Problems 19401 to 19500

Table 2.391: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	Maple	Mma	Sympy

19401	$y^{'} x - y = (x - 1) (y^{''} - x + 1)$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✗

19402	$x^{2} y y^{''} + {(y^{'} x - y)}^{2} = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]	✗	✓	✓	✗

19403	$x^{2} y^{''} - 2 x (x + 1) y^{'} + 2 (x + 1) y = x^{3}$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✗

19404	$(x^{2} + a) y^{''} - 2 y^{'} x + 2 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✗

19405	$y^{''} - \frac{2 y^{'}}{x} + (n^{2} + \frac{2}{x^{2}}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

19406	$y^{''} + 2 y^{'} x + (x^{2} + 1) y = x^{3} + 3 x$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✗

19407	$(a^{2} - x^{2}) y^{''} - \frac{a^{2} y^{'}}{x} + \frac{x^{2} y}{a} = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	✓	✓	✗

19408	$x^{4} y^{''} + 2 x^{3} y^{'} + n^{2} y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	✓	✓	✓

19409	$(- x^{2} + 1) y^{''} - 2 y^{'} x + \frac{a^{2} y}{- x^{2} + 1} = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	✓	✓	✗

19410	$(2 x - 1) y^{''} - 2 y^{'} + (3 - 2 x) y = 2 e^{x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✗

19411	$x^{2} y^{''} + y^{'} x - y = 8 x^{3}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓	✓	✓	✓

19412	$y^{''} + 2 y^{'} x + (x^{2} + 5) y = x e^{- \frac{x^{2}}{2}}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✗

19413	$x {(- x^{2} + 1)}^{2} y^{''} + (- x^{2} + 1) (3 x^{2} + 1) y^{'} + 4 x (x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✗

19414	$y^{''} + (1 - \frac{2}{x^{2}}) y = x^{2}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✗

19415	$(x^{3} - 2 x^{2}) y^{''} + 2 x^{2} y^{'} - 12 (x - 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✗

19416	$x y^{''} - 2 (x + 1) y^{'} + (x + 2) y = (x - 2) e^{2 x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✗

19417	$x^{2} y^{''} + y^{'} x - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	✓	✓	✓

19418	$x^{2} y^{''} + y^{'} x - 9 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	✓	✓	✓

19419	$(- x^{2} + 1) y^{''} - y^{'} x - a^{2} y = 0$	[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	✓	✓	✗

19420	$x y^{''} (x \cos (x) - 2 \sin (x)) + (x^{2} + 2) y^{'} \sin (x) - 2 y (x \sin (x) + \cos (x)) = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✗	✗

19421	$x y^{''} - y^{'} + 4 x^{3} y = x^{5}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✗

19422	$(x^{2} - 1) y^{''} - (4 x^{2} - 3 x - 5) y^{'} + (4 x^{2} - 6 x - 5) y = e^{2 x}$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✗

19423	$(x^{2} - 1) y^{''} + y^{'} x = m^{2} y$	[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	✓	✓	✗

19424	$y^{''} + (1 - \frac{1}{x}) y^{'} + 4 x^{2} y e^{- 2 x} = 4 (x^{3} + x^{2}) e^{- 3 x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✗

19425	$x y^{''} + (x^{2} + 1) y^{'} + 2 x y = 2 x$ i.c.	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✗	✓	✓	✗

19426	$(x + 2) y^{''} - (5 + 2 x) y^{'} + 2 y = e^{x} (x + 1)$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✗

19427	$(- x^{2} + 1) y^{''} + y^{'} x - y = x {(- x^{2} + 1)}^{3 / 2}$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✗

19428	$x^{2} y^{''} - (x^{2} + 2 x) y^{'} + (x + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✗

19429	$[\begin{matrix} x^{'} t + y = 0 \\ y^{'} t + x = 0 \end{matrix}]$	system_of_ODEs	✓	✓	✓	✓

19430	$y - y^{'} x = 0$	[_separable]	✓	✓	✓	✓

19431	$\cot (y) - y^{'} \tan (x) = 0$	[_separable]	✓	✓	✓	✓

19432	$x^{3} + x y^{2} + a^{2} y + (y^{3} + x^{2} y - a^{2} x) y^{'} = 0$	[[_1st_order, _with_linear_symmetries], _rational]	✓	✓	✓	✗

19433	$(x + 2 y^{3}) y^{'} = y$	[[_homogeneous, ‘class G‘], _rational]	✓	✓	✓	✗

19434	$\sec {(x)}^{2} \tan (y) + \sec {(y)}^{2} \tan (x) y^{'} = 0$	[_separable]	✓	✓	✓	✓

19435	$1 + y^{2} - x y^{'} y = 0$	[_separable]	✓	✓	✓	✓

19436	$y^{2} + (x^{2} + x y) y^{'} = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	✓	✓	✓

19437	$y^{'} = \frac{6 x - 2 y - 7}{2 x + 3 y - 6}$	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	✓	✓	✓

19438	$2 x + y + 1 + (4 x + 2 y - 1) y^{'} = 0$	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	✓	✓	✓

19439	$y^{'} \cos (x) + y \sin (x) = 1$	[_linear]	✓	✓	✓	✓

19440	$y^{'} + 2 x y = e^{- x^{2}}$	[_linear]	✓	✓	✓	✓

19441	$(x + 2 y^{3}) y^{'} = y$	[[_homogeneous, ‘class G‘], _rational]	✓	✓	✓	✗

19442	$y^{'} + p (x) y = q (x) y^{n}$	[_Bernoulli]	✓	✓	✓	✓

19443	$y^{'} + \sin (2 y) x = x^{3} \cos {(y)}^{2}$	[‘y=_G(x,y’)‘]	✗	✓	✓	✗

19444	$a^{2} - 2 x y - y^{2} - {(x + y)}^{2} y^{'} = 0$	[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]]	✓	✓	✓	✓

19445	$x^{2} y - (x^{3} + y^{3}) y^{'} = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	✓	✓	✓

19446	$(x y \sin (x y) + \cos (x y)) y + (x y \sin (x y) - \cos (x y)) y^{'} = 0$	[‘y=_G(x,y’)‘]	✗	✗	✗	✗

19447	$y + \frac{y^{3}}{3} + \frac{x^{2}}{2} + \frac{(x y^{2} + x) y^{'}}{4} = 0$	[_rational]	✓	✓	✓	✓

19448	$3 x^{2} y^{4} + 2 x y + (2 x^{3} y^{2} - x^{2}) y^{'} = 0$	[_rational]	✗	✗	✗	✗

19449	$y^{3} - 2 x^{2} y + (2 x y^{2} - x^{3}) y^{'} = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	✓	✓	✓

19450	$2 y^{''} + 9 y^{'} - 18 y = 0$	[[_2nd_order, _missing_x]]	✓	✓	✓	✓

19451	$y^{''''} - y^{'''} - 9 y^{''} - 11 y^{'} - 4 y = 0$	[[_high_order, _missing_x]]	✓	✓	✓	✓

19452	$y^{'''} - 8 y = 0$	[[_3rd_order, _missing_x]]	✓	✓	✓	✓

19453	$y^{'''} - 2 y^{''} + y^{'} = e^{- x}$	[[_3rd_order, _missing_y]]	✓	✓	✓	✓

19454	$y^{''} + n^{2} y = \sec (n x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

19455	$y^{'''} + y = {(e^{x} + 1)}^{2}$	[[_3rd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

19456	$y^{''} - 4 y^{'} + y = a \cos (2 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

19457	$y^{'''} + a^{2} y^{'} = \sin (a x)$	[[_3rd_order, _missing_y]]	✓	✓	✓	✓

19458	$y^{'''} - 3 y^{''} + 4 y^{'} - 2 y = e^{x} + \cos (x)$	[[_3rd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

19459	$y^{''} - 2 y^{'} + y = x \sin (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

19460	$y^{'''} + 3 y^{''} + 2 y^{'} = x^{2}$	[[_3rd_order, _missing_y]]	✓	✓	✓	✓

19461	$y^{''} - 2 y^{'} + y = x^{2} e^{3 x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

19462	$y^{''} + 4 y^{'} + 4 y = 2 \sinh (2 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✗

19463	$y^{''} + a^{2} y = \cos (a x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

19464	$y^{''} - 2 y^{'} + y = x \sin (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

19465	${y^{'}}^{3} - (x^{2} + x y + y^{2}) {y^{'}}^{2} + (x y^{3} + x^{2} y^{2} + x^{3} y) y^{'} - x^{3} y^{3} = 0$	[_quadrature]	✓	✓	✓	✓

19466	$x^{2} ({y^{'}}^{2} - y^{2}) + y^{2} = x^{4} + 2 x y^{'} y$	[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]	✗	✓	✓	✗

19467	$(a^{2} - x^{2}) {y^{'}}^{3} + b x (a^{2} - x^{2}) {y^{'}}^{2} - y^{'} - b x = 0$	[_quadrature]	✓	✓	✓	✓

19468	$(x + 2 y) {y^{'}}^{3} + 3 (x + y) {y^{'}}^{2} + (2 x + y) y^{'} = 0$	[_quadrature]	✓	✓	✓	✓

19469	$y - \frac{1}{\sqrt{{y^{'}}^{2} + 1}} = b$	[_quadrature]	✓	✓	✓	✓

19470	$y = \frac{x}{y^{'}} - a y^{'}$	[_dAlembert]	✓	✓	✓	✗

19471	${y^{'}}^{3} + m {y^{'}}^{2} = a (y + m x)$	[[_homogeneous, ‘class C‘], _dAlembert]	✓	✓	✗	✗

19472	$x {y^{'}}^{3} = a + b y^{'}$	[_quadrature]	✓	✓	✓	✓

19473	$y^{'} = \tan (x - \frac{y^{'}}{{y^{'}}^{2} + 1})$	[_quadrature]	✓	✓	✗	✗

19474	$a y {y^{'}}^{2} + (2 x - b) y^{'} - y = 0$	[[_homogeneous, ‘class C‘], _rational, _dAlembert]	✓	✓	✓	✗

19475	$y = x (y^{'} + 1) + {y^{'}}^{2}$	[[_1st_order, _with_linear_symmetries], _dAlembert]	✓	✓	✓	✓

19476	$e^{3 x} (y^{'} - 1) + e^{2 y} {y^{'}}^{3} = 0$	[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘], _dAlembert]	✓	✓	✗	✗

19477	$y = 2 y^{'} x + y^{2} {y^{'}}^{3}$	[[_1st_order, _with_linear_symmetries]]	✓	✓	✓	✗

19478	$y = - y^{'} x + x^{4} {y^{'}}^{2}$	[[_homogeneous, ‘class G‘], _rational]	✓	✓	✓	✓

19479	$y - 2 y^{'} x + a y {y^{'}}^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	✓	✓	✗

19480	$x^{2} (y - y^{'} x) = y {y^{'}}^{2}$	[[_1st_order, _with_linear_symmetries]]	✓	✓	✓	✗

19481	$x y (y - y^{'} x) = y y^{'} + x$	[_separable]	✓	✓	✓	✓

19482	$x y^{2} ({y^{'}}^{2} + 2) = 2 y^{3} y^{'} + x^{3}$	[_separable]	✓	✓	✓	✓

19483	$3 y {y^{'}}^{2} - 2 x y^{'} y + 4 y^{2} - x^{2} = 0$	[_rational]	✗	✗	✗	✗

19484	${(y y^{'} + n x)}^{2} = (y^{2} + n x^{2}) ({y^{'}}^{2} + 1)$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	✓	✓	✓

19485	$(1 - y^{2} + \frac{y^{4}}{x^{2}}) {y^{'}}^{2} - \frac{2 y y^{'}}{x} + \frac{y^{2}}{x^{2}} = 0$	[‘y=_G(x,y’)‘]	✗	✓	✗	✗

19486	$(x^{2} + y^{2}) {(y^{'} + 1)}^{2} - 2 (x + y) (y^{'} + 1) (y y^{'} + x) + {(y y^{'} + x)}^{2} = 0$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	✓	✓	✗

19487	$(- x^{2} + 1) {y^{'}}^{2} = 1 - y^{2}$	[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]	✓	✓	✓	✓

19488	$y^{2} ({y^{'}}^{2} + 1) = r^{2}$	[_quadrature]	✓	✓	✓	✓

19489	$\sin (y^{'} x) \cos (y) = \cos (y^{'} x) \sin (y) + y^{'}$	[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]	✓	✓	✓	✗

19490	$4 x {y^{'}}^{2} = {(3 x - a)}^{2}$	[_quadrature]	✓	✓	✓	✓

19491	$4 {y^{'}}^{2} x (x - a) (x - b) = {(3 x^{2} - 2 (a + b) x + a b)}^{2}$	[_quadrature]	✓	✓	✓	✓

19492	${y^{'}}^{3} - 4 x y^{'} y + 8 y^{2} = 0$	[[_1st_order, _with_linear_symmetries]]	✓	✓	✗	✗

19493	${y^{'}}^{2} + 2 y^{'} x - y = 0$	[[_1st_order, _with_linear_symmetries], _dAlembert]	✓	✓	✓	✗

19494	$x^{3} {y^{'}}^{2} + x^{2} y y^{'} + a^{3} = 0$	[[_homogeneous, ‘class G‘]]	✓	✓	✓	✗

19495	$x^{2} {y^{'}}^{3} + (2 x + y) y y^{'} + y^{2} = 0$	[[_1st_order, _with_linear_symmetries], _dAlembert]	✓	✓	✗	✗

19496	$x {y^{'}}^{2} - 2 y y^{'} + x + 2 y = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	✓	✓	✓

19497	${y^{'}}^{2} y^{2} \cos {(a)}^{2} - 2 y^{'} x y \sin {(a)}^{2} + y^{2} - x^{2} \sin {(a)}^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	✓	✓	✓

19498	$(2 x^{2} + 1) {y^{'}}^{2} + (y^{2} + 2 x y + x^{2} + 2) y^{'} + 2 y^{2} + 1 = 0$	[_rational]	✗	✗	✓	✗

19499	$x^{4} y^{'''} + 2 x^{3} y^{''} - x^{2} y^{'} + x y = 1$	[[_3rd_order, _with_linear_symmetries]]	✓	✓	✓	✓

19500	$x^{2} y^{''} - 2 y = x^{2} + \frac{1}{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓	✓	✓	✓

2.2.195 Problems 19401 to 19500