First order ode \(y'=f(x,y)\) solved using Taylor series method (not power series) which applies only when \(f(x,y)\) is found to be analytic at the expansion point.
Added May 13, 2023.
Number of problems in this table is 83
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) |
|
\[ {}y^{\prime } = y \] |
1 |
2 |
1 |
[_quadrature] |
✓ |
✓ |
0.459 |
|
|
\[ {}y^{\prime } = 4 y \] |
1 |
2 |
1 |
[_quadrature] |
✓ |
✓ |
0.543 |
|
|
\[ {}2 y^{\prime }+3 y = 0 \] |
1 |
2 |
1 |
[_quadrature] |
✓ |
✓ |
0.548 |
|
|
\[ {}2 x y+y^{\prime } = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.53 |
|
|
\[ {}y^{\prime } = x^{2} y \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.488 |
|
|
\[ {}\left (-2+x \right ) y^{\prime }+y = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.584 |
|
|
\[ {}\left (2 x -1\right ) y^{\prime }+2 y = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.601 |
|
|
\[ {}2 \left (1+x \right ) y^{\prime } = y \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.608 |
|
|
\[ {}\left (-1+x \right ) y^{\prime }+2 y = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.592 |
|
|
\[ {}2 \left (-1+x \right ) y^{\prime } = 3 y \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.584 |
|
|
\[ {}y^{\prime }-y = 0 \] |
1 |
2 |
1 |
[_quadrature] |
✓ |
✓ |
0.5 |
|
|
\[ {}y^{\prime }-x y = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.516 |
|
|
\[ {}\left (1-x \right ) y^{\prime } = y \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.556 |
|
|
\[ {}y^{\prime } = \sqrt {1-y} \] |
1 |
1 |
1 |
[_quadrature] |
✓ |
✓ |
1.026 |
|
|
\[ {}y^{\prime } = x y-x^{2} \] |
1 |
2 |
1 |
[_linear] |
✓ |
✓ |
2.225 |
|
|
\[ {}y^{\prime } = x^{2} y^{2} \] |
1 |
1 |
1 |
[_separable] |
✓ |
✓ |
0.561 |
|
|
\[ {}y^{\prime } = 3 x +\frac {y}{x} \] |
1 |
1 |
1 |
[_linear] |
✓ |
✓ |
1.091 |
|
|
\[ {}y^{\prime } = \ln \left (x y\right ) \] |
1 |
1 |
1 |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
1.115 |
|
|
\[ {}y^{\prime } = 1+y^{2} \] |
1 |
1 |
1 |
[_quadrature] |
✓ |
✓ |
1.259 |
|
|
\[ {}y^{\prime } = x^{2}+y^{2} \] |
1 |
1 |
1 |
[[_Riccati, _special]] |
✓ |
✓ |
1.905 |
|
|
\[ {}y^{\prime } = \sqrt {1+x y} \] |
1 |
1 |
1 |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
1.374 |
|
|
\[ {}y^{\prime } = \cos \left (x \right )+\sin \left (y\right ) \] |
1 |
1 |
1 |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
2.331 |
|
|
\[ {}y^{\prime } = 3 x^{2} y \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.328 |
|
|
\[ {}y^{\prime }+\left (2+x \right ) y = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.401 |
|
|
\[ {}y^{\prime }-y = 0 \] |
1 |
2 |
1 |
[_quadrature] |
✓ |
✓ |
0.368 |
|
|
\[ {}z^{\prime }-x^{2} z = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.358 |
|
|
\[ {}y^{\prime }+2 \left (-1+x \right ) y = 0 \] |
1 |
1 |
1 |
[_separable] |
✓ |
✓ |
0.348 |
|
|
\[ {}y^{\prime }-2 x y = 0 \] |
1 |
1 |
1 |
[_separable] |
✓ |
✓ |
0.415 |
|
|
\[ {}x^{\prime }+\sin \left (t \right ) x = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
2.027 |
|
|
\[ {}y^{\prime }-{\mathrm e}^{x} y = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
1.951 |
|
|
\[ {}y^{\prime }-x y = \sin \left (x \right ) \] |
1 |
2 |
1 |
[_linear] |
✓ |
✓ |
0.528 |
|
|
\[ {}w^{\prime }+x w = {\mathrm e}^{x} \] |
1 |
2 |
1 |
[_linear] |
✓ |
✓ |
0.481 |
|
|
\[ {}\left (1-x \right ) y^{\prime } = x^{2}-y \] |
1 |
2 |
1 |
[_linear] |
✓ |
✓ |
0.369 |
|
|
\[ {}x y^{\prime } = 1-x +2 y \] |
1 |
1 |
1 |
[_linear] |
✓ |
✓ |
0.432 |
|
|
\[ {}y^{\prime } = 2 x^{2}+3 y \] |
1 |
2 |
1 |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.52 |
|
|
\[ {}\left (1+x \right ) y^{\prime } = x^{2}-2 x +y \] |
1 |
2 |
1 |
[_linear] |
✓ |
✓ |
0.385 |
|
|
\[ {}y^{\prime }+x y = \cos \left (x \right ) \] |
1 |
2 |
1 |
[_linear] |
✓ |
✓ |
0.548 |
|
|
\[ {}y^{\prime }-x y = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.402 |
|
|
\[ {}\left (1+x \right ) y^{\prime } = y \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.339 |
|
|
\[ {}y^{\prime } = -2 x y \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.392 |
|
|
\[ {}y^{\prime }+4 y = 1 \] |
1 |
2 |
1 |
[_quadrature] |
✓ |
✓ |
1.404 |
|
|
\[ {}\left (-2+x \right ) y^{\prime } = x y \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
1.513 |
|
|
\[ {}y^{\prime } = 2 x y \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
1.259 |
|
|
\[ {}y^{\prime }+y = 1 \] |
1 |
2 |
1 |
[_quadrature] |
✓ |
✓ |
1.225 |
|
|
\[ {}y^{\prime }-y = 2 \] |
1 |
2 |
1 |
[_quadrature] |
✓ |
✓ |
1.184 |
|
|
\[ {}y^{\prime }+y = 0 \] |
1 |
2 |
1 |
[_quadrature] |
✓ |
✓ |
1.085 |
|
|
\[ {}y^{\prime }-y = 0 \] |
1 |
2 |
1 |
[_quadrature] |
✓ |
✓ |
0.718 |
|
|
\[ {}y^{\prime }-y = x^{2} \] |
1 |
2 |
1 |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.91 |
|
|
\[ {}y^{\prime } = \frac {1}{\sqrt {-x^{2}+1}} \] |
1 |
2 |
1 |
[_quadrature] |
✓ |
✓ |
1.063 |
|
|
\[ {}y^{\prime } = y+1 \] |
1 |
2 |
1 |
[_quadrature] |
✓ |
✓ |
0.765 |
|
|
\[ {}y^{\prime } = x -y \] |
1 |
2 |
1 |
[[_linear, ‘class A‘]] |
✓ |
✓ |
7.867 |
|
|
\[ {}y^{\prime } = -x +y^{2} \] |
1 |
1 |
1 |
[[_Riccati, _special]] |
✓ |
✓ |
4.211 |
|
|
\[ {}y^{\prime }-2 y = x^{2} \] |
1 |
1 |
1 |
[[_linear, ‘class A‘]] |
✓ |
✓ |
3.437 |
|
|
\[ {}y^{\prime } = y+x \,{\mathrm e}^{y} \] |
1 |
1 |
1 |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
3.623 |
|
|
\[ {}y^{\prime }-2 y = 0 \] |
1 |
2 |
1 |
[_quadrature] |
✓ |
✓ |
0.687 |
|
|
\[ {}y^{\prime }-2 x y = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.564 |
|
|
\[ {}y^{\prime }+\frac {2 y}{2 x -1} = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.599 |
|
|
\[ {}\left (x -3\right ) y^{\prime }-2 y = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.537 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime }-2 x y = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.451 |
|
|
\[ {}y^{\prime }+\frac {y}{-1+x} = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.584 |
|
|
\[ {}y^{\prime }+\frac {y}{-1+x} = 0 \] |
1 |
1 |
1 |
[_separable] |
✓ |
✓ |
0.532 |
|
|
\[ {}\left (1-x \right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
[_separable] |
✓ |
✓ |
0.664 |
|
|
\[ {}\left (-x^{3}+2\right ) y^{\prime }-3 x^{2} y = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.634 |
|
|
\[ {}\left (-x^{3}+2\right ) y^{\prime }+3 x^{2} y = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.55 |
|
|
\[ {}\left (1+x \right ) y^{\prime }-x y = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.624 |
|
|
\[ {}\left (1+x \right ) y^{\prime }+\left (1-x \right ) y = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.684 |
|
|
\[ {}y^{\prime }+\cos \left (y\right ) = 0 \] |
1 |
1 |
1 |
[_quadrature] |
✓ |
✓ |
0.87 |
|
|
\[ {}y^{\prime }-{\mathrm e}^{x} y = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.733 |
|
|
\[ {}y^{\prime }-\tan \left (x \right ) y = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.992 |
|
|
\[ {}y^{\prime }-{\mathrm e}^{x} y = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.621 |
|
|
\[ {}y^{\prime }+{\mathrm e}^{2 x} y = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.686 |
|
|
\[ {}y^{\prime }+y \cos \left (x \right ) = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.691 |
|
|
\[ {}y^{\prime }+\ln \left (x \right ) y = 0 \] |
1 |
1 |
1 |
[_separable] |
✓ |
✓ |
1.01 |
|
|
\[ {}y^{\prime }-{\mathrm e}^{x} y = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.969 |
|
|
\[ {}y^{\prime }+y \sqrt {x^{2}+1} = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
1.208 |
|
|
\[ {}\cos \left (x \right ) y^{\prime }+y = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
1.845 |
|
|
\[ {}y^{\prime }+\sqrt {2 x^{2}+1}\, y = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
1.302 |
|
|
\[ {}y^{\prime } = 1-x y \] |
1 |
2 |
1 |
[_linear] |
✓ |
✓ |
1.467 |
|
|
\[ {}y^{\prime } = \frac {y-x}{x +y} \] |
1 |
1 |
1 |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.197 |
|
|
\[ {}y^{\prime } = \sin \left (x \right ) y \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
1.372 |
|
|
\[ {}y^{\prime }-2 x y = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
1.181 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{y}+x y \] |
1 |
1 |
1 |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
1.368 |
|
|
\[ {}\left (1+x \right ) y^{\prime }-n y = 0 \] |
1 |
2 |
1 |
[_separable] |
✓ |
✓ |
0.543 |
|
|
|
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