Number of problems in this table is 941
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) |
|
\[ {}4 y^{2} = x^{2} {y^{\prime }}^{2} \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
3.048 |
|
|
\[ {}x y {y^{\prime }}^{2}+\left (x +y\right ) y^{\prime }+1 = 0 \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.992 |
|
|
\[ {}1+\left (2 y-x^{2}\right ) {y^{\prime }}^{2}-2 x^{2} y {y^{\prime }}^{2} = 0 \] |
2 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
4.369 |
|
|
\[ {}x \left ({y^{\prime }}^{2}-1\right ) = 2 y y^{\prime } \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.832 |
|
|
\[ {}\left (1-y^{2}\right ) {y^{\prime }}^{2} = 1 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.979 |
|
|
\[ {}x y {y^{\prime }}^{2}+\left (x y-1\right ) y^{\prime } = y \] |
2 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
9.173 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}+x y y^{\prime }-2 x^{2} = 0 \] |
2 |
2 |
4 |
separable |
[_separable] |
✓ |
✓ |
1.722 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+2 y^{2} = x^{2} \] |
2 |
6 |
6 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.738 |
|
|
\[ {}{y^{\prime }}^{3}+\left (x +y-2 x y\right ) {y^{\prime }}^{2}-2 y^{\prime } x y \left (x +y\right ) = 0 \] |
3 |
1 |
3 |
linear, quadrature, separable |
[_quadrature] |
✓ |
✓ |
1.778 |
|
|
\[ {}y {y^{\prime }}^{2}+\left (y^{2}-x^{3}-x y^{2}\right ) y^{\prime }-x y \left (x^{2}+y^{2}\right ) = 0 \] |
2 |
1 |
3 |
quadrature, first_order_ode_lie_symmetry_lookup |
[_quadrature] |
✓ |
✓ |
1.98 |
|
|
\[ {}y = y^{\prime } x \left (y^{\prime }+1\right ) \] |
2 |
3 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.954 |
|
|
\[ {}y = x +3 \ln \left (y^{\prime }\right ) \] |
0 |
2 |
2 |
dAlembert, separable |
[_separable] |
✓ |
✓ |
3.73 |
|
|
\[ {}y \left (1+{y^{\prime }}^{2}\right ) = 2 \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
2.722 |
|
|
\[ {}y {y^{\prime }}^{2}-2 x y^{\prime }+y = 0 \] |
2 |
5 |
7 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.189 |
|
|
\[ {}{y^{\prime }}^{2}+y^{2} = 1 \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.214 |
|
|
\[ {}x \left ({y^{\prime }}^{2}-1\right ) = 2 y y^{\prime } \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.735 |
|
|
\[ {}4 x -2 y y^{\prime }+x {y^{\prime }}^{2} = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.728 |
|
|
\[ {}2 x^{2} y+{y^{\prime }}^{2} = x^{3} y^{\prime } \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
12.752 |
|
|
\[ {}y {y^{\prime }}^{2} = 3 x y^{\prime }+y \] |
2 |
4 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.712 |
|
|
\[ {}8 x +1 = y {y^{\prime }}^{2} \] |
2 |
5 |
2 |
dAlembert |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
✓ |
3.06 |
|
|
\[ {}y {y^{\prime }}^{2}+2 y^{\prime }+1 = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.917 |
|
|
\[ {}\left (1+{y^{\prime }}^{2}\right ) x = \left (x +y\right ) y^{\prime } \] |
2 |
3 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.98 |
|
|
\[ {}x^{2}-3 y y^{\prime }+x {y^{\prime }}^{2} = 0 \] |
2 |
2 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
15.878 |
|
|
\[ {}y+2 x y^{\prime } = x {y^{\prime }}^{2} \] |
2 |
4 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.326 |
|
|
\[ {}x = {y^{\prime }}^{2}+y^{\prime } \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.754 |
|
|
\[ {}x = y-{y^{\prime }}^{3} \] |
3 |
4 |
3 |
dAlembert, first_order_nonlinear_p_but_linear_in_x_y |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
2.114 |
|
|
\[ {}x +2 y y^{\prime } = x {y^{\prime }}^{2} \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.735 |
|
|
\[ {}4 x -2 y y^{\prime }+x {y^{\prime }}^{2} = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.733 |
|
|
\[ {}x {y^{\prime }}^{3} = y y^{\prime }+1 \] |
3 |
4 |
3 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
152.212 |
|
|
\[ {}y \left (1+{y^{\prime }}^{2}\right ) = 2 x y^{\prime } \] |
2 |
5 |
7 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.17 |
|
|
\[ {}2 x +x {y^{\prime }}^{2} = 2 y y^{\prime } \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.827 |
|
|
\[ {}x = y y^{\prime }+{y^{\prime }}^{2} \] |
2 |
4 |
2 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
1.601 |
|
|
\[ {}4 x {y^{\prime }}^{2}+2 x y^{\prime } = y \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.678 |
|
|
\[ {}y = y^{\prime } x \left (y^{\prime }+1\right ) \] |
2 |
3 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.906 |
|
|
\[ {}2 x {y^{\prime }}^{3}+1 = y {y^{\prime }}^{2} \] |
3 |
4 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
32.186 |
|
|
\[ {}{y^{\prime }}^{3}+x y y^{\prime } = 2 y^{2} \] |
3 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
173.565 |
|
|
\[ {}3 {y^{\prime }}^{4} x = {y^{\prime }}^{3} y+1 \] |
4 |
1 |
1 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
2.234 |
|
|
\[ {}2 {y^{\prime }}^{5}+2 x y^{\prime } = y \] |
5 |
2 |
6 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
0.499 |
|
|
\[ {}\frac {1}{{y^{\prime }}^{2}}+x y^{\prime } = 2 y \] |
3 |
4 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
111.202 |
|
|
\[ {}2 y = 3 x y^{\prime }+4+2 \ln \left (y^{\prime }\right ) \] |
0 |
2 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
2.101 |
|
|
\[ {}y = x y^{\prime }+{y^{\prime }}^{2} \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.156 |
|
|
\[ {}y = x y^{\prime }+\frac {1}{y^{\prime }} \] |
2 |
3 |
3 |
clairaut |
[[_homogeneous, ‘class G‘], _rational, _Clairaut] |
✓ |
✓ |
0.181 |
|
|
\[ {}y = x y^{\prime }-\sqrt {y^{\prime }} \] |
2 |
2 |
2 |
clairaut |
[[_homogeneous, ‘class G‘], _Clairaut] |
✓ |
✓ |
0.559 |
|
|
\[ {}y = x y^{\prime }+\ln \left (y^{\prime }\right ) \] |
0 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.453 |
|
|
\[ {}y = x y^{\prime }+\frac {3}{{y^{\prime }}^{2}} \] |
3 |
4 |
4 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.516 |
|
|
\[ {}y = x y^{\prime }-{y^{\prime }}^{\frac {2}{3}} \] |
3 |
2 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
1.579 |
|
|
\[ {}y = x y^{\prime }+{\mathrm e}^{y^{\prime }} \] |
0 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.355 |
|
|
\[ {}\left (y-x y^{\prime }\right )^{2} = 1+{y^{\prime }}^{2} \] |
2 |
4 |
4 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
0.42 |
|
|
\[ {}x {y^{\prime }}^{2}-y y^{\prime }-2 = 0 \] |
2 |
3 |
3 |
clairaut |
[[_homogeneous, ‘class G‘], _rational, _Clairaut] |
✓ |
✓ |
0.205 |
|
|
\[ {}y^{2}-2 x y y^{\prime }+{y^{\prime }}^{2} \left (x^{2}-1\right ) = 0 \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.351 |
|
|
\[ {}{y^{\prime }}^{2}-y^{2} = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.22 |
|
|
\[ {}{y^{\prime }}^{2}-3 y^{\prime }+2 = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.122 |
|
|
\[ {}y = y^{\prime }+\frac {{y^{\prime }}^{2}}{2} \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.148 |
|
|
\[ {}\left (y-x y^{\prime }\right )^{2} = 1+{y^{\prime }}^{2} \] |
2 |
4 |
4 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
0.375 |
|
|
\[ {}y-x = {y^{\prime }}^{2} \left (1-\frac {2 y^{\prime }}{3}\right ) \] |
3 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
2.214 |
|
|
\[ {}x {y^{\prime }}^{3}-y {y^{\prime }}^{2}+1 = 0 \] |
3 |
4 |
4 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
1.201 |
|
|
\[ {}y = x y^{\prime }+{y^{\prime }}^{3} \] |
3 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.59 |
|
|
\[ {}x \left ({y^{\prime }}^{2}-1\right ) = 2 y^{\prime } \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.622 |
|
|
\[ {}x y^{\prime } \left (y^{\prime }+2\right ) = y \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.412 |
|
|
\[ {}x = y^{\prime } \sqrt {1+{y^{\prime }}^{2}} \] |
4 |
4 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
20.843 |
|
|
\[ {}2 {y^{\prime }}^{2} \left (y-x y^{\prime }\right ) = 1 \] |
3 |
4 |
4 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.995 |
|
|
\[ {}y = 2 x y^{\prime }+y^{2} {y^{\prime }}^{3} \] |
3 |
1 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
84.592 |
|
|
\[ {}{y^{\prime }}^{3}+y^{2} = x y y^{\prime } \] |
3 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
111.411 |
|
|
\[ {}2 x y^{\prime }-y = y^{\prime } \ln \left (y y^{\prime }\right ) \] |
0 |
1 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
2.922 |
|
|
\[ {}y = x y^{\prime }-x^{2} {y^{\prime }}^{3} \] |
3 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
92.662 |
|
|
\[ {}y \left (y-2 x y^{\prime }\right )^{3} = {y^{\prime }}^{2} \] |
3 |
1 |
6 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
103.855 |
|
|
\[ {}x y^{\prime }+y = 4 \sqrt {y^{\prime }} \] |
2 |
3 |
2 |
dAlembert |
[[_homogeneous, ‘class G‘], _dAlembert] |
✓ |
✓ |
11.989 |
|
|
\[ {}2 x y^{\prime }-y = \ln \left (y^{\prime }\right ) \] |
0 |
2 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
2.606 |
|
|
\[ {}5 y+{y^{\prime }}^{2} = x \left (x +y^{\prime }\right ) \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
4.176 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{\frac {x y^{\prime }}{y}} \] |
0 |
2 |
1 |
dAlembert, homogeneousTypeD2 |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
2.599 |
|
|
\[ {}{y^{\prime }}^{2} = a \,x^{n} \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.273 |
|
|
\[ {}{y^{\prime }}^{2} = y \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.28 |
|
|
\[ {}{y^{\prime }}^{2} = x -y \] |
2 |
2 |
1 |
dAlembert, first_order_nonlinear_p_but_linear_in_x_y |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.408 |
|
|
\[ {}{y^{\prime }}^{2} = y+x^{2} \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
2.932 |
|
|
\[ {}{y^{\prime }}^{2}+x^{2} = 4 y \] |
2 |
2 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
2.751 |
|
|
\[ {}{y^{\prime }}^{2}+3 x^{2} = 8 y \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
3.812 |
|
|
\[ {}{y^{\prime }}^{2}+x^{2} a +b y = 0 \] |
2 |
1 |
0 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
2.79 |
|
|
\[ {}{y^{\prime }}^{2} = 1+y^{2} \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.377 |
|
|
\[ {}{y^{\prime }}^{2} = 1-y^{2} \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.512 |
|
|
\[ {}{y^{\prime }}^{2} = a^{2}-y^{2} \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.484 |
|
|
\[ {}{y^{\prime }}^{2} = y^{2} a^{2} \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.398 |
|
|
\[ {}{y^{\prime }}^{2} = a +b y^{2} \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.543 |
|
|
\[ {}{y^{\prime }}^{2} = x^{2} y^{2} \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.364 |
|
|
\[ {}{y^{\prime }}^{2} = \left (y-1\right ) y^{2} \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.499 |
|
|
\[ {}{y^{\prime }}^{2} = \left (y-a \right ) \left (y-b \right ) \left (y-c \right ) \] |
2 |
2 |
5 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.505 |
|
|
\[ {}{y^{\prime }}^{2} = a^{2} y^{n} \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.709 |
|
|
\[ {}{y^{\prime }}^{2} = a^{2} \left (1-\ln \left (y\right )^{2}\right ) y^{2} \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.108 |
|
|
\[ {}{y^{\prime }}^{2}+f \left (x \right ) \left (y-a \right ) \left (y-b \right ) = 0 \] |
2 |
2 |
2 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
1.375 |
|
|
\[ {}{y^{\prime }}^{2}+f \left (x \right ) \left (y-a \right )^{2} \left (y-b \right ) = 0 \] |
2 |
2 |
2 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
0.869 |
|
|
\[ {}{y^{\prime }}^{2}+f \left (x \right ) \left (y-a \right ) \left (y-b \right ) \left (y-c \right ) = 0 \] |
2 |
2 |
2 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
1.198 |
|
|
\[ {}{y^{\prime }}^{2}+f \left (x \right ) \left (y-a \right )^{2} \left (y-b \right ) \left (y-c \right ) = 0 \] |
2 |
2 |
2 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
4.577 |
|
|
\[ {}{y^{\prime }}^{2} = f \left (x \right )^{2} \left (y-a \right ) \left (y-b \right ) \left (y-c \right )^{2} \] |
2 |
2 |
2 |
first_order_nonlinear_p_but_separable |
[_separable] |
✓ |
✓ |
5.11 |
|
|
\[ {}{y^{\prime }}^{2} = f \left (x \right )^{2} \left (y-u \left (x \right )\right )^{2} \left (y-a \right ) \left (y-b \right ) \] |
2 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
2.323 |
|
|
\[ {}{y^{\prime }}^{2}+2 y^{\prime }+x = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.255 |
|
|
\[ {}{y^{\prime }}^{2}-2 y^{\prime }+a \left (x -y\right ) = 0 \] |
2 |
2 |
2 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.232 |
|
|
\[ {}{y^{\prime }}^{2}-2 y^{\prime }-y^{2} = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.415 |
|
|
\[ {}{y^{\prime }}^{2}-5 y^{\prime }+6 = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.228 |
|
|
\[ {}{y^{\prime }}^{2}-7 y^{\prime }+12 = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.191 |
|
|
\[ {}{y^{\prime }}^{2}+a y^{\prime }+b = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.222 |
|
|
\[ {}{y^{\prime }}^{2}+a y^{\prime }+b x = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.295 |
|
|
\[ {}{y^{\prime }}^{2}+a y^{\prime }+b y = 0 \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.654 |
|
|
\[ {}{y^{\prime }}^{2}+x y^{\prime }+1 = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.368 |
|
|
\[ {}{y^{\prime }}^{2}+x y^{\prime }-y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.206 |
|
|
\[ {}{y^{\prime }}^{2}-x y^{\prime }+y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.213 |
|
|
\[ {}{y^{\prime }}^{2}-x y^{\prime }-y = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.34 |
|
|
\[ {}{y^{\prime }}^{2}+x y^{\prime }+x -y = 0 \] |
2 |
2 |
1 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.307 |
|
|
\[ {}{y^{\prime }}^{2}+\left (1-x \right ) y^{\prime }+y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.199 |
|
|
\[ {}{y^{\prime }}^{2}-\left (1+x \right ) y^{\prime }+y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.222 |
|
|
\[ {}{y^{\prime }}^{2}-\left (2-x \right ) y^{\prime }+1-y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.239 |
|
|
\[ {}{y^{\prime }}^{2}+\left (x +a \right ) y^{\prime }-y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.212 |
|
|
\[ {}{y^{\prime }}^{2}-2 x y^{\prime }+1 = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.351 |
|
|
\[ {}{y^{\prime }}^{2}+2 x y^{\prime }-3 x^{2} = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.238 |
|
|
\[ {}{y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.315 |
|
|
\[ {}{y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.279 |
|
|
\[ {}{y^{\prime }}^{2}-2 x y^{\prime }+2 y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.225 |
|
|
\[ {}{y^{\prime }}^{2}-\left (2 x +1\right ) y^{\prime }-x \left (1-x \right ) = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.326 |
|
|
\[ {}{y^{\prime }}^{2}+2 \left (1-x \right ) y^{\prime }-2 x +2 y = 0 \] |
2 |
2 |
1 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.299 |
|
|
\[ {}{y^{\prime }}^{2}+3 x y^{\prime }-y = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.36 |
|
|
\[ {}{y^{\prime }}^{2}-4 \left (1+x \right ) y^{\prime }+4 y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.201 |
|
|
\[ {}{y^{\prime }}^{2}+a x y^{\prime } = b c \,x^{2} \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.256 |
|
|
\[ {}{y^{\prime }}^{2}-a x y^{\prime }+a y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.234 |
|
|
\[ {}{y^{\prime }}^{2}+a x y^{\prime }+b \,x^{2}+c y = 0 \] |
2 |
1 |
0 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
4.085 |
|
|
\[ {}{y^{\prime }}^{2}+\left (b x +a \right ) y^{\prime }+c = b y \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.229 |
|
|
\[ {}{y^{\prime }}^{2}-2 x^{2} y^{\prime }+2 x y^{\prime } = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.228 |
|
|
\[ {}{y^{\prime }}^{2}+a \,x^{3} y^{\prime }-2 a \,x^{2} y = 0 \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
9.748 |
|
|
\[ {}{y^{\prime }}^{2}-2 a \,x^{3} y^{\prime }+4 a \,x^{2} y = 0 \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
5.714 |
|
|
\[ {}{y^{\prime }}^{2}+4 x^{5} y^{\prime }-12 x^{4} y = 0 \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
3.308 |
|
|
\[ {}{y^{\prime }}^{2}-2 y^{\prime } \cosh \left (x \right )+1 = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.265 |
|
|
\[ {}y y^{\prime }+{y^{\prime }}^{2} = \left (x +y\right ) x \] |
2 |
1 |
2 |
linear, quadrature |
[_quadrature] |
✓ |
✓ |
0.347 |
|
|
\[ {}{y^{\prime }}^{2}-y y^{\prime }+{\mathrm e}^{x} = 0 \] |
2 |
2 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
3.498 |
|
|
\[ {}{y^{\prime }}^{2}+\left (x +y\right ) y^{\prime }+x y = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.279 |
|
|
\[ {}{y^{\prime }}^{2}-2 y y^{\prime }-2 x = 0 \] |
2 |
4 |
2 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
0.417 |
|
|
\[ {}{y^{\prime }}^{2}+\left (1+2 y\right ) y^{\prime }+y \left (y-1\right ) = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
142.175 |
|
|
\[ {}{y^{\prime }}^{2}-2 \left (x -y\right ) y^{\prime }-4 x y = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.291 |
|
|
\[ {}{y^{\prime }}^{2}-\left (4 y+1\right ) y^{\prime }+\left (4 y+1\right ) y = 0 \] |
2 |
2 |
5 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.342 |
|
|
\[ {}{y^{\prime }}^{2}-2 \left (-3 y+1\right ) y^{\prime }-\left (4-9 y\right ) y = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.604 |
|
|
\[ {}{y^{\prime }}^{2}+\left (a +6 y\right ) y^{\prime }+y \left (3 a +b +9 y\right ) = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.172 |
|
|
\[ {}{y^{\prime }}^{2}+a y y^{\prime }-x a = 0 \] |
2 |
4 |
2 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
0.776 |
|
|
\[ {}{y^{\prime }}^{2}-a y y^{\prime }-x a = 0 \] |
2 |
4 |
2 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
0.467 |
|
|
\[ {}{y^{\prime }}^{2}+\left (x a +b y\right ) y^{\prime }+a b x y = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.336 |
|
|
\[ {}{y^{\prime }}^{2}-x y y^{\prime }+y^{2} \ln \left (a y\right ) = 0 \] |
2 |
2 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
4.219 |
|
|
\[ {}{y^{\prime }}^{2}-\left (1+2 x y\right ) y^{\prime }+2 x y = 0 \] |
2 |
1 |
2 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.297 |
|
|
\[ {}{y^{\prime }}^{2}-\left (4+y^{2}\right ) y^{\prime }+4+y^{2} = 0 \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.833 |
|
|
\[ {}{y^{\prime }}^{2}-\left (x -y\right ) y y^{\prime }-x y^{3} = 0 \] |
2 |
1 |
2 |
quadrature, separable |
[_separable] |
✓ |
✓ |
0.333 |
|
|
\[ {}{y^{\prime }}^{2}+x y^{2} y^{\prime }+y^{3} = 0 \] |
2 |
1 |
6 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
75.362 |
|
|
\[ {}{y^{\prime }}^{2}-2 x^{3} y^{2} y^{\prime }-4 x^{2} y^{3} = 0 \] |
2 |
1 |
6 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
18.745 |
|
|
\[ {}{y^{\prime }}^{2}-x y \left (x^{2}+y^{2}\right ) y^{\prime }+x^{4} y^{4} = 0 \] |
2 |
2 |
3 |
separable |
[_separable] |
✓ |
✓ |
0.554 |
|
|
\[ {}{y^{\prime }}^{2}+2 x y^{3} y^{\prime }+y^{4} = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
4.079 |
|
|
\[ {}{y^{\prime }}^{2}+2 y y^{\prime } \cot \left (x \right )-y^{2} = 0 \] |
2 |
2 |
3 |
separable |
[_separable] |
✓ |
✓ |
4.494 |
|
|
\[ {}{y^{\prime }}^{2}-3 x y^{\frac {2}{3}} y^{\prime }+9 y^{\frac {5}{3}} = 0 \] |
2 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
6.318 |
|
|
\[ {}{y^{\prime }}^{2} = {\mathrm e}^{4 x -2 y} \left (y^{\prime }-1\right ) \] |
2 |
2 |
2 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.48 |
|
|
\[ {}2 {y^{\prime }}^{2}+x y^{\prime }-2 y = 0 \] |
2 |
3 |
1 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.306 |
|
|
\[ {}2 {y^{\prime }}^{2}-\left (1-x \right ) y^{\prime }-y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.204 |
|
|
\[ {}2 {y^{\prime }}^{2}-2 x^{2} y^{\prime }+3 x y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
7.216 |
|
|
\[ {}2 {y^{\prime }}^{2}+2 \left (6 y-1\right ) y^{\prime }+3 y \left (6 y-1\right ) = 0 \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.972 |
|
|
\[ {}3 {y^{\prime }}^{2}-2 x y^{\prime }+y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.305 |
|
|
\[ {}3 {y^{\prime }}^{2}+4 x y^{\prime }+x^{2}-y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
1.894 |
|
|
\[ {}4 {y^{\prime }}^{2} = 9 x \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.227 |
|
|
\[ {}4 {y^{\prime }}^{2}+2 x \,{\mathrm e}^{-2 y} y^{\prime }-{\mathrm e}^{-2 y} = 0 \] |
2 |
2 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
11.534 |
|
|
\[ {}4 {y^{\prime }}^{2}+2 \,{\mathrm e}^{2 x -2 y} y^{\prime }-{\mathrm e}^{2 x -2 y} = 0 \] |
2 |
2 |
2 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.481 |
|
|
\[ {}5 {y^{\prime }}^{2}+3 x y^{\prime }-y = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.383 |
|
|
\[ {}5 {y^{\prime }}^{2}+6 x y^{\prime }-2 y = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.349 |
|
|
\[ {}9 {y^{\prime }}^{2}+3 x y^{4} y^{\prime }+y^{5} = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
82.548 |
|
|
\[ {}x {y^{\prime }}^{2} = a \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.234 |
|
|
\[ {}x {y^{\prime }}^{2} = -x^{2}+a \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.574 |
|
|
\[ {}x {y^{\prime }}^{2} = y \] |
2 |
3 |
3 |
dAlembert, first_order_nonlinear_p_but_separable |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.632 |
|
|
\[ {}x {y^{\prime }}^{2}+x -2 y = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.435 |
|
|
\[ {}x {y^{\prime }}^{2}+y^{\prime } = y \] |
2 |
4 |
1 |
dAlembert |
[_rational, _dAlembert] |
✓ |
✓ |
0.362 |
|
|
\[ {}x {y^{\prime }}^{2}+2 y^{\prime }-y = 0 \] |
2 |
4 |
1 |
dAlembert |
[_rational, _dAlembert] |
✓ |
✓ |
0.366 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y^{\prime }-y = 0 \] |
2 |
4 |
1 |
dAlembert |
[_rational, _dAlembert] |
✓ |
✓ |
0.329 |
|
|
\[ {}x {y^{\prime }}^{2}+4 y^{\prime }-2 y = 0 \] |
2 |
4 |
1 |
dAlembert |
[_rational, _dAlembert] |
✓ |
✓ |
0.349 |
|
|
\[ {}x {y^{\prime }}^{2}+x y^{\prime }-y = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.375 |
|
|
\[ {}x {y^{\prime }}^{2}-\left (x^{2}+1\right ) y^{\prime }+x = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.241 |
|
|
\[ {}x {y^{\prime }}^{2}+y y^{\prime }+a = 0 \] |
2 |
2 |
2 |
dAlembert |
[[_homogeneous, ‘class G‘], _dAlembert] |
✓ |
✓ |
0.438 |
|
|
\[ {}x {y^{\prime }}^{2}-y y^{\prime }+a = 0 \] |
2 |
3 |
3 |
clairaut |
[[_homogeneous, ‘class G‘], _rational, _Clairaut] |
✓ |
✓ |
0.263 |
|
|
\[ {}x {y^{\prime }}^{2}-y y^{\prime }+x a = 0 \] |
2 |
2 |
1 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.434 |
|
|
\[ {}x {y^{\prime }}^{2}+y y^{\prime }+x^{3} = 0 \] |
2 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
8.079 |
|
|
\[ {}x {y^{\prime }}^{2}-y y^{\prime }+a y = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.448 |
|
|
\[ {}x {y^{\prime }}^{2}+y y^{\prime }-y^{4} = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
2.631 |
|
|
\[ {}x {y^{\prime }}^{2}+\left (-y+a \right ) y^{\prime }+b = 0 \] |
2 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
0.294 |
|
|
\[ {}x {y^{\prime }}^{2}+\left (x -y\right ) y^{\prime }+1-y = 0 \] |
2 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _dAlembert] |
✓ |
✓ |
0.258 |
|
|
\[ {}x {y^{\prime }}^{2}+\left (a +x -y\right ) y^{\prime }-y = 0 \] |
2 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _dAlembert] |
✓ |
✓ |
0.279 |
|
|
\[ {}x {y^{\prime }}^{2}-\left (3 x -y\right ) y^{\prime }+y = 0 \] |
2 |
4 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.56 |
|
|
\[ {}x {y^{\prime }}^{2}+a +b x -y-b y = 0 \] |
2 |
3 |
1 |
dAlembert |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
✓ |
1.33 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y y^{\prime }+a = 0 \] |
2 |
2 |
3 |
dAlembert |
[[_homogeneous, ‘class G‘], _rational, _dAlembert] |
✓ |
✓ |
0.435 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y y^{\prime }+x a = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.293 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y y^{\prime }+x +2 y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.315 |
|
|
\[ {}x {y^{\prime }}^{2}-3 y y^{\prime }+9 x^{2} = 0 \] |
2 |
2 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
5.328 |
|
|
\[ {}x {y^{\prime }}^{2}-\left (2 x +3 y\right ) y^{\prime }+6 y = 0 \] |
2 |
1 |
2 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.382 |
|
|
\[ {}x {y^{\prime }}^{2}-a y y^{\prime }+b = 0 \] |
2 |
2 |
2 |
dAlembert |
[[_homogeneous, ‘class G‘], _rational, _dAlembert] |
✓ |
✓ |
54.294 |
|
|
\[ {}x {y^{\prime }}^{2}+a y y^{\prime }+b x = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.714 |
|
|
\[ {}x {y^{\prime }}^{2}-\left (x y+1\right ) y^{\prime }+y = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.249 |
|
|
\[ {}x {y^{\prime }}^{2}+\left (1-x \right ) y y^{\prime }-y^{2} = 0 \] |
2 |
1 |
2 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.321 |
|
|
\[ {}x {y^{\prime }}^{2}+\left (1-x^{2} y\right ) y^{\prime }-x y = 0 \] |
2 |
1 |
2 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.332 |
|
|
\[ {}\left (1+x \right ) {y^{\prime }}^{2} = y \] |
2 |
3 |
3 |
dAlembert, first_order_nonlinear_p_but_separable |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
✓ |
0.651 |
|
|
\[ {}\left (1+x \right ) {y^{\prime }}^{2}-\left (x +y\right ) y^{\prime }+y = 0 \] |
2 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _dAlembert] |
✓ |
✓ |
0.295 |
|
|
\[ {}\left (a -x \right ) {y^{\prime }}^{2}+y y^{\prime }-b = 0 \] |
2 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.297 |
|
|
\[ {}2 x {y^{\prime }}^{2}+\left (2 x -y\right ) y^{\prime }+1-y = 0 \] |
2 |
3 |
1 |
dAlembert |
[_rational, _dAlembert] |
✓ |
✓ |
0.555 |
|
|
\[ {}3 x {y^{\prime }}^{2}-6 y y^{\prime }+x +2 y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.287 |
|
|
\[ {}\left (1+3 x \right ) {y^{\prime }}^{2}-3 \left (y+2\right ) y^{\prime }+9 = 0 \] |
2 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.311 |
|
|
\[ {}\left (3 x +5\right ) {y^{\prime }}^{2}-\left (3+3 y\right ) y^{\prime }+y = 0 \] |
2 |
3 |
2 |
dAlembert |
[_rational, _dAlembert] |
✓ |
✓ |
1.172 |
|
|
\[ {}4 x {y^{\prime }}^{2} = \left (a -3 x \right )^{2} \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.289 |
|
|
\[ {}4 x {y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.264 |
|
|
\[ {}4 x {y^{\prime }}^{2}-3 y y^{\prime }+3 = 0 \] |
2 |
2 |
4 |
dAlembert |
[[_homogeneous, ‘class G‘], _rational, _dAlembert] |
✓ |
✓ |
0.332 |
|
|
\[ {}4 x {y^{\prime }}^{2}+4 y y^{\prime } = 1 \] |
2 |
2 |
2 |
dAlembert |
[[_homogeneous, ‘class G‘], _rational, _dAlembert] |
✓ |
✓ |
0.421 |
|
|
\[ {}4 x {y^{\prime }}^{2}+4 y y^{\prime }-y^{4} = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
2.632 |
|
|
\[ {}4 \left (2-x \right ) {y^{\prime }}^{2}+1 = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.267 |
|
|
\[ {}16 x {y^{\prime }}^{2}+8 y y^{\prime }+y^{6} = 0 \] |
2 |
2 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
3.387 |
|
|
\[ {}x^{2} {y^{\prime }}^{2} = a^{2} \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.255 |
|
|
\[ {}x^{2} {y^{\prime }}^{2} = y^{2} \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.419 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+x^{2}-y^{2} = 0 \] |
2 |
4 |
1 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.098 |
|
|
\[ {}x^{2} {y^{\prime }}^{2} = \left (x -y\right )^{2} \] |
2 |
1 |
2 |
linear |
[_linear] |
✓ |
✓ |
0.527 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+y^{2}-y^{4} = 0 \] |
2 |
2 |
5 |
first_order_nonlinear_p_but_separable |
[_separable] |
✓ |
✓ |
0.937 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-x y^{\prime }+y \left (1-y\right ) = 0 \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.51 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+2 a x y^{\prime }+a^{2}+x^{2}-2 a y = 0 \] |
2 |
0 |
1 |
unknown |
[_rational] |
✗ |
N/A |
1.644 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-2 x y y^{\prime }-x +y \left (y+1\right ) = 0 \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
3.608 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-2 x y y^{\prime }-x^{4}+\left (-x^{2}+1\right ) y^{2} = 0 \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
3.108 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-\left (1+2 x y\right ) y^{\prime }+1+y^{2} = 0 \] |
2 |
4 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
0.423 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-\left (a +2 x y\right ) y^{\prime }+y^{2} = 0 \] |
2 |
4 |
3 |
clairaut |
[[_homogeneous, ‘class G‘], _rational, _Clairaut] |
✓ |
✓ |
0.525 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-x \left (x -2 y\right ) y^{\prime }+y^{2} = 0 \] |
2 |
5 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.512 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+2 x \left (y+2 x \right ) y^{\prime }-4 a +y^{2} = 0 \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
4.254 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+x \left (x^{3}-2 y\right ) y^{\prime }-\left (2 x^{3}-y\right ) y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
9.777 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+3 x y y^{\prime }+2 y^{2} = 0 \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.542 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-3 x y y^{\prime }+x^{3}+2 y^{2} = 0 \] |
2 |
2 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
16.458 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+4 x y y^{\prime }-5 y^{2} = 0 \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.556 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-4 x \left (y+2\right ) y^{\prime }+4 \left (y+2\right ) y = 0 \] |
2 |
2 |
5 |
separable |
[_separable] |
✓ |
✓ |
3.259 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-5 x y y^{\prime }+6 y^{2} = 0 \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.499 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+x \left (x^{2}+x y-2 y\right ) y^{\prime }+\left (1-x \right ) \left (x^{2}-y\right ) y = 0 \] |
2 |
0 |
0 |
unknown |
[_rational] |
✗ |
N/A |
7.028 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+\left (y+2 x \right ) y y^{\prime }+y^{2} = 0 \] |
2 |
3 |
6 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.978 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+\left (2 x -y\right ) y y^{\prime }+y^{2} = 0 \] |
2 |
3 |
6 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.935 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+\left (a +b \,x^{2} y^{3}\right ) y^{\prime }+a b y^{3} = 0 \] |
2 |
1 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.434 |
|
|
\[ {}\left (-x^{2}+1\right ) {y^{\prime }}^{2} = 1-y^{2} \] |
2 |
2 |
4 |
first_order_nonlinear_p_but_separable |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
1.449 |
|
|
\[ {}\left (-x^{2}+1\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+4 x^{2} = 0 \] |
2 |
0 |
3 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
143.288 |
|
|
\[ {}\left (a^{2}+x^{2}\right ) {y^{\prime }}^{2} = b^{2} \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.341 |
|
|
\[ {}\left (a^{2}-x^{2}\right ) {y^{\prime }}^{2}+b^{2} = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.347 |
|
|
\[ {}\left (a^{2}-x^{2}\right ) {y^{\prime }}^{2} = b^{2} \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.466 |
|
|
\[ {}\left (a^{2}-x^{2}\right ) {y^{\prime }}^{2} = x^{2} \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.342 |
|
|
\[ {}\left (a^{2}-x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+x^{2} = 0 \] |
2 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
90.2 |
|
|
\[ {}\left (a^{2}-x^{2}\right ) {y^{\prime }}^{2}-2 x y y^{\prime }-y^{2} = 0 \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.523 |
|
|
\[ {}\left (a^{2}+x^{2}\right ) {y^{\prime }}^{2}-2 x y y^{\prime }+b +y^{2} = 0 \] |
2 |
6 |
4 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
2.242 |
|
|
\[ {}4 x^{2} {y^{\prime }}^{2}-4 x y y^{\prime } = 8 x^{3}-y^{2} \] |
2 |
2 |
2 |
linear |
[_linear] |
✓ |
✓ |
0.633 |
|
|
\[ {}a \,x^{2} {y^{\prime }}^{2}-2 a x y y^{\prime }+a \left (1-a \right ) x^{2}+y^{2} = 0 \] |
2 |
8 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.332 |
|
|
\[ {}\left (-a^{2}+1\right ) x^{2} {y^{\prime }}^{2}-2 x y y^{\prime }-a^{2} x^{2}+y^{2} = 0 \] |
2 |
8 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
240.952 |
|
|
\[ {}x^{3} {y^{\prime }}^{2} = a \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.247 |
|
|
\[ {}x^{3} {y^{\prime }}^{2}+x y^{\prime }-y = 0 \] |
2 |
2 |
0 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
16.725 |
|
|
\[ {}x^{3} {y^{\prime }}^{2}+x^{2} y y^{\prime }+a = 0 \] |
2 |
2 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
5.416 |
|
|
\[ {}x \left (-x^{2}+1\right ) {y^{\prime }}^{2}-2 \left (-x^{2}+1\right ) y y^{\prime }+x \left (1-y^{2}\right ) = 0 \] |
2 |
0 |
3 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
6.466 |
|
|
\[ {}4 x \left (a -x \right ) \left (-x +b \right ) {y^{\prime }}^{2} = \left (a b -2 x \left (a +b \right )+2 x^{2}\right )^{2} \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.593 |
|
|
\[ {}x^{4} {y^{\prime }}^{2}-x y^{\prime }-y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
4.308 |
|
|
\[ {}x^{4} {y^{\prime }}^{2}+2 x^{3} y y^{\prime }-4 = 0 \] |
2 |
2 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
4.475 |
|
|
\[ {}x^{4} {y^{\prime }}^{2}+x y^{2} y^{\prime }-y^{3} = 0 \] |
2 |
2 |
6 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
38.073 |
|
|
\[ {}x^{2} \left (a^{2}-x^{2}\right ) {y^{\prime }}^{2}+1 = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.56 |
|
|
\[ {}3 x^{4} {y^{\prime }}^{2}-x y-y = 0 \] |
2 |
2 |
5 |
first_order_nonlinear_p_but_separable |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
0.737 |
|
|
\[ {}4 x^{5} {y^{\prime }}^{2}+12 x^{4} y y^{\prime }+9 = 0 \] |
2 |
2 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
5.716 |
|
|
\[ {}x^{6} {y^{\prime }}^{2}-2 x y^{\prime }-4 y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
4.532 |
|
|
\[ {}x^{8} {y^{\prime }}^{2}+3 x y^{\prime }+9 y = 0 \] |
2 |
2 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
4.646 |
|
|
\[ {}y {y^{\prime }}^{2} = a \] |
2 |
2 |
6 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.413 |
|
|
\[ {}y {y^{\prime }}^{2} = x \,a^{2} \] |
2 |
5 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.892 |
|
|
\[ {}y {y^{\prime }}^{2} = {\mathrm e}^{2 x} \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
4.884 |
|
|
\[ {}y {y^{\prime }}^{2}+2 a x y^{\prime }-a y = 0 \] |
2 |
5 |
5 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.69 |
|
|
\[ {}y {y^{\prime }}^{2}-4 a^{2} x y^{\prime }+a^{2} y = 0 \] |
2 |
5 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.92 |
|
|
\[ {}y {y^{\prime }}^{2}+a x y^{\prime }+b y = 0 \] |
2 |
4 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
16.224 |
|
|
\[ {}y {y^{\prime }}^{2}-\left (-2 b x +a \right ) y^{\prime }-b y = 0 \] |
2 |
5 |
7 |
dAlembert |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
✓ |
0.815 |
|
|
\[ {}y {y^{\prime }}^{2}+x^{3} y^{\prime }-x^{2} y = 0 \] |
2 |
2 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
6.483 |
|
|
\[ {}y {y^{\prime }}^{2}+\left (x -y\right ) y^{\prime }-x = 0 \] |
2 |
2 |
3 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.442 |
|
|
\[ {}y {y^{\prime }}^{2}-\left (x +y\right ) y^{\prime }+y = 0 \] |
2 |
4 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.855 |
|
|
\[ {}y {y^{\prime }}^{2}-\left (x y+1\right ) y^{\prime }+x = 0 \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.249 |
|
|
\[ {}y {y^{\prime }}^{2}+\left (x -y^{2}\right ) y^{\prime }-x y = 0 \] |
2 |
2 |
3 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.379 |
|
|
\[ {}y {y^{\prime }}^{2}+y = a \] |
2 |
2 |
5 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.975 |
|
|
\[ {}\left (x +y\right ) {y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
2 |
5 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.695 |
|
|
\[ {}\left (2 x -y\right ) {y^{\prime }}^{2}-2 \left (1-x \right ) y^{\prime }+2-y = 0 \] |
2 |
5 |
4 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.658 |
|
|
\[ {}2 y {y^{\prime }}^{2}+\left (5-4 x \right ) y^{\prime }+2 y = 0 \] |
2 |
5 |
7 |
dAlembert |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
✓ |
0.544 |
|
|
\[ {}9 y {y^{\prime }}^{2}+4 x^{3} y^{\prime }-4 x^{2} y = 0 \] |
2 |
2 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
6.322 |
|
|
\[ {}\left (1-a y\right ) {y^{\prime }}^{2} = a y \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.126 |
|
|
\[ {}\left (x^{2}-a y\right ) {y^{\prime }}^{2}-2 x y y^{\prime } = 0 \] |
2 |
1 |
2 |
quadrature, first_order_ode_lie_symmetry_calculated |
[_quadrature] |
✓ |
✓ |
0.816 |
|
|
\[ {}x y {y^{\prime }}^{2}+\left (x +y\right ) y^{\prime }+1 = 0 \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.343 |
|
|
\[ {}x y {y^{\prime }}^{2}+\left (x^{2}+y^{2}\right ) y^{\prime }+x y = 0 \] |
2 |
2 |
3 |
separable |
[_separable] |
✓ |
✓ |
0.481 |
|
|
\[ {}x y {y^{\prime }}^{2}+\left (x^{2}-y^{2}\right ) y^{\prime }-x y = 0 \] |
2 |
1 |
3 |
separable |
[_separable] |
✓ |
✓ |
0.465 |
|
|
\[ {}x y {y^{\prime }}^{2}-\left (x^{2}-y^{2}\right ) y^{\prime }-x y = 0 \] |
2 |
2 |
3 |
separable |
[_separable] |
✓ |
✓ |
0.448 |
|
|
\[ {}x y {y^{\prime }}^{2}+\left (a +x^{2}-y^{2}\right ) y^{\prime }-x y = 0 \] |
2 |
0 |
0 |
unknown |
[_rational] |
✗ |
N/A |
233.874 |
|
|
\[ {}x y {y^{\prime }}^{2}-\left (a -b \,x^{2}+y^{2}\right ) y^{\prime }-b x y = 0 \] |
2 |
1 |
0 |
first_order_ode_lie_symmetry_calculated |
[_rational] |
✓ |
✓ |
66.336 |
|
|
\[ {}x y {y^{\prime }}^{2}+\left (3 x^{2}-2 y^{2}\right ) y^{\prime }-6 x y = 0 \] |
2 |
1 |
3 |
separable |
[_separable] |
✓ |
✓ |
0.702 |
|
|
\[ {}x \left (x -2 y\right ) {y^{\prime }}^{2}-2 x y y^{\prime }-2 x y+y^{2} = 0 \] |
2 |
5 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.077 |
|
|
\[ {}x \left (x -2 y\right ) {y^{\prime }}^{2}+6 x y y^{\prime }-2 x y+y^{2} = 0 \] |
2 |
9 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
221.604 |
|
|
\[ {}y^{2} {y^{\prime }}^{2} = a^{2} \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.414 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}-a^{2}+y^{2} = 0 \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.568 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}-3 x y^{\prime }+y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
4.924 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}-6 x^{3} y^{\prime }+4 x^{2} y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
5.02 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}-4 a y y^{\prime }+4 a^{2}-4 x a +y^{2} = 0 \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
2.564 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}-\left (1+x \right ) y y^{\prime }+x = 0 \] |
2 |
2 |
4 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.453 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}+2 x y y^{\prime }+x^{2} = 0 \] |
2 |
4 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.355 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}+2 x y y^{\prime }+a -y^{2} = 0 \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
8.827 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}-2 x y y^{\prime }-x^{2}+2 y^{2} = 0 \] |
2 |
6 |
6 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.358 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+a -x^{2}+2 y^{2} = 0 \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
2.385 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}+2 a x y y^{\prime }+\left (a -1\right ) b +x^{2} a +\left (1-a \right ) y^{2} = 0 \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
3.305 |
|
|
\[ {}\left (1-y^{2}\right ) {y^{\prime }}^{2} = 1 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.752 |
|
|
\[ {}\left (a^{2}-y^{2}\right ) {y^{\prime }}^{2} = y^{2} \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.974 |
|
|
\[ {}\left (a^{2}-2 a x y+y^{2}\right ) {y^{\prime }}^{2}+2 a y y^{\prime }+y^{2} = 0 \] |
2 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
4.385 |
|
|
\[ {}\left (\left (1-a \right ) x^{2}+y^{2}\right ) {y^{\prime }}^{2}+2 a x y y^{\prime }+x^{2}+\left (1-a \right ) y^{2} = 0 \] |
2 |
4 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.087 |
|
|
\[ {}\left (\left (-4 a^{2}+1\right ) x^{2}+y^{2}\right ) {y^{\prime }}^{2}-8 a^{2} x y y^{\prime }+x^{2}+\left (-4 a^{2}+1\right ) y^{2} = 0 \] |
2 |
8 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
183.437 |
|
|
\[ {}\left (\left (-a^{2}+1\right ) x^{2}+y^{2}\right ) {y^{\prime }}^{2}+2 a^{2} x y y^{\prime }+x^{2}+\left (-a^{2}+1\right ) y^{2} = 0 \] |
2 |
4 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.123 |
|
|
\[ {}\left (x +y\right )^{2} {y^{\prime }}^{2} = y^{2} \] |
2 |
1 |
3 |
homogeneousTypeD2 |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.948 |
|
|
\[ {}\left (x +y\right )^{2} {y^{\prime }}^{2}-\left (x^{2}-x y-2 y^{2}\right ) y^{\prime }-\left (x -y\right ) y = 0 \] |
2 |
1 |
4 |
homogeneousTypeD2 |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.136 |
|
|
\[ {}\left (a^{2}-\left (x -y\right )^{2}\right ) {y^{\prime }}^{2}+2 a^{2} y^{\prime }+a^{2}-\left (x -y\right )^{2} = 0 \] |
2 |
6 |
4 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
11.89 |
|
|
\[ {}2 y^{2} {y^{\prime }}^{2}+2 x y y^{\prime }-1+x^{2}+y^{2} = 0 \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
74.107 |
|
|
\[ {}3 y^{2} {y^{\prime }}^{2}-2 x y y^{\prime }-x^{2}+4 y^{2} = 0 \] |
2 |
6 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
2.616 |
|
|
\[ {}4 y^{2} {y^{\prime }}^{2}+2 \left (1+3 x \right ) x y y^{\prime }+3 x^{3} = 0 \] |
2 |
2 |
4 |
separable |
[_separable] |
✓ |
✓ |
0.59 |
|
|
\[ {}\left (x^{2}-4 y^{2}\right ) {y^{\prime }}^{2}+6 x y y^{\prime }-4 x^{2}+y^{2} = 0 \] |
2 |
1 |
2 |
homogeneousTypeD2 |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.957 |
|
|
\[ {}9 y^{2} {y^{\prime }}^{2}-3 x y^{\prime }+y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
3.781 |
|
|
\[ {}\left (2-3 y\right )^{2} {y^{\prime }}^{2} = 4-4 y \] |
2 |
2 |
7 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.474 |
|
|
\[ {}\left (-a^{2}+1\right ) y^{2} {y^{\prime }}^{2}-3 a^{2} x y y^{\prime }-a^{2} x^{2}+y^{2} = 0 \] |
2 |
8 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
11.996 |
|
|
\[ {}\left (-b +a \right ) y^{2} {y^{\prime }}^{2}-2 b x y y^{\prime }-a b -b \,x^{2}+a y^{2} = 0 \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
3.382 |
|
|
\[ {}a^{2} \left (b^{2}-\left (c x -a y\right )^{2}\right ) {y^{\prime }}^{2}+2 a \,b^{2} c y^{\prime }+c^{2} \left (b^{2}-\left (c x -a y\right )^{2}\right ) = 0 \] |
2 |
6 |
4 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
19.815 |
|
|
\[ {}x y^{2} {y^{\prime }}^{2}-y^{3} y^{\prime }+x \,a^{2} = 0 \] |
2 |
2 |
6 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
5.448 |
|
|
\[ {}x y^{2} {y^{\prime }}^{2}+\left (a -x^{3}-y^{3}\right ) y^{\prime }+x^{2} y = 0 \] |
2 |
0 |
9 |
unknown |
[_rational] |
✗ |
N/A |
7.855 |
|
|
\[ {}2 x y^{2} {y^{\prime }}^{2}-y^{3} y^{\prime }-a = 0 \] |
2 |
2 |
8 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.208 |
|
|
\[ {}4 x^{2} y^{2} {y^{\prime }}^{2} = \left (x^{2}+y^{2}\right )^{2} \] |
2 |
1 |
4 |
homogeneousTypeD2 |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.309 |
|
|
\[ {}4 y^{3} {y^{\prime }}^{2}-4 x y^{\prime }+y = 0 \] |
2 |
2 |
6 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
5.898 |
|
|
\[ {}3 x y^{4} {y^{\prime }}^{2}-y^{5} y^{\prime }+1 = 0 \] |
2 |
1 |
12 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.103 |
|
|
\[ {}9 x y^{4} {y^{\prime }}^{2}-3 y^{5} y^{\prime }-a = 0 \] |
2 |
1 |
12 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
2.038 |
|
|
\[ {}9 \left (-x^{2}+1\right ) y^{4} {y^{\prime }}^{2}+6 x y^{5} y^{\prime }+4 x^{2} = 0 \] |
2 |
0 |
9 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
3.399 |
|
|
\[ {}{y^{\prime }}^{3} = b x +a \] |
3 |
3 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.587 |
|
|
\[ {}{y^{\prime }}^{3} = a \,x^{n} \] |
3 |
3 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.45 |
|
|
\[ {}{y^{\prime }}^{3}+x -y = 0 \] |
3 |
4 |
3 |
dAlembert, first_order_nonlinear_p_but_linear_in_x_y |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.728 |
|
|
\[ {}{y^{\prime }}^{3} = \left (a +b y+c y^{2}\right ) f \left (x \right ) \] |
3 |
3 |
3 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
1.733 |
|
|
\[ {}{y^{\prime }}^{3} = \left (y-a \right )^{2} \left (y-b \right )^{2} \] |
3 |
3 |
5 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.214 |
|
|
\[ {}{y^{\prime }}^{3}+f \left (x \right ) \left (y-a \right )^{2} \left (y-b \right )^{2} = 0 \] |
3 |
3 |
3 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
2.263 |
|
|
\[ {}{y^{\prime }}^{3}+f \left (x \right ) \left (y-a \right )^{2} \left (y-b \right )^{2} \left (y-c \right )^{2} = 0 \] |
3 |
3 |
3 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
188.713 |
|
|
\[ {}{y^{\prime }}^{3}+y^{\prime }+a -b x = 0 \] |
3 |
3 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.565 |
|
|
\[ {}{y^{\prime }}^{3}+y^{\prime }-y = 0 \] |
3 |
3 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.901 |
|
|
\[ {}{y^{\prime }}^{3}+y^{\prime } = {\mathrm e}^{y} \] |
3 |
3 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.155 |
|
|
\[ {}{y^{\prime }}^{3}-7 y^{\prime }+6 = 0 \] |
3 |
1 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.286 |
|
|
\[ {}{y^{\prime }}^{3}-x y^{\prime }+a y = 0 \] |
3 |
4 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
169.119 |
|
|
\[ {}{y^{\prime }}^{3}+2 x y^{\prime }-y = 0 \] |
3 |
4 |
4 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
3.142 |
|
|
\[ {}{y^{\prime }}^{3}-2 x y^{\prime }-y = 0 \] |
3 |
4 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
99.775 |
|
|
\[ {}{y^{\prime }}^{3}-a x y^{\prime }+x^{3} = 0 \] |
3 |
3 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.445 |
|
|
\[ {}{y^{\prime }}^{3}+a x y^{\prime }-a y = 0 \] |
3 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.434 |
|
|
\[ {}{y^{\prime }}^{3}-\left (b x +a \right ) y^{\prime }+b y = 0 \] |
3 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.503 |
|
|
\[ {}{y^{\prime }}^{3}-2 y y^{\prime }+y^{2} = 0 \] |
3 |
3 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
2.445 |
|
|
\[ {}{y^{\prime }}^{3}-a x y y^{\prime }+2 a y^{2} = 0 \] |
3 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
16.117 |
|
|
\[ {}{y^{\prime }}^{3}-x y^{4} y^{\prime }-y^{5} = 0 \] |
3 |
1 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
17.514 |
|
|
\[ {}{y^{\prime }}^{3}+{\mathrm e}^{3 x -2 y} \left (y^{\prime }-1\right ) = 0 \] |
3 |
2 |
3 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.973 |
|
|
\[ {}{y^{\prime }}^{3}+{\mathrm e}^{-2 y} \left ({\mathrm e}^{2 x}+{\mathrm e}^{3 x}\right ) y^{\prime }-{\mathrm e}^{3 x -2 y} = 0 \] |
3 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
104.449 |
|
|
\[ {}{y^{\prime }}^{3}+{y^{\prime }}^{2}-y = 0 \] |
3 |
3 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.296 |
|
|
\[ {}{y^{\prime }}^{3}-{y^{\prime }}^{2}+y^{2} = 0 \] |
3 |
3 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.492 |
|
|
\[ {}{y^{\prime }}^{3}-{y^{\prime }}^{2}+x y^{\prime }-y = 0 \] |
3 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.56 |
|
|
\[ {}{y^{\prime }}^{3}-a {y^{\prime }}^{2}+b y+a b x = 0 \] |
3 |
4 |
1 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
167.053 |
|
|
\[ {}{y^{\prime }}^{3}+\operatorname {a0} {y^{\prime }}^{2}+\operatorname {a1} y^{\prime }+\operatorname {a2} +\operatorname {a3} y = 0 \] |
3 |
3 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.897 |
|
|
\[ {}{y^{\prime }}^{3}+\left (1-3 x \right ) {y^{\prime }}^{2}-x \left (1-3 x \right ) y^{\prime }-1-x^{3} = 0 \] |
3 |
3 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.723 |
|
|
\[ {}{y^{\prime }}^{3}-y {y^{\prime }}^{2}+y^{2} = 0 \] |
3 |
3 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.787 |
|
|
\[ {}{y^{\prime }}^{3}+\left (\cos \left (x \right ) \cot \left (x \right )-y\right ) {y^{\prime }}^{2}-\left (1+y \cos \left (x \right ) \cot \left (x \right )\right ) y^{\prime }+y = 0 \] |
3 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.78 |
|
|
\[ {}{y^{\prime }}^{3}+\left (2 x -y^{2}\right ) {y^{\prime }}^{2}-2 x y^{2} y^{\prime } = 0 \] |
3 |
1 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.313 |
|
|
\[ {}{y^{\prime }}^{3}-\left (y^{2}+2 x \right ) {y^{\prime }}^{2}+\left (x^{2}-y^{2}+2 x y^{2}\right ) y^{\prime }-\left (x^{2}-y^{2}\right ) y^{2} = 0 \] |
3 |
1 |
3 |
linear, quadrature |
[_quadrature] |
✓ |
✓ |
0.622 |
|
|
\[ {}{y^{\prime }}^{3}-\left (x^{2}+x y+y^{2}\right ) {y^{\prime }}^{2}+x y \left (x^{2}+x y+y^{2}\right ) y^{\prime }-x^{3} y^{3} = 0 \] |
3 |
1 |
3 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.481 |
|
|
\[ {}{y^{\prime }}^{3}-\left (x^{2}+x y^{2}+y^{4}\right ) {y^{\prime }}^{2}+x y^{2} \left (x^{2}+x y^{2}+y^{4}\right ) y^{\prime }-x^{3} y^{6} = 0 \] |
3 |
1 |
5 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.987 |
|
|
\[ {}2 {y^{\prime }}^{3}+x y^{\prime }-2 y = 0 \] |
3 |
4 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
100.448 |
|
|
\[ {}2 {y^{\prime }}^{3}+{y^{\prime }}^{2}-y = 0 \] |
3 |
3 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.233 |
|
|
\[ {}3 {y^{\prime }}^{3}-x^{4} y^{\prime }+2 x^{3} y = 0 \] |
3 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
86.383 |
|
|
\[ {}4 {y^{\prime }}^{3}+4 y^{\prime } = x \] |
3 |
3 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.579 |
|
|
\[ {}8 {y^{\prime }}^{3}+12 {y^{\prime }}^{2} = 27 x +27 y \] |
3 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.527 |
|
|
\[ {}x {y^{\prime }}^{3}-y {y^{\prime }}^{2}+a = 0 \] |
3 |
4 |
4 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.745 |
|
|
\[ {}x {y^{\prime }}^{3}-\left (x +x^{2}+y\right ) {y^{\prime }}^{2}+\left (x^{2}+y+x y\right ) y^{\prime }-x y = 0 \] |
3 |
1 |
3 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.448 |
|
|
\[ {}x {y^{\prime }}^{3}-2 y {y^{\prime }}^{2}+4 x^{2} = 0 \] |
3 |
1 |
11 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
93.679 |
|
|
\[ {}2 x {y^{\prime }}^{3}-3 y {y^{\prime }}^{2}-x = 0 \] |
3 |
4 |
5 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.613 |
|
|
\[ {}4 x {y^{\prime }}^{3}-6 y {y^{\prime }}^{2}-x +3 y = 0 \] |
3 |
6 |
5 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
13.632 |
|
|
\[ {}8 x {y^{\prime }}^{3}-12 y {y^{\prime }}^{2}+9 y = 0 \] |
3 |
6 |
5 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
23.288 |
|
|
\[ {}x^{2} {y^{\prime }}^{3}-2 x y {y^{\prime }}^{2}+y^{2} y^{\prime }+1 = 0 \] |
3 |
8 |
5 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
3.786 |
|
|
\[ {}\left (a^{2}-x^{2}\right ) {y^{\prime }}^{3}+b x \left (a^{2}-x^{2}\right ) {y^{\prime }}^{2}-y^{\prime }-b x = 0 \] |
3 |
1 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.54 |
|
|
\[ {}x {y^{\prime }}^{3}-3 x^{2} y {y^{\prime }}^{2}+x \left (x^{5}+3 y^{2}\right ) y^{\prime }-2 x^{5} y-y^{3} = 0 \] |
3 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
315.692 |
|
|
\[ {}2 x^{3} {y^{\prime }}^{3}+6 x^{2} y {y^{\prime }}^{2}-\left (1-6 x y\right ) y y^{\prime }+2 y^{3} = 0 \] |
3 |
1 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
21.441 |
|
|
\[ {}x^{4} {y^{\prime }}^{3}-x^{3} y {y^{\prime }}^{2}-x^{2} y^{2} y^{\prime }+x y^{3} = 1 \] |
3 |
1 |
6 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
18.007 |
|
|
\[ {}x^{6} {y^{\prime }}^{3}-x y^{\prime }-y = 0 \] |
3 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
100.911 |
|
|
\[ {}y {y^{\prime }}^{3}-3 x y^{\prime }+3 y = 0 \] |
3 |
4 |
4 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
173.355 |
|
|
\[ {}2 y {y^{\prime }}^{3}-3 x y^{\prime }+2 y = 0 \] |
3 |
7 |
7 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
154.286 |
|
|
\[ {}\left (2 y+x \right ) {y^{\prime }}^{3}+3 \left (x +y\right ) {y^{\prime }}^{2}+\left (y+2 x \right ) y^{\prime } = 0 \] |
3 |
1 |
4 |
quadrature, homogeneousTypeD2 |
[_quadrature] |
✓ |
✓ |
1.062 |
|
|
\[ {}y^{2} {y^{\prime }}^{3}-x y^{\prime }+y = 0 \] |
3 |
1 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
81.401 |
|
|
\[ {}y^{2} {y^{\prime }}^{3}+2 x y^{\prime }-y = 0 \] |
3 |
1 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
87.318 |
|
|
\[ {}4 y^{2} {y^{\prime }}^{3}-2 x y^{\prime }+y = 0 \] |
3 |
1 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
47.616 |
|
|
\[ {}16 y^{2} {y^{\prime }}^{3}+2 x y^{\prime }-y = 0 \] |
3 |
1 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
86.735 |
|
|
\[ {}x y^{2} {y^{\prime }}^{3}-y^{3} {y^{\prime }}^{2}+x \left (x^{2}+1\right ) y^{\prime }-x^{2} y = 0 \] |
3 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
175.365 |
|
|
\[ {}y^{3} {y^{\prime }}^{3}-\left (1-3 x \right ) y^{2} {y^{\prime }}^{2}+3 x^{2} y y^{\prime }+x^{3}-y^{2} = 0 \] |
3 |
0 |
10 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
134.116 |
|
|
\[ {}y^{4} {y^{\prime }}^{3}-6 x y^{\prime }+2 y = 0 \] |
3 |
1 |
10 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
143.295 |
|
|
\[ {}{y^{\prime }}^{4} = \left (y-a \right )^{3} \left (y-b \right )^{2} \] |
4 |
4 |
6 |
quadrature |
[_quadrature] |
✓ |
✓ |
4.075 |
|
|
\[ {}{y^{\prime }}^{4}+f \left (x \right ) \left (y-a \right )^{3} \left (y-b \right )^{2} = 0 \] |
4 |
4 |
4 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
4.665 |
|
|
\[ {}{y^{\prime }}^{4}+f \left (x \right ) \left (y-a \right )^{3} \left (y-b \right )^{3} = 0 \] |
4 |
4 |
4 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
12.732 |
|
|
\[ {}{y^{\prime }}^{4}+f \left (x \right ) \left (y-a \right )^{3} \left (y-b \right )^{3} \left (y-c \right )^{2} = 0 \] |
4 |
4 |
1 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
668.064 |
|
|
\[ {}{y^{\prime }}^{4}+x y^{\prime }-3 y = 0 \] |
4 |
2 |
1 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
1.38 |
|
|
\[ {}{y^{\prime }}^{4}-4 x^{2} y {y^{\prime }}^{2}+16 x y^{2} y^{\prime }-16 y^{3} = 0 \] |
4 |
0 |
3 |
unknown |
[[_homogeneous, ‘class G‘]] |
✗ |
N/A |
1.309 |
|
|
\[ {}{y^{\prime }}^{4}+4 y {y^{\prime }}^{3}+6 y^{2} {y^{\prime }}^{2}-\left (1-4 y^{3}\right ) y^{\prime }-\left (3-y^{3}\right ) y = 0 \] |
4 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.856 |
|
|
\[ {}2 {y^{\prime }}^{4}-y y^{\prime }-2 = 0 \] |
4 |
1 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.622 |
|
|
\[ {}{y^{\prime }}^{4} x -2 y {y^{\prime }}^{3}+12 x^{3} = 0 \] |
4 |
0 |
5 |
unknown |
[[_1st_order, _with_linear_symmetries]] |
✗ |
N/A |
2.131 |
|
|
\[ {}3 {y^{\prime }}^{5}-y y^{\prime }+1 = 0 \] |
5 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.233 |
|
|
\[ {}{y^{\prime }}^{6} = \left (y-a \right )^{4} \left (y-b \right )^{3} \] |
6 |
6 |
8 |
quadrature |
[_quadrature] |
✓ |
✓ |
91.425 |
|
|
\[ {}{y^{\prime }}^{6}+f \left (x \right ) \left (y-a \right )^{4} \left (y-b \right )^{3} = 0 \] |
6 |
6 |
1 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
99.567 |
|
|
\[ {}{y^{\prime }}^{6}+f \left (x \right ) \left (y-a \right )^{5} \left (y-b \right )^{3} = 0 \] |
6 |
6 |
1 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
213.078 |
|
|
\[ {}{y^{\prime }}^{6}+f \left (x \right ) \left (y-a \right )^{5} \left (y-b \right )^{4} = 0 \] |
6 |
6 |
1 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
778.25 |
|
|
\[ {}x^{2} \left ({y^{\prime }}^{6}+3 y^{4}+3 y^{2}+1\right ) = a^{2} \] |
6 |
0 |
0 |
unknown |
[_rational] |
✗ |
N/A |
15.077 |
|
|
\[ {}2 \sqrt {a y^{\prime }}+x y^{\prime }-y = 0 \] |
2 |
2 |
1 |
clairaut |
[[_homogeneous, ‘class G‘], _Clairaut] |
✓ |
✓ |
2.11 |
|
|
\[ {}\left (x -y\right ) \sqrt {y^{\prime }} = a \left (y^{\prime }+1\right ) \] |
2 |
3 |
2 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
3.405 |
|
|
|
|||||||||
|
\[ {}\sqrt {1+{y^{\prime }}^{2}}+a y^{\prime } = x \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
2.527 |
|
|
\[ {}\sqrt {1+{y^{\prime }}^{2}}+a y^{\prime } = y \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
2.367 |
|
|
\[ {}\sqrt {1+{y^{\prime }}^{2}} = x y^{\prime } \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.731 |
|
|
\[ {}\sqrt {a^{2}+b^{2} {y^{\prime }}^{2}}+x y^{\prime }-y = 0 \] |
2 |
3 |
1 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
4.38 |
|
|
\[ {}a \sqrt {1+{y^{\prime }}^{2}}+x y^{\prime }-y = 0 \] |
2 |
3 |
1 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
2.182 |
|
|
\[ {}a x \sqrt {1+{y^{\prime }}^{2}}+x y^{\prime }-y = 0 \] |
2 |
2 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.764 |
|
|
\[ {}\sqrt {\left (x^{2} a +y^{2}\right ) \left (1+{y^{\prime }}^{2}\right )}-y y^{\prime }-x a = 0 \] |
2 |
8 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
246.119 |
|
|
\[ {}a \left (1+{y^{\prime }}^{3}\right )^{\frac {1}{3}}+x y^{\prime }-y = 0 \] |
3 |
7 |
1 |
clairaut |
[_Clairaut] |
✓ |
✓ |
183.375 |
|
|
\[ {}\cos \left (y^{\prime }\right )+x y^{\prime } = y \] |
0 |
2 |
2 |
clairaut |
[_Clairaut] |
✓ |
✓ |
0.425 |
|
|
\[ {}a \cos \left (y^{\prime }\right )+b y^{\prime }+x = 0 \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.26 |
|
|
\[ {}\sin \left (y^{\prime }\right )+y^{\prime } = x \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.269 |
|
|
\[ {}y^{\prime } \sin \left (y^{\prime }\right )+\cos \left (y^{\prime }\right ) = y \] |
0 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.821 |
|
|
\[ {}{y^{\prime }}^{2} \left (x +\sin \left (y^{\prime }\right )\right ) = y \] |
0 |
3 |
2 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
0.898 |
|
|
\[ {}\left (1+{y^{\prime }}^{2}\right ) \sin \left (-y+x y^{\prime }\right )^{2} = 1 \] |
0 |
6 |
6 |
clairaut |
[_Clairaut] |
✓ |
✓ |
3.925 |
|
|
\[ {}\left (1+{y^{\prime }}^{2}\right ) \left (\arctan \left (y^{\prime }\right )+x a \right )+y^{\prime } = 0 \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✗ |
1.71 |
|
|
\[ {}{\mathrm e}^{y^{\prime }-y}-{y^{\prime }}^{2}+1 = 0 \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.203 |
|
|
\[ {}\ln \left (y^{\prime }\right )+x y^{\prime }+a = 0 \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.567 |
|
|
\[ {}\ln \left (y^{\prime }\right )+x y^{\prime }+a = y \] |
0 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.879 |
|
|
\[ {}\ln \left (y^{\prime }\right )+x y^{\prime }+a +b y = 0 \] |
0 |
2 |
1 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
3.114 |
|
|
\[ {}\ln \left (y^{\prime }\right )+4 x y^{\prime }-2 y = 0 \] |
0 |
2 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
1.312 |
|
|
\[ {}\ln \left (y^{\prime }\right )+a \left (-y+x y^{\prime }\right ) = 0 \] |
0 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.828 |
|
|
\[ {}a \left (\ln \left (y^{\prime }\right )-y^{\prime }\right )-x +y = 0 \] |
0 |
2 |
2 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.364 |
|
|
\[ {}y \ln \left (y^{\prime }\right )+y^{\prime }-y \ln \left (y\right )-x y = 0 \] |
0 |
1 |
1 |
separable, homogeneousTypeD2 |
[_separable] |
✓ |
✓ |
3.098 |
|
|
\[ {}y^{\prime } \ln \left (y^{\prime }\right )-\left (1+x \right ) y^{\prime }+y = 0 \] |
0 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.868 |
|
|
\[ {}y^{\prime } \ln \left (y^{\prime }+\sqrt {a +{y^{\prime }}^{2}}\right )-\sqrt {1+{y^{\prime }}^{2}}-x y^{\prime }+y = 0 \] |
0 |
2 |
0 |
clairaut |
[_Clairaut] |
✓ |
✓ |
7.899 |
|
|
\[ {}\ln \left (\cos \left (y^{\prime }\right )\right )+y^{\prime } \tan \left (y^{\prime }\right ) = y \] |
0 |
0 |
2 |
unknown |
[_dAlembert] |
✗ |
N/A |
1.551 |
|
|
\[ {}y^{2} \left (1+{y^{\prime }}^{2}\right ) = R^{2} \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.811 |
|
|
\[ {}y = x y^{\prime }+\frac {a y^{\prime }}{\sqrt {1+{y^{\prime }}^{2}}} \] |
4 |
7 |
1 |
clairaut |
[_Clairaut] |
✓ |
✓ |
60.916 |
|
|
\[ {}y = x {y^{\prime }}^{2}+{y^{\prime }}^{2} \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
✓ |
0.309 |
|
|
\[ {}{y^{\prime }}^{2}-5 y^{\prime }+6 = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.249 |
|
|
\[ {}{y^{\prime }}^{2}-\frac {a^{2}}{x^{2}} = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.291 |
|
|
\[ {}{y^{\prime }}^{2} = \frac {1-x}{x} \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.854 |
|
|
\[ {}{y^{\prime }}^{2}+\frac {2 x y^{\prime }}{y}-1 = 0 \] |
2 |
5 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.858 |
|
|
\[ {}y = a y^{\prime }+b {y^{\prime }}^{2} \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
2.165 |
|
|
\[ {}x = a y^{\prime }+b {y^{\prime }}^{2} \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.504 |
|
|
\[ {}y = \sqrt {1+{y^{\prime }}^{2}}+a y^{\prime } \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
2.484 |
|
|
\[ {}x = \sqrt {1+{y^{\prime }}^{2}}+a y^{\prime } \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
2.289 |
|
|
\[ {}y^{\prime }-\frac {\sqrt {1+{y^{\prime }}^{2}}}{x} = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.679 |
|
|
\[ {}x^{2} \left (1+{y^{\prime }}^{2}\right )^{3}-a^{2} = 0 \] |
6 |
6 |
6 |
quadrature |
[_quadrature] |
✓ |
✓ |
5.812 |
|
|
\[ {}1+{y^{\prime }}^{2} = \frac {\left (x +a \right )^{2}}{2 x a +x^{2}} \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.461 |
|
|
\[ {}y = x y^{\prime }+y^{\prime }-{y^{\prime }}^{2} \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.26 |
|
|
\[ {}y = x y^{\prime }+\sqrt {b^{2}-a^{2} {y^{\prime }}^{2}} \] |
2 |
3 |
1 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
4.908 |
|
|
\[ {}y = x y^{\prime }+a x \sqrt {1+{y^{\prime }}^{2}} \] |
2 |
2 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.856 |
|
|
\[ {}x -y y^{\prime } = a {y^{\prime }}^{2} \] |
2 |
4 |
2 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
90.835 |
|
|
\[ {}x +y y^{\prime } = a \sqrt {1+{y^{\prime }}^{2}} \] |
2 |
4 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
38.324 |
|
|
\[ {}y y^{\prime } = x +y^{2}-y^{2} {y^{\prime }}^{2} \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
3.847 |
|
|
\[ {}y-\frac {1}{\sqrt {1+{y^{\prime }}^{2}}} = x +\frac {y^{\prime }}{\sqrt {1+{y^{\prime }}^{2}}} \] |
2 |
3 |
1 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
91.131 |
|
|
\[ {}y-2 x y^{\prime } = x {y^{\prime }}^{2} \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.322 |
|
|
\[ {}\frac {y-x y^{\prime }}{y^{2}+y^{\prime }} = \frac {y-x y^{\prime }}{1+x^{2} y^{\prime }} \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.897 |
|
|
\[ {}y {y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
2 |
5 |
7 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.514 |
|
|
\[ {}\left (-x^{2}+1\right ) {y^{\prime }}^{2}+1 = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.335 |
|
|
\[ {}y = x y^{\prime }+{y^{\prime }}^{4} \] |
4 |
4 |
4 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
1.519 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+x y y^{\prime }-6 y^{2} = 0 \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.877 |
|
|
\[ {}x {y^{\prime }}^{2}+\left (y-1-x^{2}\right ) y^{\prime }-x \left (y-1\right ) = 0 \] |
2 |
1 |
2 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.586 |
|
|
\[ {}4 x -2 y y^{\prime }+x {y^{\prime }}^{2} = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.341 |
|
|
\[ {}3 x^{4} {y^{\prime }}^{2}-x y^{\prime }-y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
4.763 |
|
|
\[ {}8 y {y^{\prime }}^{2}-2 x y^{\prime }+y = 0 \] |
2 |
5 |
5 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.699 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}+3 x y^{\prime }-y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
5.545 |
|
|
\[ {}{y^{\prime }}^{2}-x y^{\prime }+y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.297 |
|
|
\[ {}16 y^{3} {y^{\prime }}^{2}-4 x y^{\prime }+y = 0 \] |
2 |
2 |
6 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
7.541 |
|
|
\[ {}x {y^{\prime }}^{5}-y {y^{\prime }}^{4}+\left (x^{2}+1\right ) {y^{\prime }}^{3}-2 x y {y^{\prime }}^{2}+\left (x +y^{2}\right ) y^{\prime }-y = 0 \] |
5 |
3 |
6 |
clairaut |
[[_homogeneous, ‘class G‘], _rational, _Clairaut] |
✓ |
✓ |
0.808 |
|
|
\[ {}x {y^{\prime }}^{2}-y y^{\prime }-y = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.482 |
|
|
\[ {}y = 2 x y^{\prime }+y^{2} {y^{\prime }}^{3} \] |
3 |
1 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
94.018 |
|
|
\[ {}{y^{\prime }}^{2}-x y^{\prime }-y = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.438 |
|
|
\[ {}y = x \left (y^{\prime }+1\right )+{y^{\prime }}^{2} \] |
2 |
2 |
1 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.404 |
|
|
\[ {}y = 2 y^{\prime }+\sqrt {1+{y^{\prime }}^{2}} \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
16.452 |
|
|
\[ {}y {y^{\prime }}^{2}-x y^{\prime }+3 y = 0 \] |
2 |
4 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.709 |
|
|
\[ {}y = x y^{\prime }-2 {y^{\prime }}^{2} \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.317 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}+3 x y^{\prime }-y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
4.674 |
|
|
\[ {}4 x -2 y y^{\prime }+x {y^{\prime }}^{2} = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.368 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y y^{\prime }+x +2 y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.423 |
|
|
\[ {}\left (3 y-1\right )^{2} {y^{\prime }}^{2} = 4 y \] |
2 |
2 |
7 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.476 |
|
|
\[ {}y = -x y^{\prime }+x^{4} {y^{\prime }}^{2} \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
4.43 |
|
|
\[ {}2 y = {y^{\prime }}^{2}+4 x y^{\prime } \] |
2 |
3 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.429 |
|
|
\[ {}y \left (3-4 y\right )^{2} {y^{\prime }}^{2} = 4-4 y \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.737 |
|
|
\[ {}{y^{\prime }}^{3}-4 x^{4} y^{\prime }+8 x^{3} y = 0 \] |
3 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
117.097 |
|
|
\[ {}\left (1+{y^{\prime }}^{2}\right ) \left (x -y\right )^{2} = \left (x +y y^{\prime }\right )^{2} \] |
2 |
5 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
4.073 |
|
|
\[ {}y^{\prime } \left (y^{\prime }+y\right ) = \left (x +y\right ) x \] |
2 |
1 |
1 |
linear, quadrature |
[_quadrature] |
✓ |
✓ |
0.649 |
|
|
\[ {}\left (x y^{\prime }+y\right )^{2} = y^{2} y^{\prime } \] |
2 |
2 |
6 |
homogeneous |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
5.827 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-3 x y y^{\prime }+2 y^{2} = 0 \] |
2 |
2 |
2 |
homogeneous |
[_separable] |
✓ |
✓ |
0.859 |
|
|
\[ {}y {y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
2 |
2 |
7 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.971 |
|
|
\[ {}x +y y^{\prime } = a {y^{\prime }}^{2} \] |
2 |
4 |
2 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
0.789 |
|
|
\[ {}{y^{\prime }}^{2}-y^{2} a^{2} = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.069 |
|
|
\[ {}{y^{\prime }}^{2} = 4 x^{2} \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.316 |
|
|
\[ {}{y^{\prime }}^{2} = a^{2}-y^{2} \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.984 |
|
|
\[ {}x y^{\prime }+y = x^{4} {y^{\prime }}^{2} \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
7.297 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-y^{2} = 0 \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.528 |
|
|
\[ {}x {y^{\prime }}^{2}-\left (2 x +3 y\right ) y^{\prime }+6 y = 0 \] |
2 |
1 |
2 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.503 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-5 x y y^{\prime }+6 y^{2} = 0 \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.602 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+x y^{\prime }-y^{2}-y = 0 \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.618 |
|
|
\[ {}x {y^{\prime }}^{2}+\left (1-x^{2} y\right ) y^{\prime }-x y = 0 \] |
2 |
1 |
2 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.421 |
|
|
\[ {}{y^{\prime }}^{2}-\left (x^{2} y+3\right ) y^{\prime }+3 x^{2} y = 0 \] |
2 |
1 |
2 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.398 |
|
|
\[ {}x {y^{\prime }}^{2}-\left (x y+1\right ) y^{\prime }+y = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.282 |
|
|
\[ {}{y^{\prime }}^{2}-x^{2} y^{2} = 0 \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.465 |
|
|
\[ {}\left (x +y\right )^{2} {y^{\prime }}^{2} = y^{2} \] |
2 |
1 |
3 |
homogeneousTypeD2 |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.195 |
|
|
\[ {}y {y^{\prime }}^{2}+\left (x -y^{2}\right ) y^{\prime }-x y = 0 \] |
2 |
2 |
3 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.532 |
|
|
\[ {}{y^{\prime }}^{2}-x y \left (x +y\right ) y^{\prime }+x^{3} y^{3} = 0 \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.546 |
|
|
\[ {}\left (4 x -y\right ) {y^{\prime }}^{2}+6 \left (x -y\right ) y^{\prime }+2 x -5 y = 0 \] |
2 |
1 |
3 |
quadrature, homogeneousTypeD2 |
[_quadrature] |
✓ |
✓ |
0.836 |
|
|
\[ {}\left (x -y\right )^{2} {y^{\prime }}^{2} = y^{2} \] |
2 |
1 |
3 |
homogeneousTypeD2 |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.11 |
|
|
\[ {}x y {y^{\prime }}^{2}+\left (x y^{2}-1\right ) y^{\prime }-y = 0 \] |
2 |
2 |
3 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.536 |
|
|
\[ {}\left (x^{2}+y^{2}\right )^{2} {y^{\prime }}^{2} = 4 x^{2} y^{2} \] |
2 |
1 |
5 |
homogeneousTypeD2 |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.7 |
|
|
\[ {}\left (x +y\right )^{2} {y^{\prime }}^{2}+\left (2 y^{2}+x y-x^{2}\right ) y^{\prime }+y \left (y-x \right ) = 0 \] |
2 |
1 |
4 |
homogeneousTypeD2 |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.439 |
|
|
\[ {}x y \left (x^{2}+y^{2}\right ) \left ({y^{\prime }}^{2}-1\right ) = y^{\prime } \left (x^{4}+x^{2} y^{2}+y^{4}\right ) \] |
2 |
1 |
6 |
homogeneousTypeD2 |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.103 |
|
|
\[ {}x {y^{\prime }}^{3}-\left (x +x^{2}+y\right ) {y^{\prime }}^{2}+\left (x^{2}+y+x y\right ) y^{\prime }-x y = 0 \] |
3 |
1 |
3 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.477 |
|
|
\[ {}x y {y^{\prime }}^{2}+\left (x +y\right ) y^{\prime }+1 = 0 \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.403 |
|
|
\[ {}4 x -2 y y^{\prime }+x {y^{\prime }}^{2} = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.368 |
|
|
\[ {}3 x^{4} {y^{\prime }}^{2}-x y^{\prime }-y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
4.311 |
|
|
\[ {}{y^{\prime }}^{2}-x y^{\prime }-y = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.498 |
|
|
\[ {}{y^{\prime }}^{2}-x y^{\prime }+y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.262 |
|
|
\[ {}{y^{\prime }}^{2}+4 x^{5} y^{\prime }-12 x^{4} y = 0 \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
4.935 |
|
|
\[ {}4 y^{3} {y^{\prime }}^{2}-4 x y^{\prime }+y = 0 \] |
2 |
2 |
6 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
6.27 |
|
|
\[ {}4 y^{3} {y^{\prime }}^{2}+4 x y^{\prime }+y = 0 \] |
2 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
22.549 |
|
|
\[ {}{y^{\prime }}^{3}+x {y^{\prime }}^{2}-y = 0 \] |
3 |
5 |
4 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
151.773 |
|
|
\[ {}y^{4} {y^{\prime }}^{3}-6 x y^{\prime }+2 y = 0 \] |
3 |
1 |
10 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
114.33 |
|
|
\[ {}{y^{\prime }}^{2}+x^{3} y^{\prime }-2 x^{2} y = 0 \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
5.641 |
|
|
\[ {}{y^{\prime }}^{2}+4 x^{5} y^{\prime }-12 x^{4} y = 0 \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
4.579 |
|
|
\[ {}2 x {y^{\prime }}^{3}-6 y {y^{\prime }}^{2}+x^{4} = 0 \] |
3 |
1 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
108.481 |
|
|
\[ {}{y^{\prime }}^{2}-x y^{\prime }+y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.292 |
|
|
\[ {}y = x y^{\prime }+k {y^{\prime }}^{2} \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.302 |
|
|
\[ {}x^{8} {y^{\prime }}^{2}+3 x y^{\prime }+9 y = 0 \] |
2 |
2 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
4.955 |
|
|
\[ {}x^{4} {y^{\prime }}^{2}+2 x^{3} y y^{\prime }-4 = 0 \] |
2 |
2 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
5.307 |
|
|
\[ {}4 x -2 y y^{\prime }+x {y^{\prime }}^{2} = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.375 |
|
|
\[ {}3 x^{4} {y^{\prime }}^{2}-x y^{\prime }-y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
4.55 |
|
|
\[ {}x {y^{\prime }}^{2}+\left (x -y\right ) y^{\prime }+1-y = 0 \] |
2 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _dAlembert] |
✓ |
✓ |
0.401 |
|
|
\[ {}y^{\prime } \left (x y^{\prime }-y+k \right )+a = 0 \] |
2 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
0.422 |
|
|
\[ {}x^{6} {y^{\prime }}^{3}-3 x y^{\prime }-3 y = 0 \] |
3 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
23.949 |
|
|
\[ {}y = x^{6} {y^{\prime }}^{3}-x y^{\prime } \] |
3 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
94.164 |
|
|
\[ {}{y^{\prime }}^{4} x -2 y {y^{\prime }}^{3}+12 x^{3} = 0 \] |
4 |
0 |
5 |
unknown |
[[_1st_order, _with_linear_symmetries]] |
✗ |
N/A |
3.405 |
|
|
\[ {}x {y^{\prime }}^{3}-y {y^{\prime }}^{2}+1 = 0 \] |
3 |
4 |
4 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
1.012 |
|
|
\[ {}{y^{\prime }}^{2}-x y^{\prime }-y = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.461 |
|
|
\[ {}2 {y^{\prime }}^{3}+x y^{\prime }-2 y = 0 \] |
3 |
4 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
94.181 |
|
|
\[ {}2 {y^{\prime }}^{2}+x y^{\prime }-2 y = 0 \] |
2 |
3 |
1 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.528 |
|
|
\[ {}{y^{\prime }}^{3}+2 x y^{\prime }-y = 0 \] |
3 |
4 |
4 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
6.592 |
|
|
\[ {}4 x {y^{\prime }}^{2}-3 y y^{\prime }+3 = 0 \] |
2 |
2 |
4 |
dAlembert |
[[_homogeneous, ‘class G‘], _rational, _dAlembert] |
✓ |
✓ |
0.581 |
|
|
\[ {}{y^{\prime }}^{3}-x y^{\prime }+2 y = 0 \] |
3 |
4 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
92.384 |
|
|
\[ {}5 {y^{\prime }}^{2}+6 x y^{\prime }-2 y = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.694 |
|
|
\[ {}2 x {y^{\prime }}^{2}+\left (2 x -y\right ) y^{\prime }+1-y = 0 \] |
2 |
3 |
1 |
dAlembert |
[_rational, _dAlembert] |
✓ |
✓ |
1.139 |
|
|
\[ {}5 {y^{\prime }}^{2}+3 x y^{\prime }-y = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.63 |
|
|
\[ {}{y^{\prime }}^{2}+3 x y^{\prime }-y = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.612 |
|
|
\[ {}y = x y^{\prime }+x^{3} {y^{\prime }}^{2} \] |
2 |
2 |
0 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
25.161 |
|
|
\[ {}x^{3} {y^{\prime }}^{2}+x^{2} y y^{\prime }+4 = 0 \] |
2 |
2 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
8.249 |
|
|
\[ {}6 x {y^{\prime }}^{2}-\left (3 x +2 y\right ) y^{\prime }+y = 0 \] |
2 |
1 |
2 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.651 |
|
|
\[ {}9 {y^{\prime }}^{2}+3 x y^{4} y^{\prime }+y^{5} = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
79.343 |
|
|
\[ {}4 y^{3} {y^{\prime }}^{2}-4 x y^{\prime }+y = 0 \] |
2 |
2 |
6 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
9.931 |
|
|
\[ {}x^{6} {y^{\prime }}^{2}-2 x y^{\prime }-4 y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
8.614 |
|
|
\[ {}5 {y^{\prime }}^{2}+6 x y^{\prime }-2 y = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.564 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}-\left (1+x \right ) y y^{\prime }+x = 0 \] |
2 |
2 |
4 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.669 |
|
|
\[ {}4 x^{5} {y^{\prime }}^{2}+12 x^{4} y y^{\prime }+9 = 0 \] |
2 |
2 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
10.965 |
|
|
\[ {}4 y^{2} {y^{\prime }}^{3}-2 x y^{\prime }+y = 0 \] |
3 |
1 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
83.928 |
|
|
\[ {}{y^{\prime }}^{4}+x y^{\prime }-3 y = 0 \] |
4 |
2 |
1 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
3.161 |
|
|
\[ {}x^{2} {y^{\prime }}^{3}-2 x y {y^{\prime }}^{2}+y^{2} y^{\prime }+1 = 0 \] |
3 |
8 |
5 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
9.514 |
|
|
\[ {}16 x {y^{\prime }}^{2}+8 y y^{\prime }+y^{6} = 0 \] |
2 |
2 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
8.799 |
|
|
\[ {}x {y^{\prime }}^{2}-\left (x^{2}+1\right ) y^{\prime }+x = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.295 |
|
|
\[ {}{y^{\prime }}^{3}-2 x y^{\prime }-y = 0 \] |
3 |
4 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
101.202 |
|
|
\[ {}9 x y^{4} {y^{\prime }}^{2}-3 y^{5} y^{\prime }-1 = 0 \] |
2 |
1 |
12 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
7.156 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-\left (1+2 x y\right ) y^{\prime }+1+y^{2} = 0 \] |
2 |
4 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
1.059 |
|
|
\[ {}x^{6} {y^{\prime }}^{2} = 16 y+8 x y^{\prime } \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
8.246 |
|
|
\[ {}x^{2} {y^{\prime }}^{2} = \left (x -y\right )^{2} \] |
2 |
1 |
2 |
linear |
[_linear] |
✓ |
✓ |
1.242 |
|
|
\[ {}\left (y^{\prime }+1\right )^{2} \left (y-x y^{\prime }\right ) = 1 \] |
3 |
4 |
4 |
clairaut |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
2.13 |
|
|
\[ {}{y^{\prime }}^{3}-{y^{\prime }}^{2}+x y^{\prime }-y = 0 \] |
3 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
1.22 |
|
|
\[ {}x {y^{\prime }}^{2}+\left (1-x \right ) y y^{\prime }-y^{2} = 0 \] |
2 |
1 |
2 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.806 |
|
|
\[ {}y {y^{\prime }}^{2}-\left (x +y\right ) y^{\prime }+y = 0 \] |
2 |
4 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.45 |
|
|
\[ {}x {y^{\prime }}^{2}+\left (k -x -y\right ) y^{\prime }+y = 0 \] |
2 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _dAlembert] |
✓ |
✓ |
0.583 |
|
|
\[ {}x {y^{\prime }}^{3}-2 y {y^{\prime }}^{2}+4 x^{2} = 0 \] |
3 |
1 |
11 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
117.751 |
|
|
\[ {}\frac {{y^{\prime }}^{2}}{4}-x y^{\prime }+y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.258 |
|
|
\[ {}y = x y^{\prime }+x^{2} {y^{\prime }}^{2} \] |
2 |
2 |
2 |
separable |
[_separable] |
✓ |
✓ |
2.337 |
|
|
\[ {}y = x {y^{\prime }}^{2}+{y^{\prime }}^{2} \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
✓ |
0.378 |
|
|
\[ {}x f^{\prime }-f = \frac {{f^{\prime }}^{2} \left (1-{f^{\prime }}^{\lambda }\right )^{2}}{\lambda ^{2}} \] |
0 |
2 |
3 |
clairaut |
[_Clairaut] |
✓ |
✓ |
41.631 |
|
|
\[ {}y = x {y^{\prime }}^{2} \] |
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 |
|
|
\[ {}\frac {y^{\prime } y}{1+\frac {\sqrt {1+{y^{\prime }}^{2}}}{2}} = -x \] |
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 |
|
|
\[ {}{y^{\prime }}^{2}+y^{2} = \sec \left (x \right )^{4} \] |
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 |
|
|
\[ {}h^{2}+\frac {2 a h}{\sqrt {1+{h^{\prime }}^{2}}} = b^{2} \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
11.224 |
|
|
\[ {}x \sin \left (x \right ) {y^{\prime }}^{2} = 0 \] |
2 |
2 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.195 |
|
|
\[ {}y {y^{\prime }}^{2} = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.258 |
|
|
\[ {}{y^{\prime }}^{n} = 0 \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.125 |
|
|
\[ {}x {y^{\prime }}^{n} = 0 \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.122 |
|
|
\[ {}{y^{\prime }}^{2} = x \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.398 |
|
|
\[ {}{y^{\prime }}^{2} = x +y \] |
2 |
2 |
1 |
dAlembert, first_order_nonlinear_p_but_linear_in_x_y |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.018 |
|
|
\[ {}{y^{\prime }}^{2} = \frac {y}{x} \] |
2 |
3 |
3 |
dAlembert, first_order_nonlinear_p_but_separable |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.82 |
|
|
\[ {}{y^{\prime }}^{2} = \frac {y^{2}}{x} \] |
2 |
2 |
3 |
first_order_nonlinear_p_but_separable |
[_separable] |
✓ |
✓ |
2.366 |
|
|
\[ {}{y^{\prime }}^{2} = \frac {y^{3}}{x} \] |
2 |
2 |
2 |
first_order_nonlinear_p_but_separable |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
1.619 |
|
|
\[ {}{y^{\prime }}^{3} = \frac {y^{2}}{x} \] |
3 |
3 |
10 |
first_order_nonlinear_p_but_separable |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.214 |
|
|
\[ {}{y^{\prime }}^{2} = \frac {1}{x y} \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
4.559 |
|
|
\[ {}{y^{\prime }}^{2} = \frac {1}{x y^{3}} \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
4.829 |
|
|
\[ {}{y^{\prime }}^{2} = \frac {1}{x^{2} y^{3}} \] |
2 |
2 |
2 |
separable |
[_separable] |
✓ |
✓ |
1.224 |
|
|
\[ {}{y^{\prime }}^{4} = \frac {1}{x y^{3}} \] |
4 |
4 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
12.116 |
|
|
\[ {}{y^{\prime }}^{2} = \frac {1}{x^{3} y^{4}} \] |
2 |
6 |
6 |
separable |
[_separable] |
✓ |
✓ |
1.374 |
|
|
\[ {}{y^{\prime }}^{2}+a y+b \,x^{2} = 0 \] |
2 |
1 |
0 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
6.334 |
|
|
\[ {}{y^{\prime }}^{2}+y^{2}-a^{2} = 0 \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.372 |
|
|
\[ {}{y^{\prime }}^{2}+y^{2}-f \left (x \right )^{2} = 0 \] |
2 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
2.067 |
|
|
\[ {}{y^{\prime }}^{2}-y^{3}+y^{2} = 0 \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.365 |
|
|
\[ {}{y^{\prime }}^{2}-4 y^{3}+a y+b = 0 \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
17.666 |
|
|
\[ {}{y^{\prime }}^{2}+a^{2} y^{2} \left (\ln \left (y\right )^{2}-1\right ) = 0 \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
3.986 |
|
|
\[ {}{y^{\prime }}^{2}-2 y^{\prime }-y^{2} = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.912 |
|
|
\[ {}{y^{\prime }}^{2}+a y^{\prime }+b x = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.733 |
|
|
\[ {}{y^{\prime }}^{2}+a y^{\prime }+b y = 0 \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
2.322 |
|
|
\[ {}{y^{\prime }}^{2}+\left (-2+x \right ) y^{\prime }-y+1 = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.513 |
|
|
|
|||||||||
|
\[ {}{y^{\prime }}^{2}+\left (x +a \right ) y^{\prime }-y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.466 |
|
|
\[ {}{y^{\prime }}^{2}-\left (1+x \right ) y^{\prime }+y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.458 |
|
|
\[ {}{y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.687 |
|
|
\[ {}{y^{\prime }}^{2}-2 x y^{\prime }+y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.692 |
|
|
\[ {}{y^{\prime }}^{2}+a x y^{\prime }-b \,x^{2}-c = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.967 |
|
|
\[ {}{y^{\prime }}^{2}+a x y^{\prime }+b y+c \,x^{2} = 0 \] |
2 |
1 |
0 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
8.793 |
|
|
\[ {}{y^{\prime }}^{2}+\left (x a +b \right ) y^{\prime }-a y+c = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.502 |
|
|
\[ {}{y^{\prime }}^{2}-2 x^{2} y^{\prime }+2 x y = 0 \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
52.138 |
|
|
\[ {}{y^{\prime }}^{2}+a \,x^{3} y^{\prime }-2 a \,x^{2} y = 0 \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
23.797 |
|
|
\[ {}{y^{\prime }}^{2}+\left (y^{\prime }-y\right ) {\mathrm e}^{x} = 0 \] |
2 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
13.042 |
|
|
\[ {}{y^{\prime }}^{2}-2 y y^{\prime }-2 x = 0 \] |
2 |
4 |
2 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
0.981 |
|
|
\[ {}{y^{\prime }}^{2}-\left (4 y+1\right ) y^{\prime }+\left (4 y+1\right ) y = 0 \] |
2 |
2 |
5 |
quadrature |
[_quadrature] |
✓ |
✓ |
3.441 |
|
|
\[ {}{y^{\prime }}^{2}+a y y^{\prime }-b x -c = 0 \] |
2 |
4 |
1 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
91.843 |
|
|
\[ {}{y^{\prime }}^{2}+\left (b x +a y\right ) y^{\prime }+a b x y = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.919 |
|
|
\[ {}{y^{\prime }}^{2}-x y y^{\prime }+y^{2} \ln \left (a y\right ) = 0 \] |
2 |
2 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
8.968 |
|
|
\[ {}{y^{\prime }}^{2}+2 y y^{\prime } \cot \left (x \right )-y^{2} = 0 \] |
2 |
2 |
3 |
separable |
[_separable] |
✓ |
✓ |
8.635 |
|
|
\[ {}{y^{\prime }}^{2}+2 f \left (x \right ) y y^{\prime }+g \left (x \right ) y^{2}+h \left (x \right ) = 0 \] |
2 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
14.407 |
|
|
\[ {}{y^{\prime }}^{2}+y \left (y-x \right ) y^{\prime }-x y^{3} = 0 \] |
2 |
1 |
2 |
quadrature, separable |
[_separable] |
✓ |
✓ |
0.816 |
|
|
\[ {}{y^{\prime }}^{2}-2 x^{3} y^{2} y^{\prime }-4 x^{2} y^{3} = 0 \] |
2 |
1 |
6 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
35.307 |
|
|
\[ {}{y^{\prime }}^{2}-3 x y^{\frac {2}{3}} y^{\prime }+9 y^{\frac {5}{3}} = 0 \] |
2 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
14.772 |
|
|
\[ {}2 {y^{\prime }}^{2}+\left (-1+x \right ) y^{\prime }-y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.484 |
|
|
\[ {}2 {y^{\prime }}^{2}-2 x^{2} y^{\prime }+3 x y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
14.767 |
|
|
\[ {}3 {y^{\prime }}^{2}-2 x y^{\prime }+y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.743 |
|
|
\[ {}3 {y^{\prime }}^{2}+4 x y^{\prime }+x^{2}-y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
3.991 |
|
|
\[ {}a {y^{\prime }}^{2}+b y^{\prime }-y = 0 \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
2.518 |
|
|
\[ {}a {y^{\prime }}^{2}+b \,x^{2} y^{\prime }+c x y = 0 \] |
2 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
44.898 |
|
|
\[ {}a {y^{\prime }}^{2}+y y^{\prime }-x = 0 \] |
2 |
4 |
2 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
91.692 |
|
|
\[ {}a {y^{\prime }}^{2}-y y^{\prime }-x = 0 \] |
2 |
4 |
2 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
1.307 |
|
|
\[ {}x {y^{\prime }}^{2}-y = 0 \] |
2 |
3 |
3 |
dAlembert, first_order_nonlinear_p_but_separable |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.808 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y+x = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.03 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y^{\prime }-y = 0 \] |
2 |
4 |
1 |
dAlembert |
[_rational, _dAlembert] |
✓ |
✓ |
0.774 |
|
|
\[ {}x {y^{\prime }}^{2}+4 y^{\prime }-2 y = 0 \] |
2 |
4 |
1 |
dAlembert |
[_rational, _dAlembert] |
✓ |
✓ |
0.786 |
|
|
\[ {}x {y^{\prime }}^{2}+x y^{\prime }-y = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.789 |
|
|
\[ {}x {y^{\prime }}^{2}+y y^{\prime }+a = 0 \] |
2 |
2 |
2 |
dAlembert |
[[_homogeneous, ‘class G‘], _dAlembert] |
✓ |
✓ |
1.003 |
|
|
\[ {}x {y^{\prime }}^{2}+y y^{\prime }-x^{2} = 0 \] |
2 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
75.885 |
|
|
\[ {}x {y^{\prime }}^{2}+y y^{\prime }+x^{3} = 0 \] |
2 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
18.308 |
|
|
\[ {}x {y^{\prime }}^{2}+y y^{\prime }-y^{4} = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
5.524 |
|
|
\[ {}x {y^{\prime }}^{2}+\left (-3 x +y\right ) y^{\prime }+y = 0 \] |
2 |
4 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.282 |
|
|
\[ {}x {y^{\prime }}^{2}-y y^{\prime }+a = 0 \] |
2 |
3 |
3 |
clairaut |
[[_homogeneous, ‘class G‘], _rational, _Clairaut] |
✓ |
✓ |
0.625 |
|
|
\[ {}x {y^{\prime }}^{2}-y y^{\prime }+a y = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.109 |
|
|
\[ {}x {y^{\prime }}^{2}+2 y y^{\prime }-x = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.729 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y y^{\prime }+a = 0 \] |
2 |
2 |
3 |
dAlembert |
[[_homogeneous, ‘class G‘], _rational, _dAlembert] |
✓ |
✓ |
1.007 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y y^{\prime }-x = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.658 |
|
|
\[ {}4 x -2 y y^{\prime }+x {y^{\prime }}^{2} = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.582 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y y^{\prime }+x +2 y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.724 |
|
|
\[ {}x {y^{\prime }}^{2}+a y y^{\prime }+b x = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.891 |
|
|
\[ {}\left (1+x \right ) {y^{\prime }}^{2}-\left (x +y\right ) y^{\prime }+y = 0 \] |
2 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _dAlembert] |
✓ |
✓ |
0.747 |
|
|
\[ {}\left (1+3 x \right ) {y^{\prime }}^{2}-3 \left (y+2\right ) y^{\prime }+9 = 0 \] |
2 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.732 |
|
|
\[ {}\left (3 x +5\right ) {y^{\prime }}^{2}-\left (x +3 y\right ) y^{\prime }+y = 0 \] |
2 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _dAlembert] |
✓ |
✓ |
0.777 |
|
|
\[ {}a x {y^{\prime }}^{2}+\left (b x -a y+c \right ) y^{\prime }-b y = 0 \] |
2 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _dAlembert] |
✓ |
✓ |
0.829 |
|
|
\[ {}a x {y^{\prime }}^{2}-\left (a y+b x -a -b \right ) y^{\prime }+b y = 0 \] |
2 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _dAlembert] |
✓ |
✓ |
0.886 |
|
|
\[ {}\left (\operatorname {a2} x +\operatorname {c2} \right ) {y^{\prime }}^{2}+\left (\operatorname {a1} x +\operatorname {b1} y+\operatorname {c1} \right ) y^{\prime }+\operatorname {a0} x +\operatorname {b0} y+\operatorname {c0} = 0 \] |
2 |
4 |
2 |
dAlembert |
[_rational, _dAlembert] |
✓ |
✓ |
128.933 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+y^{2}-y^{4} = 0 \] |
2 |
2 |
5 |
first_order_nonlinear_p_but_separable |
[_separable] |
✓ |
✓ |
2.795 |
|
|
\[ {}\left (x y^{\prime }+a \right )^{2}-2 a y+x^{2} = 0 \] |
2 |
0 |
1 |
unknown |
[_rational] |
✗ |
N/A |
4.225 |
|
|
\[ {}\left (x y^{\prime }+y+2 x \right )^{2}-4 x y-4 x^{2}-4 a = 0 \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
8.199 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-2 x y y^{\prime }-x +y \left (y+1\right ) = 0 \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
8.697 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-2 x y y^{\prime }-x^{4}+\left (-x^{2}+1\right ) y^{2} = 0 \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
6.124 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-\left (2 x y+a \right ) y^{\prime }+y^{2} = 0 \] |
2 |
4 |
3 |
clairaut |
[[_homogeneous, ‘class G‘], _rational, _Clairaut] |
✓ |
✓ |
1.856 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+3 x y y^{\prime }+2 y^{2} = 0 \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
1.616 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+3 x y y^{\prime }+3 y^{2} = 0 \] |
2 |
2 |
3 |
separable |
[_separable] |
✓ |
✓ |
1.979 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+4 x y y^{\prime }-5 y^{2} = 0 \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
1.934 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-4 x \left (y+2\right ) y^{\prime }+4 y \left (y+2\right ) = 0 \] |
2 |
2 |
5 |
separable |
[_separable] |
✓ |
✓ |
7.426 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+\left (x^{2} y-2 x y+x^{3}\right ) y^{\prime }+\left (y^{2}-x^{2} y\right ) \left (1-x \right ) = 0 \] |
2 |
1 |
2 |
linear, separable |
[_linear] |
✓ |
✓ |
1.335 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-y \left (-2 x +y\right ) y^{\prime }+y^{2} = 0 \] |
2 |
3 |
6 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
2.429 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+\left (a \,x^{2} y^{3}+b \right ) y^{\prime }+a b y^{3} = 0 \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.109 |
|
|
\[ {}\left (x^{2}+1\right ) {y^{\prime }}^{2}-2 x y y^{\prime }+y^{2}-1 = 0 \] |
2 |
4 |
4 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
0.944 |
|
|
\[ {}\left (x^{2}-1\right ) {y^{\prime }}^{2}-1 = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.809 |
|
|
\[ {}\left (x^{2}-1\right ) {y^{\prime }}^{2}-y^{2}+1 = 0 \] |
2 |
2 |
4 |
first_order_nonlinear_p_but_separable |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
2.639 |
|
|
\[ {}\left (-a^{2}+x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+y^{2} = 0 \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
1.438 |
|
|
\[ {}\left (-a^{2}+x^{2}\right ) {y^{\prime }}^{2}-2 x y y^{\prime }-x^{2} = 0 \] |
2 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
132.964 |
|
|
\[ {}\left (x^{2}+a \right ) {y^{\prime }}^{2}-2 x y y^{\prime }+y^{2}+b = 0 \] |
2 |
5 |
4 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
3.489 |
|
|
\[ {}\left (2 x^{2}+1\right ) {y^{\prime }}^{2}+\left (y^{2}+2 x y+x^{2}+2\right ) y^{\prime }+2 y^{2}+1 = 0 \] |
2 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
26.482 |
|
|
\[ {}\left (a^{2}-1\right ) x^{2} {y^{\prime }}^{2}+2 x y y^{\prime }-y^{2}+a^{2} x^{2} = 0 \] |
2 |
8 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
213.142 |
|
|
\[ {}a \,x^{2} {y^{\prime }}^{2}-2 a x y y^{\prime }+y^{2}-a \left (a -1\right ) x^{2} = 0 \] |
2 |
8 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.766 |
|
|
\[ {}x^{3} {y^{\prime }}^{2}+x^{2} y y^{\prime }+a = 0 \] |
2 |
2 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
6.274 |
|
|
\[ {}x \left (x^{2}-1\right ) {y^{\prime }}^{2}+2 \left (-x^{2}+1\right ) y y^{\prime }+x y^{2}-x = 0 \] |
2 |
0 |
3 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
8.06 |
|
|
\[ {}x^{4} {y^{\prime }}^{2}-x y^{\prime }-y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
4.196 |
|
|
\[ {}x^{2} \left (-a^{2}+x^{2}\right ) {y^{\prime }}^{2}-1 = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.685 |
|
|
\[ {}{\mathrm e}^{-2 x} {y^{\prime }}^{2}-\left (y^{\prime }-1\right )^{2}+{\mathrm e}^{-2 y} = 0 \] |
2 |
0 |
2 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
12.216 |
|
|
\[ {}\left ({y^{\prime }}^{2}+y^{2}\right ) \cos \left (x \right )^{4}-a^{2} = 0 \] |
2 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
11.382 |
|
|
\[ {}\operatorname {d0} \left (x \right ) {y^{\prime }}^{2}+2 \operatorname {b0} \left (x \right ) y y^{\prime }+\operatorname {c0} \left (x \right ) y^{2}+2 \operatorname {d0} \left (x \right ) y^{\prime }+2 \operatorname {e0} \left (x \right ) y+\operatorname {f0} \left (x \right ) = 0 \] |
2 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
77.711 |
|
|
\[ {}y {y^{\prime }}^{2}-1 = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.422 |
|
|
\[ {}y {y^{\prime }}^{2}-{\mathrm e}^{2 x} = 0 \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
4.508 |
|
|
\[ {}y {y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
2 |
5 |
7 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.859 |
|
|
\[ {}y {y^{\prime }}^{2}+2 x y^{\prime }-9 y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.352 |
|
|
\[ {}y {y^{\prime }}^{2}-2 x y^{\prime }+y = 0 \] |
2 |
5 |
7 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.612 |
|
|
\[ {}y {y^{\prime }}^{2}-4 x y^{\prime }+y = 0 \] |
2 |
4 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.174 |
|
|
\[ {}y {y^{\prime }}^{2}-4 a^{2} x y^{\prime }+a^{2} y = 0 \] |
2 |
5 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.326 |
|
|
\[ {}y {y^{\prime }}^{2}+a x y^{\prime }+b y = 0 \] |
2 |
4 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
22.196 |
|
|
\[ {}y {y^{\prime }}^{2}+x^{3} y^{\prime }-x^{2} y = 0 \] |
2 |
2 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
5.781 |
|
|
\[ {}y {y^{\prime }}^{2}-\left (y-x \right ) y^{\prime }-x = 0 \] |
2 |
2 |
3 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.495 |
|
|
\[ {}\left (x +y\right ) {y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
2 |
5 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.971 |
|
|
\[ {}\left (-2 x +y\right ) {y^{\prime }}^{2}-2 \left (-1+x \right ) y^{\prime }+y-2 = 0 \] |
2 |
5 |
4 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.936 |
|
|
\[ {}2 y {y^{\prime }}^{2}-\left (4 x -5\right ) y^{\prime }+2 y = 0 \] |
2 |
5 |
7 |
dAlembert |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
✓ |
0.817 |
|
|
\[ {}4 y {y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
2 |
5 |
7 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.879 |
|
|
\[ {}9 y {y^{\prime }}^{2}+4 x^{3} y^{\prime }-4 x^{2} y = 0 \] |
2 |
2 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
5.589 |
|
|
\[ {}a y {y^{\prime }}^{2}+\left (2 x -b \right ) y^{\prime }-y = 0 \] |
2 |
5 |
5 |
dAlembert |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
✓ |
1.192 |
|
|
\[ {}\left (a y+b \right ) \left (1+{y^{\prime }}^{2}\right )-c = 0 \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
6.11 |
|
|
\[ {}\left (b_{2} y+a_{2} x +c_{2} \right ) {y^{\prime }}^{2}+\left (a_{1} x +b_{1} y+c_{1} \right ) y^{\prime }+a_{0} x +b_{0} y+c_{0} = 0 \] |
2 |
3 |
2 |
dAlembert |
[_rational, _dAlembert] |
✓ |
✗ |
9.348 |
|
|
\[ {}\left (a y-x^{2}\right ) {y^{\prime }}^{2}+2 x y {y^{\prime }}^{2}-y^{2} = 0 \] |
2 |
0 |
0 |
unknown |
[_rational] |
✗ |
N/A |
2.563 |
|
|
\[ {}x y {y^{\prime }}^{2}+\left (x^{2}+y^{2}\right ) y^{\prime }+x y = 0 \] |
2 |
2 |
3 |
separable |
[_separable] |
✓ |
✓ |
0.627 |
|
|
\[ {}x y {y^{\prime }}^{2}+\left (x^{22}-y^{2}+a \right ) y^{\prime }-x y = 0 \] |
2 |
0 |
0 |
unknown |
[_rational] |
✗ |
N/A |
24.905 |
|
|
\[ {}\left (2 x y-x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+2 x y-y^{2} = 0 \] |
2 |
5 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.207 |
|
|
\[ {}\left (2 x y-x^{2}\right ) {y^{\prime }}^{2}-6 x y y^{\prime }-y^{2}+2 x y = 0 \] |
2 |
9 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
219.059 |
|
|
\[ {}a x y {y^{\prime }}^{2}-\left (a y^{2}+b \,x^{2}+c \right ) y^{\prime }+b x y = 0 \] |
2 |
1 |
0 |
first_order_ode_lie_symmetry_calculated |
[_rational] |
✓ |
✓ |
100.549 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}+y^{2}-a^{2} = 0 \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.729 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}-6 x^{3} y^{\prime }+4 x^{2} y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
6.134 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}-4 a y y^{\prime }+4 a^{2}-4 x a +y^{2} = 0 \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
5.074 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}+2 x y y^{\prime }+a y^{2}+b x +c = 0 \] |
2 |
0 |
2 |
unknown |
[_rational] |
✗ |
N/A |
5.374 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+a -x^{2}+2 y^{2} = 0 \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
5.46 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}+2 a x y y^{\prime }+\left (a -1\right ) b +x^{2} a +\left (1-a \right ) y^{2} = 0 \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
7.128 |
|
|
\[ {}\left (y^{2}-a^{2}\right ) {y^{\prime }}^{2}+y^{2} = 0 \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.174 |
|
|
\[ {}\left (y^{2}-2 x a +a^{2}\right ) {y^{\prime }}^{2}+2 a y y^{\prime }+y^{2} = 0 \] |
2 |
0 |
2 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
7.22 |
|
|
\[ {}\left (y^{2}-a^{2} x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+\left (-a^{2}+1\right ) x^{2} = 0 \] |
2 |
10 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.904 |
|
|
\[ {}\left (\left (1-a \right ) x^{2}+y^{2}\right ) {y^{\prime }}^{2}+2 a x y y^{\prime }+\left (1-a \right ) y^{2}+x^{2} = 0 \] |
2 |
4 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.012 |
|
|
\[ {}\left (y-x \right )^{2} \left (1+{y^{\prime }}^{2}\right )-a^{2} \left (y^{\prime }+1\right )^{2} = 0 \] |
2 |
6 |
4 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
18.796 |
|
|
\[ {}3 y^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+4 y^{2}-x^{2} = 0 \] |
2 |
6 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
3.49 |
|
|
\[ {}\left (3 y-2\right ) {y^{\prime }}^{2}-4+4 y = 0 \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.652 |
|
|
\[ {}\left (-a^{2}+1\right ) y^{2} {y^{\prime }}^{2}-2 a^{2} x y y^{\prime }+y^{2}-a^{2} x^{2} = 0 \] |
2 |
7 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
10.343 |
|
|
\[ {}\left (-b +a \right ) y^{2} {y^{\prime }}^{2}-2 b x y y^{\prime }-a b -b \,x^{2}+a y^{2} = 0 \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
6.755 |
|
|
\[ {}\left (a y^{2}+b x +c \right ) {y^{\prime }}^{2}-b y y^{\prime }+d y^{2} = 0 \] |
2 |
0 |
3 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
13.713 |
|
|
\[ {}\left (a y-b x \right )^{2} \left (a^{2} {y^{\prime }}^{2}+b^{2}\right )-c^{2} \left (a y^{\prime }+b \right )^{2} = 0 \] |
2 |
6 |
4 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
30.862 |
|
|
\[ {}\left (\operatorname {b2} y+\operatorname {a2} x +\operatorname {c2} \right )^{2} {y^{\prime }}^{2}+\left (\operatorname {a1} x +\operatorname {b1} y+\operatorname {c1} \right ) y^{\prime }+\operatorname {b0} y+\operatorname {a0} +\operatorname {c0} = 0 \] |
2 |
0 |
0 |
unknown |
[_rational] |
✗ |
N/A |
90.787 |
|
|
\[ {}x y^{2} {y^{\prime }}^{2}-\left (y^{3}+x^{3}-a \right ) y^{\prime }+x^{2} y = 0 \] |
2 |
0 |
9 |
unknown |
[_rational] |
✗ |
N/A |
15.87 |
|
|
\[ {}x y^{2} {y^{\prime }}^{2}-2 y^{3} y^{\prime }+2 x y^{2}-x^{3} = 0 \] |
2 |
1 |
4 |
separable, homogeneousTypeD2 |
[_separable] |
✓ |
✓ |
1.033 |
|
|
\[ {}x^{2} \left (x y^{2}-1\right ) {y^{\prime }}^{2}+2 x^{2} y^{2} \left (y-x \right ) y^{\prime }-y^{2} \left (x^{2} y-1\right ) = 0 \] |
2 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
30.803 |
|
|
\[ {}\left (y^{4}-a^{2} x^{2}\right ) {y^{\prime }}^{2}+2 a^{2} x y y^{\prime }+y^{2} \left (y^{2}-a^{2}\right ) = 0 \] |
2 |
0 |
2 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
12.389 |
|
|
\[ {}\left (y^{4}+x^{2} y^{2}-x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }-y^{2} = 0 \] |
2 |
0 |
4 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
8.423 |
|
|
\[ {}9 y^{4} \left (x^{2}-1\right ) {y^{\prime }}^{2}-6 x y^{5} y^{\prime }-4 x^{2} = 0 \] |
2 |
0 |
9 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
7.985 |
|
|
\[ {}x^{2} \left (y^{4} x^{2}-1\right ) {y^{\prime }}^{2}+2 x^{3} y^{3} \left (-x^{2}+y^{2}\right ) y^{\prime }-y^{2} \left (x^{4} y^{2}-1\right ) = 0 \] |
2 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
26.609 |
|
|
\[ {}\left (a^{2} \sqrt {x^{2}+y^{2}}-x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+a^{2} \sqrt {x^{2}+y^{2}}-y^{2} = 0 \] |
2 |
1 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
22.525 |
|
|
\[ {}\left (a \left (x^{2}+y^{2}\right )^{\frac {3}{2}}-x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+a \left (x^{2}+y^{2}\right )^{\frac {3}{2}}-y^{2} = 0 \] |
2 |
0 |
3 |
unknown |
[[_1st_order, _with_linear_symmetries]] |
✗ |
N/A |
39.219 |
|
|
\[ {}{y^{\prime }}^{2} \sin \left (y\right )+2 x y^{\prime } \cos \left (y\right )^{3}-\sin \left (y\right ) \cos \left (y\right )^{4} = 0 \] |
2 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
99.033 |
|
|
\[ {}{y^{\prime }}^{2} \left (a \cos \left (y\right )+b \right )-c \cos \left (y\right )+d = 0 \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
11.155 |
|
|
\[ {}f \left (x^{2}+y^{2}\right ) \left (1+{y^{\prime }}^{2}\right )-\left (-y+x y^{\prime }\right )^{2} = 0 \] |
2 |
1 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
4.023 |
|
|
\[ {}\left (x^{2}+y^{2}\right ) f \left (\frac {x}{\sqrt {x^{2}+y^{2}}}\right ) \left (1+{y^{\prime }}^{2}\right )-\left (-y+x y^{\prime }\right )^{2} = 0 \] |
2 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘]] |
✓ |
✓ |
6.807 |
|
|
\[ {}\left (x^{2}+y^{2}\right ) f \left (\frac {y}{\sqrt {x^{2}+y^{2}}}\right ) \left (1+{y^{\prime }}^{2}\right )-\left (-y+x y^{\prime }\right )^{2} = 0 \] |
2 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘]] |
✓ |
✓ |
7.245 |
|
|
\[ {}{y^{\prime }}^{3}-\left (y-a \right )^{2} \left (y-b \right )^{2} = 0 \] |
3 |
3 |
5 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.117 |
|
|
\[ {}{y^{\prime }}^{3}-f \left (x \right ) \left (a y^{2}+b y+c \right )^{2} = 0 \] |
3 |
3 |
3 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
2.878 |
|
|
\[ {}{y^{\prime }}^{3}+y^{\prime }-y = 0 \] |
3 |
3 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.805 |
|
|
\[ {}{y^{\prime }}^{3}+x y^{\prime }-y = 0 \] |
3 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.293 |
|
|
\[ {}{y^{\prime }}^{3}-\left (x +5\right ) y^{\prime }+y = 0 \] |
3 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.324 |
|
|
\[ {}{y^{\prime }}^{3}-a x y^{\prime }+x^{3} = 0 \] |
3 |
3 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.045 |
|
|
\[ {}{y^{\prime }}^{3}-2 y y^{\prime }+y^{2} = 0 \] |
3 |
3 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
2.474 |
|
|
\[ {}{y^{\prime }}^{2}-a x y y^{\prime }+2 a y^{2} = 0 \] |
2 |
2 |
3 |
separable |
[_separable] |
✓ |
✓ |
6.857 |
|
|
\[ {}{y^{\prime }}^{3}-\left (x^{2}+x y+y^{2}\right ) {y^{\prime }}^{2}+\left (x y^{3}+x^{2} y^{2}+x^{3} y\right ) y^{\prime }-x^{3} y^{3} = 0 \] |
3 |
1 |
3 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.309 |
|
|
\[ {}{y^{\prime }}^{3}-x y^{4} y^{\prime }-y^{5} = 0 \] |
3 |
1 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
15.946 |
|
|
\[ {}{y^{\prime }}^{3}+a {y^{\prime }}^{2}+b y+a b x = 0 \] |
3 |
4 |
1 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
118.026 |
|
|
\[ {}{y^{\prime }}^{3}+x {y^{\prime }}^{2}-y = 0 \] |
3 |
5 |
4 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
163.223 |
|
|
\[ {}{y^{\prime }}^{3}-y {y^{\prime }}^{2}+y^{2} = 0 \] |
3 |
3 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.626 |
|
|
\[ {}{y^{\prime }}^{2}-\left (x^{2}+x y^{2}+y^{4}\right ) {y^{\prime }}^{2}+\left (x y^{6}+y^{4} x^{2}+x^{3} y^{2}\right ) y^{\prime }-x^{3} y^{6} = 0 \] |
2 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
21.569 |
|
|
\[ {}a {y^{\prime }}^{3}+b {y^{\prime }}^{2}+c y^{\prime }-y-d = 0 \] |
3 |
3 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
65.237 |
|
|
\[ {}x {y^{\prime }}^{3}-y {y^{\prime }}^{2}+a = 0 \] |
3 |
4 |
4 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.572 |
|
|
\[ {}4 x {y^{\prime }}^{3}-6 y {y^{\prime }}^{2}+3 y-x = 0 \] |
3 |
6 |
5 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
10.048 |
|
|
\[ {}8 x {y^{\prime }}^{3}-12 y {y^{\prime }}^{2}+9 y = 0 \] |
3 |
6 |
5 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
19.878 |
|
|
\[ {}\left (-a^{2}+x^{2}\right ) {y^{\prime }}^{3}+b x \left (-a^{2}+x^{2}\right ) {y^{\prime }}^{2}+y^{\prime }+b x = 0 \] |
3 |
1 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.499 |
|
|
\[ {}x^{3} {y^{\prime }}^{3}-3 x^{2} y {y^{\prime }}^{2}+\left (3 x y^{2}+x^{6}\right ) y^{\prime }-y^{3}-2 x^{5} y = 0 \] |
3 |
1 |
0 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
179.494 |
|
|
\[ {}2 \left (x y^{\prime }+y\right )^{3}-y y^{\prime } = 0 \] |
3 |
1 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
17.451 |
|
|
\[ {}{y^{\prime }}^{3} \sin \left (x \right )-\left (y \sin \left (x \right )-\cos \left (x \right )^{2}\right ) {y^{\prime }}^{2}-\left (y \cos \left (x \right )^{2}+\sin \left (x \right )\right ) y^{\prime }+y \sin \left (x \right ) = 0 \] |
3 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.508 |
|
|
\[ {}2 y {y^{\prime }}^{3}-y {y^{\prime }}^{2}+2 x y^{\prime }-x = 0 \] |
3 |
4 |
3 |
dAlembert, quadrature |
[_quadrature] |
✓ |
✓ |
0.523 |
|
|
\[ {}y^{2} {y^{\prime }}^{3}+2 x y^{\prime }-y = 0 \] |
3 |
1 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
81.011 |
|
|
\[ {}16 y^{2} {y^{\prime }}^{3}+2 x y^{\prime }-y = 0 \] |
3 |
1 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
81.368 |
|
|
\[ {}x y^{2} {y^{\prime }}^{3}-y^{3} {y^{\prime }}^{2}+x \left (x^{2}+1\right ) y^{\prime }-x^{2} y = 0 \] |
3 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
146.769 |
|
|
\[ {}x^{7} y^{2} {y^{\prime }}^{3}-\left (3 x^{6} y^{3}-1\right ) {y^{\prime }}^{2}+3 x^{5} y^{4} y^{\prime }-x^{4} y^{5} = 0 \] |
3 |
1 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
126.033 |
|
|
\[ {}{y^{\prime }}^{4}-\left (y-a \right )^{3} \left (y-b \right )^{2} = 0 \] |
4 |
4 |
6 |
quadrature |
[_quadrature] |
✓ |
✓ |
3.864 |
|
|
\[ {}{y^{\prime }}^{4}+3 \left (-1+x \right ) {y^{\prime }}^{2}-3 \left (2 y-1\right ) y^{\prime }+3 x = 0 \] |
4 |
3 |
4 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
1.138 |
|
|
\[ {}{y^{\prime }}^{4}-4 y \left (x y^{\prime }-2 y\right )^{2} = 0 \] |
4 |
0 |
3 |
unknown |
[[_homogeneous, ‘class G‘]] |
✗ |
N/A |
1.073 |
|
|
\[ {}{y^{\prime }}^{6}-\left (y-a \right )^{4} \left (y-b \right )^{3} = 0 \] |
6 |
6 |
8 |
quadrature |
[_quadrature] |
✓ |
✓ |
73.431 |
|
|
\[ {}x^{2} \left (1+{y^{\prime }}^{2}\right )^{3}-a^{2} = 0 \] |
6 |
6 |
6 |
quadrature |
[_quadrature] |
✓ |
✓ |
3.214 |
|
|
\[ {}{y^{\prime }}^{r}-a y^{s}-b \,x^{\frac {r s}{r -s}} = 0 \] |
0 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
1.43 |
|
|
\[ {}{y^{\prime }}^{n}-f \left (x \right )^{n} \left (y-a \right )^{n +1} \left (y-b \right )^{n -1} = 0 \] |
0 |
1 |
1 |
separable, first_order_nonlinear_p_but_separable |
[_separable] |
✓ |
✓ |
40.158 |
|
|
\[ {}{y^{\prime }}^{n}-f \left (x \right ) g \left (y\right ) = 0 \] |
0 |
1 |
1 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
0.72 |
|
|
\[ {}a {y^{\prime }}^{m}+b {y^{\prime }}^{n}-y = 0 \] |
0 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.252 |
|
|
\[ {}x^{n -1} {y^{\prime }}^{n}-n x y^{\prime }+y = 0 \] |
0 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
1.272 |
|
|
\[ {}\sqrt {1+{y^{\prime }}^{2}}+x y^{\prime }-y = 0 \] |
2 |
2 |
1 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
1.257 |
|
|
\[ {}\sqrt {1+{y^{\prime }}^{2}}+x {y^{\prime }}^{2}+y = 0 \] |
4 |
6 |
5 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
210.911 |
|
|
\[ {}a x \sqrt {1+{y^{\prime }}^{2}}+x y^{\prime }-y = 0 \] |
2 |
2 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.607 |
|
|
\[ {}y \sqrt {1+{y^{\prime }}^{2}}-a y y^{\prime }-x a = 0 \] |
2 |
1 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
180.667 |
|
|
\[ {}a y \sqrt {1+{y^{\prime }}^{2}}-2 x y y^{\prime }+y^{2}-x^{2} = 0 \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[_rational] |
✓ |
✓ |
19.386 |
|
|
\[ {}f \left (x^{2}+y^{2}\right ) \sqrt {1+{y^{\prime }}^{2}}-x y^{\prime }+y = 0 \] |
2 |
0 |
1 |
unknown |
[[_1st_order, _with_linear_symmetries]] |
✗ |
N/A |
6.225 |
|
|
\[ {}a \left (1+{y^{\prime }}^{3}\right )^{\frac {1}{3}}+b x y^{\prime }-y = 0 \] |
3 |
4 |
3 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
4.359 |
|
|
\[ {}\ln \left (y^{\prime }\right )+x y^{\prime }+a y+b = 0 \] |
0 |
2 |
1 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
2.369 |
|
|
\[ {}\ln \left (y^{\prime }\right )+a \left (-y+x y^{\prime }\right ) = 0 \] |
0 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.636 |
|
|
\[ {}y \ln \left (y^{\prime }\right )+y^{\prime }-y \ln \left (y\right )-x y = 0 \] |
0 |
1 |
1 |
separable, homogeneousTypeD2 |
[_separable] |
✓ |
✓ |
1.886 |
|
|
\[ {}\sin \left (y^{\prime }\right )+y^{\prime }-x = 0 \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.199 |
|
|
\[ {}a \cos \left (y^{\prime }\right )+b y^{\prime }+x = 0 \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.16 |
|
|
\[ {}{y^{\prime }}^{2} \sin \left (y^{\prime }\right )-y = 0 \] |
0 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.272 |
|
|
\[ {}\left (1+{y^{\prime }}^{2}\right ) \sin \left (-y+x y^{\prime }\right )^{2}-1 = 0 \] |
0 |
6 |
6 |
clairaut |
[_Clairaut] |
✓ |
✓ |
2.947 |
|
|
\[ {}\left (1+{y^{\prime }}^{2}\right ) \left (\arctan \left (y^{\prime }\right )+x a \right )+y^{\prime } = 0 \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✗ |
1.307 |
|
|
\[ {}a \,x^{n} f \left (y^{\prime }\right )+x y^{\prime }-y = 0 \] |
0 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
0.537 |
|
|
\[ {}\left (-y+x y^{\prime }\right )^{n} f \left (y^{\prime }\right )+y g \left (y^{\prime }\right )+x h \left (y^{\prime }\right ) = 0 \] |
0 |
0 |
0 |
unknown |
[‘x=_G(y,y’)‘] |
✗ |
N/A |
2.117 |
|
|
\[ {}f \left (x {y^{\prime }}^{2}\right )+2 x y^{\prime }-y = 0 \] |
0 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
0.347 |
|
|
\[ {}f \left (x -\frac {3 {y^{\prime }}^{2}}{2}\right )+{y^{\prime }}^{3}-y = 0 \] |
0 |
0 |
2 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
0.781 |
|
|
\[ {}y^{\prime } f \left (x y y^{\prime }-y^{2}\right )-x^{2} y^{\prime }+x y = 0 \] |
0 |
0 |
0 |
unknown |
[NONE] |
✗ |
N/A |
0.613 |
|
|
\[ {}\phi \left (f \left (x , y, y^{\prime }\right ), g \left (x , y, y^{\prime }\right )\right ) = 0 \] |
0 |
0 |
0 |
unknown |
[NONE] |
✗ |
N/A |
1.004 |
|
|
\[ {}{y^{\prime }}^{2}+\left (x +y\right ) y^{\prime }+x y = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.577 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y y^{\prime }-x = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.416 |
|
|
\[ {}{y^{\prime }}^{2}+y^{2} = 1 \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.601 |
|
|
\[ {}\left (2 x y^{\prime }-y\right )^{2} = 8 x^{3} \] |
2 |
2 |
2 |
linear |
[_linear] |
✓ |
✓ |
0.988 |
|
|
\[ {}\left (x^{2}+1\right ) {y^{\prime }}^{2} = 1 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.446 |
|
|
|
|||||||||
|
\[ {}{y^{\prime }}^{3}-\left (y^{2}+2 x \right ) {y^{\prime }}^{2}+\left (x^{2}-y^{2}+2 x y^{2}\right ) y^{\prime }-\left (x^{2}-y^{2}\right ) y^{2} = 0 \] |
3 |
1 |
3 |
linear, quadrature |
[_quadrature] |
✓ |
✓ |
0.855 |
|
|
\[ {}2 x y^{\prime }-y+\ln \left (y^{\prime }\right ) = 0 \] |
0 |
2 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
1.593 |
|
|
\[ {}4 x {y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.349 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y y^{\prime }-x = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.381 |
|
|
\[ {}y = -x y^{\prime }+x^{4} {y^{\prime }}^{2} \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
5.934 |
|
|
\[ {}{y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.498 |
|
|
\[ {}x +y^{\prime } y \left (2 {y^{\prime }}^{2}+3\right ) = 0 \] |
3 |
7 |
5 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
140.131 |
|
|
\[ {}a^{2} y {y^{\prime }}^{2}-2 x y^{\prime }+y = 0 \] |
2 |
5 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.736 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y y^{\prime }-x = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.397 |
|
|
\[ {}{y^{\prime }}^{3}-4 x y y^{\prime }+8 y^{2} = 0 \] |
3 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
136.206 |
|
|
\[ {}\left (-y+x y^{\prime }\right )^{2} = 1+{y^{\prime }}^{2} \] |
2 |
4 |
4 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
0.709 |
|
|
\[ {}4 \,{\mathrm e}^{2 y} {y^{\prime }}^{2}+2 x y^{\prime }-1 = 0 \] |
2 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
4.438 |
|
|
\[ {}4 \,{\mathrm e}^{2 y} {y^{\prime }}^{2}+2 \,{\mathrm e}^{2 x} y^{\prime }-{\mathrm e}^{2 x} = 0 \] |
2 |
2 |
2 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.338 |
|
|
\[ {}{\mathrm e}^{2 y} {y^{\prime }}^{3}+\left ({\mathrm e}^{2 x}+{\mathrm e}^{3 x}\right ) y^{\prime }-{\mathrm e}^{3 x} = 0 \] |
3 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
204.26 |
|
|
\[ {}x y^{2} {y^{\prime }}^{2}-y^{3} y^{\prime }+x = 0 \] |
2 |
2 |
6 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
10.509 |
|
|
\[ {}\left (x^{2}+y^{2}\right ) \left (y^{\prime }+1\right )^{2}-2 \left (x +y\right ) \left (y^{\prime }+1\right ) \left (x +y y^{\prime }\right )+\left (x +y y^{\prime }\right )^{2} = 0 \] |
2 |
5 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
3.897 |
|
|
\[ {}y = 2 x y^{\prime }+y^{2} {y^{\prime }}^{3} \] |
3 |
1 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
133.768 |
|
|
\[ {}a^{2} y {y^{\prime }}^{2}-2 x y^{\prime }+y = 0 \] |
2 |
5 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.736 |
|
|
\[ {}\left (x -y^{\prime }-y\right )^{2} = x^{2} \left (2 x y-x^{2} y^{\prime }\right ) \] |
2 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
5.846 |
|
|
\[ {}y^{2} \left (1+{y^{\prime }}^{2}\right ) = a^{2} \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.152 |
|
|
\[ {}y y^{\prime } = \left (-b +x \right ) {y^{\prime }}^{2}+a \] |
2 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.422 |
|
|
\[ {}x^{3} {y^{\prime }}^{2}+x^{2} y y^{\prime }+1 = 0 \] |
2 |
2 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
6.652 |
|
|
\[ {}3 x {y^{\prime }}^{2}-6 y y^{\prime }+x +2 y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.42 |
|
|
\[ {}y = \left (1+x \right ) {y^{\prime }}^{2} \] |
2 |
3 |
3 |
dAlembert, first_order_nonlinear_p_but_separable |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
✓ |
1.473 |
|
|
\[ {}\left (-y+x y^{\prime }\right ) \left (x +y y^{\prime }\right ) = a^{2} y^{\prime } \] |
2 |
1 |
0 |
first_order_ode_lie_symmetry_calculated |
[_rational] |
✓ |
✓ |
72.443 |
|
|
\[ {}{y^{\prime }}^{2}+2 y y^{\prime } \cot \left (x \right ) = y^{2} \] |
2 |
2 |
3 |
separable |
[_separable] |
✓ |
✓ |
6.134 |
|
|
\[ {}\left (x^{2}+1\right ) {y^{\prime }}^{2}-2 x y y^{\prime }+y^{2}-1 = 0 \] |
2 |
4 |
4 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
0.564 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-2 \left (x y+2 y^{\prime }\right ) y^{\prime }+y^{2} = 0 \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.957 |
|
|
\[ {}y = x y^{\prime }+\frac {y {y^{\prime }}^{2}}{x^{2}} \] |
2 |
2 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
8.293 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+y^{2} = x^{2} y^{2}+x^{4} \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
3.427 |
|
|
\[ {}y = x y^{\prime }+\frac {1}{y^{\prime }} \] |
2 |
3 |
3 |
clairaut |
[[_homogeneous, ‘class G‘], _rational, _Clairaut] |
✓ |
✓ |
0.363 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y y^{\prime }-x = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.373 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-2 \left (x y-2\right ) y^{\prime }+y^{2} = 0 \] |
2 |
4 |
3 |
clairaut |
[[_homogeneous, ‘class G‘], _Clairaut] |
✓ |
✓ |
0.776 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-\left (-1+x \right )^{2} = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.37 |
|
|
\[ {}8 \left (y^{\prime }+1\right )^{3} = 27 \left (x +y\right ) \left (1-y^{\prime }\right )^{3} \] |
3 |
4 |
4 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
473.104 |
|
|
\[ {}4 {y^{\prime }}^{2} = 9 x \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.282 |
|
|
\[ {}y \left (3-4 y\right )^{2} {y^{\prime }}^{2} = 4-4 y \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.886 |
|
|
\[ {}{x^{\prime }}^{2}+x t = \sqrt {t +1} \] |
2 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
1.834 |
|
|
\[ {}{y^{\prime }}^{2}-4 y = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.732 |
|
|
\[ {}x y {y^{\prime }}^{2}-\left (x^{2}+y^{2}\right ) y^{\prime }+x y = 0 \] |
2 |
1 |
3 |
separable |
[_separable] |
✓ |
✓ |
1.464 |
|
|
\[ {}{y^{\prime }}^{2} = 9 y^{4} \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.51 |
|
|
\[ {}{y^{\prime }}^{2}+x^{2} = 1 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.323 |
|
|
\[ {}x = {y^{\prime }}^{3}-y^{\prime }+2 \] |
3 |
3 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.693 |
|
|
\[ {}y = {y^{\prime }}^{4}-{y^{\prime }}^{3}-2 \] |
4 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.596 |
|
|
\[ {}{y^{\prime }}^{2}+y^{2} = 4 \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.616 |
|
|
\[ {}{y^{\prime }}^{3}-{\mathrm e}^{2 x} y^{\prime } = 0 \] |
3 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.772 |
|
|
\[ {}y = 5 x y^{\prime }-{y^{\prime }}^{2} \] |
2 |
3 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.71 |
|
|
\[ {}y = x y^{\prime }+{y^{\prime }}^{2} \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.594 |
|
|
\[ {}y = x y^{\prime }+{y^{\prime }}^{2} \] |
2 |
2 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.797 |
|
|
\[ {}y = x^{2}+2 x y^{\prime }+\frac {{y^{\prime }}^{2}}{2} \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
3.135 |
|
|
\[ {}y \left (1+{y^{\prime }}^{2}\right ) = a \] |
2 |
2 |
5 |
quadrature |
[_quadrature] |
✓ |
✓ |
2.602 |
|
|
\[ {}{y^{\prime }}^{2}+\left (x +a \right ) y^{\prime }-y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.456 |
|
|
\[ {}{y^{\prime }}^{2}-2 x y^{\prime }+y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.609 |
|
|
\[ {}{y^{\prime }}^{2}+2 y y^{\prime } \cot \left (x \right )-y^{2} = 0 \] |
2 |
2 |
3 |
separable |
[_separable] |
✓ |
✓ |
7.544 |
|
|
\[ {}{y^{\prime \prime \prime }}^{2}+{y^{\prime \prime }}^{2} = 1 \] |
2 |
2 |
3 |
unknown |
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]] |
✗ |
N/A |
0.0 |
|
|
\[ {}6 y^{\prime \prime } y^{\prime \prime \prime \prime }-5 {y^{\prime \prime \prime }}^{2} = 0 \] |
2 |
2 |
2 |
unknown |
[[_high_order, _missing_x], [_high_order, _missing_y], [_high_order, _with_linear_symmetries], [_high_order, _reducible, _mu_poly_yn]] |
✗ |
N/A |
0.0 |
|
|
\[ {}\sinh \left (x \right ) {y^{\prime }}^{2}+3 y = 0 \] |
2 |
2 |
7 |
first_order_nonlinear_p_but_separable |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
0.704 |
|
|
\[ {}{y^{\prime }}^{2} \sqrt {y} = \sin \left (x \right ) \] |
2 |
2 |
2 |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
4.825 |
|
|
\[ {}{y^{\prime }}^{2}+x y {y^{\prime }}^{2} = \ln \left (x \right ) \] |
2 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
2.12 |
|
|
\[ {}{y^{\prime }}^{2}-y^{\prime }-x y^{\prime }+y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.322 |
|
|
\[ {}y {y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
2 |
5 |
7 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.759 |
|
|
\[ {}x y \left (1-{y^{\prime }}^{2}\right ) = \left (x^{2}-y^{2}-a^{2}\right ) y^{\prime } \] |
2 |
1 |
0 |
first_order_ode_lie_symmetry_calculated |
[_rational] |
✓ |
✓ |
71.363 |
|
|
\[ {}y = 2 x y^{\prime }+{y^{\prime }}^{2} \] |
2 |
3 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.43 |
|
|
\[ {}y = x {y^{\prime }}^{2}+{y^{\prime }}^{2} \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
✓ |
0.406 |
|
|
\[ {}y = x \left (y^{\prime }+1\right )+{y^{\prime }}^{2} \] |
2 |
2 |
1 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.448 |
|
|
\[ {}y = y {y^{\prime }}^{2}+2 x y^{\prime } \] |
2 |
5 |
7 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.701 |
|
|
\[ {}y = y y^{\prime }+y^{\prime }-{y^{\prime }}^{2} \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.45 |
|
|
\[ {}y = x y^{\prime }+\sqrt {1-{y^{\prime }}^{2}} \] |
2 |
2 |
1 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
1.396 |
|
|
\[ {}y = x y^{\prime }+\frac {1}{y^{\prime }} \] |
2 |
3 |
3 |
clairaut |
[[_homogeneous, ‘class G‘], _rational, _Clairaut] |
✓ |
✓ |
0.372 |
|
|
\[ {}y = x y^{\prime }-\frac {1}{{y^{\prime }}^{2}} \] |
3 |
4 |
4 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.758 |
|
|
\[ {}y = x {y^{\prime }}^{2}+{y^{\prime }}^{2} \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
✓ |
0.393 |
|
|
\[ {}{y^{\prime }}^{2}-4 y = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.437 |
|
|
\[ {}{y^{\prime }}^{2}-9 x y = 0 \] |
2 |
2 |
5 |
first_order_nonlinear_p_but_separable |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
0.653 |
|
|
\[ {}{y^{\prime }}^{2} = x^{6} \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.417 |
|
|
\[ {}{y^{\prime }}^{2}+y = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.197 |
|
|
\[ {}t y^{\prime }-{y^{\prime }}^{3} = y \] |
3 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.749 |
|
|
\[ {}t y^{\prime }-y-2 \left (t y^{\prime }-y\right )^{2} = y^{\prime }+1 \] |
2 |
4 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
1.021 |
|
|
\[ {}t y^{\prime }-y-1 = {y^{\prime }}^{2}-y^{\prime } \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.456 |
|
|
\[ {}1+y-t y^{\prime } = \ln \left (y^{\prime }\right ) \] |
0 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
1.332 |
|
|
\[ {}1-2 t y^{\prime }+2 y = \frac {1}{{y^{\prime }}^{2}} \] |
3 |
4 |
4 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
1.589 |
|
|
\[ {}y = -t y^{\prime }+\frac {{y^{\prime }}^{5}}{5} \] |
5 |
2 |
1 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
0.66 |
|
|
\[ {}y = t {y^{\prime }}^{2}+3 {y^{\prime }}^{2}-2 {y^{\prime }}^{3} \] |
3 |
5 |
4 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
59.272 |
|
|
\[ {}y = t \left (2-y^{\prime }\right )+2 {y^{\prime }}^{2}+1 \] |
2 |
3 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.661 |
|
|
\[ {}y = t y^{\prime }+3 {y^{\prime }}^{4} \] |
4 |
4 |
4 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
2.53 |
|
|
\[ {}-t y^{\prime }+y = -2 {y^{\prime }}^{3} \] |
3 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.726 |
|
|
\[ {}-t y^{\prime }+y = -4 {y^{\prime }}^{2} \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.48 |
|
|
\[ {}\cos \left (y^{\prime }\right ) = 0 \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.224 |
|
|
\[ {}{\mathrm e}^{y^{\prime }} = 1 \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.209 |
|
|
\[ {}\sin \left (y^{\prime }\right ) = x \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.291 |
|
|
\[ {}\ln \left (y^{\prime }\right ) = x \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.603 |
|
|
\[ {}\tan \left (y^{\prime }\right ) = 0 \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.209 |
|
|
\[ {}{\mathrm e}^{y^{\prime }} = x \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.238 |
|
|
\[ {}\tan \left (y^{\prime }\right ) = x \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.312 |
|
|
\[ {}4 {y^{\prime }}^{2}-9 x = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.444 |
|
|
\[ {}{y^{\prime }}^{2}-2 y y^{\prime } = y^{2} \left ({\mathrm e}^{2 x}-1\right ) \] |
2 |
2 |
3 |
separable |
[_separable] |
✓ |
✓ |
1.556 |
|
|
\[ {}{y^{\prime }}^{2}-2 x y^{\prime }-8 x^{2} = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.635 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+3 x y y^{\prime }+2 y^{2} = 0 \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
2.367 |
|
|
\[ {}{y^{\prime }}^{2}-\left (y+2 x \right ) y^{\prime }+x^{2}+x y = 0 \] |
2 |
1 |
2 |
linear, quadrature |
[_quadrature] |
✓ |
✓ |
1.46 |
|
|
\[ {}{y^{\prime }}^{3}+\left (2+x \right ) {\mathrm e}^{y} = 0 \] |
3 |
3 |
3 |
first_order_nonlinear_p_but_separable |
[[_1st_order, _with_exponential_symmetries]] |
✓ |
✓ |
3.904 |
|
|
\[ {}{y^{\prime }}^{3} = y {y^{\prime }}^{2}-x^{2} y^{\prime }+x^{2} y \] |
3 |
1 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.882 |
|
|
\[ {}{y^{\prime }}^{2}-y y^{\prime }+{\mathrm e}^{x} = 0 \] |
2 |
2 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
9.56 |
|
|
\[ {}{y^{\prime }}^{2}-4 x y^{\prime }+2 y+2 x^{2} = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
3.668 |
|
|
\[ {}y = {y^{\prime }}^{2} {\mathrm e}^{y^{\prime }} \] |
0 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
2.111 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{\frac {y^{\prime }}{y}} \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.197 |
|
|
\[ {}x = \ln \left (y^{\prime }\right )+\sin \left (y^{\prime }\right ) \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✗ |
5.153 |
|
|
\[ {}x = {y^{\prime }}^{2}-2 y^{\prime }+2 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.63 |
|
|
\[ {}y = y^{\prime } \ln \left (y^{\prime }\right ) \] |
0 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
3.011 |
|
|
\[ {}y = \left (y^{\prime }-1\right ) {\mathrm e}^{y^{\prime }} \] |
0 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.575 |
|
|
\[ {}x {y^{\prime }}^{2} = {\mathrm e}^{\frac {1}{y^{\prime }}} \] |
0 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.648 |
|
|
\[ {}x \left (1+{y^{\prime }}^{2}\right )^{\frac {3}{2}} = a \] |
6 |
6 |
6 |
quadrature |
[_quadrature] |
✓ |
✓ |
14.369 |
|
|
\[ {}y^{\frac {2}{5}}+{y^{\prime }}^{\frac {2}{5}} = a^{\frac {2}{5}} \] |
2 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
3.309 |
|
|
\[ {}x = y^{\prime }+\sin \left (y^{\prime }\right ) \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.668 |
|
|
\[ {}y = y^{\prime } \left (1+y^{\prime } \cos \left (y^{\prime }\right )\right ) \] |
0 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.714 |
|
|
\[ {}y = \arcsin \left (y^{\prime }\right )+\ln \left (1+{y^{\prime }}^{2}\right ) \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✗ |
1.055 |
|
|
\[ {}y = 2 x y^{\prime }+\ln \left (y^{\prime }\right ) \] |
0 |
2 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
2.528 |
|
|
\[ {}y = x \left (y^{\prime }+1\right )+{y^{\prime }}^{2} \] |
2 |
2 |
1 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.702 |
|
|
\[ {}y = 2 x y^{\prime }+\sin \left (y^{\prime }\right ) \] |
0 |
2 |
2 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
1.829 |
|
|
\[ {}y = x {y^{\prime }}^{2}-\frac {1}{y^{\prime }} \] |
3 |
4 |
3 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
151.895 |
|
|
\[ {}y = \frac {3 x y^{\prime }}{2}+{\mathrm e}^{y^{\prime }} \] |
0 |
2 |
2 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
1.561 |
|
|
\[ {}y = x y^{\prime }+\frac {a}{{y^{\prime }}^{2}} \] |
3 |
4 |
4 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.572 |
|
|
\[ {}y = x y^{\prime }+{y^{\prime }}^{2} \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.279 |
|
|
\[ {}x {y^{\prime }}^{2}-y y^{\prime }-y^{\prime }+1 = 0 \] |
2 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
0.306 |
|
|
\[ {}y = x y^{\prime }+a \sqrt {1+{y^{\prime }}^{2}} \] |
2 |
3 |
1 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
1.637 |
|
|
\[ {}x = \frac {y}{y^{\prime }}+\frac {1}{{y^{\prime }}^{2}} \] |
2 |
3 |
3 |
clairaut |
[[_homogeneous, ‘class G‘], _rational, _Clairaut] |
✓ |
✓ |
0.299 |
|
|
\[ {}y^{2} \left (1+{y^{\prime }}^{2}\right )-4 y y^{\prime }-4 x = 0 \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
2.936 |
|
|
\[ {}{y^{\prime }}^{2}-4 y = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.335 |
|
|
\[ {}{y^{\prime }}^{3}-4 x y y^{\prime }+8 y^{2} = 0 \] |
3 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
90.18 |
|
|
\[ {}{y^{\prime }}^{2}-y^{2} = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.23 |
|
|
\[ {}\left (x y^{\prime }+y\right )^{2}+3 x^{5} \left (x y^{\prime }-2 y\right ) = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
69.164 |
|
|
\[ {}y \left (y-2 x y^{\prime }\right )^{2} = 2 y^{\prime } \] |
2 |
2 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.633 |
|
|
\[ {}8 {y^{\prime }}^{3}-12 {y^{\prime }}^{2} = 27 y-27 x \] |
3 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.437 |
|
|
\[ {}\left (y^{\prime }-1\right )^{2} = y^{2} \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.35 |
|
|
\[ {}y = {y^{\prime }}^{2}-x y^{\prime }+x \] |
2 |
3 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.348 |
|
|
\[ {}\left (x y^{\prime }+y\right )^{2} = y^{2} y^{\prime } \] |
2 |
3 |
6 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.498 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}+y^{2} = 1 \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.394 |
|
|
\[ {}{y^{\prime }}^{2}-y y^{\prime }+{\mathrm e}^{x} = 0 \] |
2 |
2 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
3.18 |
|
|
\[ {}3 x {y^{\prime }}^{2}-6 y y^{\prime }+x +2 y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.298 |
|
|
\[ {}y = x y^{\prime }+\sqrt {a^{2} {y^{\prime }}^{2}+b^{2}} \] |
2 |
3 |
1 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
2.647 |
|
|
\[ {}4 x^{2} {y^{\prime }}^{2}-y^{2} = x y^{3} \] |
2 |
2 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
45.149 |
|
|
\[ {}y^{\prime }+x {y^{\prime }}^{2}-y = 0 \] |
2 |
4 |
1 |
dAlembert |
[_rational, _dAlembert] |
✓ |
✓ |
0.367 |
|
|
\[ {}{y^{\prime }}^{4} = 1 \] |
4 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.375 |
|
|
\[ {}x y = y^{\prime } \ln \left (\frac {y^{\prime }}{x}\right ) \] |
0 |
2 |
2 |
separable |
[_separable] |
✓ |
✓ |
1.428 |
|
|
|
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|
|
|||||||||
|
|