Number of problems in this table is 43
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ODE |
CAS classification |
Program classification |
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\[ {}2 y y^{\prime \prime } = y^{3}+2 {y^{\prime }}^{2} \] |
[[_2nd_order, _missing_x]] |
second_order_ode_missing_x |
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\[ {}\left (1+{y^{\prime }}^{2}\right ) \left (\arctan \left (y^{\prime }\right )+a x \right )+y^{\prime } = 0 \] |
[_quadrature] |
quadrature |
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\[ {}y^{\prime }-x^{a} y^{3}+3 y^{2}-x^{-a} y-x^{-2 a}+a \,x^{-1-a} = 0 \] |
[_Abel] |
abelFirstKind |
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\[ {}\frac {1}{y}-\frac {x y^{\prime }}{y^{2}} = 0 \] |
[_separable] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
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\[ {}{y^{\prime \prime }}^{3} = 12 y^{\prime } \left (x y^{\prime \prime }-2 y^{\prime }\right ) \] |
[[_2nd_order, _missing_y]] |
second_order_ode_missing_y |
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\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+1+{y^{\prime }}^{2} = x \] |
[[_2nd_order, _missing_y]] |
second_order_ode_missing_y |
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\[ {}y^{\prime }-x^{a} y^{3}+3 y^{2}-x^{-a} y-x^{-2 a}+a \,x^{-1-a} = 0 \] |
[_Abel] |
abelFirstKind |
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\[ {}y^{\prime }-\sqrt {\left (b_{4} y^{4}+b_{3} y^{3}+b_{2} y^{2}+b_{1} y+b_{0} \right ) \left (a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0} \right )} = 0 \] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
exactWithIntegrationFactor |
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\[ {}y^{\prime }-\sqrt {\frac {a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}{b_{4} y^{4}+b_{3} y^{3}+b_{2} y^{2}+b_{1} y+b_{0}}} = 0 \] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
exactWithIntegrationFactor |
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\[ {}y^{\prime }-\sqrt {\frac {b_{4} y^{4}+b_{3} y^{3}+b_{2} y^{2}+b_{1} y+b_{0}}{a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}} = 0 \] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
exactWithIntegrationFactor |
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\[ {}\left (b_{2} y+a_{2} x +c_{2} \right ) {y^{\prime }}^{2}+\left (a_{1} x +b_{1} y+c_{1} \right ) y^{\prime }+a_{0} x +b_{0} y+c_{0} = 0 \] |
[_rational, _dAlembert] |
dAlembert |
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\[ {}\left (1+{y^{\prime }}^{2}\right ) \left (\arctan \left (y^{\prime }\right )+a x \right )+y^{\prime } = 0 \] |
[_quadrature] |
quadrature |
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\[ {}x^{3} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }+\ln \left (x \right )+2 x y^{\prime }-y-2 x^{3} = 0 \] |
[[_3rd_order, _linear, _nonhomogeneous]] |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
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\[ {}y^{\prime \prime }+a y y^{\prime }+b y^{3} = 0 \] |
[[_2nd_order, _missing_x]] |
second_order_ode_missing_x |
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\[ {}y^{\prime \prime }+a y {y^{\prime }}^{2}+b y = 0 \] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
second_order_ode_missing_x |
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\[ {}2 \left (y-a \right ) \left (y-b \right ) \left (y-c \right ) y^{\prime \prime }-\left (\left (y-a \right )^{2} \left (y-b \right ) \left (y-c \right )+\left (y-b \right ) \left (y-c \right )\right ) {y^{\prime }}^{2}+\left (y-a \right )^{2} \left (y-b \right )^{2} \left (y-c \right )^{2} \left (A_{0} +\frac {B_{0}}{\left (y-a \right )^{2}}+\frac {C_{1}}{\left (y-b \right )^{2}}+\frac {D_{0}}{\left (y-c \right )^{2}}\right ) = 0 \] |
[[_2nd_order, _missing_x]] |
second_order_ode_missing_x |
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\[ {}\left (4 y^{3}-a y-b \right ) \left (y^{\prime \prime }+f y^{\prime }\right )-\left (6 y^{2}-\frac {a}{2}\right ) {y^{\prime }}^{2} = 0 \] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
second_order_ode_missing_x |
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\[ {}a \,x^{2} \left (-1+x \right )^{2} \left (y^{\prime }+\lambda y^{2}\right )+x^{2} b +c x +s = 0 \] |
[_rational, _Riccati] |
riccati |
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\[ {}x^{2} y^{\prime } = \left (\alpha \,x^{2 n}+\beta \,x^{n}+\gamma \right ) y^{2}+\left (a \,x^{n}+b \right ) x y+c \,x^{2} \] |
[_rational, _Riccati] |
riccati |
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\[ {}\left (x^{2} a +b x +c \right ) y^{\prime } = y^{2}+\left (a x +\mu \right ) y-\lambda ^{2} x^{2}+\lambda \left (b -\mu \right ) x +c \lambda \] |
[_rational, _Riccati] |
riccati |
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\[ {}\left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime } = y^{2}+\left (a_{1} x +b_{1} \right ) y-\lambda \left (\lambda +a_{1} -a_{2} \right ) x^{2}+\lambda \left (b_{2} -b_{1} \right ) x +\lambda c_{2} \] |
[_rational, _Riccati] |
riccati |
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\[ {}\left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime } = y^{2}+\left (a_{1} x +b_{1} \right ) y+a_{0} x^{2}+b_{0} x +c_{0} \] |
[_rational, _Riccati] |
riccati |
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\[ {}\left (c_{2} x^{2}+b_{2} x +a_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (b_{1} x +a_{1} \right ) y+a_{0} = 0 \] |
[_rational, _Riccati] |
riccati |
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\[ {}x^{2} \left (x +a \right ) \left (y^{\prime }+\lambda y^{2}\right )+x \left (b x +c \right ) y+\alpha x +\beta = 0 \] |
[_rational, _Riccati] |
riccati |
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\[ {}x^{2} \left (x^{2}+a \right ) \left (y^{\prime }+\lambda y^{2}\right )+x \left (x^{2} b +c \right ) y+s = 0 \] |
[_rational, _Riccati] |
riccati |
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\[ {}x^{2} \left (a \,x^{n}-1\right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (p \,x^{n}+q \right ) x y+r \,x^{n}+s = 0 \] |
[_rational, _Riccati] |
riccati |
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\[ {}y^{\prime } = y^{2}+a \,{\mathrm e}^{8 \lambda x}+b \,{\mathrm e}^{6 \lambda x}+c \,{\mathrm e}^{4 \lambda x}-\lambda ^{2} \] |
[_Riccati] |
riccati |
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\[ {}2 y^{\prime } = \left (\lambda +a -a \sin \left (\lambda x \right )\right ) y^{2}+\lambda -a -a \sin \left (\lambda x \right ) \] |
[_Riccati] |
riccati |
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\[ {}y^{\prime } = y^{2}+\lambda \arccos \left (x \right )^{n} y-a^{2}+a \lambda \arccos \left (x \right )^{n} \] |
[_Riccati] |
riccati |
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\[ {}2 \left (a \,x^{3}+x^{2} b +c x +d \right ) y^{\prime \prime }+\left (3 x^{2} a +2 b x +c \right ) y^{\prime }+\lambda y = 0 \] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
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\[ {}\cos \left (\theta \right ) v^{\prime }+v = 3 \] |
[_separable] |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
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\[ {}y^{\prime \prime }+y = \delta \left (t \right ) \] |
[[_2nd_order, _linear, _nonhomogeneous]] |
second_order_laplace |
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\[ {}y^{\prime \prime }+4 y^{\prime }-12 y = \delta \left (t \right ) \] |
[[_2nd_order, _linear, _nonhomogeneous]] |
second_order_laplace |
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\[ {}y^{\prime \prime }-12 y^{\prime }+45 y = \delta \left (t \right ) \] |
[[_2nd_order, _linear, _nonhomogeneous]] |
second_order_laplace |
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\[ {}y^{\prime \prime \prime \prime }-16 y = \delta \left (t \right ) \] |
[[_high_order, _linear, _nonhomogeneous]] |
higher_order_laplace |
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\[ {}y^{\prime }+y \cot \left (x \right ) = y^{4} \] |
[_Bernoulli] |
bernoulli, first_order_ode_lie_symmetry_lookup |
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\[ {}x^{2} y^{\prime } \cos \left (y\right )+1 = 0 \] |
[_separable] |
exact, separable, first_order_ode_lie_symmetry_lookup |
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\[ {}x^{2} y^{\prime }+\cos \left (2 y\right ) = 1 \] |
[_separable] |
exact, separable, first_order_ode_lie_symmetry_lookup |
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\[ {}x^{2} y^{\prime }+\sin \left (2 y\right ) = 1 \] |
[_separable] |
exact, separable, first_order_ode_lie_symmetry_lookup |
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\[ {}2 x y^{\prime }+y = \left (x^{2}+1\right ) {\mathrm e}^{x} \] |
[_linear] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
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\[ {}x = \ln \left (y^{\prime }\right )+\sin \left (y^{\prime }\right ) \] |
[_quadrature] |
quadrature |
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\[ {}y = \arcsin \left (y^{\prime }\right )+\ln \left (1+{y^{\prime }}^{2}\right ) \] |
[_quadrature] |
quadrature |
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\[ {}y^{3} y^{\prime \prime } = -1 \] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
second_order_ode_missing_x |
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