|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}x^{\prime \prime }-x = t^{2}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.970 |
|
|
\[
{}x^{\prime \prime }-x = {\mathrm e}^{t}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.082 |
|
|
\[
{}x^{\prime \prime }+2 x^{\prime }+4 x = {\mathrm e}^{t} \cos \left (2 t \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
39.718 |
|
|
\[
{}x^{\prime \prime }-x^{\prime }+x = \sin \left (2 t \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
83.114 |
|
|
\[
{}x^{\prime \prime }+4 x^{\prime }+3 x = t \sin \left (t \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.919 |
|
|
\[
{}x^{\prime \prime }+x = \cos \left (t \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
6.847 |
|
|
\[
{}x^{2} y^{\prime \prime }-\frac {x^{2} {y^{\prime }}^{2}}{2 y}+4 x y^{\prime }+4 y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.206 |
|
|
\[
{}y^{\prime }+c y = a
\] |
[_quadrature] |
✓ |
0.947 |
|
|
\[
{}y^{\prime \prime }+\frac {y^{\prime }}{x}+k^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.098 |
|
|
\[
{}y^{\prime \prime } \sin \left (x \right )+y^{\prime } \cos \left (x \right )+n y \sin \left (x \right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.519 |
|
|
\[
{}y^{\prime } = \frac {\sqrt {1-y^{2}}\, \arcsin \left (y\right )}{x}
\] |
[_separable] |
✓ |
386.878 |
|
|
\[
{}v^{\prime \prime } = \left (\frac {1}{v}+{v^{\prime }}^{4}\right )^{{1}/{3}}
\] |
[[_2nd_order, _missing_x]] |
✗ |
9.939 |
|
|
\[
{}v^{\prime }+u^{2} v = \sin \left (u \right )
\] |
[_linear] |
✓ |
2.353 |
|
|
\[
{}\sqrt {y^{\prime }+y} = \left (y^{\prime \prime }+2 x \right )^{{1}/{4}}
\] |
[NONE] |
✗ |
31.644 |
|
|
\[
{}v^{\prime }+\frac {2 v}{u} = 3
\] |
[_linear] |
✓ |
1.904 |
|
|
\[
{}\sin \left (x \right ) \cos \left (y\right )^{2}+\cos \left (x \right )^{2} y^{\prime } = 0
\] |
[_separable] |
✓ |
3.579 |
|
|
\[
{}y^{\prime }+\sqrt {\frac {1-y^{2}}{-x^{2}+1}} = 0
\] |
unknown |
✓ |
1240.170 |
|
|
\[
{}y-x y^{\prime } = b \left (1+x^{2} y^{\prime }\right )
\] |
[_separable] |
✓ |
126.757 |
|
|
\[
{}x^{\prime } = k \left (A -n x\right ) \left (M -m x\right )
\] |
[_quadrature] |
✓ |
18.167 |
|
|
\[
{}y^{\prime } = 1+\frac {1}{x}-\frac {1}{y^{2}+2}-\frac {1}{x \left (y^{2}+2\right )}
\] |
[_separable] |
✓ |
213.631 |
|
|
\[
{}y^{2} = x \left (y-x \right ) y^{\prime }
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
358.378 |
|
|
\[
{}2 x^{2} y+y^{3}-x^{3} y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
761.594 |
|
|
\[
{}2 a x +b y+\left (2 c y+b x +e \right ) y^{\prime } = g
\] |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.390 |
|
|
\[
{}\sec \left (x \right )^{2} \tan \left (y\right ) y^{\prime }+\sec \left (y\right )^{2} \tan \left (x \right ) = 0
\] |
[_separable] |
✓ |
38.081 |
|
|
\[
{}x +y^{\prime } y = m y
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
10.129 |
|
|
\[
{}\frac {2 x}{y^{3}}+\left (\frac {1}{y^{2}}-\frac {3 x^{2}}{y^{4}}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
4.920 |
|
|
\[
{}\left (T+\frac {1}{\sqrt {t^{2}-T^{2}}}\right ) T^{\prime } = \frac {T}{t \sqrt {t^{2}-T^{2}}}-t
\] |
[_exact] |
✓ |
3.570 |
|
|
\[
{}y^{\prime }+x y = x
\] |
[_separable] |
✓ |
1.480 |
|
|
\[
{}y^{\prime }+\frac {y}{x} = \sin \left (x \right )
\] |
[_linear] |
✓ |
1.271 |
|
|
\[
{}y^{\prime }+\frac {y}{x} = \frac {\sin \left (x \right )}{y^{3}}
\] |
[_Bernoulli] |
✓ |
35.321 |
|
|
\[
{}p^{\prime } = \frac {p+a \,t^{3}-2 p t^{2}}{t \left (-t^{2}+1\right )}
\] |
[_linear] |
✓ |
1.490 |
|
|
\[
{}\left (T \ln \left (t \right )-1\right ) T = t T^{\prime }
\] |
[_Bernoulli] |
✓ |
2.372 |
|
|
\[
{}y^{\prime }+y \cos \left (x \right ) = \frac {\sin \left (2 x \right )}{2}
\] |
[_linear] |
✓ |
2.431 |
|
|
\[
{}y-y^{\prime } \cos \left (x \right ) = y^{2} \cos \left (x \right ) \left (-\sin \left (x \right )+1\right )
\] |
[_Bernoulli] |
✓ |
7.110 |
|
|
\[
{}x {y^{\prime }}^{2}-y+2 y^{\prime } = 0
\] |
[_rational, _dAlembert] |
✓ |
1.105 |
|
|
\[
{}2 {y^{\prime }}^{3}+{y^{\prime }}^{2}-y = 0
\] |
[_quadrature] |
✓ |
74.751 |
|
|
\[
{}y^{\prime } = {\mathrm e}^{z -y^{\prime }}
\] |
[_quadrature] |
✓ |
0.263 |
|
|
\[
{}\sqrt {t^{2}+T} = T^{\prime }
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
4.627 |
|
|
\[
{}\left (x^{2}-1\right ) {y^{\prime }}^{2} = 1
\] |
[_quadrature] |
✓ |
0.379 |
|
|
\[
{}y^{\prime } = \left (x +y\right )^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
1.159 |
|
|
\[
{}\theta ^{\prime \prime } = -p^{2} \theta
\] |
[[_2nd_order, _missing_x]] |
✓ |
3.096 |
|
|
\[
{}\sec \left (\theta \right )^{2} = \frac {m s^{\prime }}{k}
\] |
[_quadrature] |
✓ |
0.320 |
|
|
\[
{}y^{\prime \prime } = \frac {m \sqrt {1+{y^{\prime }}^{2}}}{k}
\] |
[[_2nd_order, _missing_x]] |
✓ |
0.594 |
|
|
\[
{}\phi ^{\prime \prime } = \frac {4 \pi n c}{\sqrt {v_{0}^{2}+\frac {2 e \left (\phi -V_{0} \right )}{m}}}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
10.487 |
|
|
\[
{}y^{\prime } = x \left (y^{2} a +b \right )
\] |
[_separable] |
✓ |
2.421 |
|
|
\[
{}n^{\prime } = \left (n^{2}+1\right ) x
\] |
[_separable] |
✓ |
1.722 |
|
|
\[
{}v^{\prime }+\frac {2 v}{u} = 3 v
\] |
[_separable] |
✓ |
1.215 |
|
|
\[
{}\sqrt {-u^{2}+1}\, v^{\prime } = 2 u \sqrt {1-v^{2}}
\] |
[_separable] |
✓ |
2.654 |
|
|
\[
{}\sqrt {1+v^{\prime }} = \frac {{\mathrm e}^{u}}{2}
\] |
[_quadrature] |
✓ |
0.263 |
|
|
\[
{}\frac {y^{\prime }}{x} = y \sin \left (x^{2}-1\right )-\frac {2 y}{\sqrt {x}}
\] |
[_separable] |
✓ |
2.792 |
|
|
\[
{}y^{\prime } = 1+\frac {2 y}{x -y}
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.403 |
|
|
\[
{}v^{\prime }+2 v u = 2 u
\] |
[_separable] |
✓ |
1.124 |
|
|
\[
{}1+v^{2}+\left (u^{2}+1\right ) v v^{\prime } = 0
\] |
[_separable] |
✓ |
2.764 |
|
|
\[
{}u \ln \left (u \right ) v^{\prime }+\sin \left (v\right )^{2} = 1
\] |
[_separable] |
✓ |
3.989 |
|
|
\[
{}4 y {y^{\prime }}^{3}-2 x^{2} {y^{\prime }}^{2}+4 x y y^{\prime }+x^{3} = 16 y^{2}
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
123.131 |
|
|
\[
{}\theta ^{\prime \prime }-p^{2} \theta = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
6.349 |
|
|
\[
{}y^{\prime \prime }+y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
3.865 |
|
|
\[
{}y^{\prime \prime }+12 y = 7 y^{\prime }
\] |
[[_2nd_order, _missing_x]] |
✓ |
0.901 |
|
|
\[
{}r^{\prime \prime }-a^{2} r = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
6.772 |
|
|
\[
{}y^{\prime \prime \prime \prime }-a^{4} y = 0
\] |
[[_high_order, _missing_x]] |
✓ |
0.151 |
|
|
\[
{}v^{\prime \prime }-6 v^{\prime }+13 v = {\mathrm e}^{-2 u}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
11.856 |
|
|
\[
{}y^{\prime \prime }+4 y^{\prime }-y = \sin \left (t \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.709 |
|
|
\[
{}y^{\prime \prime }+3 y = \sin \left (x \right )+\frac {\sin \left (3 x \right )}{3}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
4.831 |
|
|
\[
{}5 x^{\prime }+x = \sin \left (3 t \right )
\] |
[[_linear, ‘class A‘]] |
✓ |
1.667 |
|
|
\[
{}x^{\prime \prime \prime \prime }-6 x^{\prime \prime \prime }+11 x^{\prime \prime }-6 x^{\prime } = {\mathrm e}^{-3 t}
\] |
[[_high_order, _missing_y]] |
✓ |
0.121 |
|
|
\[
{}x^{4} y^{\prime \prime \prime \prime }+x^{3} y^{\prime \prime \prime }-20 x^{2} y^{\prime \prime }+20 x y^{\prime } = 17 x^{6}
\] |
[[_high_order, _missing_y]] |
✓ |
0.579 |
|
|
\[
{}t^{4} x^{\prime \prime \prime \prime }-2 t^{3} x^{\prime \prime \prime }-20 t^{2} x^{\prime \prime }+12 x^{\prime } t +16 x = \cos \left (3 \ln \left (t \right )\right )
\] |
[[_high_order, _exact, _linear, _nonhomogeneous]] |
✓ |
0.807 |
|
|
\[
{}y^{\prime \prime \prime }-y^{\prime \prime }-y^{\prime }+y = 0
\] |
[[_3rd_order, _missing_x]] |
✓ |
0.194 |
|
|
\[
{}y^{\prime \prime \prime \prime }-3 y^{\prime \prime \prime }+3 y^{\prime \prime }-y^{\prime } = {\mathrm e}^{2 x}
\] |
[[_high_order, _missing_y]] |
✓ |
0.122 |
|
|
\[
{}y^{\prime \prime \prime }-y^{\prime \prime }+y^{\prime }-y = \cos \left (x \right )
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
0.632 |
|
|
\[
{}x^{2} y^{\prime \prime }+3 x y^{\prime }+y = \frac {1}{x}
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
2.380 |
|
|
\[
{}y^{\prime \prime } = c \left (1+{y^{\prime }}^{2}\right )
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.478 |
|
|
\[
{}y^{\prime \prime } = c \left (1+{y^{\prime }}^{2}\right )^{{3}/{2}}
\] |
[[_2nd_order, _missing_x]] |
✓ |
0.494 |
|
|
\[
{}x^{3} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }+x y^{\prime }+y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
0.177 |
|
|
\[
{}y^{\prime \prime } = -m^{2} y
\] |
[[_2nd_order, _missing_x]] |
✓ |
2.640 |
|
|
\[
{}1+{y^{\prime }}^{2}+\frac {m y^{\prime \prime }}{\sqrt {1+{y^{\prime }}^{2}}} = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
72.381 |
|
|
\[
{}y = x y^{\prime }+y^{\prime }-{y^{\prime }}^{3}
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.347 |
|
|
\[
{}x y^{\prime \prime }+2 y^{\prime } = x y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.963 |
|
|
\[
{}y-2 x y^{\prime }-{y^{\prime }}^{2} y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
2.152 |
|
|
\[
{}y^{\prime }+\frac {x y}{x^{2}+1} = \frac {1}{x \left (x^{2}+1\right )}
\] |
[_linear] |
✓ |
1.317 |
|
|
\[
{}y^{\prime \prime \prime }+\frac {3 y^{\prime \prime }}{x} = 0
\] |
[[_3rd_order, _missing_y]] |
✓ |
0.397 |
|
|
\[
{}x^{2} y^{\prime \prime }-5 x y^{\prime }+5 y = \frac {1}{x}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.520 |
|
|
\[
{}y^{\prime \prime }+4 y^{\prime }+3 y = 2 \,{\mathrm e}^{2 x}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.887 |
|
|
\[
{}v^{\prime \prime }+\frac {2 v^{\prime }}{r} = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.429 |
|
|
\[
{}y^{\prime \prime }-2 y^{\prime } y = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.740 |
|
|
\[
{}y^{\prime \prime }-{y^{\prime }}^{2}-y {y^{\prime }}^{3} = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
0.388 |
|
|
\[
{}\left (1+{y^{\prime }}^{2}\right )^{{3}/{2}} = r y^{\prime \prime }
\] |
[[_2nd_order, _missing_x]] |
✓ |
65.100 |
|
|
\[
{}y^{\prime \prime \prime } y^{\prime }-3 {y^{\prime \prime }}^{2}+3 y^{\prime \prime } {y^{\prime }}^{2}-2 {y^{\prime }}^{4}-x {y^{\prime }}^{5} = 0
\] |
[[_3rd_order, _missing_y], [_3rd_order, _with_exponential_symmetries], [_3rd_order, _with_linear_symmetries]] |
✗ |
0.168 |
|
|
\[
{}\left (1+y^{2}\right ) y^{\prime \prime }-2 {y^{\prime }}^{2} y-2 \left (1+y^{2}\right ) y^{\prime } = y^{2} \left (1+y^{2}\right )
\] |
[[_2nd_order, _missing_x]] |
✗ |
3.918 |
|
|
\[
{}y^{2} y^{\prime \prime \prime }-\left (3 y^{\prime } y+2 x y^{2}\right ) y^{\prime \prime }+\left (2 {y^{\prime }}^{2}+2 x y y^{\prime }+3 x^{2} y^{2}\right ) y^{\prime }+x^{3} y^{3} = 0
\] |
|
✗ |
0.116 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime }+y = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.640 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime } = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.692 |
|
|
\[
{}x^{3} v^{\prime \prime \prime }+2 x^{2} v^{\prime \prime }+v = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
0.188 |
|
|
\[
{}v^{\prime \prime }+\frac {2 x v^{\prime }}{x^{2}+1}+\frac {v}{\left (x^{2}+1\right )^{2}} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.016 |
|
|
\[
{}y^{\prime }+\frac {y}{x} = -x^{2}+1
\] |
[_linear] |
✓ |
0.921 |
|
|
\[
{}y^{\prime }+\cot \left (x \right ) y = \csc \left (x \right )^{2}
\] |
[_linear] |
✓ |
1.683 |
|
|
\[
{}y^{\prime } = x -y
\] |
[[_linear, ‘class A‘]] |
✓ |
0.834 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime }+x^{2} y = x^{3}-x^{2} \arctan \left (x \right )
\] |
[_linear] |
✗ |
40.885 |
|
|
\[
{}y^{\prime }+\frac {x y}{x^{2}+1} = \frac {1}{x \left (x^{2}+1\right )}
\] |
[_linear] |
✓ |
1.262 |
|
|
\[
{}x \left (-x^{2}+1\right ) y^{\prime }+\left (x^{2}-1\right ) y = x^{3}
\] |
[_linear] |
✓ |
1.073 |
|