|
# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}y^{\prime \prime }-2 a y^{\prime }+\left (a^{2}+b^{2}\right ) y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.247 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-6 y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.178 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = 0 \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.242 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = 0 \] |
kovacic, second_order_euler_ode, exact linear second order ode, second_order_integrable_as_is, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.474 |
|
|
\[ {}x^{2} y^{\prime \prime }+5 x y^{\prime }+4 y = 0 \] |
kovacic, second_order_euler_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2 |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.839 |
|
|
\[ {}y^{\prime } = \frac {{\mathrm e}^{x}-\sin \left (y\right )}{x \cos \left (y\right )} \] |
exact |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
1.505 |
|
|
\[ {}y^{\prime } = \frac {1-y^{2}}{2 x y+2} \] |
exact |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.023 |
|
|
\[ {}y^{\prime } = \frac {\left (1-y \,{\mathrm e}^{x y}\right ) {\mathrm e}^{-x y}}{x} \] |
exact |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
1.167 |
|
|
\[ {}y^{\prime } = \frac {x^{2} \left (1-y^{2}\right )+y \,{\mathrm e}^{\frac {y}{x}}}{x \left ({\mathrm e}^{\frac {y}{x}}+2 x^{2} y\right )} \] |
exactWithIntegrationFactor |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
37.297 |
|
|
\[ {}y^{\prime } = \frac {\cos \left (x \right )-2 x y^{2}}{2 x^{2} y} \] |
exact, bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
33.947 |
|
|
\[ {}y^{\prime } = \sin \left (x \right ) \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.143 |
|
|
\[ {}y^{\prime } = \frac {1}{x^{\frac {2}{3}}} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.094 |
|
|
\[ {}y^{\prime \prime } = x \,{\mathrm e}^{x} \] |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_ode_quadrature, second_order_linear_constant_coeff |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.664 |
|
|
\[ {}y^{\prime \prime } = x^{n} \] |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_ode_quadrature, second_order_linear_constant_coeff |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.662 |
|
|
\[ {}y^{\prime } = \ln \left (x \right ) x^{2} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.26 |
|
|
\[ {}y^{\prime \prime } = \cos \left (x \right ) \] |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_ode_quadrature, second_order_linear_constant_coeff |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
1.073 |
|
|
\[ {}y^{\prime \prime \prime } = 6 x \] |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _quadrature]] |
✓ |
✓ |
0.235 |
|
|
\[ {}y^{\prime \prime } = x \,{\mathrm e}^{x} \] |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_ode_quadrature, second_order_linear_constant_coeff |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.981 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-6 y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.194 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }-8 y = 0 \] |
kovacic, second_order_euler_ode, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2 |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.826 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = \ln \left (x \right ) x^{2} \] |
kovacic, second_order_euler_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.636 |
|
|
\[ {}y^{\prime } = 2 x y \] |
separable |
[_separable] |
✓ |
✓ |
0.128 |
|
|
\[ {}y^{\prime } = \frac {y^{2}}{x^{2}+1} \] |
separable |
[_separable] |
✓ |
✓ |
0.11 |
|
|
\[ {}{\mathrm e}^{x +y} y^{\prime }-1 = 0 \] |
separable |
[_separable] |
✓ |
✓ |
0.105 |
|
|
\[ {}y^{\prime } = \frac {y}{x \ln \left (x \right )} \] |
separable |
[_separable] |
✓ |
✓ |
0.1 |
|
|
\[ {}y-\left (-1+x \right ) y^{\prime } = 0 \] |
separable |
[_separable] |
✓ |
✓ |
0.119 |
|
|
\[ {}y^{\prime } = \frac {2 x \left (y-1\right )}{x^{2}+3} \] |
separable |
[_separable] |
✓ |
✓ |
0.259 |
|
|
\[ {}y-x y^{\prime } = 3-2 x^{2} y^{\prime } \] |
separable |
[_separable] |
✓ |
✓ |
0.19 |
|
|
\[ {}y^{\prime } = \frac {\cos \left (x -y\right )}{\sin \left (x \right ) \sin \left (y\right )}-1 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.547 |
|
|
\[ {}y^{\prime } = \frac {x \left (y^{2}-1\right )}{2 \left (-2+x \right ) \left (-1+x \right )} \] |
separable |
[_separable] |
✓ |
✓ |
0.187 |
|
|
\[ {}y^{\prime } = \frac {x^{2} y-32}{-x^{2}+16}+2 \] |
separable |
[_separable] |
✓ |
✓ |
0.254 |
|
|
\[ {}\left (x -a \right ) \left (-b +x \right ) y^{\prime }-y+c = 0 \] |
separable |
[_separable] |
✓ |
✓ |
0.269 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime }+y^{2} = -1 \] |
separable |
[_separable] |
✓ |
✓ |
0.321 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime }+x y = x a \] |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.208 |
|
|
\[ {}y^{\prime } = 1-\frac {\sin \left (x +y\right )}{\sin \left (y\right ) \cos \left (x \right )} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.422 |
|
|
\[ {}y^{\prime } = y^{3} \sin \left (x \right ) \] |
separable |
[_separable] |
✓ |
✓ |
0.288 |
|
|
\[ {}y^{\prime } = \frac {2 \sqrt {y-1}}{3} \] |
separable |
[_quadrature] |
✓ |
✓ |
0.395 |
|
|
\[ {}m v^{\prime } = m g -k v^{2} \] |
separable |
[_quadrature] |
✓ |
✓ |
1.526 |
|
|
\[ {}y^{\prime }+y = 4 \,{\mathrm e}^{x} \] |
linear |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.184 |
|
|
\[ {}y^{\prime }+\frac {2 y}{x} = 5 x^{2} \] |
linear |
[_linear] |
✓ |
✓ |
0.155 |
|
|
\[ {}x^{2} y^{\prime }-4 x y = x^{7} \sin \left (x \right ) \] |
linear |
[_linear] |
✓ |
✓ |
0.178 |
|
|
\[ {}y^{\prime }+2 x y = 2 x^{3} \] |
linear |
[_linear] |
✓ |
✓ |
0.166 |
|
|
\[ {}y^{\prime }+\frac {2 x y}{-x^{2}+1} = 4 x \] |
linear |
[_linear] |
✓ |
✓ |
0.214 |
|
|
\[ {}y^{\prime }+\frac {2 x y}{x^{2}+1} = \frac {4}{\left (x^{2}+1\right )^{2}} \] |
linear |
[_linear] |
✓ |
✓ |
0.195 |
|
|
\[ {}2 \cos \left (x \right )^{2} y^{\prime }+y \sin \left (2 x \right ) = 4 \cos \left (x \right )^{4} \] |
linear |
[_linear] |
✓ |
✓ |
0.198 |
|
|
\[ {}y^{\prime }+\frac {y}{x \ln \left (x \right )} = 9 x^{2} \] |
linear |
[_linear] |
✓ |
✓ |
0.165 |
|
|
\[ {}y^{\prime }-y \tan \left (x \right ) = 8 \sin \left (x \right )^{3} \] |
linear |
[_linear] |
✓ |
✓ |
0.188 |
|
|
\[ {}t x^{\prime }+2 x = 4 \,{\mathrm e}^{t} \] |
linear |
[_linear] |
✓ |
✓ |
0.174 |
|
|
\[ {}y^{\prime } = \sin \left (x \right ) \left (y \sec \left (x \right )-2\right ) \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.619 |
|
|
\[ {}1-y \sin \left (x \right )-\cos \left (x \right ) y^{\prime } = 0 \] |
linear |
[_linear] |
✓ |
✓ |
0.188 |
|
|
\[ {}y^{\prime }-\frac {y}{x} = 2 \ln \left (x \right ) x^{2} \] |
linear |
[_linear] |
✓ |
✓ |
0.169 |
|
|
\[ {}y^{\prime }+\alpha y = {\mathrm e}^{\beta x} \] |
linear |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.213 |
|
|
\[ {}y^{\prime }+\frac {m y}{x} = \ln \left (x \right ) \] |
linear |
[_linear] |
✓ |
✓ |
0.222 |
|
|
\[ {}y^{\prime }+\frac {2 y}{x} = 4 x \] |
linear |
[_linear] |
✓ |
✓ |
0.383 |
|
|
\[ {}y^{\prime } \sin \left (x \right )-y \cos \left (x \right ) = \sin \left (2 x \right ) \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.749 |
|
|
\[ {}x^{\prime }+\frac {2 x}{4-t} = 5 \] |
linear, homogeneousTypeMapleC, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.122 |
|
|
\[ {}y-{\mathrm e}^{x}+y^{\prime } = 0 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.675 |
|
|
\[
{}y^{\prime }-2 y = \left \{\begin {array}{cc} 1 & x \le 1 \\ 0 & 1 |
linear, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.357 |
|
|
\[ {}y^{\prime }-2 y = \left \{\begin {array}{cc} 1-x & x <1 \\ 0 & 1\le x \end {array}\right . \] |
linear, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.064 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{x} = 9 x \] |
kovacic, second_order_ode_missing_y, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.858 |
|
|
\[ {}y^{\prime }+\frac {y}{x} = \cos \left (x \right ) \] |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.568 |
|
|
\[ {}y^{\prime }+y = {\mathrm e}^{-2 x} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.488 |
|
|
\[ {}y^{\prime }+y \cot \left (x \right ) = 2 \cos \left (x \right ) \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.724 |
|
|
\[ {}-y+x y^{\prime } = \ln \left (x \right ) x^{2} \] |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.58 |
|
|
\[ {}y^{\prime } = \frac {x^{2}+x y+y^{2}}{x^{2}} \] |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
0.319 |
|
|
\[ {}\left (3 x -y\right ) y^{\prime } = 3 y \] |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.332 |
|
|
\[ {}y^{\prime } = \frac {\left (x +y\right )^{2}}{2 x^{2}} \] |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
0.339 |
|
|
\[ {}\sin \left (\frac {y}{x}\right ) \left (-y+x y^{\prime }\right ) = x \cos \left (\frac {y}{x}\right ) \] |
homogeneous |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.664 |
|
|
\[ {}x y^{\prime } = \sqrt {16 x^{2}-y^{2}}+y \] |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.809 |
|
|
\[ {}-y+x y^{\prime } = \sqrt {9 x^{2}+y^{2}} \] |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.483 |
|
|
\[ {}y \left (x^{2}-y^{2}\right )-x \left (x^{2}-y^{2}\right ) y^{\prime } = 0 \] |
homogeneous |
[_separable] |
✓ |
✓ |
0.135 |
|
|
\[ {}x y^{\prime }+y \ln \left (x \right ) = y \ln \left (y\right ) \] |
homogeneous |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.69 |
|
|
\[ {}y^{\prime } = \frac {y^{2}+2 x y-2 x^{2}}{x^{2}-x y+y^{2}} \] |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.586 |
|
|
\[ {}2 x y y^{\prime }-x^{2} {\mathrm e}^{-\frac {y^{2}}{x^{2}}}-2 y^{2} = 0 \] |
homogeneous |
[[_homogeneous, ‘class A‘]] |
✓ |
✓ |
0.325 |
|
|
\[ {}x^{2} y^{\prime } = y^{2}+3 x y+x^{2} \] |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
0.296 |
|
|
\[ {}y y^{\prime } = \sqrt {x^{2}+y^{2}}-x \] |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.119 |
|
|
\[ {}2 x \left (y+2 x \right ) y^{\prime } = y \left (4 x -y\right ) \] |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.352 |
|
|
\[ {}x y^{\prime } = x \tan \left (\frac {y}{x}\right )+y \] |
homogeneous |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.881 |
|
|
\[ {}y^{\prime } = \frac {x \sqrt {x^{2}+y^{2}}+y^{2}}{x y} \] |
homogeneous |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.543 |
|
|
\[ {}y^{\prime } = \frac {4 y-2 x}{x +y} \] |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.953 |
|
|
\[ {}y^{\prime } = \frac {2 x -y}{x +4 y} \] |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.803 |
|
|
\[ {}y^{\prime } = \frac {y-\sqrt {x^{2}+y^{2}}}{x} \] |
homogeneous |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.087 |
|
|
\[ {}-y+x y^{\prime } = \sqrt {4 x^{2}-y^{2}} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.148 |
|
|
\[ {}y^{\prime } = \frac {a y+x}{x a -y} \] |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.393 |
|
|
\[ {}y^{\prime } = \frac {x +\frac {y}{2}}{\frac {x}{2}-y} \] |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.944 |
|
|
\[ {}y^{\prime }-\frac {y}{x} = \frac {4 x^{2} \cos \left (x \right )}{y} \] |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _Bernoulli] |
✓ |
✓ |
2.286 |
|
|
\[ {}y^{\prime }+\frac {y \tan \left (x \right )}{2} = 2 y^{3} \sin \left (x \right ) \] |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
8.359 |
|
|
\[ {}y^{\prime }-\frac {3 y}{2 x} = 6 y^{\frac {1}{3}} x^{2} \ln \left (x \right ) \] |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
2.007 |
|
|
\[ {}y^{\prime }+\frac {2 y}{x} = 6 \sqrt {x^{2}+1}\, \sqrt {y} \] |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.845 |
|
|
\[ {}y^{\prime }+\frac {2 y}{x} = 6 x^{4} y^{2} \] |
riccati, bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
0.689 |
|
|
\[ {}2 x \left (y^{\prime }+x^{2} y^{3}\right )+y = 0 \] |
bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
0.687 |
|
|
\[ {}\left (x -a \right ) \left (-b +x \right ) \left (y^{\prime }-\sqrt {y}\right ) = 2 \left (-a +b \right ) y \] |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
2.076 |
|
|
\[ {}y^{\prime }+\frac {6 y}{x} = \frac {3 y^{\frac {2}{3}} \cos \left (x \right )}{x} \] |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
82.877 |
|
|
\[ {}y^{\prime }+4 x y = 4 x^{3} \sqrt {y} \] |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.561 |
|
|
\[ {}y^{\prime }-\frac {y}{2 x \ln \left (x \right )} = 2 x y^{3} \] |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
0.931 |
|
|
\[ {}y^{\prime }-\frac {y}{\left (\pi -1\right ) x} = \frac {3 x y^{\pi }}{1-\pi } \] |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
0.887 |
|
|
\[ {}2 y^{\prime }+y \cot \left (x \right ) = \frac {8 \cos \left (x \right )^{3}}{y} \] |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
14.635 |
|
|
\[ {}\left (1-\sqrt {3}\right ) y^{\prime }+y \sec \left (x \right ) = y^{\sqrt {3}} \sec \left (x \right ) \] |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
187.417 |
|
|
\[ {}y^{\prime }+\frac {2 x y}{x^{2}+1} = x y^{2} \] |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
1.042 |
|
|
\[ {}y^{\prime }+y \cot \left (x \right ) = y^{3} \sin \left (x \right )^{3} \] |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.64 |
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|
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