 0$	[[_homogeneous, ‘class D‘], _rational, _Bernoulli]	✓	✓	✓	✓

4431	$2 x + y \cos (x y) + x \cos (x y) y^{'} = 0$	[_exact, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]	✓	✓	✓	✗

4432	$y y^{''} - y^{2} y^{'} - {y^{'}}^{2} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _with_potential_symmetries], [_2nd_order, _reducible, _mu_xy]]	✓	✓	✓	✗

4433	$2 y^{'} + x = 4 \sqrt{y}$	[[_1st_order, _with_linear_symmetries], _Chini]	✓	✓	✓	✗

4434	$2 {y^{'}}^{3} - 3 {y^{'}}^{2} + x = y$	[[_homogeneous, ‘class C‘], _dAlembert]	✓	✓	✗	✗

4435	$y^{'} - 6 x e^{x - y} - 1 = 0$	[[_1st_order, _with_linear_symmetries]]	✓	✓	✓	✓

4436	$(1 + y^{2}) y^{''} + {y^{'}}^{3} + y^{'} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓	✓	✓	✗

4437	$y \sin (x) + \cos {(x)}^{2} - y^{'} \cos (x) = 0$	[_linear]	✓	✓	✓	✓

4438	$y (6 y^{2} - x - 1) + 2 y^{'} x = 0$	[_rational, _Bernoulli]	✓	✓	✓	✓

4439	$y^{'} (x - \ln (y^{'})) = 1$	[_quadrature]	✓	✓	✓	✓

4440	$(\cos (x) + 1) y^{'} + \sin (x) (\sin (x) + \sin (x) \cos (x) - y) = 0$	[_linear]	✓	✓	✓	✓

4441	$x + \sin {(\frac{y}{x})}^{2} (y - y^{'} x) = 0$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	✓	✓	✗

4442	$2 x y^{4} e^{y} + 2 x y^{3} + y + (x^{2} y^{4} e^{y} - x^{2} y^{2} - 3 x) y^{'} = 0$	[‘x=_G(y,y’)‘]	✓	✓	✓	✗

4443	$x y^{3} - 1 + x^{2} y^{2} y^{'} = 0$	[[_homogeneous, ‘class G‘], _rational, _Bernoulli]	✓	✓	✓	✓

4444	$y^{'''} - 2 y^{''} + y^{'} - 2 y = 0$	[[_3rd_order, _missing_x]]	✓	✓	✓	✓

4445	$y^{'''} + y^{''} + 9 y^{'} + 9 y = 0$	[[_3rd_order, _missing_x]]	✓	✓	✓	✓

4446	$y^{'''} + y^{''} - y^{'} - y = 0$	[[_3rd_order, _missing_x]]	✓	✓	✓	✓

4447	$y^{'''} + 8 y = 0$	[[_3rd_order, _missing_x]]	✓	✓	✓	✓

4448	$y^{'''} - 8 y = 0$	[[_3rd_order, _missing_x]]	✓	✓	✓	✓

4449	$y^{''''} + 4 y = 0$	[[_high_order, _missing_x]]	✓	✓	✓	✓

4450	$y^{''''} + 18 y^{''} + 81 y = 0$	[[_high_order, _missing_x]]	✓	✓	✓	✓

4451	$y^{''''} - 4 y^{''} + 16 y = 0$	[[_high_order, _missing_x]]	✓	✓	✓	✓

4452	$y^{''''} - 2 y^{'''} + 2 y^{''} - 2 y^{'} + y = 0$	[[_high_order, _missing_x]]	✓	✓	✓	✓

4453	$y^{''''} - 5 y^{'''} + 5 y^{''} + 5 y^{'} - 6 y = 0$	[[_high_order, _missing_x]]	✓	✓	✓	✓

4454	$y^{(5)} - 6 y^{''''} + 9 y^{'''} = 0$	[[_high_order, _missing_x]]	✓	✓	✓	✓

4455	$y^{(6)} - 64 y = 0$	[[_high_order, _missing_x]]	✓	✓	✓	✓

4456	$y^{''} + 6 y^{'} + 10 y = 3 x e^{- 3 x} - 2 e^{3 x} \cos (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4457	$y^{''} - 8 y^{'} + 17 y = e^{4 x} (x^{2} - 3 x \sin (x))$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4458	$y^{''} - 2 y^{'} + 2 y = (x + e^{x}) \sin (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4459	$y^{''} + 4 y = \sinh (x) \sin (2 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4460	$y^{''} + 2 y^{'} + 2 y = \cosh (x) \sin (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✗

4461	$y^{'''} + y^{'} = \sin (x) + x \cos (x)$	[[_3rd_order, _missing_y]]	✓	✓	✓	✓

4462	$y^{'''} - 2 y^{''} + 4 y^{'} - 8 y = e^{2 x} \sin (2 x) + 2 x^{2}$	[[_3rd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4463	$y^{'''} - 4 y^{''} + 3 y^{'} = x^{2} + x e^{2 x}$	[[_3rd_order, _missing_y]]	✓	✓	✓	✓

4464	$y^{''''} + 2 y^{''} = 7 x - 3 \cos (x)$	[[_high_order, _missing_y]]	✓	✓	✓	✓

4465	$y^{''''} + 5 y^{''} + 4 y = \sin (x) \cos (2 x)$	[[_high_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4466	$y^{(5)} - 3 y^{'''} + y = 9 e^{2 x}$	[[_high_order, _with_linear_symmetries]]	✓	✓	✓	✓

4467	$y^{'''} - 3 y^{''} + 3 y^{'} - y = 48 x e^{x}$	[[_3rd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4468	$y^{'''} - 3 y^{'} = 9 x^{2}$	[[_3rd_order, _missing_y]]	✓	✓	✓	✓

4469	$y^{(5)} + 4 y^{'''} = 7 + x$	[[_high_order, _missing_y]]	✓	✓	✓	✓

4470	$y^{''} - y^{'} - 2 y = 36 x e^{2 x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4471	$y^{''''} + 16 y = 64 \cos (2 x)$	[[_high_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4472	$y^{''''} + 4 y^{''} - y = 44 \sin (3 x)$	[[_high_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4473	$y^{'''} + y^{''} + 5 y^{'} + 5 y = 5 \cos (2 x)$	[[_3rd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4474	$y^{''} + 3 y^{'} + 5 y = 5 e^{- x} \sin (2 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4475	$y^{''''} - y = 4 e^{- x}$	[[_high_order, _with_linear_symmetries]]	✓	✓	✓	✓

4476	$y^{''} + 4 y = 8 \sin {(x)}^{2}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4477	$y^{'''} - y^{''} + y^{'} - y = 4 \sin (x)$	[[_3rd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4478	$y^{''''} - y^{''} = 2 e^{x}$	[[_high_order, _missing_y]]	✓	✓	✓	✓

4479	$y^{''} - 4 y^{'} + 4 y = e^{x} (x + 1) + 2 e^{2 x} + 3 e^{3 x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4480	$y^{''} - 2 y^{'} + 5 y = 4 e^{x} \cos (2 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4481	$y^{''} + 4 y = 4 \sin (2 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4482	$y^{''} - y = 12 x^{2} e^{x} + 3 e^{2 x} + 10 \cos (3 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4483	$y^{''} + y = 2 \sin (x) - 3 \cos (2 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4484	$y^{''} - y^{'} = e^{x} (x^{2} + 10)$	[[_2nd_order, _missing_y]]	✓	✓	✓	✓

4485	$y^{''} - 4 y = 96 x^{2} e^{2 x} + 4 e^{- 2 x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4486	$y^{''} + 2 y^{'} + 2 y = 5 \cos (x) + 10 \sin (2 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4487	$y^{''} - 2 y^{'} + 2 y = 4 x - 2 + 2 e^{x} \sin (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4488	$y^{''} - 4 y^{'} + 4 y = 4 x e^{2 x} \sin (2 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4489	$y^{'''} - y^{''} + y^{'} - y = 15 \sin (2 x)$	[[_3rd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4490	$y^{'''} + 3 y^{''} - 4 y = 40 \sin (2 x)$	[[_3rd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4491	$y^{'''} - y^{''} + y^{'} - y = 2 e^{x} + 5 e^{2 x}$	[[_3rd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4492	$y^{'''} - 6 y^{''} + 11 y^{'} - 6 y = 10 e^{x} \sin (x)$	[[_3rd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4493	$y^{'''} - 2 y^{'} - 4 y = 50 e^{2 x} + 50 \sin (x)$	[[_3rd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4494	$y^{'''} - 3 y^{''} + 4 y = 12 e^{2 x} + 4 e^{3 x}$	[[_3rd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4495	$y^{''''} - 8 y^{''} + 16 y = 32 e^{2 x} + 16 x^{3}$	[[_high_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4496	$y^{''''} - 18 y^{''} + 81 y = 72 e^{3 x} + 729 x^{2}$	[[_high_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4497	$y^{''} - y = \frac{1}{x} - \frac{2}{x^{3}}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4498	$y^{''} - y = \frac{1}{\sinh (x)}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4499	$y^{''} - 2 y^{'} + y = \frac{e^{x}}{x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

4500	$y^{''} + 3 y^{'} + 2 y = \sin (e^{x})$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

2.2.45 Problems 4401 to 4500