|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}x y^{\prime } = y \cos \left (\frac {y}{x}\right )
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
2.971 |
|
|
\[
{}y+\sqrt {x y}-x y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
7.419 |
|
|
\[
{}x y^{\prime }-\sqrt {x^{2}-y^{2}}-y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
87.083 |
|
|
\[
{}x +y-\left (x -y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.644 |
|
|
\[
{}x^{2}+2 x y-y^{2}+\left (y^{2}+2 x y-x^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
7.629 |
|
|
\[
{}-y+x y^{\prime } = y^{\prime } y
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.422 |
|
|
\[
{}y^{2}+\left (x^{2}-x y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
37.053 |
|
|
\[
{}y^{2}+x y+x^{2} = x^{2} y^{\prime }
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
2.066 |
|
|
\[
{}\frac {1}{x^{2}-x y+y^{2}} = \frac {y^{\prime }}{2 y^{2}-x y}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
24.231 |
|
|
\[
{}y^{\prime } = \frac {2 x y}{3 x^{2}-y^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
3.793 |
|
|
\[
{}y^{\prime } = \frac {x}{y}+\frac {y}{x}
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
4.249 |
|
|
\[
{}x y^{\prime } = y+\sqrt {y^{2}-x^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
4.185 |
|
|
\[
{}y+\left (2 \sqrt {x y}-x \right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
91.224 |
|
|
\[
{}x y^{\prime } = y \ln \left (\frac {y}{x}\right )
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
2.943 |
|
|
\[
{}y^{\prime } \left (y^{\prime }+y\right ) = \left (x +y\right ) x
\] |
[_quadrature] |
✓ |
1.476 |
|
|
\[
{}\left (x y^{\prime }+y\right )^{2} = y^{2} y^{\prime }
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
150.530 |
|
|
\[
{}x^{2} {y^{\prime }}^{2}-3 x y y^{\prime }+2 y^{2} = 0
\] |
[_separable] |
✓ |
2.687 |
|
|
\[
{}-y+x y^{\prime } = \sqrt {y^{2}+x^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
6.727 |
|
|
\[
{}y {y^{\prime }}^{2}+2 x y^{\prime }-y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
1.880 |
|
|
\[
{}y^{\prime }+\frac {x +2 y}{x} = 0
\] |
[_linear] |
✓ |
1.928 |
|
|
\[
{}y^{\prime } = \frac {y}{x +y}
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.201 |
|
|
\[
{}x y^{\prime } = x +\frac {y}{2}
\] |
[_linear] |
✓ |
6.028 |
|
|
\[
{}y^{\prime } = \frac {x +y-2}{y-x -4}
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.524 |
|
|
\[
{}2 x -4 y+6+\left (x +y-2\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.225 |
|
|
\[
{}y^{\prime } = \frac {2 y-x +5}{2 x -y-4}
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.194 |
|
|
\[
{}y^{\prime } = -\frac {4 x +3 y+15}{2 x +y+7}
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
4.576 |
|
|
\[
{}y^{\prime } = \frac {x +3 y-5}{x -y-1}
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.813 |
|
|
\[
{}y^{\prime } = \frac {2 \left (y+2\right )^{2}}{\left (x +y+1\right )^{2}}
\] |
[[_homogeneous, ‘class C‘], _rational] |
✓ |
1.755 |
|
|
\[
{}2 x +y+1-\left (4 x +2 y-3\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.381 |
|
|
\[
{}x -y-1+\left (y-x +2\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.422 |
|
|
\[
{}\left (4 y+x \right ) y^{\prime } = 2 x +3 y-5
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.714 |
|
|
\[
{}y+2 = \left (2 x +y-4\right ) y^{\prime }
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.936 |
|
|
\[
{}\left (1+y^{\prime }\right ) \ln \left (\frac {x +y}{x +3}\right ) = \frac {x +y}{x +3}
\] |
[[_homogeneous, ‘class C‘], _exact, _dAlembert] |
✓ |
7.069 |
|
|
\[
{}y^{\prime } = \frac {x -2 y+5}{y-2 x -4}
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.187 |
|
|
\[
{}y^{\prime } = \frac {3 x -y+1}{2 x +y+4}
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
5.474 |
|
|
\[
{}2 x y^{\prime }+\left (x^{2} y^{4}+1\right ) y = 0
\] |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
2.121 |
|
|
\[
{}2 x y^{\prime } \left (x -y^{2}\right )+y^{3} = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
0.574 |
|
|
\[
{}x^{3} \left (y^{\prime }-x \right ) = y^{2}
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
0.284 |
|
|
\[
{}2 x^{2} y^{\prime } = y^{3}+x y
\] |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
0.608 |
|
|
\[
{}y+x \left (2 x y+1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
0.435 |
|
|
\[
{}2 y^{\prime }+x = 4 \sqrt {y}
\] |
[[_1st_order, _with_linear_symmetries], _Chini] |
✓ |
1.102 |
|
|
\[
{}y^{\prime } = y^{2}-\frac {2}{x^{2}}
\] |
[[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]] |
✓ |
0.561 |
|
|
\[
{}2 x y^{\prime }+y = y^{2} \sqrt {x -x^{2} y^{2}}
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
10.380 |
|
|
\[
{}\frac {2 x y y^{\prime }}{3} = \sqrt {x^{6}-y^{4}}+y^{2}
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
3.840 |
|
|
\[
{}2 y+\left (x^{2} y+1\right ) x y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
0.381 |
|
|
\[
{}y \left (x y+1\right )+\left (1-x y\right ) x y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
0.431 |
|
|
\[
{}y \left (x^{2} y^{2}+1\right )+\left (x^{2} y^{2}-1\right ) x y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
0.582 |
|
|
\[
{}\left (x^{2}-y^{4}\right ) y^{\prime }-x y = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
1.572 |
|
|
\[
{}y \left (1+\sqrt {x^{2} y^{4}-1}\right )+2 x y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
0.845 |
|
|
\[
{}x \left (2-9 x y^{2}\right )+y \left (4 y^{2}-6 x^{3}\right ) y^{\prime } = 0
\] |
[_exact, _rational] |
✓ |
1.832 |
|
|
\[
{}\frac {y}{x}+\left (y^{3}+\ln \left (x \right )\right ) y^{\prime } = 0
\] |
[_exact, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
1.384 |
|
|
\[
{}2 x +3+\left (2 y-2\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
2.520 |
|
|
\[
{}2 x +4 y+\left (2 x -2 y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
5.141 |
|
|
\[
{}y^{\prime \prime }+2 y^{\prime }-y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.061 |
|
|
\[
{}y^{\prime \prime }+\frac {y^{\prime }}{x}-\frac {y}{x^{2}} = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.231 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+x y^{\prime }+y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
2.506 |
|
|
\[
{}y^{\prime \prime }-\cot \left (x \right ) y^{\prime }+y \cos \left (x \right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.158 |
|
|
\[
{}y^{\prime \prime }+\frac {y^{\prime }}{x}+x^{2} y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.940 |
|
|
\[
{}x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+2 x \left (-x^{2}+1\right ) y^{\prime }-2 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.239 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.898 |
|
|
\[
{}y^{\prime \prime \prime }-2 x y^{\prime \prime }+4 x^{2} y^{\prime }+8 x^{3} y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.048 |
|
|
\[
{}y^{\prime \prime }+x \left (1-x \right ) y^{\prime }+y \,{\mathrm e}^{x} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.308 |
|
|
\[
{}x^{2} y^{\prime \prime }+2 x y^{\prime }+4 y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
2.845 |
|
|
\[
{}x^{4} y^{\prime \prime \prime \prime }-x^{2} y^{\prime \prime }+y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
0.275 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+x y^{\prime }+y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
2.515 |
|
|
\[
{}y^{\prime \prime }+x y^{\prime }+y = 2 x \,{\mathrm e}^{x}-1
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
2.069 |
|
|
\[
{}x y^{\prime \prime }+x y^{\prime }-y = x^{2}+2 x
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.453 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime }-y = x^{2}+2 x
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
2.270 |
|
|
\[
{}x^{3} y^{\prime \prime }+x y^{\prime }-y = \cos \left (\frac {1}{x}\right )
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.279 |
|
|
\[
{}x \left (x +1\right ) y^{\prime \prime }+\left (x +2\right ) y^{\prime }-y = x +\frac {1}{x}
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
1.766 |
|
|
\[
{}2 x y^{\prime \prime }+\left (-2+x \right ) y^{\prime }-y = x^{2}-1
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.399 |
|
|
\[
{}x^{2} \left (x +1\right ) y^{\prime \prime }+x \left (4 x +3\right ) y^{\prime }-y = x +\frac {1}{x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
1.904 |
|
|
\[
{}x^{2} \left (\ln \left (x \right )-1\right ) y^{\prime \prime }-x y^{\prime }+y = x \left (1-\ln \left (x \right )\right )^{2}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.820 |
|
|
\[
{}x y^{\prime \prime }+2 y^{\prime }+x y = \sec \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
4.180 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+\frac {y}{4} = -\frac {x^{2}}{2}+\frac {1}{2}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.837 |
|
|
\[
{}\left (\cos \left (x \right )+\sin \left (x \right )\right ) y^{\prime \prime }-2 \cos \left (x \right ) y^{\prime }+\left (\cos \left (x \right )-\sin \left (x \right )\right ) y = \left (\cos \left (x \right )+\sin \left (x \right )\right )^{2} {\mathrm e}^{2 x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
3.746 |
|
|
\[
{}\left (\cos \left (x \right )-\sin \left (x \right )\right ) y^{\prime \prime }-2 \sin \left (x \right ) y^{\prime }+\left (\cos \left (x \right )+\sin \left (x \right )\right ) y = \left (\cos \left (x \right )-\sin \left (x \right )\right )^{2}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
2.543 |
|
|
\[
{}y^{\prime } = x^{2} \left (1+y^{2}\right )
\] |
[_separable] |
✓ |
3.082 |
|
|
\[
{}y^{\prime } = \frac {x^{2}}{1-y^{2}}
\] |
[_separable] |
✓ |
1.060 |
|
|
\[
{}y^{\prime } = \frac {3 x^{2}+4 x +2}{2 y-2}
\] |
[_separable] |
✓ |
2.156 |
|
|
\[
{}x y^{\prime }-2 \sqrt {x y} = y
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
6.038 |
|
|
\[
{}y^{\prime } = \frac {y-1+x}{x -y+3}
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.795 |
|
|
\[
{}{\mathrm e}^{x}+y+\left (x -2 \sin \left (y\right )\right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
1.859 |
|
|
\[
{}3 x +\frac {6}{y}+\left (\frac {x^{2}}{y}+\frac {3 y}{x}\right ) y^{\prime } = 0
\] |
[_rational] |
✓ |
1.362 |
|
|
\[
{}y^{2}-x y+x^{2} y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
1.771 |
|
|
\[
{}x +y-\left (x -y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.658 |
|
|
\[
{}y^{\prime } = \frac {y}{2 x}+\frac {x^{2}}{2 y}
\] |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
1.939 |
|
|
\[
{}y^{\prime } = -\frac {2}{t}+\frac {y}{t}+\frac {y^{2}}{t}
\] |
[_separable] |
✓ |
2.469 |
|
|
\[
{}y^{\prime } = -\frac {y}{t}-1-y^{2}
\] |
[_rational, _Riccati] |
✓ |
1.132 |
|
|
\[
{}x +y^{\prime } y = a {y^{\prime }}^{2}
\] |
[_dAlembert] |
✗ |
364.723 |
|
|
\[
{}{y^{\prime }}^{2}-a^{2} y^{2} = 0
\] |
[_quadrature] |
✓ |
1.230 |
|
|
\[
{}{y^{\prime }}^{2} = 4 x^{2}
\] |
[_quadrature] |
✓ |
0.465 |
|
|
\[
{}y^{\prime \prime }-2 y^{\prime }-3 y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
0.823 |
|
|
\[
{}s^{\prime \prime }+2 s^{\prime }+s = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.163 |
|
|
\[
{}y^{\prime \prime }-2 y^{\prime }+5 y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.708 |
|
|
\[
{}y^{\prime \prime }-2 y^{\prime }-3 y = 3 x +1
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.053 |
|
|
\[
{}y^{\prime \prime }-3 y^{\prime }+2 y = x \,{\mathrm e}^{2 x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.122 |
|
|
\[
{}y^{\prime \prime }+y = 4 \sin \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
2.867 |
|
|
\[
{}y^{\prime \prime }+2 x^{2} y^{\prime }+\left (x^{4}+2 x -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.152 |
|
|
\[
{}p \,x^{2} u^{\prime \prime }+q x u^{\prime }+r u = f \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
3.358 |
|