|
# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}y^{\prime \prime } = k \] |
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]] |
✓ |
✓ |
1.232 |
|
|
\[ {}y^{\prime } = -4 \sin \left (x -y\right )-4 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
39.384 |
|
|
\[ {}y^{\prime }+\sin \left (x -y\right ) = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.358 |
|
|
\[ {}y^{\prime \prime } = 4 \sin \left (x \right )-4 \] |
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]] |
✓ |
✓ |
2.115 |
|
|
\[ {}y y^{\prime \prime } = 0 \] |
second_order_ode_quadrature |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.132 |
|
|
\[ {}y y^{\prime \prime } = 1 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
0.869 |
|
|
\[ {}y y^{\prime \prime } = x \] |
unknown |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.053 |
|
|
\[ {}y^{2} y^{\prime \prime } = x \] |
unknown |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.06 |
|
|
\[ {}y^{2} y^{\prime \prime } = 0 \] |
second_order_ode_quadrature |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.135 |
|
|
\[ {}3 y y^{\prime \prime } = \sin \left (x \right ) \] |
unknown |
[NONE] |
❇ |
N/A |
0.128 |
|
|
\[ {}3 y y^{\prime \prime }+y = 5 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
1.069 |
|
|
\[ {}a y y^{\prime \prime }+b y = c \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
1.901 |
|
|
\[ {}a y^{2} y^{\prime \prime }+b y^{2} = c \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
2.931 |
|
|
\[ {}a y y^{\prime \prime }+b y = 0 \] |
second_order_ode_quadrature |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.237 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=9 x+4 y \\ y^{\prime }=-6 x-y \\ z^{\prime }=6 x+4 y+3 z \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.681 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x-3 y \\ y^{\prime }=3 x+7 y \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.481 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x-2 y \\ y^{\prime }=2 x+5 y \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.477 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=7 x+y \\ y^{\prime }=-4 x+3 y \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.495 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x+y \\ y^{\prime }=y \\ z^{\prime }=z \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.34 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=2 x+y-z \\ y^{\prime }=-x+2 z \\ z^{\prime }=-x-2 y+4 z \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.473 |
|
|
\[ {}x^{\prime } = 4 A k \left (\frac {x}{A}\right )^{\frac {3}{4}}-3 k x \] |
quadrature |
[_quadrature] |
✓ |
✓ |
1.17 |
|
|
\[ {}\frac {y^{\prime } y}{1+\frac {\sqrt {1+{y^{\prime }}^{2}}}{2}} = -x \] |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
2.651 |
|
|
\[ {}\frac {y^{\prime } y}{1+\frac {\sqrt {1+{y^{\prime }}^{2}}}{2}} = -x \] |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
2.527 |
|
|
\[ {}y^{\prime } = \frac {y \left (1+\frac {a^{2} x}{\sqrt {a^{2} \left (x^{2}+1\right )}}\right )}{\sqrt {a^{2} \left (x^{2}+1\right )}} \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
4.036 |
|
|
\[ {}y^{\prime } = x^{2}+y^{2} \] |
riccati |
[[_Riccati, _special]] |
✓ |
✓ |
1.315 |
|
|
\[ {}y^{\prime } = 2 \sqrt {y} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.315 |
|
|
\[ {}z^{\prime \prime }+3 z^{\prime }+2 z = 24 \,{\mathrm e}^{-3 t}-24 \,{\mathrm e}^{-4 t} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.63 |
|
|
\[ {}y^{\prime } = \sqrt {1-y^{2}} \] |
separable |
[_quadrature] |
✓ |
✓ |
0.821 |
|
|
\[ {}y^{\prime } = y^{2}+x^{2}-1 \] |
riccati |
[_Riccati] |
✓ |
✓ |
2.411 |
|
|
\[ {}y^{\prime } = 2 y \left (x \sqrt {y}-1\right ) \] |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.849 |
|
|
\[ {}y^{\prime \prime } = \frac {1}{y}-\frac {x y^{\prime }}{y^{2}} \] |
unknown |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.066 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.756 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.684 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.881 |
|
|
\[ {}y^{\prime \prime }-y y^{\prime } = 2 x \] |
second_order_integrable_as_is, exact nonlinear second order ode |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
12.938 |
|
|
\[ {}y^{\prime }-y^{2}-x -x^{2} = 0 \] |
riccati |
[_Riccati] |
✓ |
✓ |
9.154 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-x = 0 \] |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.337 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-2 x = 0 \] |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.918 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-3 x = 0 \] |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.99 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-x^{2}-x = 0 \] |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.596 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-x^{3}+2 = 0 \] |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.592 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-x^{4}-6 = 0 \] |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.579 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-x^{5}+24 = 0 \] |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.647 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-x = 0 \] |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.756 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-x^{2} = 0 \] |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.434 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-x^{3} = 0 \] |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.037 |
|
|
\[ {}y^{\prime \prime }-a x y^{\prime }-b x y-c x = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.597 |
|
|
\[ {}y^{\prime \prime }-a x y^{\prime }-b x y-c \,x^{2} = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.446 |
|
|
\[ {}y^{\prime \prime }-a x y^{\prime }-b x y-c \,x^{3} = 0 \] |
unknown |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
0.733 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-x y-x = 0 \] |
second_order_airy |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
18.182 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-x y-x^{2} = 0 \] |
second_order_airy |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
18.117 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-x y-x^{2}-1 = 0 \] |
second_order_airy |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
18.792 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-x y-x^{2}-1 = 0 \] |
second_order_airy |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
16.68 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-x y-x^{2}-2 = 0 \] |
second_order_airy |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
15.119 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }-x y-x^{2}-4 = 0 \] |
second_order_airy |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
15.381 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-x y-x^{3}+1 = 0 \] |
second_order_airy |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
19.355 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-x y-x^{3}-x^{2} = 0 \] |
second_order_airy |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
15.474 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-x y-x^{3}+2 = 0 \] |
second_order_airy |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
19.099 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-x y-x^{3}+2 = 0 \] |
second_order_airy |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
14.798 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }-x y-x^{3}+2 = 0 \] |
second_order_airy |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
15.169 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }-x y-x^{3}+2 = 0 \] |
second_order_airy |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
16.865 |
|
|
\[ {}y^{\prime \prime }-8 y^{\prime }-x y-x^{3}+2 = 0 \] |
second_order_airy |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
17.049 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-x y-x^{4}+3 = 0 \] |
second_order_airy |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
19.127 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-x y-x^{3} = 0 \] |
second_order_airy |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
16.58 |
|
|
\[ {}y^{\prime \prime }-x y-x^{3}+2 = 0 \] |
second_order_airy, second_order_bessel_ode |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
10.975 |
|
|
\[ {}y^{\prime \prime }-x y-x^{6}+64 = 0 \] |
second_order_airy, second_order_bessel_ode |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
14.98 |
|
|
\[ {}y^{\prime \prime }-x y-x = 0 \] |
second_order_airy, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
5.621 |
|
|
\[ {}y^{\prime \prime }-x y-x^{2} = 0 \] |
second_order_airy, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
7.583 |
|
|
\[ {}y^{\prime \prime }-x y-x^{3} = 0 \] |
second_order_airy, second_order_bessel_ode |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.599 |
|
|
\[ {}y^{\prime \prime }-x y-x^{6}-x^{3}+42 = 0 \] |
second_order_airy, second_order_bessel_ode |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
19.338 |
|
|
\[ {}y^{\prime \prime }-x^{2} y-x^{2} = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
10.475 |
|
|
\[ {}y^{\prime \prime }-x^{2} y-x^{3} = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
7.586 |
|
|
\[ {}y^{\prime \prime }-x^{2} y-x^{4} = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.766 |
|
|
\[ {}y^{\prime \prime }-x^{2} y-x^{4}+2 = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
13.415 |
|
|
\[ {}y^{\prime \prime }-2 x^{2} y-x^{4}+1 = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
36.542 |
|
|
\[ {}y^{\prime \prime }-x^{3} y-x^{3} = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
16.328 |
|
|
\[ {}y^{\prime \prime }-x^{3} y-x^{4} = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
76.621 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }-x^{2} y-x^{2} = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.956 |
|
|
\[ {}y^{\prime \prime }-x^{3} y^{\prime }-x^{3} y-x^{3} = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
1.598 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-x = 0 \] |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.172 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }-x y-x^{2} = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.187 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }-x^{2} y-x^{3}-x^{2} = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.58 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }-x^{3} y-x^{4}-x^{2} = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.812 |
|
|
\[ {}y^{\prime \prime }-\frac {y^{\prime }}{x}-x y-x^{2}-\frac {1}{x} = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.713 |
|
|
\[ {}y^{\prime \prime }-\frac {y^{\prime }}{x}-x^{2} y-x^{3}-\frac {1}{x} = 0 \] |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.931 |
|
|
\[ {}y^{\prime \prime }-\frac {y^{\prime }}{x}-x^{3} y-x^{4}-\frac {1}{x} = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.737 |
|
|
\[ {}y^{\prime \prime }-x^{3} y^{\prime }-x y-x^{3}-x^{2} = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
1.347 |
|
|
\[ {}y^{\prime \prime }-x^{3} y^{\prime }-x^{2} y-x^{3} = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.845 |
|
|
\[ {}y^{\prime \prime }-x^{3} y^{\prime }-x^{3} y-x^{4}-x^{3} = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.947 |
|
|
\[ {}y^{\prime \prime \prime }-x^{3} y^{\prime }-x^{2} y-x^{3} = 0 \] |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.062 |
|
|
\[ {}y^{\prime \prime }+c y^{\prime }+k y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.468 |
|
|
\[ {}w^{\prime } = -\frac {1}{2}-\frac {\sqrt {1-12 w}}{2} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
1.661 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.956 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.727 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.941 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.873 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.921 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.921 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.786 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.717 |
|
|
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