|
# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}y^{\prime } = f \left (x \right ) f^{\prime }\left (x \right )+f^{\prime }\left (x \right ) y \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.803 |
|
|
\[ {}y^{\prime } = f \left (x \right )+g \left (x \right ) y \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.711 |
|
|
\[ {}y^{\prime } = x^{2}-y^{2} \] |
riccati |
[_Riccati] |
✓ |
✓ |
1.545 |
|
|
\[ {}y^{\prime }+f \left (x \right )^{2} = f^{\prime }\left (x \right )+y^{2} \] |
riccati |
[_Riccati] |
✓ |
✓ |
0.612 |
|
|
\[ {}y^{\prime }+1-x = y \left (x +y\right ) \] |
riccati |
[_Riccati] |
✓ |
✓ |
1.854 |
|
|
\[ {}y^{\prime } = \left (x +y\right )^{2} \] |
riccati, homogeneousTypeC, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
0.727 |
|
|
\[ {}y^{\prime } = \left (x -y\right )^{2} \] |
riccati, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
1.586 |
|
|
\[ {}y^{\prime } = 3-3 x +3 y+\left (x -y\right )^{2} \] |
riccati, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
1.682 |
|
|
\[ {}y^{\prime } = 2 x -\left (x^{2}+1\right ) y+y^{2} \] |
riccati |
[_Riccati] |
✓ |
✓ |
2.368 |
|
|
\[ {}y^{\prime } = x \left (x^{3}+2\right )-\left (2 x^{2}-y\right ) y \] |
riccati, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
0.895 |
|
|
\[ {}y^{\prime } = 1+x \left (-x^{3}+2\right )+\left (2 x^{2}-y\right ) y \] |
riccati, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
1.597 |
|
|
\[ {}y^{\prime } = \cos \left (x \right )-\left (\sin \left (x \right )-y\right ) y \] |
riccati |
[_Riccati] |
✓ |
✓ |
4.085 |
|
|
\[ {}y^{\prime } = \cos \left (2 x \right )+\left (\sin \left (2 x \right )+y\right ) y \] |
riccati |
[_Riccati] |
✓ |
✓ |
8.227 |
|
|
\[ {}y^{\prime } = f \left (x \right )+x f \left (x \right ) y+y^{2} \] |
riccati |
[_Riccati] |
✓ |
✓ |
2.501 |
|
|
\[ {}y^{\prime } = \left (3+x -4 y\right )^{2} \] |
riccati, homogeneousTypeC, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
0.947 |
|
|
\[ {}y^{\prime } = \left (1+4 x +9 y\right )^{2} \] |
riccati, homogeneousTypeC, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
0.922 |
|
|
\[ {}y^{\prime } = 3 a +3 b x +3 b y^{2} \] |
riccati |
[_Riccati] |
✓ |
✓ |
2.067 |
|
|
\[ {}y^{\prime } = a +b y^{2} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.26 |
|
|
\[ {}y^{\prime } = x a +b y^{2} \] |
riccati |
[[_Riccati, _special]] |
✓ |
✓ |
1.648 |
|
|
\[ {}y^{\prime } = a +b x +c y^{2} \] |
riccati |
[_Riccati] |
✓ |
✓ |
1.926 |
|
|
\[ {}y^{\prime } = a \,x^{n -1}+b \,x^{2 n}+c y^{2} \] |
riccati |
[_Riccati] |
✓ |
✓ |
64.083 |
|
|
\[ {}y^{\prime } = x^{2} a +b y^{2} \] |
riccati |
[[_Riccati, _special]] |
✓ |
✓ |
1.826 |
|
|
\[ {}y^{\prime } = \operatorname {a0} +\operatorname {a1} y+\operatorname {a2} y^{2} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.44 |
|
|
\[ {}y^{\prime } = f \left (x \right )+a y+b y^{2} \] |
riccati |
[_Riccati] |
✓ |
✓ |
0.623 |
|
|
\[ {}y^{\prime } = 1+a \left (x -y\right ) y \] |
riccati |
[_Riccati] |
✓ |
✓ |
1.647 |
|
|
\[ {}y^{\prime } = f \left (x \right )+g \left (x \right ) y+a y^{2} \] |
riccati |
[_Riccati] |
✓ |
✓ |
0.804 |
|
|
\[ {}y^{\prime } = x y \left (3+y\right ) \] |
exact, riccati, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.513 |
|
|
\[ {}y^{\prime } = 1-x -x^{3}+\left (2 x^{2}+1\right ) y-x y^{2} \] |
riccati, first_order_ode_lie_symmetry_calculated |
[_Riccati] |
✓ |
✓ |
2.721 |
|
|
\[ {}y^{\prime } = x \left (2+x^{2} y-y^{2}\right ) \] |
riccati |
[_Riccati] |
✓ |
✓ |
1.99 |
|
|
\[ {}y^{\prime } = x +\left (1-2 x \right ) y-\left (1-x \right ) y^{2} \] |
riccati, first_order_ode_lie_symmetry_calculated |
[_Riccati] |
✓ |
✓ |
2.438 |
|
|
\[ {}y^{\prime } = a x y^{2} \] |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.592 |
|
|
\[ {}y^{\prime } = x^{n} \left (a +b y^{2}\right ) \] |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.323 |
|
|
\[ {}y^{\prime } = a \,x^{m}+b \,x^{n} y^{2} \] |
riccati |
[_Riccati] |
✓ |
✓ |
3.122 |
|
|
\[ {}y^{\prime } = \left (a +b y \cos \left (k x \right )\right ) y \] |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.233 |
|
|
\[ {}y^{\prime } = \sin \left (x \right ) \left (2 \sec \left (x \right )^{2}-y\right ) \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.546 |
|
|
\[ {}y^{\prime }+4 \csc \left (x \right ) = \left (3-\cot \left (x \right )\right ) y+y^{2} \sin \left (x \right ) \] |
riccati |
[_Riccati] |
✓ |
✓ |
9.078 |
|
|
\[ {}y^{\prime } = y \sec \left (x \right )+\left (\sin \left (x \right )-1\right )^{2} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
2.793 |
|
|
\[ {}y^{\prime }+\tan \left (x \right ) \left (1-y^{2}\right ) = 0 \] |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.99 |
|
|
\[ {}y^{\prime } = f \left (x \right )+g \left (x \right ) y+h \left (x \right ) y^{2} \] |
riccati |
[_Riccati] |
✓ |
✓ |
1.212 |
|
|
\[ {}y^{\prime } = \left (a +b y+c y^{2}\right ) f \left (x \right ) \] |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.484 |
|
|
\[ {}y^{\prime }+\left (x a +y\right ) y^{2} = 0 \] |
abelFirstKind |
[_Abel] |
✗ |
N/A |
1.5 |
|
|
\[ {}y^{\prime } = \left (a \,{\mathrm e}^{x}+y\right ) y^{2} \] |
abelFirstKind |
[_Abel] |
✗ |
N/A |
2.904 |
|
|
\[ {}y^{\prime }+3 a \left (y+2 x \right ) y^{2} = 0 \] |
abelFirstKind |
[_Abel] |
✗ |
N/A |
1.433 |
|
|
\[ {}y^{\prime } = y \left (a +b y^{2}\right ) \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.902 |
|
|
\[ {}y^{\prime } = \operatorname {a0} +\operatorname {a1} y+\operatorname {a2} y^{2}+\operatorname {a3} y^{3} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.193 |
|
|
\[ {}y^{\prime } = x y^{3} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.73 |
|
|
\[ {}y^{\prime }+y \left (1-x y^{2}\right ) = 0 \] |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
0.682 |
|
|
\[ {}y^{\prime } = \left (a +b x y\right ) y^{2} \] |
abelFirstKind, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _Abel] |
✓ |
✓ |
8.756 |
|
|
\[ {}y^{\prime }+2 x y \left (1+a x y^{2}\right ) = 0 \] |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
0.995 |
|
|
\[ {}y^{\prime }+\left (\tan \left (x \right )+y^{2} \sec \left (x \right )\right ) y = 0 \] |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.552 |
|
|
\[ {}y^{\prime }+y^{3} \sec \left (x \right ) \tan \left (x \right ) = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.954 |
|
|
\[ {}y^{\prime } = \operatorname {f0} \left (x \right )+\operatorname {f1} \left (x \right ) y+\operatorname {f2} \left (x \right ) y^{2}+\operatorname {f3} \left (x \right ) y^{3} \] |
abelFirstKind |
[_Abel] |
❇ |
N/A |
5.928 |
|
|
\[ {}y^{\prime } = a \,x^{\frac {n}{-n +1}}+b y^{n} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _Chini] |
✓ |
✓ |
1.043 |
|
|
\[ {}y^{\prime } = f \left (x \right ) y+g \left (x \right ) y^{k} \] |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
0.525 |
|
|
\[ {}y^{\prime } = f \left (x \right )+g \left (x \right ) y+h \left (x \right ) y^{n} \] |
unknown |
[_Chini] |
❇ |
N/A |
0.779 |
|
|
\[ {}y^{\prime } = \sqrt {{| y|}} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.488 |
|
|
\[ {}y^{\prime } = a +b y+\sqrt {\operatorname {A0} +\operatorname {B0} y} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
1.852 |
|
|
\[ {}y^{\prime } = x a +b \sqrt {y} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _Chini] |
✓ |
✓ |
3.365 |
|
|
\[ {}y^{\prime }+x^{3} = x \sqrt {x^{4}+4 y} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
1.598 |
|
|
\[ {}y^{\prime }+2 y \left (1-x \sqrt {y}\right ) = 0 \] |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.003 |
|
|
\[ {}y^{\prime } = \sqrt {a +b y^{2}} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.509 |
|
|
\[ {}y^{\prime } = y \sqrt {a +b y} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.464 |
|
|
\[ {}y^{\prime }+\left (f \left (x \right )-y\right ) g \left (x \right ) \sqrt {\left (y-a \right ) \left (y-b \right )} = 0 \] |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
1.296 |
|
|
\[ {}y^{\prime } = \sqrt {X Y} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.316 |
|
|
\[ {}y^{\prime } = \cos \left (x \right )^{2} \cos \left (y\right ) \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.817 |
|
|
\[ {}y^{\prime } = \sec \left (x \right )^{2} \cot \left (y\right ) \cos \left (y\right ) \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.894 |
|
|
\[ {}y^{\prime } = a +b \cos \left (A x +B y\right ) \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
90.74 |
|
|
\[ {}y^{\prime }+f \left (x \right )+g \left (x \right ) \sin \left (a y\right )+h \left (x \right ) \cos \left (a y\right ) = 0 \] |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
2.158 |
|
|
\[ {}y^{\prime } = a +b \cos \left (y\right ) \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.411 |
|
|
\[ {}y^{\prime }+x \left (\sin \left (2 y\right )-x^{2} \cos \left (y\right )^{2}\right ) = 0 \] |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
1.875 |
|
|
\[ {}y^{\prime }+\tan \left (x \right ) \sec \left (x \right ) \cos \left (y\right )^{2} = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.779 |
|
|
\[ {}y^{\prime } = \cot \left (x \right ) \cot \left (y\right ) \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.116 |
|
|
\[ {}y^{\prime }+\cot \left (x \right ) \cot \left (y\right ) = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.964 |
|
|
\[ {}y^{\prime } = \sin \left (x \right ) \left (\csc \left (y\right )-\cot \left (y\right )\right ) \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.105 |
|
|
\[ {}y^{\prime } = \tan \left (x \right ) \cot \left (y\right ) \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.789 |
|
|
\[ {}y^{\prime }+\tan \left (x \right ) \cot \left (y\right ) = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.756 |
|
|
\[ {}y^{\prime }+\sin \left (2 x \right ) \csc \left (2 y\right ) = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.735 |
|
|
\[ {}y^{\prime } = \tan \left (x \right ) \left (\tan \left (y\right )+\sec \left (x \right ) \sec \left (y\right )\right ) \] |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
1.905 |
|
|
\[ {}y^{\prime } = \cos \left (x \right ) \sec \left (y\right )^{2} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.069 |
|
|
\[ {}y^{\prime } = \sec \left (x \right )^{2} \sec \left (y\right )^{3} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.51 |
|
|
\[ {}y^{\prime } = a +b \sin \left (y\right ) \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.404 |
|
|
\[ {}y^{\prime } = \left (1+\cos \left (x \right ) \sin \left (y\right )\right ) \tan \left (y\right ) \] |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
2.441 |
|
|
\[ {}y^{\prime }+\csc \left (2 x \right ) \sin \left (2 y\right ) = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.963 |
|
|
\[ {}y^{\prime }+f \left (x \right )+g \left (x \right ) \tan \left (y\right ) = 0 \] |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
1.06 |
|
|
\[ {}y^{\prime } = \sqrt {a +b \cos \left (y\right )} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.765 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{y}+x \] |
first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
0.646 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{x +y} \] |
exact, separable, first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.585 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{x} \left (a +b \,{\mathrm e}^{-y}\right ) \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.281 |
|
|
\[ {}y^{\prime }+y \ln \left (x \right ) \ln \left (y\right ) = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.967 |
|
|
\[ {}y^{\prime } = x^{m -1} y^{-n +1} f \left (a \,x^{m}+b y^{n}\right ) \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.194 |
|
|
\[ {}y^{\prime } = a f \left (y\right ) \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.277 |
|
|
\[ {}y^{\prime } = f \left (a +b x +c y\right ) \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.694 |
|
|
\[ {}y^{\prime } = f \left (x \right ) g \left (y\right ) \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.615 |
|
|
\[ {}y^{\prime } = \sec \left (x \right )^{2}+y \sec \left (x \right ) \operatorname {Csx} \left (x \right ) \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.252 |
|
|
\[ {}2 y^{\prime } = 2 \sin \left (y\right )^{2} \tan \left (y\right )-x \sin \left (2 y\right ) \] |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
22.05 |
|
|
\[ {}2 y^{\prime }+x a = \sqrt {a^{2} x^{2}-4 b \,x^{2}-4 c y} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
2.02 |
|
|
\[ {}3 y^{\prime } = x +\sqrt {x^{2}-3 y} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
2.079 |
|
|
\[ {}x y^{\prime } = \sqrt {a^{2}-x^{2}} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.331 |
|
|
\[ {}x y^{\prime }+x +y = 0 \] |
exact, linear, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.751 |
|
|
\[ {}x y^{\prime }+x^{2}-y = 0 \] |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.603 |
|
|
|
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