| # |
ODE |
CAS classification |
Solved? |
Maple |
Mma |
Sympy |
time(sec) |
| \begin{align*}
w_{1}^{\prime }&=w_{2} \\
w_{2}^{\prime }&=\frac {a w_{1}}{z^{2}} \\
\end{align*} |
system_of_ODEs |
✗ |
✓ |
✓ |
✗ |
0.041 |
|
| \begin{align*}
z^{2} u^{\prime \prime }+\left (3 z +1\right ) u^{\prime }+u&=0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✗ |
0.492 |
|
| \begin{align*}
x^{\prime }+\ln \left (3\right ) x&=0 \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✓ |
1.667 |
|
| \begin{align*}
x^{\prime }+4 x&=4 \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✓ |
0.477 |
|
| \begin{align*}
x^{\prime }+\frac {\left (2 t^{3}+\sin \left (t \right )+5\right ) x}{t^{12}+5}&=0 \\
x \left (0\right ) &= 0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✗ |
27.958 |
|
| \begin{align*}
x^{\prime }&=-2 x+3 \\
x \left (0\right ) &= 1 \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✓ |
0.879 |
|
| \begin{align*}
x^{\prime }&=k x \\
x \left (0\right ) &= 1 \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✓ |
0.507 |
|
| \begin{align*}
x^{\prime }-2 x \cos \left (t \right )&=\cos \left (t \right ) \\
x \left (0\right ) &= {\frac {1}{2}} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.723 |
|
| \begin{align*}
x^{\prime }+\frac {x}{t^{2}-1}&=0 \\
x \left (-2\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.415 |
|
| \begin{align*}
x^{\prime }+\sec \left (t \right ) x&=\frac {1}{t -1} \\
x \left (\frac {\pi }{4}\right ) &= 1 \\
\end{align*} |
[_linear] |
✓ |
✓ |
✓ |
✓ |
5.237 |
|
| \begin{align*}
t x^{\prime }+x&=2 t^{2} \\
\end{align*} |
[_linear] |
✓ |
✓ |
✓ |
✓ |
2.254 |
|
| \begin{align*}
t^{2} x^{\prime }-2 t x&=t^{5} \\
x \left (0\right ) &= 0 \\
\end{align*} |
[_linear] |
✓ |
✓ |
✓ |
✗ |
2.554 |
|
| \begin{align*}
x^{\prime }&=2 t x \\
x \left (0\right ) &= 4 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.211 |
|
| \begin{align*}
x^{\prime }&=-t^{2} x \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.121 |
|
| \begin{align*}
x^{\prime }+a x&=b t \\
\end{align*} |
[[_linear, ‘class A‘]] |
✓ |
✓ |
✓ |
✓ |
0.871 |
|
| \begin{align*}
x^{\prime }&=x+2 t \\
\end{align*} |
[[_linear, ‘class A‘]] |
✓ |
✓ |
✓ |
✓ |
1.009 |
|
| \begin{align*}
x^{\prime }-2 x&=3 t \\
\end{align*} |
[[_linear, ‘class A‘]] |
✓ |
✓ |
✓ |
✓ |
0.878 |
|
| \begin{align*}
x^{\prime }+3 x&=-2 t \\
\end{align*} |
[[_linear, ‘class A‘]] |
✓ |
✓ |
✓ |
✓ |
0.865 |
|
| \begin{align*}
x^{\prime }+a x&=b t \\
x \left (t_{0} \right ) &= x_{0} \\
\end{align*} |
[[_linear, ‘class A‘]] |
✓ |
✓ |
✓ |
✓ |
1.053 |
|
| \begin{align*}
x^{\prime }-x&=\frac {t}{2} \\
x \left (0\right ) &= 1 \\
\end{align*} |
[[_linear, ‘class A‘]] |
✓ |
✓ |
✓ |
✓ |
1.193 |
|
| \begin{align*}
x^{\prime }+x&=4 t \\
x \left (1\right ) &= 0 \\
\end{align*} |
[[_linear, ‘class A‘]] |
✓ |
✓ |
✓ |
✓ |
1.447 |
|
| \begin{align*}
x^{\prime }-2 x&=2 t \\
x \left (0\right ) &= 3 \\
\end{align*} |
[[_linear, ‘class A‘]] |
✓ |
✓ |
✓ |
✓ |
1.384 |
|
| \begin{align*}
x^{\prime }+k x&=1 \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✓ |
0.400 |
|
| \begin{align*}
x^{\prime }&=\frac {x}{t^{2}+1} \\
x \left (0\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
4.208 |
|
| \begin{align*}
x^{\prime }-k^{2} x&=1 \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✓ |
0.409 |
|
| \begin{align*}
x^{\prime }+2 x&=6 t \\
\end{align*} |
[[_linear, ‘class A‘]] |
✓ |
✓ |
✓ |
✓ |
1.085 |
|
| \begin{align*}
x^{\prime }+x&=a t \\
\end{align*} |
[[_linear, ‘class A‘]] |
✓ |
✓ |
✓ |
✓ |
0.702 |
|
| \begin{align*}
x^{\prime }&=t +x^{2} \\
x \left (0\right ) &= 0 \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
17.888 |
|
| \begin{align*}
x^{\prime }&=\frac {3 x^{{1}/{3}}}{2} \\
x \left (0\right ) &= a \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✗ |
1.904 |
|
| \begin{align*}
x^{\prime }&=x^{2} \\
x \left (t_{0} \right ) &= a \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✓ |
1.661 |
|
| \begin{align*}
x^{\prime }+\frac {\sin \left (t \right ) x}{1+{\mathrm e}^{t}}&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.935 |
|
| \begin{align*}
{\mathrm e}^{x^{\prime }}&=x \\
x \left (t_{0} \right ) &= a \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✓ |
0.230 |
|
| \begin{align*}
x^{\prime }&=\sqrt {1-x^{2}} \\
x \left (\frac {\pi }{2}\right ) &= 1 \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✗ |
8.011 |
|
| \begin{align*}
x^{\prime }&=x^{{1}/{4}} \\
x \left (0\right ) &= a \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✗ |
1.454 |
|
| \begin{align*}
x^{\prime }&=x^{p} \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✓ |
1.332 |
|
| \begin{align*}
x^{\prime }&=\sin \left (x\right ) \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✗ |
28.671 |
|
| \begin{align*}
x^{\prime }&=\arctan \left (x\right ) \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✓ |
0.632 |
|
| \begin{align*}
x^{\prime }&=\ln \left (x^{2}+1\right ) \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✓ |
0.851 |
|
| \begin{align*}
x^{\prime }&=t^{2} x^{4}+1 \\
x \left (0\right ) &= 0 \\
\end{align*} |
[_Chini] |
✗ |
✗ |
✗ |
✗ |
13.087 |
|
| \begin{align*}
x^{\prime }&=2+\sin \left (x\right ) \\
x \left (0\right ) &= 0 \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✗ |
56.750 |
|
| \begin{align*}
x^{\prime }&=\sin \left (t x\right ) \\
\end{align*} |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
✗ |
✗ |
4.378 |
|
| \begin{align*}
x^{\prime }&=\left (2+x\right ) \left (1-x^{4}\right ) \\
x \left (0\right ) &= 0 \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✓ |
22.008 |
|
| \begin{align*}
x^{\prime }&=x^{3}-x \\
x \left (0\right ) &= a \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✓ |
1.983 |
|
| \begin{align*}
x^{\prime }&=\arctan \left (x\right )+t \\
\end{align*} |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
✗ |
✗ |
39.113 |
|
| \begin{align*}
x^{\prime }&={\mathrm e}^{x}-t \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
✓ |
✓ |
1.642 |
|
| \begin{align*}
x^{\prime }&=t x-t^{3} \\
x \left (a \right ) &= a^{2} \\
\end{align*} |
[_linear] |
✓ |
✓ |
✓ |
✓ |
2.247 |
|
| \begin{align*}
x^{\prime }&=t x-t^{3} \\
x \left (0\right ) &= a^{2} \\
\end{align*} |
[_linear] |
✓ |
✓ |
✓ |
✓ |
2.157 |
|
| \begin{align*}
x^{\prime }&=x^{2}-t^{2} \\
x \left (0\right ) &= 0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
5.376 |
|
| \begin{align*}
x^{\prime }&=x^{2}+1 \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✓ |
1.622 |
|
| \begin{align*}
x^{\prime }&=x^{2}-1 \\
x \left (0\right ) &= 0 \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✗ |
1.074 |
|
| \begin{align*}
x^{\prime }&=x^{2}+x \\
x \left (1\right ) &= 1 \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✓ |
0.880 |
|
| \begin{align*}
x^{\prime }&=\frac {x^{2}+x}{2 x+1} \\
x \left (0\right ) &= 1 \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✓ |
1.567 |
|
| \begin{align*}
x^{\prime }&=\frac {-x+x^{2}}{2 x-1} \\
x \left (0\right ) &= 2 \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✓ |
0.602 |
|
| \begin{align*}
x^{\prime }&=4 t^{3} x^{4} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.454 |
|
| \begin{align*}
x^{\prime }&=-t x^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.251 |
|
| \begin{align*}
x^{\prime }&={\mathrm e}^{t} \left (x^{2}+1\right ) \\
x \left (0\right ) &= 0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.306 |
|
| \begin{align*}
x^{\prime }&=\frac {t}{x} \\
x \left (\sqrt {2}\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
5.260 |
|
| \begin{align*}
x^{\prime }&=-\frac {t}{4 x^{3}} \\
x \left (1\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.030 |
|
| \begin{align*}
x^{\prime }&=-t^{2} x^{2} \\
x \left (1\right ) &= 2 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.506 |
|
| \begin{align*}
x^{\prime }&=5 t \sqrt {x} \\
x \left (0\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✗ |
6.559 |
|
| \begin{align*}
x^{\prime }&=4 t^{3} \sqrt {x} \\
x \left (0\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✗ |
5.493 |
|
| \begin{align*}
x^{\prime }&=2 t \sqrt {x} \\
x \left (a \right ) &= 0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
7.143 |
|
| \begin{align*}
x^{\prime }&=-\left (1+p \right ) t^{p} x^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.195 |
|
| \begin{align*}
x^{\prime }&=\sqrt {1-x^{2}} \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✓ |
2.892 |
|
| \begin{align*}
2 x^{2}+1&=\left (y^{5}-1\right ) y^{\prime } \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
1.488 |
|
| \begin{align*}
x +3 y+\left (3 x +y\right ) y^{\prime }&=0 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
✓ |
✓ |
8.951 |
|
| \begin{align*}
x +y+\left (x -y\right ) y^{\prime }&=0 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
✓ |
✓ |
9.401 |
|
| \begin{align*}
a \,x^{p}+b y+\left (b x +d y^{q}\right ) y^{\prime }&=0 \\
\end{align*} |
[_exact, _rational] |
✓ |
✓ |
✓ |
✗ |
2.322 |
|
| \begin{align*}
3 x^{2}-y+\left (4 y^{3}-x \right ) y^{\prime }&=0 \\
y \left (1\right ) &= 1 \\
\end{align*} |
[_exact, _rational] |
✓ |
✓ |
✗ |
✗ |
1.508 |
|
| \begin{align*}
y-x^{{1}/{3}}+\left (x +y\right ) y^{\prime }&=0 \\
y \left (0\right ) &= 1 \\
\end{align*} |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
✓ |
✗ |
2.579 |
|
| \begin{align*}
{\mathrm e}^{x}-\frac {y^{2}}{2}+\left ({\mathrm e}^{y}-y x \right ) y^{\prime }&=0 \\
y \left (0\right ) &= 0 \\
\end{align*} |
[_exact] |
✓ |
✓ |
✓ |
✗ |
2.290 |
|
| \begin{align*}
x +\sin \left (y\right )+x \cos \left (y\right ) y^{\prime }&=0 \\
y \left (2\right ) &= \pi \\
\end{align*} |
[_exact] |
✓ |
✓ |
✗ |
✓ |
2.047 |
|
| \begin{align*}
x^{2}+2 y x -y^{2}+\left (x -y\right )^{2} y^{\prime }&=0 \\
y \left (1\right ) &= 1 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
✓ |
✗ |
6.792 |
|
| \begin{align*}
x^{2}+2 y x +2 y^{2}+\left (x^{2}+4 y x +5 y^{2}\right ) y^{\prime }&=0 \\
y \left (0\right ) &= 0 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
✓ |
✗ |
22.949 |
|
| \begin{align*}
x -2 y^{3} y^{\prime }&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.325 |
|
| \begin{align*}
x^{2}+y^{2}+\left (a x y+y^{4}\right ) y^{\prime }&=0 \\
\end{align*} |
[_rational] |
✗ |
✗ |
✗ |
✗ |
59.565 |
|
| \begin{align*}
x^{2}+a_{1} x y+a_{2} y^{2}+\left (x^{2}+b_{1} x y+b_{2} y^{2}\right ) y^{\prime }&=0 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
✓ |
✗ |
0.813 |
|
| \begin{align*}
x +y^{2}+B \left (x \right ) y y^{\prime }&=0 \\
\end{align*} |
[_Bernoulli] |
✓ |
✓ |
✓ |
✓ |
1.961 |
|
| \begin{align*}
x +y^{2}+y y^{\prime } x&=0 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
✓ |
✓ |
3.204 |
|
| \begin{align*}
2 y+x +\left (x^{2}-1\right ) y^{\prime }&=0 \\
\end{align*} |
[_linear] |
✓ |
✓ |
✓ |
✓ |
1.368 |
|
| \begin{align*}
x +2 y+\left (x -1\right ) y^{\prime }&=0 \\
\end{align*} |
[_linear] |
✓ |
✓ |
✓ |
✓ |
2.193 |
|
| \begin{align*}
y^{2}+\left (y x +3 y^{3}\right ) y^{\prime }&=0 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
✓ |
✗ |
0.243 |
|
| \begin{align*}
y y^{\prime } x +1+y^{2}&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.760 |
|
| \begin{align*}
x^{\prime }&=\frac {x+2 t}{t} \\
\end{align*} |
[_linear] |
✓ |
✓ |
✓ |
✓ |
1.810 |
|
| \begin{align*}
x^{\prime }&=\frac {t x}{t^{2}+x^{2}} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
✓ |
✓ |
3.802 |
|
| \begin{align*}
x^{\prime }&=\frac {3 x^{2}-2 t^{2}}{t x} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
✓ |
✓ |
6.317 |
|
| \begin{align*}
x^{\prime }&=\frac {t^{2}+x^{2}}{2 t x} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
✓ |
✓ |
6.787 |
|
| \begin{align*}
x^{\prime }&=\frac {x-t +1}{x-t +2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
✓ |
✓ |
3.497 |
|
| \begin{align*}
x^{\prime }&=\frac {x-t}{x-t +1} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
✓ |
✓ |
3.214 |
|
| \begin{align*}
x^{\prime }&=-\frac {x+t +1}{x-t +1} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
✓ |
✓ |
8.947 |
|
| \begin{align*}
x^{\prime }-x&=t x^{2} \\
x \left (0\right ) &= a \\
\end{align*} |
[_Bernoulli] |
✓ |
✓ |
✓ |
✓ |
1.842 |
|
| \begin{align*}
x^{\prime }+2 t x&=-4 t x^{3} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
7.102 |
|
| \begin{align*}
x^{\prime }-t x&=x^{2} \\
\end{align*} |
[_Bernoulli] |
✓ |
✓ |
✓ |
✓ |
1.293 |
|
| \begin{align*}
{x^{\prime }}^{2}&=x^{2}+t^{2}-1 \\
\end{align*} |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
✗ |
✗ |
44.142 |
|
| \begin{align*}
{x^{\prime }}^{2}&=-4 x+4 \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✓ |
0.256 |
|
| \begin{align*}
{x^{\prime }}^{2}-t x+x&=0 \\
\end{align*} |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.846 |
|
| \begin{align*}
x&=t x^{\prime }-{x^{\prime }}^{2} \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
✓ |
✓ |
0.237 |
|
| \begin{align*}
x&=t x^{\prime }-{\mathrm e}^{x^{\prime }} \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
✓ |
✓ |
2.361 |
|
| \begin{align*}
x&=t x^{\prime }-\ln \left (x^{\prime }\right ) \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
✓ |
✓ |
2.947 |
|
| \begin{align*}
x&=t x^{\prime }+\frac {1}{x^{\prime }} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Clairaut] |
✓ |
✓ |
✓ |
✗ |
0.954 |
|