Problems 12101 to 12200

Table 2.245: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	Maple	Mma	Sympy

12101	$y^{'} = (λ + a \sin {(λ x)}^{2}) y^{2} + λ - a + a \sin {(λ x)}^{2}$	[_Riccati]	✓	✓	✓	✗

12102	$y^{'} = - (k + 1) x^{k} y^{2} + a x^{k + 1} \sin {(x)}^{m} y - a \sin {(x)}^{m}$	[_Riccati]	✓	✓	✓	✗

12103	$y^{'} = a \sin {(λ x + μ)}^{k} {(y - b x^{n} - c)}^{2} + b n x^{n - 1}$	[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati]	✓	✓	✓	✗

12104	$y^{'} x = a \sin {(λ x)}^{m} y^{2} + k y + a b^{2} x^{2 k} \sin {(λ x)}^{m}$	[_Riccati]	✓	✓	✓	✗

12105	$(a \sin (λ x) + b) y^{'} = y^{2} + c \sin (μ x) y - d^{2} + c d \sin (μ x)$	[_Riccati]	✗	✓	✓	✗

12106	$(a \sin (λ x) + b) (y^{'} - y^{2}) - a λ^{2} \sin (λ x) = 0$	[_Riccati]	✓	✓	✓	✗

12107	$y^{'} = α y^{2} + β + γ \cos (λ x)$	[_Riccati]	✓	✓	✓	✗

12108	$y^{'} = y^{2} - a^{2} + a λ \cos (λ x) + a^{2} \cos {(λ x)}^{2}$	[_Riccati]	✓	✓	✓	✗

12109	$y^{'} = y^{2} + λ^{2} + c \cos {(λ x + a)}^{n} \cos {(λ x + b)}^{- n - 4}$	[_Riccati]	✗	✗	✗	✗

12110	$y^{'} = y^{2} + a \cos (β x) y + a b \cos (β x) - b^{2}$	[_Riccati]	✓	✓	✓	✗

12111	$y^{'} = y^{2} + a \cos {(b x)}^{m} y + a \cos {(b x)}^{m}$	[_Riccati]	✗	✗	✗	✗

12112	$y^{'} = λ \cos (λ x) y^{2} + λ \cos {(λ x)}^{3}$	[_Riccati]	✓	✓	✗	✗

12113	$2 y^{'} = (λ + a - a \cos (λ x)) y^{2} + λ - a - a \cos (λ x)$	[_Riccati]	✓	✓	✓	✗

12114	$y^{'} = (λ + a \cos {(λ x)}^{2}) y^{2} + λ - a + a \cos {(λ x)}^{2}$	[_Riccati]	✓	✓	✓	✗

12115	$y^{'} = - (k + 1) x^{k} y^{2} + a x^{k + 1} \cos {(x)}^{m} y - a \cos {(x)}^{m}$	[_Riccati]	✓	✓	✓	✗

12116	$y^{'} = a \cos {(λ x + μ)}^{k} {(y - b x^{n} - c)}^{2} + b n x^{n - 1}$	[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati]	✓	✓	✓	✗

12117	$y^{'} x = a \cos {(λ x)}^{m} y^{2} + k y + a b^{2} x^{2 k} \cos {(λ x)}^{m}$	[_Riccati]	✓	✓	✓	✗

12118	$(a \cos (λ x) + b) y^{'} = y^{2} + c \cos (μ x) y - d^{2} + c d \cos (μ x)$	[_Riccati]	✗	✓	✓	✗

12119	$(a \cos (λ x) + b) (y^{'} - y^{2}) - a λ^{2} \cos (λ x) = 0$	[_Riccati]	✓	✓	✓	✗

12120	$y^{'} = y^{2} + a λ + a (λ - a) \tan {(λ x)}^{2}$	[_Riccati]	✗	✓	✓	✗

12121	$y^{'} = y^{2} + λ^{2} + 3 a λ + a (λ - a) \tan {(λ x)}^{2}$	[_Riccati]	✗	✓	✓	✗

12122	$y^{'} = a y^{2} + b \tan (x) y + c$	[_Riccati]	✓	✓	✓	✗

12123	$y^{'} = a y^{2} + 2 a b \tan (x) y + b (a b - 1) \tan {(x)}^{2}$	[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati]	✓	✓	✓	✓

12124	$y^{'} = y^{2} + a \tan (β x) y + a b \tan (β x) - b^{2}$	[_Riccati]	✓	✓	✓	✗

12125	$y^{'} = y^{2} + a x \tan {(b x)}^{m} y + a \tan {(b x)}^{m}$	[_Riccati]	✓	✓	✓	✗

12126	$y^{'} = - (k + 1) x^{k} y^{2} + a x^{k + 1} \tan {(x)}^{m} y - a \tan {(x)}^{m}$	[_Riccati]	✓	✓	✓	✗

12127	$y^{'} = a \tan {(λ x)}^{n} y^{2} - a b^{2} \tan {(λ x)}^{n + 2} + b λ \tan {(λ x)}^{2} + b λ$	[_Riccati]	✓	✗	✗	✗

12128	$y^{'} = a \tan {(λ x + μ)}^{k} {(y - b x^{n} - c)}^{2} + b n x^{n - 1}$	[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati]	✓	✓	✓	✗

12129	$y^{'} x = a \tan {(λ x)}^{m} y^{2} + k y + a b^{2} x^{2 k} \tan {(λ x)}^{m}$	[_Riccati]	✓	✓	✓	✗

12130	$(a \tan (λ x) + b) y^{'} = y^{2} + k \tan (μ x) y - d^{2} + k d \tan (μ x)$	[_Riccati]	✗	✓	✓	✗

12131	$y^{'} = y^{2} + a λ + a (λ - a) \cot {(λ x)}^{2}$	[_Riccati]	✗	✓	✗	✗

12132	$y^{'} = y^{2} + λ^{2} + 3 a λ + a (λ - a) \cot {(λ x)}^{2}$	[_Riccati]	✗	✓	✓	✗

12133	$y^{'} = y^{2} - 2 a b \cot (a x) y + b^{2} - a^{2}$	[_Riccati]	✗	✓	✗	✗

12134	$y^{'} = y^{2} + a \cot (β x) y + a b \cot (β x) - b^{2}$	[_Riccati]	✓	✓	✓	✗

12135	$y^{'} = y^{2} + a x \cot {(b x)}^{m} y + a \cot {(b x)}^{m}$	[_Riccati]	✓	✓	✓	✗

12136	$y^{'} = - (k + 1) x^{k} y^{2} + a x^{k + 1} \cot {(x)}^{m} y - a \cot {(x)}^{m}$	[_Riccati]	✓	✓	✗	✗

12137	$y^{'} = a \cot {(λ x + μ)}^{k} {(y - b x^{n} - c)}^{2} + b n x^{n - 1}$	[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati]	✗	✓	✓	✗

12138	$y^{'} x = a \cot {(λ x)}^{m} y^{2} + k y + a b^{2} x^{2 k} \cot {(λ x)}^{m}$	[_Riccati]	✓	✓	✓	✗

12139	$(a \cot (λ x) + b) y^{'} = y^{2} + c \cot (μ x) y - d^{2} + c d \cot (μ x)$	[_Riccati]	✗	✓	✓	✗

12140	$y^{'} = y^{2} + λ^{2} + c \sin {(λ x)}^{n} \cos {(λ x)}^{- n - 4}$	[_Riccati]	✗	✗	✗	✗

12141	$y^{'} = a \sin (λ x) y^{2} + b \sin (λ x) \cos {(λ x)}^{n}$	[_Riccati]	✓	✓	✓	✗

12142	$y^{'} = λ \sin (λ x) y^{2} + a \cos {(λ x)}^{n} y - a \cos {(λ x)}^{n - 1}$	[_Riccati]	✗	✗	✓	✗

12143	$y^{'} = a \cos (λ x) y^{2} + b \cos (λ x) \sin {(λ x)}^{n}$	[_Riccati]	✓	✓	✓	✗

12144	$y^{'} = λ \sin (λ x) y^{2} + a x^{n} \cos (λ x) y - a x^{n}$	[_Riccati]	✓	✓	✗	✗

12145	$\sin {(2 x)}^{n + 1} y^{'} = a y^{2} \sin {(x)}^{2 n} + b \cos {(x)}^{2 n}$	[_Riccati]	✗	✓	✓	✗

12146	$y^{'} = y^{2} - y \tan (x) + a (1 - a) \cot {(x)}^{2}$	[_Riccati]	✓	✓	✓	✗

12147	$y^{'} = y^{2} - m y \tan (x) + b^{2} \cos {(x)}^{2 m}$	[_Riccati]	✓	✓	✓	✗

12148	$y^{'} = y^{2} + m y \cot (x) + b^{2} \sin {(x)}^{2 m}$	[_Riccati]	✓	✓	✓	✗

12149	$y^{'} = y^{2} - 2 λ^{2} \tan {(x)}^{2} - 2 λ^{2} \cot {(λ x)}^{2}$	[_Riccati]	✗	✗	✗	✗

12150	$y^{'} = y^{2} + a λ + b λ + 2 a b + a (λ - a) \tan {(λ x)}^{2} + b (λ - b) \cot {(λ x)}^{2}$	[_Riccati]	✓	✓	✗	✗

12151	$y^{'} = y^{2} - \frac{λ^{2}}{2} - \frac{3 λ^{2} \tan {(λ x)}^{2}}{4} + a \cos {(λ x)}^{2} \sin {(λ x)}^{n}$	[_Riccati]	✓	✓	✗	✗

12152	$y^{'} = λ \sin (λ x) y^{2} + a \sin (λ x) y - a \tan (λ x)$	[_Riccati]	✓	✓	✗	✗

12153	$y^{'} = y^{2} + λ \arcsin {(x)}^{n} y - a^{2} + a λ \arcsin {(x)}^{n}$	[_Riccati]	✓	✓	✓	✓

12154	$y^{'} = y^{2} + λ x \arcsin {(x)}^{n} y + λ \arcsin {(x)}^{n}$	[_Riccati]	✓	✓	✓	✗

12155	$y^{'} = - (k + 1) x^{k} y^{2} + λ \arcsin {(x)}^{n} (x^{k + 1} y - 1)$	[_Riccati]	✓	✓	✗	✗

12156	$y^{'} = λ \arcsin {(x)}^{n} y^{2} + a y + a b - b^{2} λ \arcsin {(x)}^{n}$	[_Riccati]	✗	✓	✓	✓

12157	$y^{'} = λ \arcsin {(x)}^{n} y^{2} - b λ x^{m} \arcsin {(x)}^{n} y + b m x^{m - 1}$	[_Riccati]	✗	✗	✗	✗

12158	$y^{'} = λ \arcsin {(x)}^{n} y^{2} + β m x^{m - 1} - λ β^{2} x^{2 m} \arcsin {(x)}^{n}$	[_Riccati]	✗	✗	✗	✗

12159	$y^{'} = λ \arcsin {(x)}^{n} {(y - a x^{m} - b)}^{2} + a m x^{m - 1}$	[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati]	✗	✓	✓	✗

12160	$y^{'} x = λ \arcsin {(x)}^{n} y^{2} + k y + λ b^{2} x^{2 k} \arcsin {(x)}^{n}$	[_Riccati]	✓	✓	✓	✗

12161	$y^{'} x = (a x^{2 m} y^{2} + b x^{n} y + c) \arcsin {(x)}^{m} - n y$	[_Riccati]	✗	✗	✗	✗

12162	$y^{'} = y^{2} + λ \arccos {(x)}^{n} y - a^{2} + a λ \arccos {(x)}^{n}$	[_Riccati]	✗	✓	✓	✓

12163	$y^{'} = y^{2} + λ x \arccos {(x)}^{n} y + λ \arccos {(x)}^{n}$	[_Riccati]	✓	✓	✓	✗

12164	$y^{'} = - (k + 1) x^{k} y^{2} + λ \arccos {(x)}^{n} (x^{k + 1} y - 1)$	[_Riccati]	✓	✓	✗	✗

12165	$y^{'} = λ \arccos {(x)}^{n} y^{2} + a y + a b - b^{2} λ \arccos {(x)}^{n}$	[_Riccati]	✗	✓	✓	✓

12166	$y^{'} = λ \arccos {(x)}^{n} y^{2} - b λ x^{m} \arccos {(x)}^{n} y + b m x^{m - 1}$	[_Riccati]	✗	✗	✗	✗

12167	$y^{'} = λ \arccos {(x)}^{n} y^{2} + β m x^{m - 1} - λ β^{2} x^{2 m} \arccos {(x)}^{n}$	[_Riccati]	✗	✗	✗	✗

12168	$y^{'} = λ \arccos {(x)}^{n} {(y - a x^{m} - b)}^{2} + a m x^{m - 1}$	[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati]	✗	✓	✓	✗

12169	$y^{'} x = λ \arccos {(x)}^{n} y^{2} + k y + λ b^{2} x^{2 k} \arccos {(x)}^{n}$	[_Riccati]	✓	✓	✓	✗

12170	$y^{'} x = (a x^{2 m} y^{2} + b x^{n} y + c) \arccos {(x)}^{m} - n y$	[_Riccati]	✗	✗	✗	✗

12171	$y^{'} = y^{2} + λ \arctan {(x)}^{n} y - a^{2} + a λ \arctan {(x)}^{n}$	[_Riccati]	✓	✓	✓	✓

12172	$y^{'} = y^{2} + λ x \arctan {(x)}^{n} y + λ \arctan {(x)}^{n}$	[_Riccati]	✓	✓	✓	✗

12173	$y^{'} = - (k + 1) x^{k} y^{2} + λ \arctan {(x)}^{n} (x^{k + 1} y - 1)$	[_Riccati]	✓	✓	✗	✗

12174	$y^{'} = λ \arctan {(x)}^{n} y^{2} + a y + a b - b^{2} λ \arctan {(x)}^{n}$	[_Riccati]	✗	✓	✓	✗

12175	$y^{'} = λ \arctan {(x)}^{n} y^{2} - b λ x^{m} \arctan {(x)}^{n} y + b m x^{m - 1}$	[_Riccati]	✗	✗	✗	✗

12176	$y^{'} = λ \arctan {(x)}^{n} {(y - a x^{m} - b)}^{2} + a m x^{m - 1}$	[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati]	✗	✓	✓	✗

12177	$y^{'} x = λ \arctan {(x)}^{n} y^{2} + k y + λ b^{2} x^{2 k} \arctan {(x)}^{n}$	[_Riccati]	✓	✓	✓	✗

12178	$y^{'} = y^{2} + λ arccot {(x)}^{n} y - a^{2} + a λ arccot {(x)}^{n}$	[_Riccati]	✓	✓	✓	✗

12179	$y^{'} = y^{2} + λ x arccot {(x)}^{n} y + λ arccot {(x)}^{n}$	[_Riccati]	✓	✓	✓	✗

12180	$y^{'} = - (k + 1) x^{k} y^{2} + λ arccot {(x)}^{n} (x^{k + 1} y - 1)$	[_Riccati]	✓	✓	✗	✗

12181	$y^{'} = λ arccot {(x)}^{n} y^{2} + a y + a b - b^{2} λ arccot {(x)}^{n}$	[_Riccati]	✗	✓	✓	✗

12182	$y^{'} = λ arccot {(x)}^{n} y^{2} - b λ x^{m} arccot {(x)}^{n} y + b m x^{m - 1}$	[_Riccati]	✗	✗	✗	✗

12183	$y^{'} = λ arccot {(x)}^{n} {(y - a x^{m} - b)}^{2} + a m x^{m - 1}$	[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati]	✗	✓	✓	✗

12184	$y^{'} x = λ arccot {(x)}^{n} y^{2} + k y + λ b^{2} x^{2 k} arccot {(x)}^{n}$	[_Riccati]	✓	✓	✓	✗

12185	$y^{'} = y^{2} + f (x) y - a^{2} - a f (x)$	[_Riccati]	✓	✓	✓	✗

12186	$y^{'} = y^{2} f (x) - a y - a b - b^{2} f (x)$	[_Riccati]	✓	✓	✓	✓

12187	$y^{'} = f (x) + x f (x) y + y^{2}$	[_Riccati]	✓	✓	✓	✗

12188	$y^{'} = y^{2} f (x) - a x^{n} f (x) y + a n x^{n - 1}$	[_Riccati]	✓	✗	✗	✗

12189	$y^{'} = y^{2} f (x) + a n x^{n - 1} - a^{2} x^{2 n} f (x)$	[_Riccati]	✗	✗	✗	✗

12190	$y^{'} = - (n + 1) x^{n} y^{2} + x^{n + 1} f (x) y - f (x)$	[_Riccati]	✓	✓	✗	✗

12191	$y^{'} x = y^{2} f (x) + n y + a x^{2 n} f (x)$	[_Riccati]	✓	✓	✓	✗

12192	$y^{'} x = x^{2 n} f (x) y^{2} + (a x^{n} f (x) - n) y + b f (x)$	[_Riccati]	✗	✓	✓	✗

12193	$y^{'} = y^{2} f (x) + g (x) y - a^{2} f (x) - a g (x)$	[_Riccati]	✓	✓	✓	✗

12194	$y^{'} = y^{2} f (x) + g (x) y + a n x^{n - 1} - a x^{n} g (x) - a^{2} x^{2 n} f (x)$	[_Riccati]	✗	✗	✗	✗

12195	$y^{'} = y^{2} f (x) - a x^{n} g (x) y + a n x^{n - 1} + a^{2} x^{2 n} (g (x) - f (x))$	[_Riccati]	✗	✗	✗	✗

12196	$y^{'} = a e^{λ x} y^{2} + a e^{λ x} f (x) y + λ f (x)$	[_Riccati]	✓	✓	✓	✗

12197	$y^{'} = y^{2} f (x) - a e^{λ x} f (x) y + a λ e^{λ x}$	[_Riccati]	✓	✗	✓	✗

12198	$y^{'} = y^{2} f (x) + a λ e^{λ x} - a^{2} e^{2 λ x} f (x)$	[_Riccati]	✗	✗	✗	✗

12199	$y^{'} = y^{2} f (x) + λ y + a^{2} e^{2 λ x} f (x)$	[_Riccati]	✓	✓	✓	✗

12200	$y^{'} = y^{2} f (x) - f (x) (a e^{λ x} + b) y + a λ e^{λ x}$	[_Riccati]	✓	✗	✗	✗

2.2.122 Problems 12101 to 12200