|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}\left (6 y^{2}-3 x^{2} y+1\right ) y^{\prime }-3 x y^{2}+x = 0
\] |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
1.627 |
|
|
\[
{}\left (6 y-x \right )^{2} y^{\prime }-6 y^{2}+2 x y+a = 0
\] |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
1.509 |
|
|
\[
{}\left (a y^{2}+2 b x y+c \,x^{2}\right ) y^{\prime }+b y^{2}+2 c x y+d \,x^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
824.837 |
|
|
\[
{}\left (b \left (\beta y+\alpha x \right )^{2}-\beta \left (a x +b y\right )\right ) y^{\prime }+a \left (\beta y+\alpha x \right )^{2}-\alpha \left (a x +b y\right ) = 0
\] |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
1.813 |
|
|
\[
{}\left (a y+b x +c \right )^{2} y^{\prime }+\left (\alpha y+\beta x +\gamma \right )^{2} = 0
\] |
[[_homogeneous, ‘class C‘], _rational] |
✓ |
3.523 |
|
|
\[
{}x \left (y^{2}-3 x \right ) y^{\prime }+2 y^{3}-5 x y = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
5.211 |
|
|
\[
{}x \left (y^{2}+x^{2}-a \right ) y^{\prime }-y \left (y^{2}+x^{2}+a \right ) = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
4.456 |
|
|
\[
{}x \left (y^{2}+x y-x^{2}\right ) y^{\prime }-y^{3}+x y^{2}+x^{2} y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
87.746 |
|
|
\[
{}x \left (y^{2}+x^{2} y+x^{2}\right ) y^{\prime }-2 y^{3}-2 x^{2} y^{2}+x^{4} = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
3.293 |
|
|
\[
{}2 x \left (y^{2}+5 x^{2}\right ) y^{\prime }+y^{3}-x^{2} y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
19.870 |
|
|
\[
{}3 x y^{2} y^{\prime }+y^{3}-2 x = 0
\] |
[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli] |
✓ |
2.727 |
|
|
\[
{}\left (3 x y^{2}-x^{2}\right ) y^{\prime }+y^{3}-2 x y = 0
\] |
[[_homogeneous, ‘class G‘], _exact, _rational] |
✓ |
2.329 |
|
|
\[
{}6 x y^{2} y^{\prime }+2 y^{3}+x = 0
\] |
[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli] |
✓ |
2.791 |
|
|
\[
{}\left (6 x y^{2}+x^{2}\right ) y^{\prime }-y \left (3 y^{2}-x \right ) = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
2.270 |
|
|
\[
{}\left (x^{2} y^{2}+x \right ) y^{\prime }+y = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
4.011 |
|
|
\[
{}\left (x y-1\right )^{2} x y^{\prime }+\left (x^{2} y^{2}+1\right ) y = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
2.136 |
|
|
\[
{}\left (10 x^{3} y^{2}+x^{2} y+2 x \right ) y^{\prime }+5 x^{2} y^{3}+x y^{2} = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
3.234 |
|
|
\[
{}\left (y^{3}-3 x \right ) y^{\prime }-3 y+x^{2} = 0
\] |
[_exact, _rational] |
✓ |
1.379 |
|
|
\[
{}\left (y^{3}-x^{3}\right ) y^{\prime }-x^{2} y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
11.832 |
|
|
\[
{}\left (y^{2}+x^{2}+a \right ) y y^{\prime }+\left (y^{2}+x^{2}-a \right ) x = 0
\] |
[_exact, _rational] |
✓ |
1.756 |
|
|
\[
{}2 y^{3} y^{\prime }+x y^{2} = 0
\] |
[_separable] |
✓ |
3.471 |
|
|
\[
{}\left (2 y^{3}+y\right ) y^{\prime }-2 x^{3}-x = 0
\] |
[_separable] |
✓ |
2.371 |
|
|
\[
{}\left (2 y^{3}+5 x^{2} y\right ) y^{\prime }+5 x y^{2}+x^{3} = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
98.271 |
|
|
\[
{}\left (20 y^{3}-3 x y^{2}+6 x^{2} y+3 x^{3}\right ) y^{\prime }-y^{3}+6 x y^{2}+9 x^{2} y+4 x^{3} = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
117.218 |
|
|
\[
{}\left (\frac {y^{2}}{b}+\frac {x^{2}}{a}\right ) \left (y y^{\prime }+x \right )+\frac {\left (a -b \right ) \left (y y^{\prime }-x \right )}{a +b} = 0
\] |
[_rational] |
✗ |
4.029 |
|
|
\[
{}\left (2 a y^{3}+3 a x y^{2}-b \,x^{3}+c \,x^{2}\right ) y^{\prime }-a y^{3}+c y^{2}+3 b \,x^{2} y+2 b \,x^{3} = 0
\] |
[_rational] |
✗ |
3.423 |
|
|
\[
{}x y^{3} y^{\prime }+y^{4}-x \sin \left (x \right ) = 0
\] |
[_Bernoulli] |
✓ |
34.710 |
|
|
\[
{}\left (2 x y^{3}-x^{4}\right ) y^{\prime }-y^{4}+2 x^{3} y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
6.966 |
|
|
\[
{}\left (2 x y^{3}+y\right ) y^{\prime }+2 y^{2} = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
1.149 |
|
|
\[
{}\left (2 x y^{3}+x y+x^{2}\right ) y^{\prime }+y^{2}-x y = 0
\] |
[_rational] |
✓ |
1.655 |
|
|
\[
{}\left (3 x y^{3}-4 x y+y\right ) y^{\prime }+y^{2} \left (y^{2}-2\right ) = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
1.719 |
|
|
\[
{}\left (7 x y^{3}+y-5 x \right ) y^{\prime }+y^{4}-5 y = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
2.182 |
|
|
\[
{}\left (x^{2} y^{3}+x y\right ) y^{\prime }-1 = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
1.534 |
|
|
\[
{}\left (2 x^{2} y^{3}+x^{2} y^{2}-2 x \right ) y^{\prime }-2 y-1 = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
2.693 |
|
|
\[
{}\left (10 x^{2} y^{3}-3 y^{2}-2\right ) y^{\prime }+5 x y^{4}+x = 0
\] |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
1.615 |
|
|
\[
{}\left (a x y^{3}+c \right ) x y^{\prime }+\left (b \,x^{3} y+c \right ) y = 0
\] |
[_rational] |
✓ |
1.694 |
|
|
\[
{}\left (2 x^{3} y^{3}-x \right ) y^{\prime }+2 x^{3} y^{3}-y = 0
\] |
[_rational] |
✓ |
1.495 |
|
|
\[
{}y \left (y^{3}-2 x^{3}\right ) y^{\prime }+\left (2 y^{3}-x^{3}\right ) x = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
84.412 |
|
|
\[
{}y \left (\left (b x +a y\right )^{3}+b \,x^{3}\right ) y^{\prime }+x \left (\left (b x +a y\right )^{3}+a y^{3}\right ) = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
51.580 |
|
|
\[
{}\left (x y^{4}+2 x^{2} y^{3}+2 y+x \right ) y^{\prime }+y^{5}+y = 0
\] |
[_rational] |
✓ |
99.188 |
|
|
\[
{}a \,x^{2} y^{n} y^{\prime }-2 y^{\prime } x +y = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
1.395 |
|
|
\[
{}y^{m} x^{n} \left (a x y^{\prime }+b y\right )+\alpha x y^{\prime }+\beta y = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
2.274 |
|
|
\[
{}\left (f \left (x +y\right )+1\right ) y^{\prime }+f \left (x +y\right ) = 0
\] |
[[_homogeneous, ‘class C‘], _exact, _dAlembert] |
✓ |
1.871 |
|
|
\[
{}\frac {y^{\prime } f_{\nu }\left (x \right ) \left (-y+y^{p +1}\right )}{-1+y}-\frac {g_{\nu }\left (x \right ) \left (-y+y^{q +1}\right )}{-1+y} = 0
\] |
[_separable] |
✓ |
3.416 |
|
|
\[
{}\left (\sqrt {x y}-1\right ) x y^{\prime }-\left (\sqrt {x y}+1\right ) y = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
4.552 |
|
|
\[
{}\left (2 x^{{5}/{2}} y^{{3}/{2}}+x^{2} y-x \right ) y^{\prime }-x^{{3}/{2}} y^{{5}/{2}}+x y^{2}-y = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
48.888 |
|
|
\[
{}\left (\sqrt {x +y}+1\right ) y^{\prime }+1 = 0
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
1.497 |
|
|
\[
{}\sqrt {y^{2}-1}\, y^{\prime }-\sqrt {x^{2}-1} = 0
\] |
[_separable] |
✓ |
1.846 |
|
|
\[
{}\left (\sqrt {1+y^{2}}+a x \right ) y^{\prime }+\sqrt {x^{2}+1}+a y = 0
\] |
[_exact] |
✓ |
34.040 |
|
|
\[
{}\left (\sqrt {x^{2}+y^{2}}+x \right ) y^{\prime }-y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
5.637 |
|
|
\[
{}\left (y \sqrt {x^{2}+y^{2}}+\left (y^{2}-x^{2}\right ) \sin \left (\alpha \right )-2 x y \cos \left (\alpha \right )\right ) y^{\prime }+x \sqrt {x^{2}+y^{2}}+2 x y \sin \left (\alpha \right )+\left (y^{2}-x^{2}\right ) \cos \left (\alpha \right ) = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
165.826 |
|
|
\[
{}\left (x \sqrt {1+x^{2}+y^{2}}-y \left (x^{2}+y^{2}\right )\right ) y^{\prime }-y \sqrt {1+x^{2}+y^{2}}-x \left (x^{2}+y^{2}\right ) = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
2.686 |
|
|
\[
{}\left (\frac {\operatorname {e1} \left (x +a \right )}{\left (\left (x +a \right )^{2}+y^{2}\right )^{{3}/{2}}}+\frac {\operatorname {e2} \left (x -a \right )}{\left (\left (x -a \right )^{2}+y^{2}\right )^{{3}/{2}}}\right ) y^{\prime }-y \left (\frac {\operatorname {e1}}{\left (\left (x +a \right )^{2}+y^{2}\right )^{{3}/{2}}}+\frac {\operatorname {e2}}{\left (\left (x -a \right )^{2}+y^{2}\right )^{{3}/{2}}}\right ) = 0
\] |
unknown |
✓ |
218.575 |
|
|
\[
{}\left (x \,{\mathrm e}^{y}+{\mathrm e}^{x}\right ) y^{\prime }+{\mathrm e}^{y}+y \,{\mathrm e}^{x} = 0
\] |
[_exact] |
✓ |
1.563 |
|
|
\[
{}x \left (3 \,{\mathrm e}^{x y}+2 \,{\mathrm e}^{-x y}\right ) \left (y^{\prime } x +y\right )+1 = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
280.029 |
|
|
\[
{}\left (\ln \left (y\right )+x \right ) y^{\prime }-1 = 0
\] |
[[_1st_order, _with_exponential_symmetries]] |
✓ |
1.184 |
|
|
\[
{}\left (\ln \left (y\right )+2 x -1\right ) y^{\prime }-2 y = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
1.532 |
|
|
\[
{}x \left (2 x^{2} y \ln \left (y\right )+1\right ) y^{\prime }-2 y = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
1.487 |
|
|
\[
{}x \left (y \ln \left (x y\right )+y-a x \right ) y^{\prime }-y \left (a x \ln \left (x y\right )-y+a x \right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
1.698 |
|
|
\[
{}y^{\prime } \left (1+\sin \left (x \right )\right ) \sin \left (y\right )+\cos \left (x \right ) \left (\cos \left (y\right )-1\right ) = 0
\] |
[_separable] |
✓ |
4.668 |
|
|
\[
{}\left (x \cos \left (y\right )+\sin \left (x \right )\right ) y^{\prime }+y \cos \left (x \right )+\sin \left (y\right ) = 0
\] |
[_exact] |
✓ |
8.838 |
|
|
\[
{}x y^{\prime } \cot \left (\frac {y}{x}\right )+2 x \sin \left (\frac {y}{x}\right )-y \cot \left (\frac {y}{x}\right ) = 0
\] |
[[_homogeneous, ‘class A‘]] |
✓ |
4.388 |
|
|
\[
{}y^{\prime } \cos \left (y\right )-\cos \left (x \right ) \sin \left (y\right )^{2}-\sin \left (y\right ) = 0
\] |
unknown |
✗ |
44.154 |
|
|
\[
{}y^{\prime } \cos \left (y\right )+x \sin \left (y\right ) \cos \left (y\right )^{2}-\sin \left (y\right )^{3} = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
50.513 |
|
|
\[
{}y^{\prime } \left (\cos \left (y\right )-\sin \left (\alpha \right ) \sin \left (x \right )\right ) \cos \left (y\right )+\left (\cos \left (x \right )-\sin \left (\alpha \right ) \sin \left (y\right )\right ) \cos \left (x \right ) = 0
\] |
unknown |
✓ |
39.815 |
|
|
\[
{}x y^{\prime } \cos \left (y\right )+\sin \left (y\right ) = 0
\] |
[_separable] |
✓ |
3.949 |
|
|
\[
{}\left (x \sin \left (y\right )-1\right ) y^{\prime }+\cos \left (y\right ) = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
4.596 |
|
|
\[
{}\left (x \cos \left (y\right )+\cos \left (x \right )\right ) y^{\prime }-y \sin \left (x \right )+\sin \left (y\right ) = 0
\] |
[_exact] |
✓ |
29.114 |
|
|
\[
{}\left (x^{2} \cos \left (y\right )+2 y \sin \left (x \right )\right ) y^{\prime }+2 x \sin \left (y\right )+y^{2} \cos \left (x \right ) = 0
\] |
[_exact] |
✓ |
37.512 |
|
|
\[
{}x y^{\prime } \ln \left (x \right ) \sin \left (y\right )+\cos \left (y\right ) \left (1-x \cos \left (y\right )\right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
48.530 |
|
|
\[
{}y^{\prime } \sin \left (y\right ) \cos \left (x \right )+\cos \left (y\right ) \sin \left (x \right ) = 0
\] |
[_separable] |
✓ |
2.720 |
|
|
\[
{}3 y^{\prime } \sin \left (x \right ) \sin \left (y\right )+5 \cos \left (x \right )^{4} y = 0
\] |
[_separable] |
✓ |
4.853 |
|
|
\[
{}y^{\prime } \cos \left (a y\right )-b \left (1-c \cos \left (a y\right )\right ) \sqrt {\cos \left (a y\right )^{2}-1+c \cos \left (a y\right )} = 0
\] |
[_quadrature] |
✓ |
87.668 |
|
|
\[
{}\left (x \sin \left (x y\right )+\cos \left (x +y\right )-\sin \left (y\right )\right ) y^{\prime }+y \sin \left (x y\right )+\cos \left (x +y\right )+\cos \left (x \right ) = 0
\] |
[_exact] |
✓ |
39.335 |
|
|
\[
{}\left (x^{2} y \sin \left (x y\right )-4 x \right ) y^{\prime }+x y^{2} \sin \left (x y\right )-y = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
38.639 |
|
|
\[
{}\left (-y+y^{\prime } x \right ) \cos \left (\frac {y}{x}\right )^{2}+x = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
7.938 |
|
|
\[
{}\left (y \sin \left (\frac {y}{x}\right )-x \cos \left (\frac {y}{x}\right )\right ) x y^{\prime }-\left (x \cos \left (\frac {y}{x}\right )+y \sin \left (\frac {y}{x}\right )\right ) y = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
7.187 |
|
|
\[
{}\left (y f \left (x^{2}+y^{2}\right )-x \right ) y^{\prime }+y+x f \left (x^{2}+y^{2}\right ) = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
2.430 |
|
|
\[
{}f \left (x^{2}+a y^{2}\right ) \left (a y y^{\prime }+x \right )-y-y^{\prime } x = 0
\] |
[_exact] |
✗ |
8.521 |
|
|
\[
{}f \left (x^{c} y\right ) \left (b x y^{\prime }-a \right )-x^{a} y^{b} \left (y^{\prime } x +c y\right ) = 0
\] |
[NONE] |
✗ |
4.441 |
|
|
\[
{}{y^{\prime }}^{2}+a y+b \,x^{2} = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
2.329 |
|
|
\[
{}{y^{\prime }}^{2}+y^{2}-a^{2} = 0
\] |
[_quadrature] |
✓ |
0.681 |
|
|
\[
{}{y^{\prime }}^{2}+y^{2}-f \left (x \right )^{2} = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
3.610 |
|
|
\[
{}{y^{\prime }}^{2}-y^{3}+y^{2} = 0
\] |
[_quadrature] |
✓ |
3.812 |
|
|
\[
{}{y^{\prime }}^{2}-4 y^{3}+a y+b = 0
\] |
[_quadrature] |
✓ |
2.016 |
|
|
\[
{}{y^{\prime }}^{2}+a^{2} y^{2} \left (\ln \left (y\right )^{2}-1\right ) = 0
\] |
[_quadrature] |
✓ |
3.300 |
|
|
\[
{}{y^{\prime }}^{2}-2 y^{\prime }-y^{2} = 0
\] |
[_quadrature] |
✓ |
0.372 |
|
|
\[
{}{y^{\prime }}^{2}+a y^{\prime }+b x = 0
\] |
[_quadrature] |
✓ |
0.234 |
|
|
\[
{}{y^{\prime }}^{2}+a y^{\prime }+b y = 0
\] |
[_quadrature] |
✓ |
0.806 |
|
|
\[
{}{y^{\prime }}^{2}+\left (x -2\right ) y^{\prime }-y+1 = 0
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.484 |
|
|
\[
{}{y^{\prime }}^{2}+\left (x +a \right ) y^{\prime }-y = 0
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.481 |
|
|
\[
{}{y^{\prime }}^{2}-\left (x +1\right ) y^{\prime }+y = 0
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.448 |
|
|
\[
{}{y^{\prime }}^{2}+2 y^{\prime } x -y = 0
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
0.437 |
|
|
\[
{}{y^{\prime }}^{2}-2 y^{\prime } x +y = 0
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
0.424 |
|
|
\[
{}{y^{\prime }}^{2}+a x y^{\prime }-b \,x^{2}-c = 0
\] |
[_quadrature] |
✓ |
0.500 |
|
|
\[
{}{y^{\prime }}^{2}+a x y^{\prime }+b y+c \,x^{2} = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
3.720 |
|
|
\[
{}{y^{\prime }}^{2}+\left (a x +b \right ) y^{\prime }-a y+c = 0
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.537 |
|
|
\[
{}{y^{\prime }}^{2}-2 x^{2} y^{\prime }+2 x y = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
2.474 |
|
|
\[
{}{y^{\prime }}^{2}+a \,x^{3} y^{\prime }-2 a \,x^{2} y = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
3.270 |
|
|
\[
{}{y^{\prime }}^{2}+\left (y^{\prime }-y\right ) {\mathrm e}^{x} = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
3.753 |
|