|
# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}y^{\prime }-\sqrt {\frac {a y^{2}+b y+c}{x^{2} a +b x +c}} = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
293.068 |
|
|
\[ {}y^{\prime }-\sqrt {\frac {y^{3}+1}{x^{3}+1}} = 0 \] |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
43.512 |
|
|
\[ {}y^{\prime }-\frac {\sqrt {{| y \left (y-1\right ) \left (-1+a y\right )|}}}{\sqrt {{| x \left (-1+x \right ) \left (x a -1\right )|}}} = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.278 |
|
|
\[ {}y^{\prime }-\frac {\sqrt {1-y^{4}}}{\sqrt {-x^{4}+1}} = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
4.066 |
|
|
\[ {}y^{\prime }-\sqrt {\frac {a y^{4}+b y^{2}+1}{a \,x^{4}+b \,x^{2}+1}} = 0 \] |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
5.464 |
|
|
\[ {}y^{\prime }-\sqrt {\left (b_{4} y^{4}+b_{3} y^{3}+b_{2} y^{2}+b_{1} y+b_{0} \right ) \left (a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0} \right )} = 0 \] |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✗ |
57.172 |
|
|
\[ {}y^{\prime }-\sqrt {\frac {a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}{b_{4} y^{4}+b_{3} y^{3}+b_{2} y^{2}+b_{1} y+b_{0}}} = 0 \] |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✗ |
47.033 |
|
|
\[ {}y^{\prime }-\sqrt {\frac {b_{4} y^{4}+b_{3} y^{3}+b_{2} y^{2}+b_{1} y+b_{0}}{a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}} = 0 \] |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✗ |
43.478 |
|
|
\[ {}y^{\prime }-\operatorname {R1} \left (x , \sqrt {a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}\right ) \operatorname {R2} \left (y, \sqrt {b_{4} y^{4}+b_{3} y^{3}+b_{2} y^{2}+b_{1} y+b_{0}}\right ) = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.336 |
|
|
\[ {}y^{\prime }-\left (\frac {a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}{a_{3} y^{3}+a_{2} y^{2}+a_{1} y+a_{0}}\right )^{\frac {2}{3}} = 0 \] |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
6.01 |
|
|
\[ {}y^{\prime }-f \left (x \right ) \left (y-g \left (x \right )\right ) \sqrt {\left (y-a \right ) \left (y-b \right )} = 0 \] |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
1.742 |
|
|
\[ {}y^{\prime }-{\mathrm e}^{x -y}+{\mathrm e}^{x} = 0 \] |
exact, separable, first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.88 |
|
|
\[ {}y^{\prime }-a \cos \left (y\right )+b = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.566 |
|
|
\[ {}y^{\prime }-\cos \left (b x +a y\right ) = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
78.877 |
|
|
\[ {}y^{\prime }+a \sin \left (\alpha y+\beta x \right )+b = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
101.241 |
|
|
\[ {}y^{\prime }+f \left (x \right ) \cos \left (a y\right )+g \left (x \right ) \sin \left (a y\right )+h \left (x \right ) = 0 \] |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
3.526 |
|
|
\[ {}y^{\prime }+f \left (x \right ) \sin \left (y\right )+\left (1-f^{\prime }\left (x \right )\right ) \cos \left (y\right )-f^{\prime }\left (x \right )-1 = 0 \] |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
0.924 |
|
|
\[ {}y^{\prime }+2 \tan \left (y\right ) \tan \left (x \right )-1 = 0 \] |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
2.215 |
|
|
\[ {}y^{\prime }-a \left (1+\tan \left (y\right )^{2}\right )+\tan \left (y\right ) \tan \left (x \right ) = 0 \] |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
4.063 |
|
|
\[ {}y^{\prime }-\tan \left (x y\right ) = 0 \] |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
0.944 |
|
|
\[ {}y^{\prime }-f \left (x a +b y\right ) = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.006 |
|
|
\[ {}y^{\prime }-x^{a -1} y^{-b +1} f \left (\frac {x^{a}}{a}+\frac {y^{b}}{b}\right ) = 0 \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.996 |
|
|
\[ {}y^{\prime }-\frac {y-x f \left (x^{2}+a y^{2}\right )}{x +a y f \left (x^{2}+a y^{2}\right )} = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
2.263 |
|
|
\[ {}y^{\prime }-\frac {y a f \left (x^{c} y\right )+c \,x^{a} y^{b}}{x b f \left (x^{c} y\right )-x^{a} y^{b}} = 0 \] |
unknown |
[NONE] |
❇ |
N/A |
3.453 |
|
|
\[ {}2 y^{\prime }-3 y^{2}-4 a y-b -c \,{\mathrm e}^{-2 x a} = 0 \] |
riccati |
[_Riccati] |
✓ |
✓ |
4.635 |
|
|
\[ {}x y^{\prime }-\sqrt {a^{2}-x^{2}} = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.408 |
|
|
\[ {}x y^{\prime }+y-x \sin \left (x \right ) = 0 \] |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.964 |
|
|
\[ {}x y^{\prime }-y-\frac {x}{\ln \left (x \right )} = 0 \] |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.069 |
|
|
\[ {}x y^{\prime }-y-x^{2} \sin \left (x \right ) = 0 \] |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.191 |
|
|
\[ {}x y^{\prime }-y-\frac {x \cos \left (\ln \left (\ln \left (x \right )\right )\right )}{\ln \left (x \right )} = 0 \] |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
3.878 |
|
|
\[ {}x y^{\prime }+a y+b \,x^{n} = 0 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.228 |
|
|
\[ {}x y^{\prime }+y^{2}+x^{2} = 0 \] |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.49 |
|
|
\[ {}x y^{\prime }-y^{2}+1 = 0 \] |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.039 |
|
|
\[ {}x y^{\prime }+a y^{2}-y+b \,x^{2} = 0 \] |
riccati, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
1.899 |
|
|
\[ {}x y^{\prime }+a y^{2}-b y+c \,x^{2 b} = 0 \] |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
2.335 |
|
|
\[ {}x y^{\prime }+a y^{2}-b y-c \,x^{\beta } = 0 \] |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
2.734 |
|
|
\[ {}x y^{\prime }+x y^{2}+a = 0 \] |
riccati |
[_rational, [_Riccati, _special]] |
✓ |
✓ |
1.539 |
|
|
\[ {}x y^{\prime }+x y^{2}-y = 0 \] |
riccati, bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
1.065 |
|
|
\[ {}x y^{\prime }+x y^{2}-y-a \,x^{3} = 0 \] |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
3.173 |
|
|
\[ {}x y^{\prime }+x y^{2}-\left (2 x^{2}+1\right ) y-x^{3} = 0 \] |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
3.859 |
|
|
\[ {}x y^{\prime }+a x y^{2}+2 y+b x = 0 \] |
riccati |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
1.866 |
|
|
\[ {}x y^{\prime }+a x y^{2}+b y+c x +d = 0 \] |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
6.303 |
|
|
\[ {}x y^{\prime }+x^{a} y^{2}+\frac {\left (-b +a \right ) y}{2}+x^{b} = 0 \] |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
2.518 |
|
|
\[ {}x y^{\prime }+a \,x^{\alpha } y^{2}+b y-c \,x^{\beta } = 0 \] |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
4.059 |
|
|
\[ {}x y^{\prime }-y^{2} \ln \left (x \right )+y = 0 \] |
riccati, bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.306 |
|
|
\[ {}x y^{\prime }-y \left (2 y \ln \left (x \right )-1\right ) = 0 \] |
riccati, bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.316 |
|
|
\[ {}x y^{\prime }+f \left (x \right ) \left (-x^{2}+y^{2}\right ) = 0 \] |
riccati |
[_Riccati] |
✓ |
✓ |
1.879 |
|
|
\[ {}x y^{\prime }+y^{3}+3 x y^{2} = 0 \] |
abelFirstKind |
[_rational, _Abel] |
✗ |
N/A |
1.629 |
|
|
\[ {}x y^{\prime }-\sqrt {x^{2}+y^{2}}-y = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.572 |
|
|
\[ {}x y^{\prime }+a \sqrt {x^{2}+y^{2}}-y = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
4.931 |
|
|
\[ {}x y^{\prime }-x \sqrt {x^{2}+y^{2}}-y = 0 \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.835 |
|
|
\[ {}x y^{\prime }-x \left (y-x \right ) \sqrt {x^{2}+y^{2}}-y = 0 \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
2.317 |
|
|
\[ {}x y^{\prime }-x \,{\mathrm e}^{\frac {y}{x}}-y-x = 0 \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.657 |
|
|
\[ {}x y^{\prime }-y \ln \left (y\right ) = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.248 |
|
|
\[ {}x y^{\prime }-y \left (\ln \left (x y\right )-1\right ) = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
1.928 |
|
|
\[ {}x y^{\prime }-y \left (x \ln \left (\frac {x^{2}}{y}\right )+2\right ) = 0 \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.674 |
|
|
\[ {}x y^{\prime }-\sin \left (x -y\right ) = 0 \] |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
2.619 |
|
|
\[ {}x y^{\prime }+\left (\sin \left (y\right )-3 x^{2} \cos \left (y\right )\right ) \cos \left (y\right ) = 0 \] |
exactWithIntegrationFactor |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
3.234 |
|
|
\[ {}x y^{\prime }-x \sin \left (\frac {y}{x}\right )-y = 0 \] |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.385 |
|
|
\[ {}x y^{\prime }+x \cos \left (\frac {y}{x}\right )-y+x = 0 \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.705 |
|
|
\[ {}x y^{\prime }+x \tan \left (\frac {y}{x}\right )-y = 0 \] |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.679 |
|
|
\[ {}x y^{\prime }-y f \left (x y\right ) = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
1.169 |
|
|
\[ {}x y^{\prime }-y f \left (x^{a} y^{b}\right ) = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
1.609 |
|
|
\[ {}x y^{\prime }+a y-f \left (x \right ) g \left (x^{a} y\right ) = 0 \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.882 |
|
|
\[ {}\left (1+x \right ) y^{\prime }+y \left (y-x \right ) = 0 \] |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
1.158 |
|
|
\[ {}2 x y^{\prime }-y-2 x^{3} = 0 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.897 |
|
|
\[ {}\left (2 x +1\right ) y^{\prime }-4 \,{\mathrm e}^{-y}+2 = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.796 |
|
|
\[ {}3 x y^{\prime }-3 x \ln \left (x \right ) y^{4}-y = 0 \] |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
2.128 |
|
|
\[ {}x^{2} y^{\prime }+y-x = 0 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.943 |
|
|
\[ {}x^{2} y^{\prime }-y+x^{2} {\mathrm e}^{x -\frac {1}{x}} = 0 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.091 |
|
|
\[ {}x^{2} y^{\prime }-\left (-1+x \right ) y = 0 \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.096 |
|
|
\[ {}x^{2} y^{\prime }+y^{2}+x y+x^{2} = 0 \] |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
1.293 |
|
|
\[ {}x^{2} y^{\prime }-y^{2}-x y = 0 \] |
riccati, bernoulli, exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.117 |
|
|
\[ {}x^{2} y^{\prime }-y^{2}-x y-x^{2} = 0 \] |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
1.257 |
|
|
\[ {}x^{2} \left (y^{\prime }+y^{2}\right )+a \,x^{k}-b \left (b -1\right ) = 0 \] |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
3.506 |
|
|
\[ {}x^{2} \left (y^{\prime }+y^{2}\right )+4 x y+2 = 0 \] |
riccati, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
1.753 |
|
|
\[ {}x^{2} \left (y^{\prime }+y^{2}\right )+a x y+b = 0 \] |
riccati, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
2.507 |
|
|
\[ {}x^{2} \left (y^{\prime }-y^{2}\right )-a \,x^{2} y+x a +2 = 0 \] |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
2.374 |
|
|
\[ {}x^{2} \left (y^{\prime }+a y^{2}\right )-b = 0 \] |
riccati, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]] |
✓ |
✓ |
2.336 |
|
|
\[ {}x^{2} \left (y^{\prime }+a y^{2}\right )+b \,x^{\alpha }+c = 0 \] |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
3.549 |
|
|
\[ {}x^{2} y^{\prime }+a y^{3}-a \,x^{2} y^{2} = 0 \] |
abelFirstKind |
[_rational, _Abel] |
✗ |
N/A |
2.115 |
|
|
\[ {}x^{2} y^{\prime }+x y^{3}+a y^{2} = 0 \] |
abelFirstKind |
[_rational, _Abel] |
✗ |
N/A |
2.149 |
|
|
\[ {}x^{2} y^{\prime }+a \,x^{2} y^{3}+b y^{2} = 0 \] |
abelFirstKind |
[_rational, _Abel] |
✗ |
N/A |
3.003 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime }+x y-1 = 0 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.001 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime }+x y-\left (x^{2}+1\right ) x = 0 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.016 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime }+2 x y-2 x^{2} = 0 \] |
exact, linear, differentialType, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.036 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime }+\left (1+y^{2}\right ) \left (2 x y-1\right ) = 0 \] |
abelFirstKind |
[_rational, _Abel] |
✗ |
N/A |
66.421 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime }+x \sin \left (y\right ) \cos \left (y\right )-x \left (x^{2}+1\right ) \cos \left (y\right )^{2} = 0 \] |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
4.674 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime }-x y+a = 0 \] |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.601 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime }+2 x y-\cos \left (x \right ) = 0 \] |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
2.161 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime }+y^{2}-2 x y+1 = 0 \] |
riccati, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
1.912 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime }-y \left (y-x \right ) = 0 \] |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
1.102 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime }+a \left (y^{2}-2 x y+1\right ) = 0 \] |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
2.945 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime }+a x y^{2}+x y = 0 \] |
exact, riccati, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.3 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime }-2 x y \ln \left (y\right ) = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.702 |
|
|
\[ {}\left (x^{2}-4\right ) y^{\prime }+\left (2+x \right ) y^{2}-4 y = 0 \] |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
1.078 |
|
|
\[ {}\left (x^{2}-5 x +6\right ) y^{\prime }+3 x y-8 y+x^{2} = 0 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.147 |
|
|
\[ {}\left (x -a \right ) \left (-b +x \right ) y^{\prime }+y^{2}+k \left (y+x -a \right ) \left (y+x -b \right ) = 0 \] |
riccati, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
5.833 |
|
|
\[ {}2 x^{2} y^{\prime }-2 y^{2}-x y+2 x \,a^{2} = 0 \] |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.762 |
|
|
\[ {}2 x^{2} y^{\prime }-2 y^{2}-3 x y+2 x \,a^{2} = 0 \] |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
2.631 |
|
|
|
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