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) |
|
\[ {}y^{\prime } = 2 x +1 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.385 |
|
|
\[ {}y^{\prime } = \left (-2+x \right )^{2} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.29 |
|
|
\[ {}y^{\prime } = \sqrt {x} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.321 |
|
|
\[ {}y^{\prime } = \frac {1}{x^{2}} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.26 |
|
|
\[ {}y^{\prime } = \frac {1}{\sqrt {2+x}} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.297 |
|
|
\[ {}y^{\prime } = x \sqrt {x^{2}+9} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.384 |
|
|
\[ {}y^{\prime } = \frac {10}{x^{2}+1} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.341 |
|
|
\[ {}y^{\prime } = \cos \left (2 x \right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.352 |
|
|
\[ {}y^{\prime } = \frac {1}{\sqrt {-x^{2}+1}} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.445 |
|
|
\[ {}y^{\prime } = x \,{\mathrm e}^{-x} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.187 |
|
|
\[ {}y^{\prime } = -\sin \left (x \right )-y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.63 |
|
|
\[ {}y^{\prime } = x +y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.374 |
|
|
\[ {}y^{\prime } = -\sin \left (x \right )+y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.482 |
|
|
\[ {}y^{\prime } = x -y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.379 |
|
|
\[ {}y^{\prime } = -x +y+1 \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.482 |
|
|
\[ {}y^{\prime } = x -y+1 \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.478 |
|
|
\[ {}y^{\prime } = x^{2}-y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.381 |
|
|
\[ {}y^{\prime } = -2+x^{2}-y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.549 |
|
|
\[ {}y^{\prime } = 2 x^{2} y^{2} \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.164 |
|
|
\[ {}y^{\prime } = x \ln \left (y\right ) \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.699 |
|
|
\[ {}y^{\prime } = y^{\frac {1}{3}} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.436 |
|
|
\[ {}y^{\prime } = y^{\frac {1}{3}} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.234 |
|
|
\[ {}y y^{\prime } = -1+x \] |
1 |
1 |
1 |
exact, separable, differentialType, homogeneousTypeMapleC, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
4.47 |
|
|
\[ {}y y^{\prime } = -1+x \] |
1 |
1 |
2 |
exact, separable, differentialType, homogeneousTypeMapleC, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.789 |
|
|
\[ {}y^{\prime } = \ln \left (1+y^{2}\right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.426 |
|
|
\[ {}y^{\prime } = x^{2}-y^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
1.438 |
|
|
\[ {}2 x y+y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.871 |
|
|
\[ {}2 x y^{2}+y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.578 |
|
|
\[ {}y^{\prime } = y \sin \left (x \right ) \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.98 |
|
|
\[ {}\left (1+x \right ) y^{\prime } = 4 y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, homogeneousTypeMapleC, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.167 |
|
|
\[ {}2 \sqrt {x}\, y^{\prime } = \sqrt {1-y^{2}} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.404 |
|
|
\[ {}y^{\prime } = 3 \sqrt {x y} \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
11.688 |
|
|
\[ {}y^{\prime } = 4 \left (x y\right )^{\frac {1}{3}} \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
91.221 |
|
|
\[ {}y^{\prime } = 2 x \sec \left (y\right ) \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.791 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime } = 2 y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.298 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime } = \left (y+1\right )^{2} \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.967 |
|
|
\[ {}y^{\prime } = x y^{3} \] |
1 |
1 |
2 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.805 |
|
|
\[ {}y y^{\prime } = x \left (1+y^{2}\right ) \] |
1 |
1 |
2 |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.931 |
|
|
\[ {}y^{\prime } = \frac {1+\sqrt {x}}{1+\sqrt {y}} \] |
1 |
1 |
1 |
exact, separable, differentialType, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
170.809 |
|
|
\[ {}y^{\prime } = \frac {\left (-1+x \right ) y^{5}}{x^{2} \left (-y+2 y^{3}\right )} \] |
1 |
1 |
3 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
18.824 |
|
|
\[ {}\left (x^{2}+1\right ) \tan \left (y\right ) y^{\prime } = x \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.432 |
|
|
\[ {}y^{\prime } = 1+x +y+x y \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.866 |
|
|
\[ {}x^{2} y^{\prime } = 1-x^{2}+y^{2}-x^{2} y^{2} \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.188 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{x} y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.815 |
|
|
\[ {}y^{\prime } = 3 x^{2} \left (1+y^{2}\right ) \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.2 |
|
|
\[ {}2 y y^{\prime } = \frac {x}{\sqrt {x^{2}-16}} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
5.32 |
|
|
\[ {}y^{\prime } = -y+4 x^{3} y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.009 |
|
|
\[ {}1+y^{\prime } = 2 y \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.476 |
|
|
\[ {}\tan \left (x \right ) y^{\prime } = y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.139 |
|
|
\[ {}-y+x y^{\prime } = 2 x^{2} y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.973 |
|
|
\[ {}y^{\prime } = 2 x y^{2}+3 x^{2} y^{2} \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.709 |
|
|
\[ {}y^{\prime } = 6 \,{\mathrm e}^{2 x -y} \] |
1 |
1 |
1 |
exact, separable, first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.887 |
|
|
\[ {}2 \sqrt {x}\, y^{\prime } = \cos \left (y\right )^{2} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.51 |
|
|
\[ {}y+y^{\prime } = 2 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.481 |
|
|
\[ {}-2 y+y^{\prime } = 3 \,{\mathrm e}^{2 x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.601 |
|
|
\[ {}3 y+y^{\prime } = 2 x \,{\mathrm e}^{-3 x} \] |
1 |
1 |
1 |
exact, linear, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.821 |
|
|
\[ {}y^{\prime }-2 x y = {\mathrm e}^{x^{2}} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.777 |
|
|
\[ {}2 y+x y^{\prime } = 3 x \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.953 |
|
|
\[ {}y+2 x y^{\prime } = 10 \sqrt {x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.926 |
|
|
\[ {}y+2 x y^{\prime } = 10 \sqrt {x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.698 |
|
|
\[ {}y+3 x y^{\prime } = 12 x \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.562 |
|
|
\[ {}-y+x y^{\prime } = x \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.707 |
|
|
\[ {}-3 y+2 x y^{\prime } = 9 x^{3} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.832 |
|
|
\[ {}x y^{\prime }+y = 3 x y \] |
1 |
0 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.724 |
|
|
\[ {}3 y+x y^{\prime } = 2 x^{5} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.616 |
|
|
\[ {}y+y^{\prime } = {\mathrm e}^{x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.529 |
|
|
\[ {}-3 y+x y^{\prime } = x^{3} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.654 |
|
|
\[ {}2 x y+y^{\prime } = x \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.03 |
|
|
\[ {}y^{\prime } = \cos \left (x \right ) \left (1-y\right ) \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.056 |
|
|
\[ {}y+\left (1+x \right ) y^{\prime } = \cos \left (x \right ) \] |
1 |
1 |
1 |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.727 |
|
|
\[ {}x y^{\prime } = x^{3} \cos \left (x \right )+2 y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.968 |
|
|
\[ {}\cot \left (x \right ) y+y^{\prime } = \cos \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.103 |
|
|
\[ {}y^{\prime } = 1+x +y+x y \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.757 |
|
|
\[ {}x y^{\prime } = x^{4} \cos \left (x \right )+3 y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.804 |
|
|
\[ {}y^{\prime } = 3 x^{2} {\mathrm e}^{x^{2}}+2 x y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.689 |
|
|
\[ {}\left (2 x -3\right ) y+x y^{\prime } = 4 x^{4} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.961 |
|
|
\[ {}3 x y+\left (x^{2}+4\right ) y^{\prime } = x \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.406 |
|
|
\[ {}3 x^{3} y+\left (x^{2}+1\right ) y^{\prime } = 6 x \,{\mathrm e}^{-\frac {3 x^{2}}{2}} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
3.8 |
|
|
\[ {}\left (x +y\right ) y^{\prime } = x -y \] |
1 |
1 |
2 |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.521 |
|
|
\[ {}2 x y y^{\prime } = x^{2}+y^{2} \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.878 |
|
|
\[ {}x y^{\prime } = y+2 \sqrt {x y} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.587 |
|
|
\[ {}\left (x -y\right ) y^{\prime } = x +y \] |
1 |
1 |
1 |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.955 |
|
|
\[ {}x \left (x +y\right ) y^{\prime } = y \left (x -y\right ) \] |
1 |
1 |
1 |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.985 |
|
|
\[ {}\left (2 y+x \right ) y^{\prime } = y \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.862 |
|
|
\[ {}x y^{2} y^{\prime } = x^{3}+y^{3} \] |
1 |
1 |
3 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.965 |
|
|
\[ {}x^{2} y^{\prime } = {\mathrm e}^{\frac {y}{x}} x^{2}+x y \] |
1 |
1 |
1 |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.047 |
|
|
\[ {}x^{2} y^{\prime } = x y+y^{2} \] |
1 |
1 |
1 |
riccati, bernoulli, exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.517 |
|
|
\[ {}x y y^{\prime } = x^{2}+3 y^{2} \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
2.107 |
|
|
\[ {}\left (x^{2}-y^{2}\right ) y^{\prime } = 2 x y \] |
1 |
1 |
2 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.424 |
|
|
\[ {}x y y^{\prime } = y^{2}+x \sqrt {4 x^{2}+y^{2}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.627 |
|
|
\[ {}x y^{\prime } = y+\sqrt {x^{2}+y^{2}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.341 |
|
|
\[ {}x +y y^{\prime } = \sqrt {x^{2}+y^{2}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.68 |
|
|
\[ {}y \left (3 x +y\right )+x \left (x +y\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.957 |
|
|
\[ {}y^{\prime } = \sqrt {1+x +y} \] |
1 |
1 |
1 |
homogeneousTypeC, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
2.654 |
|
|
\[ {}y^{\prime } = \left (4 x +y\right )^{2} \] |
1 |
1 |
1 |
riccati, homogeneousTypeC, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
0.915 |
|
|
\[ {}\left (x +y\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.12 |
|
|
\[ {}2 x y+x^{2} y^{\prime } = 5 y^{3} \] |
1 |
2 |
2 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
0.892 |
|
|
\[ {}2 x y^{3}+y^{2} y^{\prime } = 6 x \] |
1 |
1 |
3 |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.352 |
|
|
\[ {}y^{\prime } = y+y^{3} \] |
1 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.649 |
|
|
\[ {}2 x y+x^{2} y^{\prime } = 5 y^{4} \] |
1 |
3 |
3 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
1.335 |
|
|
\[ {}6 y+x y^{\prime } = 3 x y^{\frac {4}{3}} \] |
1 |
1 |
1 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
1.531 |
|
|
\[ {}y^{3} {\mathrm e}^{-2 x}+2 x y^{\prime } = 2 x y \] |
1 |
2 |
2 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.37 |
|
|
\[ {}\sqrt {x^{4}+1}\, y^{2} \left (x y^{\prime }+y\right ) = x \] |
1 |
1 |
3 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
10.416 |
|
|
\[ {}y^{3}+3 y^{2} y^{\prime } = {\mathrm e}^{-x} \] |
1 |
1 |
3 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_1st_order, _with_linear_symmetries], _Bernoulli] |
✓ |
✓ |
1.696 |
|
|
\[ {}3 x y^{2} y^{\prime } = 3 x^{4}+y^{3} \] |
1 |
1 |
3 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
1.539 |
|
|
\[ {}{\mathrm e}^{y} x y^{\prime } = 2 \,{\mathrm e}^{y}+2 \,{\mathrm e}^{2 x} x^{3} \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
2.675 |
|
|
\[ {}2 x \cos \left (y\right ) \sin \left (y\right ) y^{\prime } = 4 x^{2}+\sin \left (y\right )^{2} \] |
1 |
1 |
2 |
exactWithIntegrationFactor |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
4.665 |
|
|
\[ {}\left ({\mathrm e}^{y}+x \right ) y^{\prime } = -1+x \,{\mathrm e}^{-y} \] |
1 |
1 |
2 |
exact, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
2.134 |
|
|
\[ {}2 x +3 y+\left (3 x +2 y\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.327 |
|
|
\[ {}4 x -y+\left (-x +6 y\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.744 |
|
|
\[ {}3 x^{2}+2 y^{2}+\left (4 x y+6 y^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
3 |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
2.955 |
|
|
\[ {}3 x^{2}+2 x y^{2}+\left (2 x^{2} y+4 y^{3}\right ) y^{\prime } = 0 \] |
1 |
1 |
4 |
exact |
[_exact, _rational] |
✓ |
✓ |
2.605 |
|
|
\[ {}x^{3}+\frac {y}{x}+\left (\ln \left (x \right )+y^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
3 |
exact |
[_exact] |
✓ |
✓ |
2.945 |
|
|
\[ {}1+{\mathrm e}^{x y} y+\left ({\mathrm e}^{x y} x +2 y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
3.511 |
|
|
\[ {}\cos \left (x \right )+\ln \left (y\right )+\left ({\mathrm e}^{y}+\frac {x}{y}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
5.824 |
|
|
\[ {}x +\arctan \left (y\right )+\frac {\left (x +y\right ) y^{\prime }}{1+y^{2}} = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
3.591 |
|
|
\[ {}3 x^{2} y^{3}+y^{4}+\left (3 x^{3} y^{2}+4 x y^{3}+y^{4}\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exact |
[_exact, _rational] |
✓ |
✓ |
2.685 |
|
|
\[ {}{\mathrm e}^{x} \sin \left (y\right )+\tan \left (y\right )+\left ({\mathrm e}^{x} \cos \left (y\right )+x \sec \left (y\right )^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
24.46 |
|
|
\[ {}\frac {2 x}{y}-\frac {3 y^{2}}{x^{4}}+\left (-\frac {x^{2}}{y^{2}}+\frac {1}{\sqrt {y}}+\frac {2 y}{x^{3}}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact, _rational] |
✓ |
✓ |
28.327 |
|
|
\[ {}\frac {2 x^{\frac {5}{2}}-3 y^{\frac {5}{3}}}{2 x^{\frac {5}{2}} y^{\frac {2}{3}}}+\frac {\left (-2 x^{\frac {5}{2}}+3 y^{\frac {5}{3}}\right ) y^{\prime }}{3 x^{\frac {3}{2}} y^{\frac {5}{3}}} = 0 \] |
1 |
1 |
6 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_1st_order, _with_linear_symmetries], _exact, _rational] |
✓ |
✓ |
1.786 |
|
|
\[ {}x^{3}+3 y-x y^{\prime } = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.321 |
|
|
\[ {}3 y^{2}+x y^{2}-x^{2} y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.472 |
|
|
\[ {}x y+y^{2}-x^{2} y^{\prime } = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.591 |
|
|
\[ {}{\mathrm e}^{x}+2 x y^{3}+\left (\sin \left (y\right )+3 x^{2} y^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
4.056 |
|
|
\[ {}3 y+x^{4} y^{\prime } = 2 x y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.549 |
|
|
\[ {}2 x y^{2}+x^{2} y^{\prime } = y^{2} \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.87 |
|
|
\[ {}2 x^{2} y+x^{3} y^{\prime } = 1 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.77 |
|
|
\[ {}2 x y+x^{2} y^{\prime } = y^{2} \] |
1 |
1 |
1 |
riccati, bernoulli, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
0.942 |
|
|
\[ {}2 y+x y^{\prime } = 6 x^{2} \sqrt {y} \] |
1 |
1 |
1 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
1.185 |
|
|
\[ {}y^{\prime } = 1+x^{2}+y^{2}+x^{2} y^{2} \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.974 |
|
|
\[ {}x^{2} y^{\prime } = x y+3 y^{2} \] |
1 |
1 |
1 |
riccati, bernoulli, exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.075 |
|
|
\[ {}6 x y^{3}+2 y^{4}+\left (9 x^{2} y^{2}+8 x y^{3}\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
3.483 |
|
|
\[ {}y^{\prime } = 1+x^{2}+y^{2}+y^{4} x^{2} \] |
1 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
0.565 |
|
|
\[ {}x^{3} y^{\prime } = x^{2} y-y^{3} \] |
1 |
2 |
2 |
bernoulli, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.033 |
|
|
\[ {}3 y+y^{\prime } = 3 x^{2} {\mathrm e}^{-3 x} \] |
1 |
1 |
1 |
exact, linear, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.836 |
|
|
\[ {}y^{\prime } = x^{2}-2 x y+y^{2} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
1.438 |
|
|
\[ {}{\mathrm e}^{x}+{\mathrm e}^{x y} y+\left ({\mathrm e}^{y}+{\mathrm e}^{x y} x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
3.445 |
|
|
\[ {}2 x^{2} y-x^{3} y^{\prime } = y^{3} \] |
1 |
2 |
2 |
bernoulli, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.299 |
|
|
\[ {}3 x^{5} y^{2}+x^{3} y^{\prime } = 2 y^{2} \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.846 |
|
|
\[ {}3 y+x y^{\prime } = \frac {3}{x^{\frac {3}{2}}} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.844 |
|
|
\[ {}\left (-1+x \right ) y+\left (x^{2}-1\right ) y^{\prime } = 1 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.962 |
|
|
\[ {}x y^{\prime } = 12 x^{4} y^{\frac {2}{3}}+6 y \] |
1 |
1 |
1 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
1.676 |
|
|
\[ {}{\mathrm e}^{y}+y \cos \left (x \right )+\left ({\mathrm e}^{y} x +\sin \left (x \right )\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
39.608 |
|
|
\[ {}9 x^{2} y^{2}+x^{\frac {3}{2}} y^{\prime } = y^{2} \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.177 |
|
|
\[ {}2 y+\left (1+x \right ) y^{\prime } = 3+3 x \] |
1 |
1 |
1 |
linear, homogeneousTypeMapleC, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.444 |
|
|
\[ {}9 \sqrt {x}\, y^{\frac {4}{3}}-12 x^{\frac {1}{5}} y^{\frac {3}{2}}+\left (8 x^{\frac {3}{2}} y^{\frac {1}{3}}-15 x^{\frac {6}{5}} \sqrt {y}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[[_homogeneous, ‘class G‘], _exact, _rational] |
✓ |
✓ |
0.479 |
|
|
\[ {}3 y+x^{3} y^{4}+3 x y^{\prime } = 0 \] |
1 |
1 |
3 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
1.399 |
|
|
\[ {}x y^{\prime }+y = 2 \,{\mathrm e}^{2 x} \] |
1 |
1 |
1 |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.766 |
|
|
\[ {}y+\left (2 x +1\right ) y^{\prime } = \left (2 x +1\right )^{\frac {3}{2}} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.029 |
|
|
\[ {}y^{\prime } = 3 x^{2} \left (7+y\right ) \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.994 |
|
|
\[ {}y^{\prime } = 3 x^{2} \left (7+y\right ) \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.671 |
|
|
\[ {}y^{\prime } = -x y+x y^{3} \] |
1 |
1 |
2 |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.451 |
|
|
\[ {}y^{\prime } = \frac {-3 x^{2}-2 y^{2}}{4 x y} \] |
1 |
1 |
2 |
exact, bernoulli, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.454 |
|
|
\[ {}y^{\prime } = \frac {x +3 y}{-3 x +y} \] |
1 |
1 |
2 |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.993 |
|
|
\[ {}y^{\prime } = \frac {2 x +2 x y}{x^{2}+1} \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.07 |
|
|
\[ {}y^{\prime } = \cot \left (x \right ) \left (\sqrt {y}-y\right ) \] |
1 |
1 |
1 |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
7.763 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.06 |
|
|
\[ {}y^{\prime \prime }-9 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.185 |
|
|
\[ {}y^{\prime \prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
4.809 |
|
|
\[ {}y^{\prime \prime }+25 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
5.093 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.616 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-6 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.635 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.687 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime } = 0 \] |
1 |
0 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.519 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.833 |
|
|
\[ {}y^{\prime \prime }-10 y^{\prime }+25 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.798 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.676 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+13 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.882 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
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_1, second_order_change_of_variable_on_y_method_2, linear_second_order_ode_solved_by_an_integrating_factor, second_order_ode_non_constant_coeff_transformation_on_B |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
3.006 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }-6 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.575 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+y = 0 \] |
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, second_order_ode_non_constant_coeff_transformation_on_B |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.498 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+y = 0 \] |
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], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.27 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.285 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }-15 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.296 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime } = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.936 |
|
|
\[ {}2 y^{\prime \prime }+3 y^{\prime } = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.956 |
|
|
\[ {}2 y^{\prime \prime }-y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.318 |
|
|
\[ {}4 y^{\prime \prime }+8 y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.314 |
|
|
\[ {}4 y^{\prime \prime }+4 y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.385 |
|
|
\[ {}9 y^{\prime \prime }-12 y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.401 |
|
|
\[ {}6 y^{\prime \prime }-7 y^{\prime }-20 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.334 |
|
|
\[ {}35 y^{\prime \prime }-y^{\prime }-12 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.324 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, exact linear second order ode, second_order_integrable_as_is, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.293 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }-12 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2 |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.148 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+8 x y^{\prime }-3 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.186 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime } = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.728 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = 0 \] |
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], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.418 |
|
|
\[ {}y^{\prime \prime }+y = 3 x \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.932 |
|
|
\[ {}y^{\prime \prime }-4 y = 12 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
3.01 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-3 y = 6 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.842 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = 2 x \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.994 |
|
|
\[ {}y^{\prime \prime }+2 y = 4 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.594 |
|
|
\[ {}y^{\prime \prime }+2 y = 6 x \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.67 |
|
|
\[ {}y^{\prime \prime }+2 y = 6 x +4 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.758 |
|
|
\[ {}y^{\prime \prime }-4 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.984 |
|
|
\[ {}2 y^{\prime \prime }-3 y^{\prime } = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.945 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }-10 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.304 |
|
|
\[ {}2 y^{\prime \prime }-7 y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.319 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.375 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime }+5 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.381 |
|
|
\[ {}4 y^{\prime \prime }-12 y^{\prime }+9 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.401 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+13 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.44 |
|
|
\[ {}y^{\prime \prime }+8 y^{\prime }+25 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.476 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.623 |
|
|
\[ {}9 y^{\prime \prime }+6 y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.474 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+25 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.906 |
|
|
\[ {}y^{\prime \prime }-2 i y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.474 |
|
|
\[ {}y^{\prime \prime }-i y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.506 |
|
|
\[ {}y^{\prime \prime } = \left (-2+2 i \sqrt {3}\right ) y \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
3.022 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+9 y = 0 \] |
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], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.455 |
|
|
\[ {}x^{2} y^{\prime \prime }+7 x y^{\prime }+25 y = 0 \] |
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]] |
✓ |
✓ |
1.691 |
|
|
\[ {}\frac {x^{\prime \prime }}{2}+3 x^{\prime }+4 x = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.688 |
|
|
\[ {}3 x^{\prime \prime }+30 x^{\prime }+63 x = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.693 |
|
|
\[ {}x^{\prime \prime }+8 x^{\prime }+16 x = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.893 |
|
|
\[ {}2 x^{\prime \prime }+12 x^{\prime }+50 x = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.862 |
|
|
\[ {}4 x^{\prime \prime }+20 x^{\prime }+169 x = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.981 |
|
|
\[ {}2 x^{\prime \prime }+16 x^{\prime }+40 x = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.924 |
|
|
\[ {}x^{\prime \prime }+10 x^{\prime }+125 x = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.961 |
|
|
\[ {}y^{\prime \prime }+16 y = {\mathrm e}^{3 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.674 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 3 x +4 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.579 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-6 y = 2 \sin \left (3 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.759 |
|
|
\[ {}4 y^{\prime \prime }+4 y^{\prime }+y = 3 x \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.76 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = \sin \left (x \right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.785 |
|
|
\[ {}2 y^{\prime \prime }+4 y^{\prime }+7 y = x^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.003 |
|
|
\[ {}y^{\prime \prime }-4 y = \sinh \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.197 |
|
|
\[ {}y^{\prime \prime }-4 y = \cosh \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.093 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }-3 y = 1+x \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.8 |
|
|
\[ {}y^{\prime \prime }+9 y = 2 \cos \left (3 x \right )+3 \sin \left (3 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.103 |
|
|
\[ {}y^{\prime \prime }+9 y = 2 x^{2} {\mathrm e}^{3 x}+5 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.863 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = {\mathrm e}^{x} \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.76 |
|
|
\[ {}y^{\prime \prime }+4 y = 3 x \cos \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.976 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = x \left ({\mathrm e}^{-x}-{\mathrm e}^{-2 x}\right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.869 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+13 y = x \,{\mathrm e}^{3 x} \sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.028 |
|
|
\[ {}y^{\prime \prime }+4 y = 2 x \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.961 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = {\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.975 |
|
|
\[ {}y^{\prime \prime }+9 y = \sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.267 |
|
|
\[ {}y^{\prime \prime }+y = \cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.081 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = 1+x \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.989 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = \sin \left (x \right ) \sin \left (3 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.503 |
|
|
\[ {}y^{\prime \prime }+9 y = \sin \left (x \right )^{4} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.718 |
|
|
\[ {}y^{\prime \prime }+y = x \cos \left (x \right )^{3} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.723 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = 4 \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.541 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-8 y = 3 \,{\mathrm e}^{-2 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.586 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = 2 \,{\mathrm e}^{2 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.689 |
|
|
\[ {}y^{\prime \prime }-4 y = \sinh \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.061 |
|
|
\[ {}y^{\prime \prime }+4 y = \cos \left (3 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.715 |
|
|
\[ {}y^{\prime \prime }+9 y = \sin \left (3 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.756 |
|
|
\[ {}y^{\prime \prime }+9 y = 2 \sec \left (3 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.964 |
|
|
\[ {}y^{\prime \prime }+y = \csc \left (x \right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.841 |
|
|
\[ {}y^{\prime \prime }+4 y = \sin \left (x \right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.084 |
|
|
\[ {}y^{\prime \prime }-4 y = x \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.585 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = 72 x^{5} \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, exact linear second order ode, second_order_integrable_as_is, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.015 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = x^{3} \] |
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_1, second_order_change_of_variable_on_y_method_2, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.511 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = x^{4} \] |
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]] |
✓ |
✓ |
2.28 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+3 y = 8 x^{\frac {4}{3}} \] |
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_1, second_order_change_of_variable_on_y_method_2, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.498 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+y = \ln \left (x \right ) \] |
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]] |
✓ |
✓ |
6.312 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = x^{2}-1 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.895 |
|
|
\[ {}x^{\prime \prime }+9 x = 10 \cos \left (2 t \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.181 |
|
|
\[ {}x^{\prime \prime }+4 x = 5 \sin \left (3 t \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.231 |
|
|
\[ {}x^{\prime \prime }+100 x = 225 \cos \left (5 t \right )+300 \sin \left (5 t \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.939 |
|
|
\[ {}x^{\prime \prime }+25 x = 90 \cos \left (4 t \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.28 |
|
|
\[ {}m x^{\prime \prime }+k x = F_{0} \cos \left (\omega t \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.039 |
|
|
\[ {}x^{\prime \prime }+4 x^{\prime }+4 x = 10 \cos \left (3 t \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.993 |
|
|
\[ {}x^{\prime \prime }+3 x^{\prime }+5 x = -4 \cos \left (5 t \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.577 |
|
|
\[ {}2 x^{\prime \prime }+2 x^{\prime }+x = 3 \sin \left (10 t \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.009 |
|
|
\[ {}x^{\prime \prime }+3 x^{\prime }+3 x = 8 \cos \left (10 t \right )+6 \sin \left (10 t \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.528 |
|
|
\[ {}x^{\prime \prime }+4 x^{\prime }+5 x = 10 \cos \left (3 t \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.598 |
|
|
\[ {}x^{\prime \prime }+6 x^{\prime }+13 x = 10 \sin \left (5 t \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.824 |
|
|
\[ {}x^{\prime \prime }+6 x^{\prime }+13 x = 10 \sin \left (5 t \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.03 |
|
|
\[ {}x^{\prime \prime }+2 x^{\prime }+26 x = 600 \cos \left (10 t \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.022 |
|
|
\[ {}x^{\prime \prime }+8 x^{\prime }+25 x = 200 \cos \left (t \right )+520 \sin \left (t \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.733 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-3 y \\ y^{\prime }=3 x \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.593 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=3 x-2 y \\ y^{\prime }=2 x+y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
1.081 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=2 x+4 y+3 \,{\mathrm e}^{t} \\ y^{\prime }=5 x-y-t^{2} \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
2.159 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=y+z \\ y^{\prime }=z+x \\ z^{\prime }=x+y \end {array}\right ] \] |
1 |
1 |
3 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.736 |
|
|
\[ {}\left [\begin {array}{c} x_{1}^{\prime }=x_{2} \\ x_{2}^{\prime }=2 x_{3} \\ x_{3}^{\prime }=3 x_{4} \\ x_{4}^{\prime }=4 x_{1} \end {array}\right ] \] |
1 |
1 |
4 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
3.901 |
|
|
\[ {}\left [\begin {array}{c} x_{1}^{\prime }=x_{2}+x_{3}+1 \\ x_{2}^{\prime }=x_{3}+x_{4}+t \\ x_{3}^{\prime }=x_{1}+x_{4}+t^{2} \\ x_{4}^{\prime }=x_{1}+x_{2}+t^{3} \end {array}\right ] \] |
1 |
1 |
4 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
7.091 |
|
|
|
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