# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
\[ {}a \cos \left (y^{\prime }\right )+b y^{\prime }+x = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.16 |
|
\[ {}{y^{\prime }}^{2} \sin \left (y^{\prime }\right )-y = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.272 |
|
\[ {}\left (1+{y^{\prime }}^{2}\right ) \sin \left (-y+x y^{\prime }\right )^{2}-1 = 0 \] |
clairaut |
[_Clairaut] |
✓ |
✓ |
2.947 |
|
\[ {}\left (1+{y^{\prime }}^{2}\right ) \left (\arctan \left (y^{\prime }\right )+x a \right )+y^{\prime } = 0 \] |
quadrature |
[_quadrature] |
✓ |
✗ |
1.307 |
|
\[ {}a \,x^{n} f \left (y^{\prime }\right )+x y^{\prime }-y = 0 \] |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
0.537 |
|
\[ {}\left (-y+x y^{\prime }\right )^{n} f \left (y^{\prime }\right )+y g \left (y^{\prime }\right )+x h \left (y^{\prime }\right ) = 0 \] |
unknown |
[‘x=_G(y,y’)‘] |
❇ |
N/A |
2.117 |
|
\[ {}f \left (x {y^{\prime }}^{2}\right )+2 x y^{\prime }-y = 0 \] |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
0.347 |
|
\[ {}f \left (x -\frac {3 {y^{\prime }}^{2}}{2}\right )+{y^{\prime }}^{3}-y = 0 \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
0.781 |
|
\[ {}y^{\prime } f \left (x y y^{\prime }-y^{2}\right )-x^{2} y^{\prime }+x y = 0 \] |
unknown |
[NONE] |
❇ |
N/A |
0.613 |
|
\[ {}\phi \left (f \left (x , y, y^{\prime }\right ), g \left (x , y, y^{\prime }\right )\right ) = 0 \] |
unknown |
[NONE] |
❇ |
N/A |
1.004 |
|
\[ {}y^{\prime } = F \left (\frac {y}{x +a}\right ) \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.622 |
|
\[ {}y^{\prime } = 2 x +F \left (-x^{2}+y\right ) \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
0.488 |
|
\[ {}y^{\prime } = -\frac {x a}{2}+F \left (y+\frac {x^{2} a}{4}+\frac {b x}{2}\right ) \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
0.688 |
|
\[ {}y^{\prime } = F \left (y \,{\mathrm e}^{-b x}\right ) {\mathrm e}^{b x} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
0.648 |
|
\[ {}y^{\prime } = \frac {1+2 F \left (\frac {4 x^{2} y+1}{4 x^{2}}\right ) x}{2 x^{3}} \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
0.969 |
|
\[ {}y^{\prime } = \frac {1+F \left (\frac {a x y+1}{a x}\right ) a \,x^{2}}{a \,x^{2}} \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
0.977 |
|
\[ {}y^{\prime } = -\frac {\left (x^{2} a -2 F \left (y+\frac {a \,x^{4}}{8}\right )\right ) x}{2} \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
1.083 |
|
\[ {}y^{\prime } = \frac {2 a}{y+2 F \left (y^{2}-4 x a \right ) a} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
0.882 |
|
\[ {}y^{\prime } = F \left (\ln \left (\ln \left (y\right )\right )-\ln \left (x \right )\right ) y \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
0.955 |
|
\[ {}y^{\prime } = \frac {F \left (\frac {y}{\sqrt {x^{2}+1}}\right ) x}{\sqrt {x^{2}+1}} \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.4 |
|
\[ {}y^{\prime } = \frac {\left (x^{\frac {3}{2}}+2 F \left (y-\frac {x^{3}}{6}\right )\right ) \sqrt {x}}{2} \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
1.075 |
|
\[ {}y^{\prime } = \frac {x +F \left (-\left (x -y\right ) \left (x +y\right )\right )}{y} \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
0.993 |
|
\[ {}y^{\prime } = \frac {F \left (-\frac {-1+y \ln \left (x \right )}{y}\right ) y^{2}}{x} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
1.248 |
|
\[ {}y^{\prime } = \frac {x}{-y+F \left (x^{2}+y^{2}\right )} \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.041 |
|
\[ {}y^{\prime } = \frac {F \left (\frac {a y^{2}+b \,x^{2}}{a}\right ) x}{\sqrt {a}\, y} \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.084 |
|
\[ {}y^{\prime } = \frac {6 x^{3}+5 \sqrt {x}+5 F \left (y-\frac {2 x^{3}}{5}-2 \sqrt {x}\right )}{5 x} \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
1.298 |
|
\[ {}y^{\prime } = \frac {F \left (y^{\frac {3}{2}}-\frac {3 \,{\mathrm e}^{x}}{2}\right ) {\mathrm e}^{x}}{\sqrt {y}} \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.175 |
|
\[ {}y^{\prime } = \frac {F \left (-\frac {-y^{2}+b}{x^{2}}\right ) x}{y} \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
0.985 |
|
\[ {}y^{\prime } = \frac {F \left (\frac {x y^{2}+1}{x}\right )}{y x^{2}} \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
0.931 |
|
\[ {}y^{\prime } = \frac {-2 x^{2}+x +F \left (y+x^{2}-x \right )}{x} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
1.263 |
|
\[ {}y^{\prime } = \frac {2 a}{x^{2} \left (-y+2 F \left (\frac {x y^{2}-4 a}{x}\right ) a \right )} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
2.313 |
|
\[ {}y^{\prime } = \frac {y+F \left (\frac {y}{x}\right )}{-1+x} \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class D‘]] |
✓ |
✓ |
1.589 |
|
\[ {}y^{\prime } = \frac {-x +F \left (x^{2}+y^{2}\right )}{y} \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.027 |
|
\[ {}y^{\prime } = \frac {F \left (-\frac {2 y \ln \left (x \right )-1}{y}\right ) y^{2}}{x} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
1.318 |
|
\[ {}y^{\prime } = \frac {F \left (-\left (x -y\right ) \left (x +y\right )\right ) x}{y} \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.099 |
|
\[ {}y^{\prime } = \frac {y^{2} \left (2+F \left (\frac {x^{2}-y}{y x^{2}}\right ) x^{2}\right )}{x^{3}} \] |
unknown |
[NONE] |
✗ |
N/A |
1.191 |
|
\[ {}y^{\prime } = \frac {2 F \left (y+\ln \left (2 x +1\right )\right ) x +F \left (y+\ln \left (2 x +1\right )\right )-2}{2 x +1} \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.044 |
|
\[ {}y^{\prime } = \frac {2 y^{3}}{1+2 F \left (\frac {1+4 x y^{2}}{y^{2}}\right ) y} \] |
unknown |
[‘x=_G(y,y’)‘] |
✗ |
N/A |
1.141 |
|
\[ {}y^{\prime } = -\frac {y^{2} \left (2 x -F \left (-\frac {x y-2}{2 y}\right )\right )}{4 x} \] |
first_order_ode_lie_symmetry_calculated |
[NONE] |
✓ |
✓ |
2.022 |
|
\[ {}y^{\prime } = -\left (-{\mathrm e}^{-x^{2}}+x^{2} {\mathrm e}^{-x^{2}}-F \left (y-\frac {x^{2} {\mathrm e}^{-x^{2}}}{2}\right )\right ) x \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
1.217 |
|
\[ {}y^{\prime } = \frac {2 y+F \left (\frac {y}{x^{2}}\right ) x^{3}}{x} \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
0.956 |
|
\[ {}y^{\prime } = \frac {\sqrt {y}}{\sqrt {y}+F \left (\frac {x -y}{\sqrt {y}}\right )} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
1.148 |
|
\[ {}y^{\prime } = \frac {-3 x^{2} y+F \left (x^{3} y\right )}{x^{3}} \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
0.961 |
|
\[ {}y^{\prime } = \frac {y+F \left (\frac {y}{x}\right ) x^{2}}{x} \] |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘]] |
✓ |
✓ |
0.601 |
|
\[ {}y^{\prime } = \frac {-2 x -y+F \left (x \left (x +y\right )\right )}{x} \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.066 |
|
\[ {}y^{\prime } = \frac {\left (y \,{\mathrm e}^{-\frac {x^{2}}{4}} x +2 F \left (y \,{\mathrm e}^{-\frac {x^{2}}{4}}\right )\right ) {\mathrm e}^{\frac {x^{2}}{4}}}{2} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
1.669 |
|
\[ {}y^{\prime } = \frac {x +y+F \left (-\frac {-y+x \ln \left (x \right )}{x}\right ) x^{2}}{x} \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.165 |
|
\[ {}y^{\prime } = \frac {x \left (a -1\right ) \left (1+a \right )}{y+F \left (\frac {y^{2}}{2}-\frac {a^{2} x^{2}}{2}+\frac {x^{2}}{2}\right ) a^{2}-F \left (\frac {y^{2}}{2}-\frac {a^{2} x^{2}}{2}+\frac {x^{2}}{2}\right )} \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.178 |
|
\[ {}y^{\prime } = \frac {y}{x \left (-1+F \left (x y\right ) y\right )} \] |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
1.014 |
|
\[ {}y^{\prime } = -\frac {-x^{2}+2 x^{3} y-F \left (\left (x y-1\right ) x \right )}{x^{4}} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
1.303 |
|
\[ {}y^{\prime } = \frac {F \left (\frac {\left (3+y\right ) {\mathrm e}^{\frac {3 x^{2}}{2}}}{3 y}\right ) x y^{2} {\mathrm e}^{3 x^{2}} {\mathrm e}^{-\frac {9 x^{2}}{2}}}{9} \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.406 |
|
\[ {}y^{\prime } = \frac {\left (y+1\right ) \left (\left (y-\ln \left (y+1\right )-\ln \left (x \right )\right ) x +1\right )}{y x} \] |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
1.148 |
|
\[ {}y^{\prime } = \frac {6 y}{8 y^{4}+9 y^{3}+12 y^{2}+6 y-F \left (-\frac {y^{4}}{3}-\frac {y^{3}}{2}-y^{2}-y+x \right )} \] |
unknown |
[‘x=_G(y,y’)‘] |
✗ |
N/A |
1.25 |
|
\[ {}y^{\prime } = \frac {y^{2}+2 x y+x^{2}+{\mathrm e}^{2 F \left (-\left (x -y\right ) \left (x +y\right )\right )}}{y^{2}+2 x y+x^{2}-{\mathrm e}^{2 F \left (-\left (x -y\right ) \left (x +y\right )\right )}} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
1.622 |
|
\[ {}y^{\prime } = \frac {1}{y+\sqrt {x}} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
1.198 |
|
\[ {}y^{\prime } = \frac {1}{y+2+\sqrt {1+3 x}} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
8.208 |
|
\[ {}y^{\prime } = \frac {x^{2}}{y+x^{\frac {3}{2}}} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
8.038 |
|
\[ {}y^{\prime } = \frac {x^{\frac {5}{3}}}{y+x^{\frac {4}{3}}} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
2.276 |
|
\[ {}y^{\prime } = \frac {i x^{2} \left (i-2 \sqrt {-x^{3}+6 y}\right )}{2} \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.931 |
|
\[ {}y^{\prime } = \frac {x}{y+\sqrt {x^{2}+1}} \] |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✗ |
N/A |
2.449 |
|
\[ {}y^{\prime } = \frac {\left (-1+y \ln \left (x \right )\right )^{2}}{x} \] |
riccati, first_order_ode_lie_symmetry_calculated |
[_Riccati] |
✓ |
✓ |
1.95 |
|
\[ {}y^{\prime } = \frac {x \left (-2+3 \sqrt {x^{2}+3 y}\right )}{3} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
2.971 |
|
\[ {}y^{\prime } = \frac {\left (2 y \ln \left (x \right )-1\right )^{2}}{x} \] |
riccati, first_order_ode_lie_symmetry_calculated |
[_Riccati] |
✓ |
✓ |
2.034 |
|
\[ {}y^{\prime } = \frac {{\mathrm e}^{b x}}{y \,{\mathrm e}^{-b x}+1} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
5.324 |
|
\[ {}y^{\prime } = \frac {x^{2} \left (1+2 \sqrt {x^{3}-6 y}\right )}{2} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
3.208 |
|
\[ {}y^{\prime } = \frac {{\mathrm e}^{x}}{y \,{\mathrm e}^{-x}+1} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
6.884 |
|
\[ {}y^{\prime } = \frac {{\mathrm e}^{\frac {2 x}{3}}}{y \,{\mathrm e}^{-\frac {2 x}{3}}+1} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
5.501 |
|
\[ {}y^{\prime } = \frac {1+2 x^{5} \sqrt {4 x^{2} y+1}}{2 x^{3}} \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.5 |
|
\[ {}y^{\prime } = \frac {x \left (x +2 \sqrt {x^{3}-6 y}\right )}{2} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
3.243 |
|
\[ {}y^{\prime } = \left (-\ln \left (y\right )+x^{2}\right ) y \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
1.331 |
|
\[ {}y^{\prime } = \frac {{\mathrm e}^{-x^{2}} x}{y \,{\mathrm e}^{x^{2}}+1} \] |
unknown |
[[_Abel, ‘2nd type‘, ‘class C‘], [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.121 |
|
\[ {}y^{\prime } = -\left (-\ln \left (\ln \left (y\right )\right )+\ln \left (x \right )\right ) y \] |
unknown |
[‘x=_G(y,y’)‘] |
✗ |
N/A |
0.998 |
|
\[ {}y^{\prime } = \left (-\ln \left (\ln \left (y\right )\right )+\ln \left (x \right )\right )^{2} y \] |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
0.984 |
|
\[ {}y^{\prime } = \frac {y}{\ln \left (\ln \left (y\right )\right )-\ln \left (x \right )+1} \] |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
2.066 |
|
\[ {}y^{\prime } = \frac {1+2 \sqrt {4 x^{2} y+1}\, x^{4}}{2 x^{3}} \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.386 |
|
\[ {}y^{\prime } = \frac {\left (-y^{2}+4 x a \right )^{2}}{y} \] |
unknown |
[_rational] |
✗ |
N/A |
1.089 |
|
\[ {}y^{\prime } = \frac {x \left (-2+3 x \sqrt {x^{2}+3 y}\right )}{3} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
3.1 |
|
\[ {}y^{\prime } = -\frac {x^{2} \left (x a -2 \sqrt {a \left (a \,x^{4}+8 y\right )}\right )}{2} \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
1.37 |
|
\[ {}y^{\prime } = \left (-\ln \left (y\right )+x \right ) y \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
1.007 |
|
\[ {}y^{\prime } = \frac {x^{3}+x^{2}+2 \sqrt {x^{3}-6 y}}{2 x +2} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
3.084 |
|
\[ {}y^{\prime } = \frac {\left (a y^{2}+b \,x^{2}\right )^{2} x}{a^{\frac {5}{2}} y} \] |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.277 |
|
\[ {}y^{\prime } = -\frac {x^{3} \left (\sqrt {a}\, x +\sqrt {a}-2 \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2 \left (1+x \right )} \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
2.175 |
|
\[ {}y^{\prime } = -\frac {x}{4}+\frac {1}{4}+x \sqrt {x^{2}-2 x +1+8 y} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
3.399 |
|
\[ {}y^{\prime } = -\frac {x}{2}-\frac {a}{2}+x \sqrt {x^{2}+2 x a +a^{2}+4 y} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
3.252 |
|
\[ {}y^{\prime } = \frac {\left (\ln \left (y\right )+x^{2}\right ) y}{x} \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
1.096 |
|
\[ {}y^{\prime } = \frac {2 a +x \sqrt {-y^{2}+4 x a}}{y} \] |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
1.353 |
|
\[ {}y^{\prime } = -\frac {x}{2}+1+x \sqrt {x^{2}-4 x +4 y} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
2.486 |
|
\[ {}y^{\prime } = -\frac {2 x^{2}+2 x -3 \sqrt {x^{2}+3 y}}{3 \left (1+x \right )} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
2.467 |
|
\[ {}y^{\prime } = \frac {y^{3} {\mathrm e}^{-\frac {4 x}{3}}}{y \,{\mathrm e}^{-\frac {2 x}{3}}+1} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
4.551 |
|
\[ {}y^{\prime } = \frac {\left (\ln \left (y\right )+x^{3}\right ) y}{x} \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
1.328 |
|
\[ {}y^{\prime } = -\frac {x}{4}+\frac {1}{4}+x^{2} \sqrt {x^{2}-2 x +1+8 y} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
3.495 |
|
\[ {}y^{\prime } = -\frac {x^{2}-1-4 \sqrt {x^{2}-2 x +1+8 y}}{4 \left (1+x \right )} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
2.266 |
|
\[ {}y^{\prime } = -\frac {x a}{2}-\frac {b}{2}+x \sqrt {a^{2} x^{2}+2 a b x +b^{2}+4 a y-4 c} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
2.892 |
|
\[ {}y^{\prime } = -\frac {x}{2}-\frac {a}{2}+x^{2} \sqrt {x^{2}+2 x a +a^{2}+4 y} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
3.317 |
|
\[ {}y^{\prime } = -\frac {x a}{2}-\frac {b}{2}+x^{2} \sqrt {a^{2} x^{2}+2 a b x +b^{2}+4 a y-4 c} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
3.178 |
|
\[ {}y^{\prime } = \frac {x}{2}+\frac {1}{2}+x^{2} \sqrt {x^{2}+2 x +1-4 y} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
2.891 |
|
\[ {}y^{\prime } = \frac {2 a +x^{2} \sqrt {-y^{2}+4 x a}}{y} \] |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
1.323 |
|
\[ {}y^{\prime } = -\frac {x}{2}+1+x^{2} \sqrt {x^{2}-4 x +4 y} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
2.686 |
|
\[ {}y^{\prime } = -\frac {\left (\sqrt {a}\, x^{4}+\sqrt {a}\, x^{3}-2 \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2 \left (1+x \right )} \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
2.103 |
|
\[ {}y^{\prime } = \left (-\ln \left (y\right )+1+x^{2}+x^{3}\right ) y \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
2.563 |
|
|
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