|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}1+2 x+\left (-t^{2}+4\right ) x^{\prime } = 0
\] |
[_separable] |
✓ |
1.707 |
|
|
\[
{}x^{\prime } = \cos \left (\frac {x}{t}\right )
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
2.569 |
|
|
\[
{}\left (t^{2}-x^{2}\right ) x^{\prime } = t x
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
3.411 |
|
|
\[
{}{\mathrm e}^{3 t} x^{\prime }+3 x \,{\mathrm e}^{3 t} = 2 t
\] |
[[_linear, ‘class A‘]] |
✓ |
1.769 |
|
|
\[
{}2 t +3 x+\left (3 t -x\right ) x^{\prime } = t^{2}
\] |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.343 |
|
|
\[
{}x^{\prime }+2 x = {\mathrm e}^{t}
\] |
[[_linear, ‘class A‘]] |
✓ |
1.265 |
|
|
\[
{}x^{\prime }+x \tan \left (t \right ) = 0
\] |
[_separable] |
✓ |
1.773 |
|
|
\[
{}x^{\prime }-x \tan \left (t \right ) = 4 \sin \left (t \right )
\] |
[_linear] |
✓ |
1.749 |
|
|
\[
{}t^{3} x^{\prime }+\left (-3 t^{2}+2\right ) x = t^{3}
\] |
[_linear] |
✓ |
2.230 |
|
|
\[
{}x^{\prime }+2 t x+t x^{4} = 0
\] |
[_separable] |
✓ |
2.363 |
|
|
\[
{}t x^{\prime }+x \ln \left (t \right ) = t^{2}
\] |
[_linear] |
✓ |
1.495 |
|
|
\[
{}t x^{\prime }+x g \left (t \right ) = h \left (t \right )
\] |
[_linear] |
✓ |
1.510 |
|
|
\[
{}t^{2} x^{\prime \prime }-6 t x^{\prime }+12 x = 0
\] |
[[_Emden, _Fowler]] |
✓ |
1.165 |
|
|
\[
{}x^{\prime } = -\lambda x
\] |
[_quadrature] |
✓ |
0.663 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }=x \\ y^{\prime }=x+2 y \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.369 |
|
|
\[
{}t^{2} x^{\prime \prime }-2 t x^{\prime }+2 x = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.274 |
|
|
\[
{}x^{\prime \prime }-5 x^{\prime }+6 x = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.049 |
|
|
\[
{}x^{\prime \prime }-4 x^{\prime }+4 x = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.181 |
|
|
\[
{}x^{\prime \prime }-4 x^{\prime }+5 x = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.520 |
|
|
\[
{}x^{\prime \prime }+3 x^{\prime } = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.986 |
|
|
\[
{}x^{\prime \prime }-3 x^{\prime }+2 x = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.381 |
|
|
\[
{}x^{\prime \prime }+x = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
2.772 |
|
|
\[
{}x^{\prime \prime }+2 x^{\prime }+x = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.491 |
|
|
\[
{}x^{\prime \prime }-2 x^{\prime }+2 x = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.589 |
|
|
\[
{}x^{\prime \prime }-x = t^{2}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.416 |
|
|
\[
{}x^{\prime \prime }-x = {\mathrm e}^{t}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.900 |
|
|
\[
{}x^{\prime \prime }+2 x^{\prime }+4 x = {\mathrm e}^{t} \cos \left (2 t \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
58.306 |
|
|
\[
{}x^{\prime \prime }-x^{\prime }+x = \sin \left (2 t \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
77.445 |
|
|
\[
{}x^{\prime \prime }+4 x^{\prime }+3 x = t \sin \left (t \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.754 |
|
|
\[
{}x^{\prime \prime }+x = \cos \left (t \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
3.422 |
|
|
\[
{}x^{2} y^{\prime \prime }-\frac {x^{2} {y^{\prime }}^{2}}{2 y}+4 y^{\prime } x +4 y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.161 |
|
|
\[
{}y^{\prime }+c y = a
\] |
[_quadrature] |
✓ |
0.666 |
|
|
\[
{}y^{\prime \prime }+\frac {y^{\prime }}{x}+k^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.958 |
|
|
\[
{}y^{\prime \prime } \sin \left (x \right )+y^{\prime } \cos \left (x \right )+n y \sin \left (x \right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.352 |
|
|
\[
{}y^{\prime } = \frac {\sqrt {1-y^{2}}\, \arcsin \left (y\right )}{x}
\] |
[_separable] |
✓ |
8.143 |
|
|
\[
{}v^{\prime \prime } = \left (\frac {1}{v}+{v^{\prime }}^{4}\right )^{{1}/{3}}
\] |
[[_2nd_order, _missing_x]] |
✗ |
2.791 |
|
|
\[
{}v^{\prime }+u^{2} v = \sin \left (u \right )
\] |
[_linear] |
✓ |
1.798 |
|
|
\[
{}\sqrt {y^{\prime }+y} = \left (y^{\prime \prime }+2 x \right )^{{1}/{4}}
\] |
[NONE] |
✗ |
0.330 |
|
|
\[
{}v^{\prime }+\frac {2 v}{u} = 3
\] |
[_linear] |
✓ |
2.440 |
|
|
\[
{}\sin \left (x \right ) \cos \left (y\right )^{2}+\cos \left (x \right )^{2} y^{\prime } = 0
\] |
[_separable] |
✓ |
3.309 |
|
|
\[
{}y^{\prime }+\sqrt {\frac {1-y^{2}}{-x^{2}+1}} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
25.410 |
|
|
\[
{}y-y^{\prime } x = b \left (1+x^{2} y^{\prime }\right )
\] |
[_separable] |
✓ |
0.967 |
|
|
\[
{}x^{\prime } = k \left (A -n x\right ) \left (M -m x\right )
\] |
[_quadrature] |
✓ |
2.498 |
|
|
\[
{}y^{\prime } = 1+\frac {1}{x}-\frac {1}{y^{2}+2}-\frac {1}{x \left (y^{2}+2\right )}
\] |
[_separable] |
✓ |
1.496 |
|
|
\[
{}y^{2} = x \left (y-x \right ) y^{\prime }
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
18.251 |
|
|
\[
{}2 x^{2} y+y^{3}-x^{3} y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
88.278 |
|
|
\[
{}2 a x +b y+\left (2 c y+b x +e \right ) y^{\prime } = g
\] |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.687 |
|
|
\[
{}\sec \left (x \right )^{2} \tan \left (y\right ) y^{\prime }+\sec \left (y\right )^{2} \tan \left (x \right ) = 0
\] |
[_separable] |
✓ |
37.978 |
|
|
\[
{}x +y y^{\prime } = m y
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
8.876 |
|
|
\[
{}\frac {2 x}{y^{3}}+\left (\frac {1}{y^{2}}-\frac {3 x^{2}}{y^{4}}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
4.346 |
|
|
\[
{}\left (T+\frac {1}{\sqrt {t^{2}-T^{2}}}\right ) T^{\prime } = \frac {T}{t \sqrt {t^{2}-T^{2}}}-t
\] |
[_exact] |
✓ |
3.069 |
|
|
\[
{}y^{\prime }+x y = x
\] |
[_separable] |
✓ |
1.671 |
|
|
\[
{}y^{\prime }+\frac {y}{x} = \sin \left (x \right )
\] |
[_linear] |
✓ |
1.293 |
|
|
\[
{}y^{\prime }+\frac {y}{x} = \frac {\sin \left (x \right )}{y^{3}}
\] |
[_Bernoulli] |
✓ |
33.949 |
|
|
\[
{}p^{\prime } = \frac {p+a \,t^{3}-2 p t^{2}}{t \left (-t^{2}+1\right )}
\] |
[_linear] |
✓ |
1.072 |
|
|
\[
{}\left (T \ln \left (t \right )-1\right ) T = t T^{\prime }
\] |
[_Bernoulli] |
✓ |
2.010 |
|
|
\[
{}y^{\prime }+y \cos \left (x \right ) = \frac {\sin \left (2 x \right )}{2}
\] |
[_linear] |
✓ |
2.192 |
|
|
\[
{}y-y^{\prime } \cos \left (x \right ) = y^{2} \cos \left (x \right ) \left (1-\sin \left (x \right )\right )
\] |
[_Bernoulli] |
✓ |
5.399 |
|
|
\[
{}x {y^{\prime }}^{2}-y+2 y^{\prime } = 0
\] |
[_rational, _dAlembert] |
✓ |
0.860 |
|
|
\[
{}2 {y^{\prime }}^{3}+{y^{\prime }}^{2}-y = 0
\] |
[_quadrature] |
✓ |
68.154 |
|
|
\[
{}y^{\prime } = {\mathrm e}^{z -y^{\prime }}
\] |
[_quadrature] |
✓ |
0.459 |
|
|
\[
{}\sqrt {t^{2}+T} = T^{\prime }
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
3.372 |
|
|
\[
{}\left (x^{2}-1\right ) {y^{\prime }}^{2} = 1
\] |
[_quadrature] |
✓ |
0.287 |
|
|
\[
{}y^{\prime } = \left (x +y\right )^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
1.781 |
|
|
\[
{}\theta ^{\prime \prime } = -p^{2} \theta
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.725 |
|
|
\[
{}\sec \left (\theta \right )^{2} = \frac {m s^{\prime }}{k}
\] |
[_quadrature] |
✓ |
0.296 |
|
|
\[
{}y^{\prime \prime } = \frac {m \sqrt {{y^{\prime }}^{2}+1}}{k}
\] |
[[_2nd_order, _missing_x]] |
✓ |
8.658 |
|
|
\[
{}\phi ^{\prime \prime } = \frac {4 \pi n c}{\sqrt {v_{0}^{2}+\frac {2 e \left (\phi -V_{0} \right )}{m}}}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
7.177 |
|
|
\[
{}y^{\prime } = x \left (a y^{2}+b \right )
\] |
[_separable] |
✓ |
2.468 |
|
|
\[
{}n^{\prime } = \left (n^{2}+1\right ) x
\] |
[_separable] |
✓ |
2.121 |
|
|
\[
{}v^{\prime }+\frac {2 v}{u} = 3 v
\] |
[_separable] |
✓ |
1.671 |
|
|
\[
{}\sqrt {-u^{2}+1}\, v^{\prime } = 2 u \sqrt {1-v^{2}}
\] |
[_separable] |
✓ |
1.890 |
|
|
\[
{}\sqrt {1+v^{\prime }} = \frac {{\mathrm e}^{u}}{2}
\] |
[_quadrature] |
✓ |
0.443 |
|
|
\[
{}\frac {y^{\prime }}{x} = y \sin \left (x^{2}-1\right )-\frac {2 y}{\sqrt {x}}
\] |
[_separable] |
✓ |
1.896 |
|
|
\[
{}y^{\prime } = 1+\frac {2 y}{x -y}
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.513 |
|
|
\[
{}v^{\prime }+2 v u = 2 u
\] |
[_separable] |
✓ |
1.479 |
|
|
\[
{}1+v^{2}+\left (u^{2}+1\right ) v v^{\prime } = 0
\] |
[_separable] |
✓ |
2.567 |
|
|
\[
{}u \ln \left (u \right ) v^{\prime }+\sin \left (v\right )^{2} = 1
\] |
[_separable] |
✓ |
3.036 |
|
|
\[
{}4 y {y^{\prime }}^{3}-2 x^{2} {y^{\prime }}^{2}+4 x y y^{\prime }+x^{3} = 16 y^{2}
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
110.467 |
|
|
\[
{}\theta ^{\prime \prime }-p^{2} \theta = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
3.052 |
|
|
\[
{}y^{\prime \prime }+y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
2.145 |
|
|
\[
{}y^{\prime \prime }+12 y = 7 y^{\prime }
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.070 |
|
|
\[
{}r^{\prime \prime }-a^{2} r = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
3.068 |
|
|
\[
{}y^{\prime \prime \prime \prime }-a^{4} y = 0
\] |
[[_high_order, _missing_x]] |
✓ |
0.081 |
|
|
\[
{}v^{\prime \prime }-6 v^{\prime }+13 v = {\mathrm e}^{-2 u}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
11.850 |
|
|
\[
{}y^{\prime \prime }+4 y^{\prime }-y = \sin \left (t \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
2.306 |
|
|
\[
{}y^{\prime \prime }+3 y = \sin \left (x \right )+\frac {\sin \left (3 x \right )}{3}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
4.987 |
|
|
\[
{}5 x^{\prime }+x = \sin \left (3 t \right )
\] |
[[_linear, ‘class A‘]] |
✓ |
1.572 |
|
|
\[
{}x^{\prime \prime \prime \prime }-6 x^{\prime \prime \prime }+11 x^{\prime \prime }-6 x^{\prime } = {\mathrm e}^{-3 t}
\] |
[[_high_order, _missing_y]] |
✓ |
0.118 |
|
|
\[
{}x^{4} y^{\prime \prime \prime \prime }+x^{3} y^{\prime \prime \prime }-20 x^{2} y^{\prime \prime }+20 y^{\prime } x = 17 x^{6}
\] |
[[_high_order, _missing_y]] |
✓ |
0.301 |
|
|
\[
{}t^{4} x^{\prime \prime \prime \prime }-2 t^{3} x^{\prime \prime \prime }-20 t^{2} x^{\prime \prime }+12 t x^{\prime }+16 x = \cos \left (3 \ln \left (t \right )\right )
\] |
[[_high_order, _exact, _linear, _nonhomogeneous]] |
✓ |
0.690 |
|
|
\[
{}y^{\prime \prime \prime }-y^{\prime \prime }-y^{\prime }+y = 0
\] |
[[_3rd_order, _missing_x]] |
✓ |
0.070 |
|
|
\[
{}y^{\prime \prime \prime \prime }-3 y^{\prime \prime \prime }+3 y^{\prime \prime }-y^{\prime } = {\mathrm e}^{2 x}
\] |
[[_high_order, _missing_y]] |
✓ |
0.114 |
|
|
\[
{}y^{\prime \prime \prime }-y^{\prime \prime }+y^{\prime }-y = \cos \left (x \right )
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
0.515 |
|
|
\[
{}x^{2} y^{\prime \prime }+3 y^{\prime } x +y = \frac {1}{x}
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
1.703 |
|
|
\[
{}y^{\prime \prime } = c \left ({y^{\prime }}^{2}+1\right )
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
5.362 |
|
|
\[
{}y^{\prime \prime } = c \left ({y^{\prime }}^{2}+1\right )^{{3}/{2}}
\] |
[[_2nd_order, _missing_x]] |
✓ |
3.203 |
|
|
\[
{}x^{3} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }+y^{\prime } x +y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
0.175 |
|
|
\[
{}y^{\prime \prime } = -m^{2} y
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.760 |
|
|
\[
{}1+{y^{\prime }}^{2}+\frac {m y^{\prime \prime }}{\sqrt {{y^{\prime }}^{2}+1}} = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
9.324 |
|