Solves \((y')^{\frac {n}{m}}= f(x) g(y) \) Number of problems in this table is 34
Column notations: A is ODE degree. B is Program Number of solutions generated. C is CAS Number of solutions generated.
|
# |
ODE |
A |
B |
C |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}{y^{\prime }}^{2}+f \left (x \right ) \left (y-a \right ) \left (y-b \right ) = 0 \] |
2 |
2 |
2 |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
1.375 |
|
|
\[ {}{y^{\prime }}^{2}+f \left (x \right ) \left (y-a \right )^{2} \left (y-b \right ) = 0 \] |
2 |
2 |
2 |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
0.869 |
|
|
\[ {}{y^{\prime }}^{2}+f \left (x \right ) \left (y-a \right ) \left (y-b \right ) \left (y-c \right ) = 0 \] |
2 |
2 |
2 |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
1.198 |
|
|
\[ {}{y^{\prime }}^{2}+f \left (x \right ) \left (y-a \right )^{2} \left (y-b \right ) \left (y-c \right ) = 0 \] |
2 |
2 |
2 |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
4.577 |
|
|
\[ {}{y^{\prime }}^{2} = f \left (x \right )^{2} \left (y-a \right ) \left (y-b \right ) \left (y-c \right )^{2} \] |
2 |
2 |
2 |
[_separable] |
✓ |
✓ |
5.11 |
|
|
\[ {}x {y^{\prime }}^{2} = y \] |
2 |
3 |
3 |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.632 |
|
|
\[ {}\left (1+x \right ) {y^{\prime }}^{2} = y \] |
2 |
3 |
3 |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
✓ |
0.651 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+y^{2}-y^{4} = 0 \] |
2 |
2 |
5 |
[_separable] |
✓ |
✓ |
0.937 |
|
|
\[ {}\left (-x^{2}+1\right ) {y^{\prime }}^{2} = 1-y^{2} \] |
2 |
2 |
4 |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
1.449 |
|
|
\[ {}3 x^{4} {y^{\prime }}^{2}-x y-y = 0 \] |
2 |
2 |
5 |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
0.737 |
|
|
\[ {}{y^{\prime }}^{3} = \left (a +b y+c y^{2}\right ) f \left (x \right ) \] |
3 |
3 |
3 |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
1.733 |
|
|
\[ {}{y^{\prime }}^{3}+f \left (x \right ) \left (y-a \right )^{2} \left (y-b \right )^{2} = 0 \] |
3 |
3 |
3 |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
2.263 |
|
|
\[ {}{y^{\prime }}^{3}+f \left (x \right ) \left (y-a \right )^{2} \left (y-b \right )^{2} \left (y-c \right )^{2} = 0 \] |
3 |
3 |
3 |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
188.713 |
|
|
\[ {}{y^{\prime }}^{4}+f \left (x \right ) \left (y-a \right )^{3} \left (y-b \right )^{2} = 0 \] |
4 |
4 |
4 |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
4.665 |
|
|
\[ {}{y^{\prime }}^{4}+f \left (x \right ) \left (y-a \right )^{3} \left (y-b \right )^{3} = 0 \] |
4 |
4 |
4 |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
12.732 |
|
|
\[ {}{y^{\prime }}^{4}+f \left (x \right ) \left (y-a \right )^{3} \left (y-b \right )^{3} \left (y-c \right )^{2} = 0 \] |
4 |
4 |
1 |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
668.064 |
|
|
\[ {}{y^{\prime }}^{6}+f \left (x \right ) \left (y-a \right )^{4} \left (y-b \right )^{3} = 0 \] |
6 |
6 |
1 |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
99.567 |
|
|
\[ {}{y^{\prime }}^{6}+f \left (x \right ) \left (y-a \right )^{5} \left (y-b \right )^{3} = 0 \] |
6 |
6 |
1 |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
213.078 |
|
|
\[ {}{y^{\prime }}^{6}+f \left (x \right ) \left (y-a \right )^{5} \left (y-b \right )^{4} = 0 \] |
6 |
6 |
1 |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
778.25 |
|
|
\[ {}y = x {y^{\prime }}^{2} \] |
2 |
3 |
3 |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.881 |
|
|
\[ {}{y^{\prime }}^{2} = \frac {y}{x} \] |
2 |
3 |
3 |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.82 |
|
|
\[ {}{y^{\prime }}^{2} = \frac {y^{2}}{x} \] |
2 |
2 |
3 |
[_separable] |
✓ |
✓ |
2.366 |
|
|
\[ {}{y^{\prime }}^{2} = \frac {y^{3}}{x} \] |
2 |
2 |
2 |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
1.619 |
|
|
\[ {}{y^{\prime }}^{3} = \frac {y^{2}}{x} \] |
3 |
3 |
10 |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.214 |
|
|
\[ {}x {y^{\prime }}^{2}-y = 0 \] |
2 |
3 |
3 |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.808 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+y^{2}-y^{4} = 0 \] |
2 |
2 |
5 |
[_separable] |
✓ |
✓ |
2.795 |
|
|
\[ {}\left (x^{2}-1\right ) {y^{\prime }}^{2}-y^{2}+1 = 0 \] |
2 |
2 |
4 |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
2.639 |
|
|
\[ {}{y^{\prime }}^{3}-f \left (x \right ) \left (a y^{2}+b y+c \right )^{2} = 0 \] |
3 |
3 |
3 |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
2.878 |
|
|
\[ {}{y^{\prime }}^{n}-f \left (x \right )^{n} \left (y-a \right )^{n +1} \left (y-b \right )^{n -1} = 0 \] |
0 |
1 |
1 |
[_separable] |
✓ |
✓ |
40.158 |
|
|
\[ {}{y^{\prime }}^{n}-f \left (x \right ) g \left (y\right ) = 0 \] |
0 |
1 |
1 |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
0.72 |
|
|
\[ {}y = \left (1+x \right ) {y^{\prime }}^{2} \] |
2 |
3 |
3 |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
✓ |
1.473 |
|
|
\[ {}\sinh \left (x \right ) {y^{\prime }}^{2}+3 y = 0 \] |
2 |
2 |
7 |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
0.704 |
|
|
\[ {}{y^{\prime }}^{2}-9 x y = 0 \] |
2 |
2 |
5 |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
0.653 |
|
|
\[ {}{y^{\prime }}^{3}+\left (2+x \right ) {\mathrm e}^{y} = 0 \] |
3 |
3 |
3 |
[[_1st_order, _with_exponential_symmetries]] |
✓ |
✓ |
3.904 |
|
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