|
# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}10 x^{\prime \prime }+\frac {x}{10} = 0 \] |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
6.366 |
|
|
\[ {}x^{\prime \prime }+4 x^{\prime }+3 x = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.626 |
|
|
\[ {}\frac {x^{\prime \prime }}{32}+2 x^{\prime }+x = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.11 |
|
|
\[ {}\frac {x^{\prime \prime }}{4}+2 x^{\prime }+x = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.369 |
|
|
\[ {}4 x^{\prime \prime }+2 x^{\prime }+8 x = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.602 |
|
|
\[ {}x^{\prime \prime }+4 x^{\prime }+13 x = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.007 |
|
|
\[ {}x^{\prime \prime }+4 x^{\prime }+20 x = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.061 |
|
|
\[ {}x^{\prime \prime }+x = \left \{\begin {array}{cc} 1 & 0\le t <\pi \\ 0 & \pi \le t \end {array}\right . \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
6.295 |
|
|
\[ {}x^{\prime \prime }+x = \left \{\begin {array}{cc} \cos \left (t \right ) & 0\le t <\pi \\ 0 & \pi \le t \end {array}\right . \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
10.138 |
|
|
\[ {}x^{\prime \prime }+x = \left \{\begin {array}{cc} t & 0\le t <1 \\ 2-t & 1\le t <2 \\ 0 & 2\le t \end {array}\right . \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
8.455 |
|
|
\[ {}x^{\prime \prime }+4 x^{\prime }+13 x = \left \{\begin {array}{cc} 1 & 0\le t <\pi \\ 1-t & \pi \le t <2 \pi \\ 0 & 2 \pi \le t \end {array}\right . \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
30.017 |
|
|
\[ {}x^{\prime \prime }+x = \cos \left (t \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.033 |
|
|
\[ {}x^{\prime \prime }+x = \cos \left (t \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.839 |
|
|
\[ {}x^{\prime \prime }+x = \cos \left (\frac {9 t}{10}\right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.942 |
|
|
\[ {}x^{\prime \prime }+x = \cos \left (\frac {7 t}{10}\right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.887 |
|
|
\[ {}x^{\prime \prime }+\frac {x^{\prime }}{10}+x = 3 \cos \left (2 t \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.029 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=6 \\ y^{\prime }=\cos \left (t \right ) \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.715 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x \\ y^{\prime }=1 \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.76 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=0 \\ y^{\prime }=-2 y \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.281 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x^{2} \\ y^{\prime }={\mathrm e}^{t} \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.339 |
|
|
\[ {}\left [\begin {array}{c} x_{1}^{\prime }=-3 x_{1} \\ x_{2}^{\prime }=1 \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.663 |
|
|
\[ {}\left [\begin {array}{c} x_{1}^{\prime }=-x_{1}+1 \\ x_{2}^{\prime }=x_{2} \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.655 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-3 x+6 y \\ y^{\prime }=4 x-y \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.449 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=8 x-y \\ y^{\prime }=x+6 y \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.408 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-x-2 y \\ y^{\prime }=x+y \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.577 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=4 x+2 y \\ y^{\prime }=-x+2 y \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.635 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=y \\ y^{\prime }=-x+1 \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
1.323 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=y \\ y^{\prime }=-x+\sin \left (2 t \right ) \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
1.405 |
|
|
\[ {}x^{\prime \prime }-3 x^{\prime }+4 x = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.845 |
|
|
\[ {}x^{\prime \prime }+6 x^{\prime }+9 x = 0 \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.439 |
|
|
\[ {}x^{\prime \prime }+16 x = t \sin \left (t \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.638 |
|
|
\[ {}x^{\prime \prime }+x = {\mathrm e}^{t} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.738 |
|
|
\[ {}y^{\prime } = x^{2}+y^{2} \] |
riccati |
[[_Riccati, _special]] |
✓ |
✓ |
1.366 |
|
|
\[ {}y^{\prime } = \frac {x}{y} \] |
exact, separable, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.51 |
|
|
\[ {}y^{\prime } = y+3 y^{\frac {1}{3}} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.855 |
|
|
\[ {}y^{\prime } = \sqrt {x -y} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
5.955 |
|
|
\[ {}y^{\prime } = \sqrt {x^{2}-y}-x \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
15.783 |
|
|
\[ {}y^{\prime } = \sqrt {1-y^{2}} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.566 |
|
|
\[ {}y^{\prime } = \frac {y+1}{x -y} \] |
homogeneousTypeMapleC, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.871 |
|
|
\[ {}y^{\prime } = \sin \left (y\right )-\cos \left (x \right ) \] |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
0.949 |
|
|
\[ {}y^{\prime } = 1-\cot \left (y\right ) \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.836 |
|
|
\[ {}y^{\prime } = \left (3 x -y\right )^{\frac {1}{3}}-1 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
4.842 |
|
|
\[ {}y^{\prime } = \sin \left (x y\right ) \] |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
0.863 |
|
|
\[ {}x y^{\prime }+y = \cos \left (x \right ) \] |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.331 |
|
|
\[ {}y^{\prime }+2 y = {\mathrm e}^{x} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.125 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime }+x y = 2 x \] |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.665 |
|
|
\[ {}y^{\prime } = 1+x \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.22 |
|
|
\[ {}y^{\prime } = x +y \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.951 |
|
|
\[ {}y^{\prime } = y-x \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.972 |
|
|
\[ {}y^{\prime } = \frac {x}{2}-y+\frac {3}{2} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.108 |
|
|
\[ {}y^{\prime } = \left (y-1\right )^{2} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.333 |
|
|
\[ {}y^{\prime } = \left (y-1\right ) x \] |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.275 |
|
|
\[ {}y^{\prime } = x^{2}-y^{2} \] |
riccati |
[_Riccati] |
✓ |
✓ |
1.157 |
|
|
\[ {}y^{\prime } = \cos \left (x -y\right ) \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
2.09 |
|
|
\[ {}y^{\prime } = -x^{2}+y \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.992 |
|
|
\[ {}y^{\prime } = x^{2}+2 x -y \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.021 |
|
|
\[ {}y^{\prime } = \frac {y+1}{-1+x} \] |
exact, linear, separable, homogeneousTypeD2, homogeneousTypeMapleC, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.953 |
|
|
\[ {}y^{\prime } = \frac {x +y}{x -y} \] |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.728 |
|
|
\[ {}y^{\prime } = 1-x \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.221 |
|
|
\[ {}y^{\prime } = 2 x -y \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.009 |
|
|
\[ {}y^{\prime } = y+x^{2} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.995 |
|
|
\[ {}y^{\prime } = -\frac {y}{x} \] |
exact, linear, separable, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.709 |
|
|
\[ {}y^{\prime } = 1 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.204 |
|
|
\[ {}y^{\prime } = \frac {1}{x} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.219 |
|
|
\[ {}y^{\prime } = y \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.227 |
|
|
\[ {}y^{\prime } = y^{2} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.231 |
|
|
\[ {}y^{\prime } = x^{2}-y^{2} \] |
riccati |
[_Riccati] |
✓ |
✓ |
1.845 |
|
|
\[ {}y^{\prime } = x +y^{2} \] |
riccati |
[[_Riccati, _special]] |
✓ |
✓ |
3.363 |
|
|
\[ {}y^{\prime } = x +y \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.161 |
|
|
\[ {}y^{\prime } = 2 y-2 x^{2}-3 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.28 |
|
|
\[ {}x y^{\prime } = 2 x -y \] |
exact, linear, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
2.59 |
|
|
\[ {}1+y^{2}+\left (x^{2}+1\right ) y^{\prime } = 0 \] |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.477 |
|
|
\[ {}1+y^{2}+x y y^{\prime } = 0 \] |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.681 |
|
|
\[ {}y^{\prime } \sin \left (x \right )-y \cos \left (x \right ) = 0 \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.12 |
|
|
\[ {}1+y^{2} = x y^{\prime } \] |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.455 |
|
|
\[ {}x \sqrt {1+y^{2}}+y y^{\prime } \sqrt {x^{2}+1} = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.202 |
|
|
\[ {}x \sqrt {1-y^{2}}+y \sqrt {-x^{2}+1}\, y^{\prime } = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.635 |
|
|
\[ {}{\mathrm e}^{-y} y^{\prime } = 1 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.302 |
|
|
\[ {}y \ln \left (y\right )+x y^{\prime } = 1 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
11.04 |
|
|
\[ {}y^{\prime } = a^{x +y} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.565 |
|
|
\[ {}{\mathrm e}^{y} \left (x^{2}+1\right ) y^{\prime }-2 x \left (1+{\mathrm e}^{y}\right ) = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.532 |
|
|
\[ {}2 x \sqrt {1-y^{2}} = \left (x^{2}+1\right ) y^{\prime } \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.034 |
|
|
\[ {}{\mathrm e}^{x} \sin \left (y\right )^{3}+\left ({\mathrm e}^{2 x}+1\right ) \cos \left (y\right ) y^{\prime } = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
4.908 |
|
|
\[ {}y^{2} \sin \left (x \right )+\cos \left (x \right )^{2} \ln \left (y\right ) y^{\prime } = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
4.586 |
|
|
\[ {}y^{\prime } = \sin \left (x -y\right ) \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
2.056 |
|
|
\[ {}y^{\prime } = x a +b y+c \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.722 |
|
|
\[ {}\left (x +y\right )^{2} y^{\prime } = a^{2} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.625 |
|
|
\[ {}x y^{\prime }+y = a \left (1+x y\right ) \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.595 |
|
|
\[ {}a^{2}+y^{2}+2 x \sqrt {x a -x^{2}}\, y^{\prime } = 0 \] |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
5.954 |
|
|
\[ {}y^{\prime } = \frac {y}{x} \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.441 |
|
|
\[ {}\cos \left (y^{\prime }\right ) = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.224 |
|
|
\[ {}{\mathrm e}^{y^{\prime }} = 1 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.209 |
|
|
\[ {}\sin \left (y^{\prime }\right ) = x \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.291 |
|
|
\[ {}\ln \left (y^{\prime }\right ) = x \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.603 |
|
|
\[ {}\tan \left (y^{\prime }\right ) = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.209 |
|
|
\[ {}{\mathrm e}^{y^{\prime }} = x \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.238 |
|
|
\[ {}\tan \left (y^{\prime }\right ) = x \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.312 |
|
|
\[ {}x^{2} y^{\prime } \cos \left (y\right )+1 = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✗ |
6.134 |
|
|
\[ {}x^{2} y^{\prime }+\cos \left (2 y\right ) = 1 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✗ |
10.013 |
|
|
\[ {}x^{3} y^{\prime }-\sin \left (y\right ) = 1 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
5.432 |
|
|
|
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