|
# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}y^{\prime }+y = \frac {1}{x^{2}} \] |
|
[[_linear, ‘class A‘]] |
❇ |
N/A |
0.625 |
|
|
\[ {}x y^{\prime }+y = 0 \] |
first order ode series method. Regular singular point |
[_separable] |
✓ |
✓ |
0.504 |
|
|
\[ {}y^{\prime } = \frac {1}{x} \] |
|
[_quadrature] |
❇ |
N/A |
0.284 |
|
|
\[ {}y^{\prime \prime } = \frac {1}{x} \] |
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]] |
❇ |
N/A |
0.247 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = \frac {1}{x} \] |
exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_y]] |
❇ |
N/A |
0.305 |
|
|
\[ {}y^{\prime \prime }+y = \frac {1}{x} \] |
second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
❇ |
N/A |
0.362 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = \frac {1}{x} \] |
second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
❇ |
N/A |
0.787 |
|
|
\[ {}h^{2}+\frac {2 a h}{\sqrt {1+{h^{\prime }}^{2}}} = b^{2} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
11.224 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }-24 y = 16-\left (2+x \right ) {\mathrm e}^{4 x} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.036 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }-4 y = 6 \,{\mathrm e}^{2 t -2} \] |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.359 |
|
|
\[ {}y^{\prime \prime }+y = {\mathrm e}^{a \cos \left (x \right )} \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.888 |
|
|
\[ {}y^{\prime } = \frac {y}{2 y \ln \left (y\right )+y-x} \] |
exact, differentialType, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
2.548 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x +1\right ) y^{\prime }+y \left (1+x \right ) = 0 \] |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.185 |
|
|
\[ {}x^{2} y^{\prime }+{\mathrm e}^{-y} = 0 \] |
exact, separable, first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.014 |
|
|
\[ {}y^{\prime \prime }+{\mathrm e}^{y} = 0 \] |
second_order_ode_missing_x, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
3.547 |
|
|
\[ {}y^{\prime } = \frac {x y+3 x -2 y+6}{x y-3 x -2 y+6} \] |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
1.535 |
|
|
\[ {}y^{\prime } = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.089 |
|
|
\[ {}y^{\prime } = a \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.133 |
|
|
\[ {}y^{\prime } = x \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.098 |
|
|
\[ {}y^{\prime } = 1 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.114 |
|
|
\[ {}y^{\prime } = x a \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.141 |
|
|
\[ {}y^{\prime } = a x y \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.506 |
|
|
\[ {}y^{\prime } = x a +y \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.982 |
|
|
\[ {}y^{\prime } = x a +b y \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.067 |
|
|
\[ {}y^{\prime } = y \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.216 |
|
|
\[ {}y^{\prime } = b y \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.512 |
|
|
\[ {}y^{\prime } = x a +b y^{2} \] |
riccati |
[[_Riccati, _special]] |
✓ |
✓ |
2.091 |
|
|
\[ {}c y^{\prime } = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.091 |
|
|
\[ {}c y^{\prime } = a \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.162 |
|
|
\[ {}c y^{\prime } = x a \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.151 |
|
|
\[ {}c y^{\prime } = x a +y \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.201 |
|
|
\[ {}c y^{\prime } = x a +b y \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.153 |
|
|
\[ {}c y^{\prime } = y \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.541 |
|
|
\[ {}c y^{\prime } = b y \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.664 |
|
|
\[ {}c y^{\prime } = x a +b y^{2} \] |
riccati |
[[_Riccati, _special]] |
✓ |
✓ |
2.31 |
|
|
\[ {}c y^{\prime } = \frac {x a +b y^{2}}{r} \] |
riccati |
[[_Riccati, _special]] |
✓ |
✓ |
2.399 |
|
|
\[ {}c y^{\prime } = \frac {x a +b y^{2}}{r x} \] |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
2.359 |
|
|
\[ {}c y^{\prime } = \frac {x a +b y^{2}}{r \,x^{2}} \] |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
2.62 |
|
|
\[ {}c y^{\prime } = \frac {x a +b y^{2}}{y} \] |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
1.917 |
|
|
\[ {}a \sin \left (x \right ) y x y^{\prime } = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.162 |
|
|
\[ {}f \left (x \right ) \sin \left (x \right ) y x y^{\prime } \pi = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.173 |
|
|
\[ {}y^{\prime } = \sin \left (x \right )+y \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.249 |
|
|
\[ {}y^{\prime } = \sin \left (x \right )+y^{2} \] |
riccati |
[_Riccati] |
✓ |
✓ |
6.935 |
|
|
\[ {}y^{\prime } = \cos \left (x \right )+\frac {y}{x} \] |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.673 |
|
|
\[ {}y^{\prime } = \cos \left (x \right )+\frac {y^{2}}{x} \] |
riccati |
[_Riccati] |
✓ |
✓ |
6.023 |
|
|
\[ {}y^{\prime } = x +y+b y^{2} \] |
riccati |
[_Riccati] |
✓ |
✓ |
2.653 |
|
|
\[ {}x y^{\prime } = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.092 |
|
|
\[ {}5 y^{\prime } = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.092 |
|
|
\[ {}{\mathrm e} y^{\prime } = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.092 |
|
|
\[ {}\pi y^{\prime } = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.095 |
|
|
\[ {}\sin \left (x \right ) y^{\prime } = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.112 |
|
|
\[ {}f \left (x \right ) y^{\prime } = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.095 |
|
|
\[ {}x y^{\prime } = 1 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.132 |
|
|
\[ {}x y^{\prime } = \sin \left (x \right ) \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.256 |
|
|
\[ {}\left (-1+x \right ) y^{\prime } = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.094 |
|
|
\[ {}y y^{\prime } = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.135 |
|
|
\[ {}x y y^{\prime } = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.133 |
|
|
\[ {}x y \sin \left (x \right ) y^{\prime } = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.186 |
|
|
\[ {}\pi y \sin \left (x \right ) y^{\prime } = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.171 |
|
|
\[ {}x \sin \left (x \right ) y^{\prime } = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.097 |
|
|
\[ {}x \sin \left (x \right ) {y^{\prime }}^{2} = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.195 |
|
|
\[ {}y {y^{\prime }}^{2} = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.258 |
|
|
\[ {}{y^{\prime }}^{n} = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.125 |
|
|
\[ {}x {y^{\prime }}^{n} = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.122 |
|
|
\[ {}{y^{\prime }}^{2} = x \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.398 |
|
|
\[ {}{y^{\prime }}^{2} = x +y \] |
dAlembert, first_order_nonlinear_p_but_linear_in_x_y |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.018 |
|
|
\[ {}{y^{\prime }}^{2} = \frac {y}{x} \] |
dAlembert, first_order_nonlinear_p_but_separable |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.82 |
|
|
\[ {}{y^{\prime }}^{2} = \frac {y^{2}}{x} \] |
first_order_nonlinear_p_but_separable |
[_separable] |
✓ |
✓ |
2.366 |
|
|
\[ {}{y^{\prime }}^{2} = \frac {y^{3}}{x} \] |
first_order_nonlinear_p_but_separable |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
1.619 |
|
|
\[ {}{y^{\prime }}^{3} = \frac {y^{2}}{x} \] |
first_order_nonlinear_p_but_separable |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.214 |
|
|
\[ {}{y^{\prime }}^{2} = \frac {1}{x y} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
4.559 |
|
|
\[ {}{y^{\prime }}^{2} = \frac {1}{x y^{3}} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
4.829 |
|
|
\[ {}{y^{\prime }}^{2} = \frac {1}{x^{2} y^{3}} \] |
separable |
[_separable] |
✓ |
✓ |
1.224 |
|
|
\[ {}{y^{\prime }}^{4} = \frac {1}{x y^{3}} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
12.116 |
|
|
\[ {}{y^{\prime }}^{2} = \frac {1}{x^{3} y^{4}} \] |
separable |
[_separable] |
✓ |
✓ |
1.374 |
|
|
\[ {}y^{\prime } = \sqrt {1+6 x +y} \] |
homogeneousTypeC, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
4.671 |
|
|
\[ {}y^{\prime } = \left (1+6 x +y\right )^{\frac {1}{3}} \] |
homogeneousTypeC, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
2.396 |
|
|
\[ {}y^{\prime } = \left (1+6 x +y\right )^{\frac {1}{4}} \] |
homogeneousTypeC, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
3.289 |
|
|
\[ {}y^{\prime } = \left (a +b x +y\right )^{4} \] |
homogeneousTypeC, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
14.207 |
|
|
\[ {}y^{\prime } = \left (\pi +x +7 y\right )^{\frac {7}{2}} \] |
homogeneousTypeC, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
7.107 |
|
|
\[ {}y^{\prime } = \left (a +b x +c y\right )^{6} \] |
homogeneousTypeC, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
4.806 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{x +y} \] |
exact, separable, first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.191 |
|
|
\[ {}y^{\prime } = 10+{\mathrm e}^{x +y} \] |
first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
3.434 |
|
|
\[ {}y^{\prime } = 10 \,{\mathrm e}^{x +y}+x^{2} \] |
first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
1.578 |
|
|
\[ {}y^{\prime } = x \,{\mathrm e}^{x +y}+\sin \left (x \right ) \] |
first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
3.373 |
|
|
\[ {}y^{\prime } = 5 \,{\mathrm e}^{x^{2}+20 y}+\sin \left (x \right ) \] |
first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
3.355 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }+y^{\prime }-x=t +y \\ x^{\prime }+y^{\prime }=2 x+3 y+{\mathrm e}^{t} \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.544 |
|
|
\[ {}\left [\begin {array}{c} 2 x^{\prime }+y^{\prime }-x=t +y \\ x^{\prime }+y^{\prime }=2 x+3 y+{\mathrm e}^{t} \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
3.087 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }+y^{\prime }-x=y+t +\sin \left (t \right )+\cos \left (t \right ) \\ x^{\prime }+y^{\prime }=2 x+3 y+{\mathrm e}^{t} \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.885 |
|
|
\[ {}y^{\prime \prime } = 0 \] |
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, second_order_ode_can_be_made_integrable |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
1.064 |
|
|
\[ {}{y^{\prime \prime }}^{2} = 0 \] |
second_order_ode_high_degree, second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.779 |
|
|
\[ {}{y^{\prime \prime }}^{n} = 0 \] |
second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.619 |
|
|
\[ {}a y^{\prime \prime } = 0 \] |
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, second_order_ode_can_be_made_integrable |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
1.092 |
|
|
\[ {}a {y^{\prime \prime }}^{2} = 0 \] |
second_order_ode_high_degree, second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.717 |
|
|
\[ {}a {y^{\prime \prime }}^{n} = 0 \] |
second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.572 |
|
|
\[ {}y^{\prime \prime } = 1 \] |
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, second_order_ode_can_be_made_integrable |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
1.986 |
|
|
\[ {}{y^{\prime \prime }}^{2} = 1 \] |
second_order_ode_high_degree, second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
1.793 |
|
|
\[ {}y^{\prime \prime } = x \] |
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]] |
✓ |
✓ |
1.036 |
|
|
\[ {}{y^{\prime \prime }}^{2} = x \] |
second_order_ode_high_degree, second_order_ode_missing_y |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.783 |
|
|
\[ {}{y^{\prime \prime }}^{3} = 0 \] |
second_order_ode_high_degree, second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.933 |
|
|
|
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