Column notations: A is ODE degree. B is Program Number of solutions generated. C is CAS Number of solutions generated.
|
# |
ODE |
A |
B |
C |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}y^{\prime } = \frac {\cos \left (y\right ) \sec \left (x \right )}{x} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.487 |
|
|
\[ {}y^{\prime } = x \left (\cos \left (y\right )+y\right ) \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.819 |
|
|
\[ {}y^{\prime } = \frac {\sec \left (x \right ) \left (\sin \left (y\right )+y\right )}{x} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.619 |
|
|
\[ {}y^{\prime } = \left (5+\frac {\sec \left (x \right )}{x}\right ) \left (\sin \left (y\right )+y\right ) \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
11.605 |
|
|
\[ {}y^{\prime } = y+1 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.175 |
|
|
\[ {}y^{\prime } = 1+x \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.108 |
|
|
\[ {}y^{\prime } = x \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.092 |
|
|
\[ {}y^{\prime } = y \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.103 |
|
|
\[ {}y^{\prime } = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.072 |
|
|
\[ {}y^{\prime } = 1+\frac {\sec \left (x \right )}{x} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.217 |
|
|
\[ {}y^{\prime } = x +\frac {\sec \left (x \right ) y}{x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
4.544 |
|
|
\[ {}y^{\prime } = \frac {2 y}{x} \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.167 |
|
|
\[ {}y^{\prime } = \frac {2 y}{x} \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.577 |
|
|
\[ {}y^{\prime } = \frac {\ln \left (1+y^{2}\right )}{\ln \left (x^{2}+1\right )} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.029 |
|
|
\[ {}y^{\prime } = \frac {1}{x} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.096 |
|
|
\[ {}y^{\prime } = \frac {-x y-1}{4 x^{3} y-2 x^{2}} \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.513 |
|
|
\[ {}\frac {{y^{\prime }}^{2}}{4}-x y^{\prime }+y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.258 |
|
|
\[ {}y^{\prime } = \sqrt {\frac {y+1}{y^{2}}} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
48.819 |
|
|
\[ {}y^{\prime } = \sqrt {1-x^{2}-y^{2}} \] |
1 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
0.629 |
|
|
\[ {}y^{\prime }+\frac {y}{3} = \frac {\left (1-2 x \right ) y^{4}}{3} \] |
1 |
3 |
3 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.113 |
|
|
\[ {}y^{\prime } = \sqrt {y}+x \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Chini] |
✓ |
✓ |
3.674 |
|
|
\[ {}x^{2} y^{\prime }+y^{2} = x y y^{\prime } \] |
1 |
1 |
1 |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.532 |
|
|
\[ {}y = x y^{\prime }+x^{2} {y^{\prime }}^{2} \] |
2 |
2 |
2 |
separable |
[_separable] |
✓ |
✓ |
2.337 |
|
|
\[ {}\left (x +y\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.107 |
|
|
\[ {}x y^{\prime } = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.069 |
|
|
\[ {}\frac {y^{\prime }}{x +y} = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.071 |
|
|
\[ {}\frac {y^{\prime }}{x} = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.069 |
|
|
\[ {}y^{\prime } = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.068 |
|
|
\[ {}y = x {y^{\prime }}^{2}+{y^{\prime }}^{2} \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
✓ |
0.378 |
|
|
\[ {}y^{\prime } = \frac {5 x^{2}-x y+y^{2}}{x^{2}} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
2.058 |
|
|
\[ {}2 t +3 x+\left (x+2\right ) x^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.444 |
|
|
\[ {}y^{\prime } = \frac {1}{1-y} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.278 |
|
|
\[ {}p^{\prime } = a p-b p^{2} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.123 |
|
|
\[ {}y^{2}+\frac {2}{x}+2 x y y^{\prime } = 0 \] |
1 |
1 |
2 |
exact, bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli] |
✓ |
✓ |
0.987 |
|
|
\[ {}x f^{\prime }-f = \frac {{f^{\prime }}^{2} \left (1-{f^{\prime }}^{\lambda }\right )^{2}}{\lambda ^{2}} \] |
0 |
2 |
3 |
clairaut |
[_Clairaut] |
✓ |
✓ |
41.631 |
|
|
\[ {}x y^{\prime }-2 y+b y^{2} = c \,x^{4} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.987 |
|
|
\[ {}x y^{\prime }-y+y^{2} = x^{\frac {2}{3}} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
22.616 |
|
|
\[ {}u^{\prime }+u^{2} = \frac {1}{x^{\frac {4}{5}}} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
0.265 |
|
|
\[ {}y y^{\prime }-y = x \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.423 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.323 |
|
|
\[ {}5 y^{\prime \prime }+2 y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.092 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+4 y = 1 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.799 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+4 y = \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.089 |
|
|
\[ {}y = x {y^{\prime }}^{2} \] |
2 |
3 |
3 |
dAlembert, first_order_nonlinear_p_but_separable |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.881 |
|
|
\[ {}y y^{\prime } = 1-x {y^{\prime }}^{3} \] |
3 |
4 |
3 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
152.454 |
|
|
\[ {}f^{\prime } = \frac {1}{f} \] |
1 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.257 |
|
|
\[ {}t y^{\prime \prime }+4 y^{\prime } = t^{2} \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.355 |
|
|
\[ {}\left (t^{2}+9\right ) y^{\prime \prime }+2 t y^{\prime } = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.044 |
|
|
\[ {}t^{2} y^{\prime \prime }-3 t y^{\prime }+5 y = 0 \] |
1 |
1 |
1 |
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]] |
✓ |
✓ |
2.004 |
|
|
\[ {}t y^{\prime \prime }+y^{\prime } = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.972 |
|
|
\[ {}t^{2} y^{\prime \prime }-2 y^{\prime } = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_missing_y, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.863 |
|
|
\[ {}y^{\prime \prime }+\frac {\left (t^{2}-1\right ) y^{\prime }}{t}+\frac {t^{2} y}{\left (1+{\mathrm e}^{\frac {t^{2}}{2}}\right )^{2}} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.566 |
|
|
\[ {}t y^{\prime \prime }-y^{\prime }+4 t^{3} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.806 |
|
|
\[ {}y^{\prime \prime } = 0 \] |
1 |
1 |
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]] |
✓ |
✓ |
0.726 |
|
|
\[ {}y^{\prime \prime } = 1 \] |
1 |
1 |
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]] |
✓ |
✓ |
1.167 |
|
|
\[ {}y^{\prime \prime } = f \left (t \right ) \] |
1 |
1 |
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 |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
1.664 |
|
|
\[ {}y^{\prime \prime } = k \] |
1 |
1 |
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]] |
✓ |
✓ |
1.232 |
|
|
\[ {}y^{\prime } = -4 \sin \left (x -y\right )-4 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
39.384 |
|
|
\[ {}y^{\prime }+\sin \left (x -y\right ) = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.358 |
|
|
\[ {}y^{\prime \prime } = 4 \sin \left (x \right )-4 \] |
1 |
1 |
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 |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
2.115 |
|
|
\[ {}y y^{\prime \prime } = 0 \] |
1 |
1 |
2 |
second_order_ode_quadrature |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.132 |
|
|
\[ {}y y^{\prime \prime } = 1 \] |
1 |
2 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
0.869 |
|
|
\[ {}y y^{\prime \prime } = x \] |
1 |
0 |
1 |
unknown |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.053 |
|
|
\[ {}y^{2} y^{\prime \prime } = x \] |
1 |
0 |
1 |
unknown |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.06 |
|
|
\[ {}y^{2} y^{\prime \prime } = 0 \] |
1 |
1 |
2 |
second_order_ode_quadrature |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.135 |
|
|
\[ {}3 y y^{\prime \prime } = \sin \left (x \right ) \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.128 |
|
|
\[ {}3 y y^{\prime \prime }+y = 5 \] |
1 |
2 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
1.069 |
|
|
\[ {}a y y^{\prime \prime }+b y = c \] |
1 |
2 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
1.901 |
|
|
\[ {}a y^{2} y^{\prime \prime }+b y^{2} = c \] |
1 |
2 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
2.931 |
|
|
\[ {}a y y^{\prime \prime }+b y = 0 \] |
1 |
1 |
2 |
second_order_ode_quadrature |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.237 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=9 x+4 y \\ y^{\prime }=-6 x-y \\ z^{\prime }=6 x+4 y+3 z \end {array}\right ] \] |
1 |
1 |
3 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.681 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x-3 y \\ y^{\prime }=3 x+7 y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.481 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x-2 y \\ y^{\prime }=2 x+5 y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.477 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=7 x+y \\ y^{\prime }=-4 x+3 y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.495 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x+y \\ y^{\prime }=y \\ z^{\prime }=z \end {array}\right ] \] |
1 |
1 |
3 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.34 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=2 x+y-z \\ y^{\prime }=-x+2 z \\ z^{\prime }=-x-2 y+4 z \end {array}\right ] \] |
1 |
1 |
3 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.473 |
|
|
\[ {}x^{\prime } = 4 A k \left (\frac {x}{A}\right )^{\frac {3}{4}}-3 k x \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.17 |
|
|
\[ {}\frac {y^{\prime } y}{1+\frac {\sqrt {1+{y^{\prime }}^{2}}}{2}} = -x \] |
2 |
3 |
6 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
2.651 |
|
|
\[ {}\frac {y^{\prime } y}{1+\frac {\sqrt {1+{y^{\prime }}^{2}}}{2}} = -x \] |
2 |
3 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
2.527 |
|
|
\[ {}y^{\prime } = \frac {y \left (1+\frac {a^{2} x}{\sqrt {a^{2} \left (x^{2}+1\right )}}\right )}{\sqrt {a^{2} \left (x^{2}+1\right )}} \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
4.036 |
|
|
\[ {}y^{\prime } = x^{2}+y^{2} \] |
1 |
1 |
1 |
riccati |
[[_Riccati, _special]] |
✓ |
✓ |
1.315 |
|
|
\[ {}y^{\prime } = 2 \sqrt {y} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.315 |
|
|
\[ {}z^{\prime \prime }+3 z^{\prime }+2 z = 24 \,{\mathrm e}^{-3 t}-24 \,{\mathrm e}^{-4 t} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.63 |
|
|
\[ {}y^{\prime } = \sqrt {1-y^{2}} \] |
1 |
1 |
1 |
separable |
[_quadrature] |
✓ |
✓ |
0.821 |
|
|
\[ {}y^{\prime } = x^{2}+y^{2}-1 \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.411 |
|
|
\[ {}y^{\prime } = 2 y \left (x \sqrt {y}-1\right ) \] |
1 |
1 |
1 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.849 |
|
|
\[ {}y^{\prime \prime } = \frac {1}{y}-\frac {x y^{\prime }}{y^{2}} \] |
1 |
0 |
1 |
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 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.756 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.684 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.881 |
|
|
\[ {}y^{\prime \prime }-y y^{\prime } = 2 x \] |
1 |
1 |
1 |
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 |
|
|
\[ {}y^{\prime }-y^{2}-x -x^{2} = 0 \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
9.154 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.337 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-2 x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.918 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-3 x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.99 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-x^{2}-x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.596 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-x^{3}+2 = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.592 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-x^{4}-6 = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.579 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-x^{5}+24 = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.647 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.756 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-x^{2} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.434 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-x^{3} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.037 |
|
|
\[ {}y^{\prime \prime }-a x y^{\prime }-b x y-c x = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.597 |
|
|
\[ {}y^{\prime \prime }-a x y^{\prime }-b x y-c \,x^{2} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.446 |
|
|
\[ {}y^{\prime \prime }-a x y^{\prime }-b x y-c \,x^{3} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
0.733 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-x y-x = 0 \] |
1 |
1 |
1 |
second_order_airy |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
18.182 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-x y-x^{2} = 0 \] |
1 |
1 |
1 |
second_order_airy |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
18.117 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-x y-x^{2}-1 = 0 \] |
1 |
1 |
1 |
second_order_airy |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
18.792 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-x y-x^{2}-1 = 0 \] |
1 |
1 |
1 |
second_order_airy |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
16.68 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-x y-x^{2}-2 = 0 \] |
1 |
1 |
1 |
second_order_airy |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
15.119 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }-x y-x^{2}-4 = 0 \] |
1 |
1 |
1 |
second_order_airy |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
15.381 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-x y-x^{3}+1 = 0 \] |
1 |
1 |
1 |
second_order_airy |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
19.355 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-x y-x^{3}-x^{2} = 0 \] |
1 |
1 |
1 |
second_order_airy |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
15.474 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-x y-x^{3}+2 = 0 \] |
1 |
1 |
1 |
second_order_airy |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
19.099 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-x y-x^{3}+2 = 0 \] |
1 |
1 |
1 |
second_order_airy |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
14.798 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }-x y-x^{3}+2 = 0 \] |
1 |
1 |
1 |
second_order_airy |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
15.169 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }-x y-x^{3}+2 = 0 \] |
1 |
1 |
1 |
second_order_airy |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
16.865 |
|
|
\[ {}y^{\prime \prime }-8 y^{\prime }-x y-x^{3}+2 = 0 \] |
1 |
1 |
1 |
second_order_airy |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
17.049 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-x y-x^{4}+3 = 0 \] |
1 |
1 |
1 |
second_order_airy |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
19.127 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-x y-x^{3} = 0 \] |
1 |
1 |
1 |
second_order_airy |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
16.58 |
|
|
\[ {}y^{\prime \prime }-x y-x^{3}+2 = 0 \] |
1 |
1 |
1 |
second_order_airy, second_order_bessel_ode |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
10.975 |
|
|
\[ {}y^{\prime \prime }-x y-x^{6}+64 = 0 \] |
1 |
1 |
1 |
second_order_airy, second_order_bessel_ode |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
14.98 |
|
|
\[ {}y^{\prime \prime }-x y-x = 0 \] |
1 |
1 |
1 |
second_order_airy, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
5.621 |
|
|
\[ {}y^{\prime \prime }-x y-x^{2} = 0 \] |
1 |
1 |
1 |
second_order_airy, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
7.583 |
|
|
\[ {}y^{\prime \prime }-x y-x^{3} = 0 \] |
1 |
1 |
1 |
second_order_airy, second_order_bessel_ode |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.599 |
|
|
\[ {}y^{\prime \prime }-x y-x^{6}-x^{3}+42 = 0 \] |
1 |
1 |
1 |
second_order_airy, second_order_bessel_ode |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
19.338 |
|
|
\[ {}y^{\prime \prime }-x^{2} y-x^{2} = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
10.475 |
|
|
\[ {}y^{\prime \prime }-x^{2} y-x^{3} = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
7.586 |
|
|
\[ {}y^{\prime \prime }-x^{2} y-x^{4} = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.766 |
|
|
\[ {}y^{\prime \prime }-x^{2} y-x^{4}+2 = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
13.415 |
|
|
\[ {}y^{\prime \prime }-2 x^{2} y-x^{4}+1 = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
36.542 |
|
|
\[ {}y^{\prime \prime }-x^{3} y-x^{3} = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
16.328 |
|
|
\[ {}y^{\prime \prime }-x^{3} y-x^{4} = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
76.621 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }-x^{2} y-x^{2} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.956 |
|
|
\[ {}y^{\prime \prime }-x^{3} y^{\prime }-x^{3} y-x^{3} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
1.598 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-x = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.172 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }-x y-x^{2} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.187 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }-x^{2} y-x^{3}-x^{2} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.58 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }-x^{3} y-x^{4}-x^{2} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.812 |
|
|
\[ {}y^{\prime \prime }-\frac {y^{\prime }}{x}-x y-x^{2}-\frac {1}{x} = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.713 |
|
|
\[ {}y^{\prime \prime }-\frac {y^{\prime }}{x}-x^{2} y-x^{3}-\frac {1}{x} = 0 \] |
1 |
1 |
1 |
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 |
|
|
\[ {}y^{\prime \prime }-\frac {y^{\prime }}{x}-x^{3} y-x^{4}-\frac {1}{x} = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.737 |
|
|
\[ {}y^{\prime \prime }-x^{3} y^{\prime }-x y-x^{3}-x^{2} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
1.347 |
|
|
\[ {}y^{\prime \prime }-x^{3} y^{\prime }-x^{2} y-x^{3} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.845 |
|
|
\[ {}y^{\prime \prime }-x^{3} y^{\prime }-x^{3} y-x^{4}-x^{3} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.947 |
|
|
\[ {}y^{\prime \prime \prime }-x^{3} y^{\prime }-x^{2} y-x^{3} = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.062 |
|
|
\[ {}y^{\prime \prime }+c y^{\prime }+k y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.468 |
|
|
\[ {}w^{\prime } = -\frac {1}{2}-\frac {\sqrt {1-12 w}}{2} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.661 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.956 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.727 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.941 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.873 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.921 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.921 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.786 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.717 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
7.818 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.169 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.121 |
|
|
\[ {}y^{\prime \prime \prime }+y^{\prime }+y = x \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
67.554 |
|
|
\[ {}x^{4} y^{\prime \prime }+x^{3} y^{\prime }-4 x^{2} y = 1 \] |
1 |
1 |
1 |
kovacic, 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, _with_linear_symmetries]] |
✓ |
✓ |
1.953 |
|
|
\[ {}x^{4} y^{\prime \prime }+x^{3} y^{\prime }-4 x^{2} y = x \] |
1 |
1 |
1 |
kovacic, 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, _with_linear_symmetries]] |
✓ |
✓ |
1.736 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-4 y = x \] |
1 |
1 |
1 |
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, _with_linear_symmetries]] |
✓ |
✓ |
1.913 |
|
|
\[ {}x^{4} y^{\prime \prime \prime }+x^{3} y^{\prime \prime }+x^{2} y^{\prime }+x y = 0 \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.77 |
|
|
\[ {}x^{4} y^{\prime \prime \prime }+x^{3} y^{\prime \prime }+x^{2} y^{\prime }+x y = x \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.071 |
|
|
\[ {}5 x^{5} y^{\prime \prime \prime \prime }+4 x^{4} y^{\prime \prime \prime }+x^{2} y^{\prime }+x y = 0 \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
62.378 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+1+{y^{\prime }}^{2} = 0 \] |
1 |
1 |
1 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
1.336 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+1+{y^{\prime }}^{2} = x \] |
1 |
1 |
1 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✗ |
94.947 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+1+x {y^{\prime }}^{2} = 1 \] |
1 |
1 |
1 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
1.063 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+y {y^{\prime }}^{2} = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.071 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+{y^{\prime }}^{2} = 0 \] |
1 |
1 |
1 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
2.509 |
|
|
\[ {}y^{\prime \prime }+\sin \left (y\right ) {y^{\prime }}^{2} = 0 \] |
1 |
1 |
1 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.299 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+{y^{\prime }}^{3} = 0 \] |
1 |
2 |
2 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
2.088 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{-\frac {y}{x}} \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.303 |
|
|
\[ {}y^{\prime } = 2 x^{2} \sin \left (\frac {y}{x}\right )^{2}+\frac {y}{x} \] |
1 |
2 |
1 |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘]] |
✓ |
✓ |
4.247 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+y = 8 \sqrt {x}\, \left (1+\ln \left (x \right )\right ) \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.844 |
|
|
\[ {}v v^{\prime } = \frac {2 v^{2}}{r^{3}}+\frac {\lambda r}{3} \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
1.752 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.487 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = 1 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.492 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = 1+x \] |
1 |
0 |
0 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
❇ |
N/A |
1.358 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = x \] |
1 |
0 |
0 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
❇ |
N/A |
1.158 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = x^{2}+x +1 \] |
1 |
0 |
0 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
❇ |
N/A |
1.42 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = x^{2} \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.279 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = x^{2}+1 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.556 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = x^{4} \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.329 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = \sin \left (x \right ) \] |
1 |
0 |
0 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
❇ |
N/A |
1.541 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = \sin \left (x \right )+1 \] |
1 |
0 |
0 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
❇ |
N/A |
1.593 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = x \sin \left (x \right ) \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.678 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = \cos \left (x \right )+\sin \left (x \right ) \] |
1 |
0 |
0 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
❇ |
N/A |
1.996 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (\cos \left (x \right )-1\right ) y^{\prime }+{\mathrm e}^{x} y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Complex roots |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
16.961 |
|
|
\[ {}\left (-2+x \right ) y^{\prime \prime }+\frac {y^{\prime }}{x}+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.357 |
|
|
\[ {}\left (-2+x \right ) y^{\prime \prime }+\frac {y^{\prime }}{x}+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.547 |
|
|
\[ {}\left (1+x \right ) \left (3 x -1\right ) y^{\prime \prime }+\cos \left (x \right ) y^{\prime }-3 x y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
17.183 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+x y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Lienard] |
✓ |
✓ |
1.501 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+3 x y^{\prime }-x y = x^{2}+2 x \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.738 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = 1 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.22 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+2 x y^{\prime }-x y = 1 \] |
1 |
0 |
0 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _linear, _nonhomogeneous]] |
❇ |
N/A |
1.352 |
|
|
\[ {}y^{\prime \prime }+\left (x -6\right ) y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.036 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (3 x^{2}+2 x \right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.627 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = x^{2}+\cos \left (x \right ) \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.145 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = \cos \left (x \right ) \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.615 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = x^{3}+\cos \left (x \right ) \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.303 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = x^{3} \cos \left (x \right ) \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.842 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = x^{3} \cos \left (x \right )+\sin \left (x \right )^{2} \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
5.192 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = \ln \left (x \right ) \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.628 |
|
|
\[ {}2 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+x \left (11 x^{2}+11 x +9\right ) y^{\prime }+\left (7 x^{2}+10 x +6\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
6.01 |
|
|
\[ {}x^{2} \left (x +3\right ) y^{\prime \prime }+5 x \left (1+x \right ) y^{\prime }-\left (1-4 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.873 |
|
|
\[ {}x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }-x \left (4 x^{2}+3\right ) y^{\prime }+\left (-2 x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.339 |
|
|
\[ {}{y^{\prime }}^{2}+y^{2} = \sec \left (x \right )^{4} \] |
2 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
12.38 |
|
|
\[ {}\left (y-2 x y^{\prime }\right )^{2} = {y^{\prime }}^{3} \] |
3 |
8 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
276.817 |
|
|
\[ {}x^{2} y^{\prime \prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Complex roots |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.102 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime }-y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.472 |
|
|
\[ {}4 x y^{\prime \prime }+2 y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.714 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime }-y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.36 |
|
|
\[ {}x y^{\prime \prime }+\left (1+x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.688 |
|
|
\[ {}x \left (-1+x \right ) y^{\prime \prime }+3 x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
3.53 |
|
|
\[ {}x^{2} \left (x^{2}-2 x +1\right ) y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+\left (x +4\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.018 |
|
|
\[ {}2 x^{2} \left (2+x \right ) y^{\prime \prime }+5 x^{2} y^{\prime }+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.253 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+x y^{\prime }+\left (x -5\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.365 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+2 x y^{\prime }-x y = \sin \left (x \right ) \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.121 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+2 x y^{\prime }-x y = x \sin \left (x \right ) \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.728 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+2 x y^{\prime }-x y = \sin \left (x \right ) \cos \left (x \right ) \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.309 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+2 x y^{\prime }-x y = x^{3}+x \sin \left (x \right ) \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.411 |
|
|
\[ {}\cos \left (x \right ) y^{\prime \prime }+2 x y^{\prime }-x y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
63.965 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.919 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-x y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.385 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.74 |
|
|
\[ {}\left (x^{2}-x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.196 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2}+6 x \right ) y^{\prime }+x y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.009 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+\left (x^{2}-8\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.206 |
|
|
\[ {}x^{2} y^{\prime \prime }-9 x y^{\prime }+25 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.118 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }-\left (x^{2}+\frac {5}{4}\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.864 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.55 |
|
|
\[ {}x y^{\prime \prime }+\left (2-x \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.73 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+3 x y^{\prime }-y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.329 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+5 x y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Complex roots |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.525 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+4 x^{4} y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.769 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.93 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }+y^{\prime }+y = x \,{\mathrm e}^{x} \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
11.66 |
|
|
\[ {}y^{\prime } = y \left (1-y^{2}\right ) \] |
1 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
2.602 |
|
|
\[ {}\frac {x y^{\prime \prime }}{1-x}+y = \frac {1}{1-x} \] |
1 |
1 |
1 |
second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.831 |
|
|
\[ {}\frac {x y^{\prime \prime }}{1-x}+x y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.949 |
|
|
\[ {}\frac {x y^{\prime \prime }}{1-x}+y = \cos \left (x \right ) \] |
1 |
1 |
1 |
second_order_bessel_ode, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
5.763 |
|
|
\[ {}\frac {x y^{\prime \prime }}{-x^{2}+1}+y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.057 |
|
|
\[ {}y^{\prime \prime } = \left (x^{2}+3\right ) y \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.413 |
|
|
\[ {}y^{\prime \prime }+\left (-1+x \right ) y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.198 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x+2 y+2 t +1 \\ y^{\prime }=5 x+y+3 t -1 \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
2.147 |
|
|
\[ {}y^{\prime \prime }+20 y^{\prime }+500 y = 100000 \cos \left (100 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.781 |
|
|
\[ {}y^{\prime \prime } \sin \left (2 x \right )^{2}+y^{\prime } \sin \left (4 x \right )-4 y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
10.211 |
|
|
\[ {}y^{\prime \prime } = A y^{\frac {2}{3}} \] |
1 |
2 |
3 |
second_order_ode_missing_x, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
2.175 |
|
|
\[ {}y^{\prime \prime }+2 x y^{\prime }+\left (x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.978 |
|
|
\[ {}y^{\prime \prime }+2 \cot \left (x \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.969 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.065 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+\left (-8 x^{2}+4 x \right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y = 4 \sqrt {x}\, {\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.559 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x +2\right ) y^{\prime }+\left (2+x \right ) y = 6 x^{3} {\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.158 |
|
|
\[ {}y^{\prime }+y = \frac {1}{x} \] |
1 |
0 |
0 |
|
[[_linear, ‘class A‘]] |
❇ |
N/A |
0.693 |
|
|
\[ {}y^{\prime }+y = \frac {1}{x^{2}} \] |
1 |
0 |
0 |
|
[[_linear, ‘class A‘]] |
❇ |
N/A |
0.625 |
|
|
\[ {}x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
first order ode series method. Regular singular point |
[_separable] |
✓ |
✓ |
0.504 |
|
|
\[ {}y^{\prime } = \frac {1}{x} \] |
1 |
0 |
0 |
|
[_quadrature] |
❇ |
N/A |
0.284 |
|
|
\[ {}y^{\prime \prime } = \frac {1}{x} \] |
1 |
0 |
0 |
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 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = \frac {1}{x} \] |
1 |
0 |
0 |
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 |
|
|
\[ {}y^{\prime \prime }+y = \frac {1}{x} \] |
1 |
0 |
0 |
second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
❇ |
N/A |
0.362 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = \frac {1}{x} \] |
1 |
0 |
0 |
second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
❇ |
N/A |
0.787 |
|
|
\[ {}h^{2}+\frac {2 a h}{\sqrt {1+{h^{\prime }}^{2}}} = b^{2} \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
11.224 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }-24 y = 16-\left (2+x \right ) {\mathrm e}^{4 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.036 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }-4 y = 6 \,{\mathrm e}^{2 t -2} \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.359 |
|
|
\[ {}y^{\prime \prime }+y = {\mathrm e}^{a \cos \left (x \right )} \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.888 |
|
|
\[ {}y^{\prime } = \frac {y}{2 y \ln \left (y\right )+y-x} \] |
1 |
1 |
1 |
exact, differentialType, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
2.548 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x +1\right ) y^{\prime }+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.185 |
|
|
\[ {}x^{2} y^{\prime }+{\mathrm e}^{-y} = 0 \] |
1 |
1 |
1 |
exact, separable, first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.014 |
|
|
\[ {}y^{\prime \prime }+{\mathrm e}^{y} = 0 \] |
1 |
2 |
1 |
second_order_ode_missing_x, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
3.547 |
|
|
\[ {}y^{\prime } = \frac {x y+3 x -2 y+6}{x y-3 x -2 y+6} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
1.535 |
|
|
|
|||||||||
|
|
|||||||||
|
|