|
# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}y^{\prime \prime \prime }+8 y^{\prime \prime }+16 y^{\prime } = 0 \] |
higher_order_laplace |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.469 |
|
|
\[ {}y^{\prime \prime \prime }+6 y^{\prime \prime }+13 y^{\prime } = 0 \] |
higher_order_laplace |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.538 |
|
|
\[ {}y^{\prime \prime \prime }-6 y^{\prime \prime }+13 y^{\prime } = 0 \] |
higher_order_laplace |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.54 |
|
|
\[ {}y^{\prime \prime \prime }+4 y^{\prime \prime }+29 y^{\prime } = 0 \] |
higher_order_laplace |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.529 |
|
|
\[ {}y^{\prime \prime \prime }+6 y^{\prime \prime }+25 y^{\prime } = 0 \] |
higher_order_laplace |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.541 |
|
|
\[ {}y^{\prime \prime \prime }-6 y^{\prime \prime }+10 y^{\prime } = 0 \] |
higher_order_laplace |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.52 |
|
|
\[ {}y^{\prime \prime \prime \prime }+13 y^{\prime \prime }+36 y = 0 \] |
higher_order_laplace |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.692 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+3 y = 9 t \] |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.537 |
|
|
\[ {}4 y^{\prime \prime }+16 y^{\prime }+17 y = 17 t -1 \] |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.532 |
|
|
\[ {}4 y^{\prime \prime }+5 y^{\prime }+4 y = 3 \,{\mathrm e}^{-t} \] |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.654 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = t^{2} {\mathrm e}^{2 t} \] |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.447 |
|
|
\[ {}y^{\prime \prime }+9 y = {\mathrm e}^{-2 t} \] |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.475 |
|
|
\[ {}2 y^{\prime \prime }-3 y^{\prime }+17 y = 17 t -1 \] |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.635 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = {\mathrm e}^{-t} \] |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.431 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+5 y = 2+t \] |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.51 |
|
|
\[ {}2 y^{\prime }+y = {\mathrm e}^{-\frac {t}{2}} \] |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.454 |
|
|
\[ {}y^{\prime \prime }+8 y^{\prime }+20 y = \sin \left (2 t \right ) \] |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.599 |
|
|
\[ {}4 y^{\prime \prime }-4 y^{\prime }+y = t^{2} \] |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.453 |
|
|
\[ {}2 y^{\prime \prime }+y^{\prime }-y = 4 \sin \left (t \right ) \] |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.592 |
|
|
\[ {}y^{\prime }-y = {\mathrm e}^{2 t} \] |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.408 |
|
|
\[ {}3 y^{\prime \prime }+5 y^{\prime }-2 y = 7 \,{\mathrm e}^{-2 t} \] |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.447 |
|
|
\[ {}y^{\prime }+y = \operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -2\right ) \] |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.832 |
|
|
\[ {}y^{\prime }-2 y = 4 t \left (\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -2\right )\right ) \] |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.905 |
|
|
\[ {}y^{\prime \prime }+9 y = 24 \sin \left (t \right ) \left (\operatorname {Heaviside}\left (t \right )+\operatorname {Heaviside}\left (t -\pi \right )\right ) \] |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.84 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = \operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (-1+t \right ) \] |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.198 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+2 y = 5 \cos \left (t \right ) \left (\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )\right ) \] |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.081 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime }+6 y = 36 t \left (\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (-1+t \right )\right ) \] |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.909 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+13 y = 39 \operatorname {Heaviside}\left (t \right )-507 \left (t -2\right ) \operatorname {Heaviside}\left (t -2\right ) \] |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.249 |
|
|
\[ {}y^{\prime \prime }+4 y = 3 \operatorname {Heaviside}\left (t \right )-3 \operatorname {Heaviside}\left (t -4\right )+\left (2 t -5\right ) \operatorname {Heaviside}\left (t -4\right ) \] |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.873 |
|
|
\[ {}4 y^{\prime \prime }+4 y^{\prime }+5 y = 25 t \left (\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )\right ) \] |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.298 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+3 y = \operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (-1+t \right )+\operatorname {Heaviside}\left (t -2\right )-\operatorname {Heaviside}\left (t -3\right ) \] |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.49 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime } = \left \{\begin {array}{cc} 4 & 0\le t <1 \\ 6 & 1\le t \end {array}\right . \] |
second_order_laplace |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.482 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = \left \{\begin {array}{cc} 0 & 0\le t <1 \\ 1 & 1\le t <2 \\ -1 & 2\le t \end {array}\right . \] |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.882 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = \left \{\begin {array}{cc} 1 & 0\le t <2 \\ -1 & 2\le t \end {array}\right . \] |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.442 |
|
|
\[ {}y^{\prime \prime }+y = \left \{\begin {array}{cc} t & 0\le t <\pi \\ -t & \pi \le t \end {array}\right . \] |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.574 |
|
|
\[ {}y^{\prime \prime }+4 y = \left \{\begin {array}{cc} 8 t & 0\le t <\frac {\pi }{2} \\ 8 \pi & \frac {\pi }{2}\le t \end {array}\right . \] |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.757 |
|
|
\[ {}y^{\prime \prime }+4 \pi ^{2} y = 3 \delta \left (t -\frac {1}{3}\right )-\delta \left (-1+t \right ) \] |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.138 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+2 y = 3 \delta \left (-1+t \right ) \] |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.684 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+29 y = 5 \delta \left (t -\pi \right )-5 \delta \left (t -2 \pi \right ) \] |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.166 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = 1-\delta \left (-1+t \right ) \] |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.707 |
|
|
\[ {}4 y^{\prime \prime }+4 y^{\prime }+y = {\mathrm e}^{-\frac {t}{2}} \delta \left (-1+t \right ) \] |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.546 |
|
|
\[ {}y^{\prime \prime }-7 y^{\prime }+6 y = \delta \left (-1+t \right ) \] |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.575 |
|
|
\[ {}10 Q^{\prime }+100 Q = \operatorname {Heaviside}\left (-1+t \right )-\operatorname {Heaviside}\left (t -2\right ) \] |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.082 |
|
|
\[ {}y^{\prime \prime \prime }+y^{\prime \prime }+4 y^{\prime }+4 y = 8 \] |
higher_order_laplace |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
1.167 |
|
|
\[ {}y^{\prime \prime \prime }-2 y^{\prime \prime }-y^{\prime }+2 y = 4 t \] |
higher_order_laplace |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.648 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime \prime }+4 y^{\prime }-4 y = 8 \,{\mathrm e}^{2 t}-5 \,{\mathrm e}^{t} \] |
higher_order_laplace |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.227 |
|
|
\[ {}y^{\prime \prime \prime }-5 y^{\prime \prime }+y^{\prime }-y = -t^{2}+2 t -10 \] |
higher_order_laplace |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
73.314 |
|
|
\[ {}y^{\prime \prime \prime \prime }-5 y^{\prime \prime }+4 y = 12 \operatorname {Heaviside}\left (t \right )-12 \operatorname {Heaviside}\left (-1+t \right ) \] |
higher_order_laplace |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.646 |
|
|
\[ {}y^{\prime \prime \prime \prime }-16 y = 32 \operatorname {Heaviside}\left (t \right )-32 \operatorname {Heaviside}\left (t -\pi \right ) \] |
higher_order_laplace |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.746 |
|
|
\[ {}t^{2} y^{\prime \prime }+3 t y^{\prime }+y = t^{7} \] |
kovacic, second_order_euler_ode, exact linear second order ode, second_order_integrable_as_is, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.401 |
|
|
\[ {}t^{2} y^{\prime \prime }-6 t y^{\prime }+\sin \left (2 t \right ) y = \ln \left (t \right ) \] |
unknown |
[[_2nd_order, _linear, _nonhomogeneous]] |
❇ |
N/A |
3.346 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+\frac {y}{t} = t \] |
unknown |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
0.377 |
|
|
\[ {}y^{\prime \prime }+t y^{\prime }-y \ln \left (t \right ) = \cos \left (2 t \right ) \] |
unknown |
[[_2nd_order, _linear, _nonhomogeneous]] |
❇ |
N/A |
0.88 |
|
|
\[ {}t^{3} y^{\prime \prime }-2 t y^{\prime }+y = t^{4} \] |
unknown |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
0.51 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 1 \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.401 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+5 y = {\mathrm e}^{t} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.525 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-7 y = 4 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.491 |
|
|
\[ {}y^{\prime \prime \prime }+3 y^{\prime \prime }+3 y^{\prime }+y = 5 \] |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.205 |
|
|
\[ {}3 y^{\prime \prime }+5 y^{\prime }-2 y = 3 t^{2} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.449 |
|
|
\[ {}y^{\prime \prime \prime } = 2 y^{\prime \prime }-4 y^{\prime }+\sin \left (t \right ) \] |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
1.964 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x-2 y \\ y^{\prime }=3 x-4 y \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.272 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=\frac {5 x}{4}+\frac {3 y}{4} \\ y^{\prime }=\frac {x}{2}-\frac {3 y}{2} \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.433 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }-x+2 y=0 \\ y^{\prime }+y-x=0 \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.327 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }+5 x-2 y=0 \\ y^{\prime }+2 x-y=0 \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.43 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }-3 x+2 y=0 \\ y^{\prime }-x+3 y=0 \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.405 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }+x-z=0 \\ x+y^{\prime }-y=0 \\ z^{\prime }+x+2 y-3 z=0 \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.345 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-\frac {x}{2}+2 y-3 z \\ y^{\prime }=y-\frac {z}{2} \\ z^{\prime }=-2 x+z \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.865 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }+y^{\prime }=y \\ x^{\prime }-y^{\prime }=x \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.333 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }+2 y^{\prime }=t \\ x^{\prime }-y^{\prime }=x+y \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.685 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }-y^{\prime }=x+y-t \\ 2 x^{\prime }+3 y^{\prime }=2 x+6 \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.708 |
|
|
\[ {}\left [\begin {array}{c} 2 x^{\prime }-y^{\prime }=t \\ 3 x^{\prime }+2 y^{\prime }=y \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.636 |
|
|
\[ {}\left [\begin {array}{c} 5 x^{\prime }-3 y^{\prime }=x+y \\ 3 x^{\prime }-y^{\prime }=t \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.704 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }-4 y^{\prime }=0 \\ 2 x^{\prime }-3 y^{\prime }=t +y \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.625 |
|
|
\[ {}\left [\begin {array}{c} 3 x^{\prime }+2 y^{\prime }=\sin \left (t \right ) \\ x^{\prime }-2 y^{\prime }=x+y+t \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.893 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-4 x+9 y+12 \,{\mathrm e}^{-t} \\ y^{\prime }=-5 x+2 y \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.824 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-7 x+6 y+6 \,{\mathrm e}^{-t} \\ y^{\prime }=-12 x+5 y+37 \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
1.575 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-7 x+10 y+18 \,{\mathrm e}^{t} \\ y^{\prime }=-10 x+9 y+37 \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
2.152 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-14 x+39 y+78 \sinh \left (t \right ) \\ y^{\prime }=-6 x+16 y+6 \cosh \left (t \right ) \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
2.599 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=2 x+4 y-2 z-2 \sinh \left (t \right ) \\ y^{\prime }=4 x+2 y-2 z+10 \cosh \left (t \right ) \\ z^{\prime }=-x+3 y+z+5 \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
10.472 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=2 x+6 y-2 z+50 \,{\mathrm e}^{t} \\ y^{\prime }=6 x+2 y-2 z+21 \,{\mathrm e}^{-t} \\ z^{\prime }=-x+6 y+z+9 \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
1.638 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-2 x-2 y+4 z \\ y^{\prime }=-2 x+y+2 z \\ z^{\prime }=-4 x-2 y+6 z+{\mathrm e}^{2 t} \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
1.344 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=3 x-2 y+3 z \\ y^{\prime }=x-y+2 z+2 \,{\mathrm e}^{-t} \\ z^{\prime }=-2 x+2 y-2 z \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.945 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=7 x+y-1-6 \,{\mathrm e}^{t} \\ y^{\prime }=-4 x+3 y+4 \,{\mathrm e}^{t}-3 \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.606 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=3 x-2 y+24 \sin \left (t \right ) \\ y^{\prime }=9 x-3 y+12 \cos \left (t \right ) \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
1.343 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=7 x-4 y+10 \,{\mathrm e}^{t} \\ y^{\prime }=3 x+14 y+6 \,{\mathrm e}^{2 t} \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.618 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-7 x+4 y+6 \,{\mathrm e}^{3 t} \\ y^{\prime }=-5 x+2 y+6 \,{\mathrm e}^{2 t} \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.687 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-3 x-3 y+z \\ y^{\prime }=2 y+2 z+29 \,{\mathrm e}^{-t} \\ z^{\prime }=5 x+y+z+39 \,{\mathrm e}^{t} \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
31.909 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=2 x+y-z+5 \sin \left (t \right ) \\ y^{\prime }=y+z-10 \cos \left (t \right ) \\ z^{\prime }=x+z+2 \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
1.967 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-3 x+3 y+z+5 \sin \left (2 t \right ) \\ y^{\prime }=x-5 y-3 z+5 \cos \left (2 t \right ) \\ z^{\prime }=-3 x+7 y+3 z+23 \,{\mathrm e}^{t} \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
2.759 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-3 x+y-3 z+2 \,{\mathrm e}^{t} \\ y^{\prime }=4 x-y+2 z+4 \,{\mathrm e}^{t} \\ z^{\prime }=4 x-2 y+3 z+4 \,{\mathrm e}^{t} \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
1.853 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x+5 y+10 \sinh \left (t \right ) \\ y^{\prime }=19 x-13 y+24 \sinh \left (t \right ) \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
29.035 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=9 x-3 y-6 t \\ y^{\prime }=-x+11 y+10 t \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.793 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.411 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+x y = 0 \] |
reduction_of_order |
[_Lienard] |
✓ |
✓ |
0.493 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = x^{\frac {3}{2}} {\mathrm e}^{x} \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.948 |
|
|
\[ {}y^{\prime \prime }+4 y = 2 \sec \left (2 x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.069 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{x}+\left (1-\frac {1}{4 x^{2}}\right ) y = x \] |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.148 |
|
|
\[ {}y^{\prime \prime }+y = f \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.295 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x -\frac {1}{2}\right ) y^{\prime }+\frac {y}{2} = 0 \] |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.02 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-y = 0 \] |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.618 |
|
|
|
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