Problem 12001
| ODE |
\[ \boxed {x^{\prime }+\left (a +\frac {1}{t}\right ) x=b} \] With initial conditions \begin {align*} [x \left (1\right ) = x_{0}] \end {align*}
|
|
program solution |
\[ x = \frac {{\mathrm e}^{-t a} b \,{\mathrm e}^{t a} t a +{\mathrm e}^{-t a} {\mathrm e}^{a} a^{2} x_{0} -{\mathrm e}^{-t a} {\mathrm e}^{a} a b -{\mathrm e}^{-t a} b \,{\mathrm e}^{t a}+{\mathrm e}^{-t a} {\mathrm e}^{a} b}{t \,a^{2}} \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = \frac {\left (x_{0} a^{2}-a b +b \right ) {\mathrm e}^{-a \left (t -1\right )}+b \left (a t -1\right )}{t \,a^{2}} \] |
Problem 12002
| ODE |
\[ \boxed {T^{\prime }+k \left (T-\mu -a \cos \left (\omega \left (t -\phi \right )\right )\right )=0} \] |
|
program solution |
\[ T = -\frac {{\mathrm e}^{-t k} \left (a \omega \sin \left (\omega \left (-t +\phi \right )\right ) k \,{\mathrm e}^{t k}-\cos \left (\omega \left (-t +\phi \right )\right ) {\mathrm e}^{t k} a \,k^{2}-{\mathrm e}^{t k} k^{2} \mu -{\mathrm e}^{t k} \mu \,\omega ^{2}-c_{1} k^{2}-c_{1} \omega ^{2}\right )}{k^{2}+\omega ^{2}} \] Verified OK.
|
|
Maple solution | \[ T \left (t \right ) = \frac {\cos \left (\omega \left (-t +\phi \right )\right ) a \,k^{2}-\sin \left (\omega \left (-t +\phi \right )\right ) a k \omega +\left (k^{2}+\omega ^{2}\right ) \left ({\mathrm e}^{-k t} c_{1} +\mu \right )}{k^{2}+\omega ^{2}} \] |
Problem 12003
| ODE |
\[ \boxed {2 x y+\left (x^{2}+2 y\right ) y^{\prime }=\sec \left (x \right )^{2}} \] |
|
program solution |
\[ -\tan \left (x \right )+x^{2} y+y^{2} = c_{1} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (x \right ) &= -\frac {x^{2}}{2}-\frac {\sqrt {x^{4}+4 \tan \left (x \right )-4 c_{1}}}{2} \\ y \left (x \right ) &= -\frac {x^{2}}{2}+\frac {\sqrt {x^{4}+4 \tan \left (x \right )-4 c_{1}}}{2} \\ \end{align*} |
Problem 12004
| ODE |
\[ \boxed {y \,{\mathrm e}^{x}+y x \,{\mathrm e}^{x}+\left (x \,{\mathrm e}^{x}+2\right ) y^{\prime }=-1} \] |
|
program solution |
\[ y = \frac {-x +c_{1}}{x \,{\mathrm e}^{x}+2} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {c_{1} -x}{{\mathrm e}^{x} x +2} \] |
Problem 12005
| ODE |
\[ \boxed {\left (\cos \left (y\right ) x +\cos \left (x \right )\right ) y^{\prime }+\sin \left (y\right )-\sin \left (x \right ) y=0} \] |
|
program solution |
\[ y \cos \left (x \right )+\sin \left (y\right ) x = c_{1} \] Verified OK.
|
|
Maple solution | \[ \cos \left (x \right ) y \left (x \right )+x \sin \left (y \left (x \right )\right )+c_{1} = 0 \] |
Problem 12006
| ODE |
\[ \boxed {{\mathrm e}^{x} \sin \left (y\right )+y+\left ({\mathrm e}^{x} \cos \left (y\right )+x +{\mathrm e}^{y}\right ) y^{\prime }=0} \] |
|
program solution |
\[ x y+{\mathrm e}^{x} \sin \left (y\right )+{\mathrm e}^{y} = c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) x +{\mathrm e}^{x} \sin \left (y \left (x \right )\right )+{\mathrm e}^{y \left (x \right )}+c_{1} = 0 \] |
Problem 12007
| ODE |
\[ \boxed {{\mathrm e}^{-y} \sec \left (x \right )-{\mathrm e}^{-y} y^{\prime }=-2 \cos \left (x \right )} \] |
|
program solution |
\[ y = -\ln \left (-\frac {4 \sin \left (\frac {x}{2}\right ) \cos \left (\frac {x}{2}\right )^{2}-4 \cos \left (\frac {x}{2}\right )^{3}+c_{1} \sin \left (\frac {x}{2}\right )-2 \sin \left (\frac {x}{2}\right ) x -c_{1} \cos \left (\frac {x}{2}\right )+2 x \cos \left (\frac {x}{2}\right )}{\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )}\right ) \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \ln \left (-\frac {\left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^{2}}{\left (-4 \cos \left (\frac {x}{2}\right )^{2}+c_{1} +2 x \right ) \left (2 \cos \left (\frac {x}{2}\right )^{2}-1\right )}\right ) \] |
Problem 12008
| ODE |
\[ \boxed {2 y y^{\prime }=-V^{\prime }\left (x \right )} \] |
|
program solution |
\[ -V \left (x \right )-y^{2} = c_{1} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (x \right ) &= \sqrt {-V \left (x \right )+c_{1}} \\ y \left (x \right ) &= -\sqrt {-V \left (x \right )+c_{1}} \\ \end{align*} |
Problem 12009
| ODE |
\[ \boxed {\left (\frac {1}{y}-a \right ) y^{\prime }=-\frac {2}{x}+b} \] |
|
program solution |
\[ y = -\frac {\operatorname {LambertW}\left (-\frac {a \,{\mathrm e}^{b x +c_{1}}}{x^{2}}\right )}{a} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = -\frac {\operatorname {LambertW}\left (-\frac {a \,{\mathrm e}^{b x} c_{1}}{x^{2}}\right )}{a} \] |
Problem 12010
| ODE |
\[ \boxed {x y+y^{2}-y^{\prime } x^{2}=-x^{2}} \] |
|
program solution |
\[ y = \frac {\left (-\cos \left (\ln \left (x \right )\right ) c_{3} +\sin \left (\ln \left (x \right )\right )\right ) x}{\sin \left (\ln \left (x \right )\right ) c_{3} +\cos \left (\ln \left (x \right )\right )} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \tan \left (\ln \left (x \right )+c_{1} \right ) x \] |
Problem 12011
| ODE |
\[ \boxed {x^{\prime }-\frac {x^{2}+t \sqrt {x^{2}+t^{2}}}{x t}=0} \] |
|
program solution |
\[ -\frac {\sqrt {x^{2}+t^{2}}}{t} = -\ln \left (t \right )+c_{1} \] Verified OK.
|
|
Maple solution | \[ \frac {t \ln \left (t \right )-c_{1} t -\sqrt {t^{2}+x \left (t \right )^{2}}}{t} = 0 \] |
Problem 12012
| ODE |
\[ \boxed {x^{\prime }-k x+x^{2}=0} \] |
|
program solution |
\[ x = \frac {k c_{1} {\mathrm e}^{t k}}{-1+c_{1} {\mathrm e}^{t k}} \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = \frac {k}{1+{\mathrm e}^{-k t} c_{1} k} \] |
Problem 12013
| ODE |
\[ \boxed {x^{\prime \prime }-3 x^{\prime }+2 x=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 2, x^{\prime }\left (0\right ) = 6] \end {align*}
|
|
program solution |
\[ x = 4 \,{\mathrm e}^{2 t}-2 \,{\mathrm e}^{t} \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = -2 \,{\mathrm e}^{t}+4 \,{\mathrm e}^{2 t} \] |
Problem 12014
| ODE |
\[ \boxed {y^{\prime \prime }-4 y^{\prime }+4 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 3] \end {align*}
|
|
program solution |
\[ y = 3 \,{\mathrm e}^{2 x} x \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = 3 x \,{\mathrm e}^{2 x} \] |
Problem 12015
| ODE |
\[ \boxed {z^{\prime \prime }-4 z^{\prime }+13 z=0} \] With initial conditions \begin {align*} [z \left (0\right ) = 7, z^{\prime }\left (0\right ) = 42] \end {align*}
|
|
program solution |
\[ z = \frac {7 \,{\mathrm e}^{2 t} \left (3 \cos \left (3 t \right )+4 \sin \left (3 t \right )\right )}{3} \] Verified OK.
|
|
Maple solution | \[ z \left (t \right ) = \frac {7 \,{\mathrm e}^{2 t} \left (4 \sin \left (3 t \right )+3 \cos \left (3 t \right )\right )}{3} \] |
Problem 12016
| ODE |
\[ \boxed {y^{\prime \prime }+y^{\prime }-6 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = 8] \end {align*}
|
|
program solution |
\[ y = {\mathrm e}^{2 t}-2 \,{\mathrm e}^{-3 t} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \left ({\mathrm e}^{5 t}-2\right ) {\mathrm e}^{-3 t} \] |
Problem 12017
| ODE |
\[ \boxed {y^{\prime \prime }-4 y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 13, y^{\prime }\left (0\right ) = 0] \end {align*}
|
|
program solution |
\[ y = 13 \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = 13 \] |
Problem 12018
| ODE |
\[ \boxed {\theta ^{\prime \prime }+4 \theta =0} \] With initial conditions \begin {align*} [\theta \left (0\right ) = 0, \theta ^{\prime }\left (0\right ) = 10] \end {align*}
|
|
program solution |
\[ \theta = 5 \sin \left (2 t \right ) \] Verified OK.
|
|
Maple solution | \[ \theta \left (t \right ) = 5 \sin \left (2 t \right ) \] |
Problem 12019
| ODE |
\[ \boxed {y^{\prime \prime }+2 y^{\prime }+10 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = 0] \end {align*}
|
|
program solution |
\[ y = {\mathrm e}^{-t} \left (3 \cos \left (3 t \right )+\sin \left (3 t \right )\right ) \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = {\mathrm e}^{-t} \left (\sin \left (3 t \right )+3 \cos \left (3 t \right )\right ) \] |
Problem 12020
| ODE |
\[ \boxed {2 z^{\prime \prime }+7 z^{\prime }-4 z=0} \] With initial conditions \begin {align*} [z \left (0\right ) = 0, z^{\prime }\left (0\right ) = 9] \end {align*}
|
|
program solution |
\[ z = 2 \left ({\mathrm e}^{\frac {9 t}{2}}-1\right ) {\mathrm e}^{-4 t} \] Verified OK.
|
|
Maple solution | \[ z \left (t \right ) = 2 \left ({\mathrm e}^{\frac {9 t}{2}}-1\right ) {\mathrm e}^{-4 t} \] |
Problem 12021
| ODE |
\[ \boxed {y^{\prime \prime }+2 y^{\prime }+y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = -1] \end {align*}
|
|
program solution |
\[ y = -t \,{\mathrm e}^{-t} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = -t \,{\mathrm e}^{-t} \] |
Problem 12022
| ODE |
\[ \boxed {x^{\prime \prime }+6 x^{\prime }+10 x=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 3, x^{\prime }\left (0\right ) = 1] \end {align*}
|
|
program solution |
\[ x = {\mathrm e}^{-3 t} \left (3 \cos \left (t \right )+10 \sin \left (t \right )\right ) \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = {\mathrm e}^{-3 t} \left (10 \sin \left (t \right )+3 \cos \left (t \right )\right ) \] |
Problem 12023
| ODE |
\[ \boxed {4 x^{\prime \prime }-20 x^{\prime }+21 x=0} \] With initial conditions \begin {align*} [x \left (0\right ) = -4, x^{\prime }\left (0\right ) = -12] \end {align*}
|
|
program solution |
\[ x = -3 \,{\mathrm e}^{\frac {7 t}{2}}-{\mathrm e}^{\frac {3 t}{2}} \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = -{\mathrm e}^{\frac {3 t}{2}}-3 \,{\mathrm e}^{\frac {7 t}{2}} \] |
Problem 12024
| ODE |
\[ \boxed {y^{\prime \prime }+y^{\prime }-2 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 4, y^{\prime }\left (0\right ) = -4] \end {align*}
|
|
program solution |
\[ y = \frac {4 \,{\mathrm e}^{t}}{3}+\frac {8 \,{\mathrm e}^{-2 t}}{3} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {4 \left ({\mathrm e}^{3 t}+2\right ) {\mathrm e}^{-2 t}}{3} \] |
Problem 12025
| ODE |
\[ \boxed {y^{\prime \prime }-4 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 10, y^{\prime }\left (0\right ) = 0] \end {align*}
|
|
program solution |
\[ y = 5 \,{\mathrm e}^{2 t}+5 \,{\mathrm e}^{-2 t} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = 5 \,{\mathrm e}^{2 t}+5 \,{\mathrm e}^{-2 t} \] |
Problem 12026
| ODE |
\[ \boxed {y^{\prime \prime }+4 y^{\prime }+4 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 27, y^{\prime }\left (0\right ) = -54] \end {align*}
|
|
program solution |
\[ y = 27 \,{\mathrm e}^{-2 t} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = 27 \,{\mathrm e}^{-2 t} \] |
Problem 12027
| ODE |
\[ \boxed {y^{\prime \prime }+\omega ^{2} y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
|
|
program solution |
\[ y = -\frac {\sqrt {-\omega ^{2}}\, \left (-{\mathrm e}^{-\sqrt {-\omega ^{2}}\, t}+{\mathrm e}^{\sqrt {-\omega ^{2}}\, t}\right )}{2 \omega ^{2}} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {\sin \left (\omega t \right )}{\omega } \] |
Problem 12028
| ODE |
\[ \boxed {x^{\prime \prime }-4 x=t^{2}} \] |
|
program solution |
\[ x = c_{1} {\mathrm e}^{-2 t}+\frac {c_{2} {\mathrm e}^{2 t}}{4}-\frac {t^{2}}{4}-\frac {1}{8} \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = c_{2} {\mathrm e}^{2 t}+{\mathrm e}^{-2 t} c_{1} -\frac {t^{2}}{4}-\frac {1}{8} \] |
Problem 12029
| ODE |
\[ \boxed {x^{\prime \prime }-4 x^{\prime }=t^{2}} \] |
|
program solution |
\[ x = -\frac {t^{3}}{12}-\frac {t^{2}}{16}-\frac {c_{1}}{4}-\frac {t}{32}-\frac {1}{128}+c_{2} {\mathrm e}^{4 t} \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = -\frac {t^{2}}{16}-\frac {t^{3}}{12}+\frac {c_{1} {\mathrm e}^{4 t}}{4}-\frac {t}{32}+c_{2} \] |
Problem 12030
| ODE |
\[ \boxed {x^{\prime \prime }+x^{\prime }-2 x=3 \,{\mathrm e}^{-t}} \] |
|
program solution |
\[ x = c_{1} {\mathrm e}^{-2 t}+\frac {c_{2} {\mathrm e}^{t}}{3}-\frac {3 \,{\mathrm e}^{-t}}{2} \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = -\frac {\left (-2 c_{2} {\mathrm e}^{3 t}+3 \,{\mathrm e}^{t}-2 c_{1} \right ) {\mathrm e}^{-2 t}}{2} \] |
Problem 12031
| ODE |
\[ \boxed {x^{\prime \prime }+x^{\prime }-2 x={\mathrm e}^{t}} \] |
|
program solution |
\[ x = c_{1} {\mathrm e}^{-2 t}+\frac {c_{2} {\mathrm e}^{t}}{3}+\frac {{\mathrm e}^{t} \left (-1+3 t \right )}{9} \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = \frac {{\mathrm e}^{-2 t} \left (\left (t +3 c_{2} \right ) {\mathrm e}^{3 t}+3 c_{1} \right )}{3} \] |
Problem 12032
| ODE |
\[ \boxed {x^{\prime \prime }+2 x^{\prime }+x={\mathrm e}^{-t}} \] |
|
program solution |
\[ x = {\mathrm e}^{-t} \left (c_{2} t +c_{1} \right )+\frac {t^{2} {\mathrm e}^{-t}}{2} \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = {\mathrm e}^{-t} \left (c_{2} +c_{1} t +\frac {1}{2} t^{2}\right ) \] |
Problem 12033
| ODE |
\[ \boxed {x^{\prime \prime }+\omega ^{2} x=\sin \left (\alpha t \right )} \] |
|
program solution |
\[ x = c_{1} {\mathrm e}^{\sqrt {-\omega ^{2}}\, t}+\frac {c_{2} \sqrt {-\omega ^{2}}\, {\mathrm e}^{-\sqrt {-\omega ^{2}}\, t}}{2 \omega ^{2}}-\frac {\sin \left (\alpha t \right )}{\alpha ^{2}-\omega ^{2}} \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = \sin \left (\omega t \right ) c_{2} +\cos \left (\omega t \right ) c_{1} +\frac {\sin \left (\alpha t \right )}{-\alpha ^{2}+\omega ^{2}} \] |
Problem 12034
| ODE |
\[ \boxed {x^{\prime \prime }+\omega ^{2} x=\sin \left (\omega t \right )} \] |
|
program solution |
\[ x = c_{1} {\mathrm e}^{\sqrt {-\omega ^{2}}\, t}+\frac {c_{2} \sqrt {-\omega ^{2}}\, {\mathrm e}^{-\sqrt {-\omega ^{2}}\, t}}{2 \omega ^{2}}-\frac {t \cos \left (\omega t \right )}{2 \omega } \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = \frac {\sin \left (\omega t \right ) \left (2 c_{2} \omega ^{2}+1\right )-\omega \cos \left (\omega t \right ) \left (-2 c_{1} \omega +t \right )}{2 \omega ^{2}} \] |
Problem 12035
| ODE |
\[ \boxed {x^{\prime \prime }+2 x^{\prime }+10 x={\mathrm e}^{-t}} \] |
|
program solution |
\[ x = c_{1} {\mathrm e}^{-t} \cos \left (3 t \right )+\frac {c_{2} {\mathrm e}^{-t} \sin \left (3 t \right )}{3}+\frac {{\mathrm e}^{-t}}{9} \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = \frac {{\mathrm e}^{-t} \left (9 c_{2} \sin \left (3 t \right )+9 c_{1} \cos \left (3 t \right )+1\right )}{9} \] |
Problem 12036
| ODE |
\[ \boxed {x^{\prime \prime }+2 x^{\prime }+10 x={\mathrm e}^{-t} \cos \left (3 t \right )} \] |
|
program solution |
\[ x = c_{1} {\mathrm e}^{-t} \cos \left (3 t \right )+\frac {c_{2} {\mathrm e}^{-t} \sin \left (3 t \right )}{3}+\frac {t \,{\mathrm e}^{-t} \sin \left (3 t \right )}{6} \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = \frac {\left (\left (6 c_{1} +\frac {1}{3}\right ) \cos \left (3 t \right )+\sin \left (3 t \right ) \left (t +6 c_{2} \right )\right ) {\mathrm e}^{-t}}{6} \] |
Problem 12037
| ODE |
\[ \boxed {x^{\prime \prime }+6 x^{\prime }+10 x=\cos \left (t \right ) {\mathrm e}^{-2 t}} \] |
|
program solution |
\[ x = {\mathrm e}^{-3 t} \left (c_{1} \cos \left (t \right )+c_{2} \sin \left (t \right )\right )+\frac {\cos \left (t \right ) {\mathrm e}^{-2 t}}{5}+\frac {2 \sin \left (t \right ) {\mathrm e}^{-2 t}}{5} \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = \left (\sin \left (t \right ) c_{2} +\cos \left (t \right ) c_{1} \right ) {\mathrm e}^{-3 t}+\frac {{\mathrm e}^{-2 t} \left (\cos \left (t \right )+2 \sin \left (t \right )\right )}{5} \] |
Problem 12038
| ODE |
\[ \boxed {x^{\prime \prime }+4 x^{\prime }+4 x={\mathrm e}^{2 t}} \] |
|
program solution |
\[ x = {\mathrm e}^{-2 t} \left (c_{2} t +c_{1} \right )+\frac {{\mathrm e}^{2 t}}{16} \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = \left (c_{1} t +c_{2} \right ) {\mathrm e}^{-2 t}+\frac {{\mathrm e}^{2 t}}{16} \] |
Problem 12039
| ODE |
\[ \boxed {x^{\prime \prime }+x^{\prime }-2 x=12 \,{\mathrm e}^{-t}-6 \,{\mathrm e}^{t}} \] |
|
program solution |
\[ x = c_{1} {\mathrm e}^{-2 t}+\frac {c_{2} {\mathrm e}^{t}}{3}-6 \,{\mathrm e}^{-t}+\frac {2 \,{\mathrm e}^{t}}{3}-2 t \,{\mathrm e}^{t} \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = -2 \left (\left (t -\frac {c_{2}}{2}-\frac {1}{3}\right ) {\mathrm e}^{3 t}-\frac {c_{1}}{2}+3 \,{\mathrm e}^{t}\right ) {\mathrm e}^{-2 t} \] |
Problem 12040
| ODE |
\[ \boxed {x^{\prime \prime }+4 x=289 t \,{\mathrm e}^{t} \sin \left (2 t \right )} \] |
|
program solution |
\[ x = c_{1} \cos \left (2 t \right )+\frac {c_{2} \sin \left (2 t \right )}{2}+76 \,{\mathrm e}^{t} \cos \left (2 t \right )-2 \,{\mathrm e}^{t} \sin \left (2 t \right )-68 t \,{\mathrm e}^{t} \cos \left (2 t \right )+17 t \,{\mathrm e}^{t} \sin \left (2 t \right ) \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = \left (\left (-68 t +76\right ) {\mathrm e}^{t}+c_{1} \right ) \cos \left (2 t \right )+17 \sin \left (2 t \right ) \left ({\mathrm e}^{t} \left (t -\frac {2}{17}\right )+\frac {c_{2}}{17}\right ) \] |
Problem 12041
| ODE |
\[ \boxed {x^{\prime \prime }+\omega ^{2} x=\cos \left (\alpha t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
|
|
program solution |
\[ x = \frac {-2 \cos \left (\alpha t \right )+{\mathrm e}^{\sqrt {-\omega ^{2}}\, t}+{\mathrm e}^{-\sqrt {-\omega ^{2}}\, t}}{2 \alpha ^{2}-2 \omega ^{2}} \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = \frac {\cos \left (\omega t \right )-\cos \left (\alpha t \right )}{\alpha ^{2}-\omega ^{2}} \] |
Problem 12042
| ODE |
\[ \boxed {x^{\prime \prime }+\omega ^{2} x=\cos \left (\omega t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
|
|
program solution |
\[ x = \frac {2 \sin \left (\omega t \right ) \omega ^{2} t +\omega \cos \left (\omega t \right )+\sin \left (\omega t \right ) \sqrt {-\omega ^{2}}-{\mathrm e}^{\sqrt {-\omega ^{2}}\, t} \omega }{4 \omega ^{3}} \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = \frac {\sin \left (\omega t \right ) t}{2 \omega } \] |
Problem 12043
| ODE |
\[ \boxed {x^{\prime \prime \prime }-6 x^{\prime \prime }+11 x^{\prime }-6 x={\mathrm e}^{-t}} \] |
|
program solution |
\[ x = {\mathrm e}^{t} c_{1} +c_{2} {\mathrm e}^{2 t}+{\mathrm e}^{3 t} c_{3} -\frac {{\mathrm e}^{-t}}{24} \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = -\frac {{\mathrm e}^{-t}}{24}+c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{3 t} \] |
Problem 12044
| ODE |
\[ \boxed {y^{\prime \prime \prime }-3 y^{\prime \prime }+2 y=\sin \left (x \right )} \] |
|
program solution |
\[ y = c_{1} {\mathrm e}^{x}+{\mathrm e}^{-\left (\sqrt {3}-1\right ) x} c_{2} +{\mathrm e}^{\left (1+\sqrt {3}\right ) x} c_{3} +\frac {\cos \left (x \right )}{26}+\frac {5 \sin \left (x \right )}{26} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = c_{3} {\mathrm e}^{-\left (\sqrt {3}-1\right ) x}+c_{1} {\mathrm e}^{x}+c_{2} {\mathrm e}^{\left (1+\sqrt {3}\right ) x}+\frac {5 \sin \left (x \right )}{26}+\frac {\cos \left (x \right )}{26} \] |
Problem 12045
| ODE |
\[ \boxed {x^{\prime \prime \prime \prime }-4 x^{\prime \prime \prime }+8 x^{\prime \prime }-8 x^{\prime }+4 x=\sin \left (t \right )} \] |
|
program solution |
\[ x = \left (c_{4} t +c_{3} \right ) {\mathrm e}^{\left (1-i\right ) t}+{\mathrm e}^{\left (1+i\right ) t} \left (c_{2} t +c_{1} \right )+\frac {4 \cos \left (t \right )}{25}-\frac {3 \sin \left (t \right )}{25} \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = \left (\left (c_{3} t +c_{1} \right ) \cos \left (t \right )+\sin \left (t \right ) \left (c_{4} t +c_{2} \right )\right ) {\mathrm e}^{t}+\frac {4 \cos \left (t \right )}{25}-\frac {3 \sin \left (t \right )}{25} \] |
Problem 12046
| ODE |
\[ \boxed {x^{\prime \prime \prime \prime }-5 x^{\prime \prime }+4 x={\mathrm e}^{t}} \] |
|
program solution |
\[ x = c_{1} {\mathrm e}^{-t}+c_{2} {\mathrm e}^{-2 t}+{\mathrm e}^{t} c_{3} +{\mathrm e}^{2 t} c_{4} -\frac {t \,{\mathrm e}^{t}}{6} \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = -\frac {{\mathrm e}^{-2 t} \left (\left (t -6 c_{1} \right ) {\mathrm e}^{3 t}-6 c_{3} {\mathrm e}^{t}-6 c_{4} {\mathrm e}^{4 t}-6 c_{2} \right )}{6} \] |
Problem 12047
| ODE |
\[ \boxed {t^{2} y^{\prime \prime }-\left (t^{2}+2 t \right ) y^{\prime }+\left (t +2\right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= t \end {align*}
|
|
program solution |
\[ y = c_{1} t +c_{2} t \,{\mathrm e}^{t} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = t \left (c_{1} +c_{2} {\mathrm e}^{t}\right ) \] |
Problem 12048
| ODE |
\[ \boxed {\left (x -1\right ) y^{\prime \prime }-y^{\prime } x +y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{x} \end {align*}
|
|
program solution |
\[ y = c_{1} {\mathrm e}^{x}-c_{2} {\mathrm e}^{x} x \,{\mathrm e}^{-x} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = c_{2} {\mathrm e}^{x}+c_{1} x \] |
Problem 12049
| ODE |
\[ \boxed {\left (\cos \left (t \right ) t -\sin \left (t \right )\right ) x^{\prime \prime }-x^{\prime } t \sin \left (t \right )-x \sin \left (t \right )=0} \] Given that one solution of the ode is \begin {align*} x_1 &= t \end {align*}
|
|
program solution |
\[ x = c_{1} t +c_{2} t \left (\int \frac {1}{t^{2} \left (\cos \left (t \right ) t -\sin \left (t \right )\right )}d t \right ) \] Verified OK.
|
|
Maple solution | \[ \text {No solution found} \] |
Problem 12050
| ODE |
\[ \boxed {\left (-t^{2}+t \right ) x^{\prime \prime }+\left (-t^{2}+2\right ) x^{\prime }+\left (-t +2\right ) x=0} \] Given that one solution of the ode is \begin {align*} x_1 &= {\mathrm e}^{-t} \end {align*}
|
|
program solution |
\[ x = c_{1} {\mathrm e}^{-t}+\frac {c_{2} {\mathrm e}^{-t} {\mathrm e}^{t}}{t} \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = \frac {c_{2} {\mathrm e}^{-t} t +c_{1}}{t} \] |
Problem 12051
| ODE |
\[ \boxed {y^{\prime \prime }-y^{\prime } x +y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= x \end {align*}
|
|
program solution |
\[ y = c_{1} x +c_{2} x \left (-\frac {{\mathrm e}^{\frac {x^{2}}{2}}}{x}-\frac {i \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, x}{2}\right )}{2}\right ) \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = c_{2} {\mathrm e}^{\frac {x^{2}}{2}}+\frac {\left (i c_{2} \sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (\frac {i \sqrt {2}\, x}{2}\right )+2 c_{1} \right ) x}{2} \] |
Problem 12052
| ODE |
\[ \boxed {\tan \left (t \right ) x^{\prime \prime }-3 x^{\prime }+\left (\tan \left (t \right )+3 \cot \left (t \right )\right ) x=0} \] Given that one solution of the ode is \begin {align*} x_1 &= \sin \left (t \right ) \end {align*}
|
|
program solution |
\[ x = c_{1} \sin \left (t \right )-c_{2} \sin \left (t \right ) \cos \left (t \right ) \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = \sin \left (t \right ) \left (c_{1} +c_{2} \cos \left (t \right )\right ) \] |
Problem 12053
| ODE |
\[ \boxed {y^{\prime \prime }-y^{\prime }-6 y={\mathrm e}^{x}} \] |
|
program solution |
\[ y = c_{1} {\mathrm e}^{-2 x}+\frac {{\mathrm e}^{3 x} c_{2}}{5}-\frac {{\mathrm e}^{x}}{6} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {\left (6 c_{1} {\mathrm e}^{5 x}-{\mathrm e}^{3 x}+6 c_{2} \right ) {\mathrm e}^{-2 x}}{6} \] |
Problem 12054
| ODE |
\[ \boxed {x^{\prime \prime }-x=\frac {1}{t}} \] |
|
program solution |
\[ x = c_{1} {\mathrm e}^{-t}+\frac {c_{2} {\mathrm e}^{t}}{2}-\frac {\operatorname {expIntegral}_{1}\left (t \right ) {\mathrm e}^{t}}{2}+\frac {\operatorname {expIntegral}_{1}\left (-t \right ) {\mathrm e}^{-t}}{2} \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = \frac {\operatorname {expIntegral}_{1}\left (-t \right ) {\mathrm e}^{-t}}{2}+c_{2} {\mathrm e}^{-t}+{\mathrm e}^{t} \left (c_{1} -\frac {\operatorname {expIntegral}_{1}\left (t \right )}{2}\right ) \] |
Problem 12055
| ODE |
\[ \boxed {y^{\prime \prime }+4 y=\cot \left (2 x \right )} \] |
|
program solution |
\[ y = c_{1} \cos \left (2 x \right )+\frac {c_{2} \sin \left (2 x \right )}{2}+\frac {\sin \left (2 x \right ) \ln \left (\csc \left (2 x \right )-\cot \left (2 x \right )\right )}{4} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = c_{2} \sin \left (2 x \right )+\cos \left (2 x \right ) c_{1} +\frac {\sin \left (2 x \right ) \ln \left (\csc \left (2 x \right )-\cot \left (2 x \right )\right )}{4} \] |
Problem 12056
| ODE |
\[ \boxed {t^{2} x^{\prime \prime }-2 x=t^{3}} \] |
|
program solution |
\[ x = t^{2} \left (\frac {t}{4}-\frac {c_{1}}{3 t^{3}}\right )+c_{2} t^{2} \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = c_{2} t^{2}+\frac {t^{3}}{4}+\frac {c_{1}}{t} \] |
Problem 12057
| ODE |
\[ \boxed {x^{\prime \prime }-4 x^{\prime }=\tan \left (t \right )} \] |
|
program solution |
\[ x = {\mathrm e}^{4 t} \left (-\left (\int \left (\ln \left (\cos \left (t \right )\right )-c_{1} \right ) {\mathrm e}^{-4 t}d t \right )+c_{2} \right ) \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = \int \left (\int \tan \left (t \right ) {\mathrm e}^{-4 t}d t +c_{1} \right ) {\mathrm e}^{4 t}d t +c_{2} \] |
Problem 12058
| ODE |
\[ \boxed {\left (\tan \left (x \right )^{2}-1\right ) y^{\prime \prime }-4 y^{\prime } \tan \left (x \right )^{3}+2 y \sec \left (x \right )^{4}=\left (\tan \left (x \right )^{2}-1\right ) \left (1-2 \sin \left (x \right )^{2}\right )} \] Given that one solution of the ode is \begin {align*} y_1 &= \sec \left (x \right )^{2} \end {align*}
|
|
program solution |
\[ y = \tan \left (x \right ) c_{2} +\sec \left (x \right )^{2} c_{1} -\frac {\cos \left (x \right )^{2}}{4}+\frac {x \tan \left (x \right )}{2}+\frac {1}{2} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {\left (4 c_{1} +2 x \right ) \tan \left (x \right )}{4}+\sec \left (x \right )^{2} c_{2} -\frac {\cos \left (x \right )^{2}}{4}+\frac {1}{2} \] |
Problem 12059
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }-4 y^{\prime } x +6 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 0, y^{\prime }\left (1\right ) = 1] \end {align*}
|
|
program solution |
\[ y = x^{2} \left (x -1\right ) \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = x^{2} \left (-1+x \right ) \] |
Problem 12060
| ODE |
\[ \boxed {4 x^{2} y^{\prime \prime }+y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1, y^{\prime }\left (1\right ) = 0] \end {align*}
|
|
program solution |
\[ y = \left (-\frac {\ln \left (x \right )}{2}+1\right ) \sqrt {x} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \sqrt {x}\, \left (1-\frac {\ln \left (x \right )}{2}\right ) \] |
Problem 12061
| ODE |
\[ \boxed {t^{2} x^{\prime \prime }-5 x^{\prime } t +10 x=0} \] With initial conditions \begin {align*} [x \left (1\right ) = 2, x^{\prime }\left (1\right ) = 1] \end {align*}
|
|
program solution |
\[ x = \left (1-\frac {5 i}{2}\right ) t^{3-i}+\left (1+\frac {5 i}{2}\right ) t^{3+i} \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = t^{3} \left (-5 \sin \left (\ln \left (t \right )\right )+2 \cos \left (\ln \left (t \right )\right )\right ) \] |
Problem 12062
| ODE |
\[ \boxed {t^{2} x^{\prime \prime }+x^{\prime } t -x=0} \] With initial conditions \begin {align*} [x \left (1\right ) = 1, x^{\prime }\left (1\right ) = 1] \end {align*}
|
|
program solution |
\[ x = t \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = t \] |
Problem 12063
| ODE |
\[ \boxed {x^{2} z^{\prime \prime }+3 x z^{\prime }+4 z=0} \] With initial conditions \begin {align*} [z \left (1\right ) = 0, z^{\prime }\left (1\right ) = 5] \end {align*}
|
|
program solution |
\[ z = \frac {5 i \sqrt {3}\, \left (x^{-i \sqrt {3}}-x^{i \sqrt {3}}\right )}{6 x} \] Verified OK.
|
|
Maple solution | \[ z \left (x \right ) = \frac {5 \sqrt {3}\, \sin \left (\sqrt {3}\, \ln \left (x \right )\right )}{3 x} \] |
Problem 12064
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x -3 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1, y^{\prime }\left (1\right ) = -1] \end {align*}
|
|
program solution |
\[ y = \frac {1}{x} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {1}{x} \] |
Problem 12065
| ODE |
\[ \boxed {4 t^{2} x^{\prime \prime }+8 x^{\prime } t +5 x=0} \] With initial conditions \begin {align*} [x \left (1\right ) = 2, x^{\prime }\left (1\right ) = 0] \end {align*}
|
|
program solution |
\[ x = \left (1+\frac {i}{2}\right ) t^{-\frac {1}{2}-i}+\left (1-\frac {i}{2}\right ) t^{-\frac {1}{2}+i} \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = \frac {\sin \left (\ln \left (t \right )\right )+2 \cos \left (\ln \left (t \right )\right )}{\sqrt {t}} \] |
Problem 12066
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }-5 y^{\prime } x +5 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = -2, y^{\prime }\left (1\right ) = 1] \end {align*}
|
|
program solution |
\[ y = \frac {3}{4} x^{5}-\frac {11}{4} x \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {3}{4} x^{5}-\frac {11}{4} x \] |
Problem 12067
| ODE |
\[ \boxed {3 x^{2} z^{\prime \prime }+5 x z^{\prime }-z=0} \] With initial conditions \begin {align*} [z \left (1\right ) = 2, z^{\prime }\left (1\right ) = -1] \end {align*}
|
|
program solution |
\[ z = \frac {3 x^{\frac {4}{3}}+5}{4 x} \] Verified OK.
|
|
Maple solution | \[ z \left (x \right ) = \frac {3 x^{\frac {4}{3}}+5}{4 x} \] |
Problem 12068
| ODE |
\[ \boxed {t^{2} x^{\prime \prime }+3 x^{\prime } t +13 x=0} \] With initial conditions \begin {align*} [x \left (1\right ) = -1, x^{\prime }\left (1\right ) = 2] \end {align*}
|
|
program solution |
\[ x = \frac {-i t^{2 i \sqrt {3}} \sqrt {3}+i \sqrt {3}\, t^{-2 i \sqrt {3}}-6 t^{2 i \sqrt {3}}-6 t^{-2 i \sqrt {3}}}{12 t} \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = \frac {\sqrt {3}\, \sin \left (2 \sqrt {3}\, \ln \left (t \right )\right )-6 \cos \left (2 \sqrt {3}\, \ln \left (t \right )\right )}{6 t} \] |
Problem 12069
| ODE |
\[ \boxed {a y^{\prime \prime }+\left (b -a \right ) y^{\prime }+c y=0} \] |
|
program solution |
\[ y = c_{1} {\mathrm e}^{\frac {z \left (\sqrt {\frac {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}{a^{2}}}\, a +a -b \right )}{2 a}}-\frac {c_{2} a^{2} {\mathrm e}^{-\frac {z \left (\sqrt {\frac {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}{a^{2}}}\, a -a +b \right )}{2 a}} \sqrt {\frac {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}{a^{2}}}}{a^{2}+\left (-2 b -4 c \right ) a +b^{2}} \] Verified OK.
|
|
Maple solution | \[ y \left (z \right ) = c_{1} {\mathrm e}^{\frac {\left (a -b +\sqrt {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}\right ) z}{2 a}}+{\mathrm e}^{-\frac {\left (-a +b +\sqrt {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}\right ) z}{2 a}} c_{2} \] |
Problem 12070
| ODE |
\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +n \left (n +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\). |
|
program solution |
\[ y = \left (1-\frac {1}{2} n^{2} x^{2}-\frac {1}{2} n \,x^{2}+\frac {1}{24} n^{4} x^{4}+\frac {1}{12} x^{4} n^{3}-\frac {5}{24} n^{2} x^{4}-\frac {1}{4} n \,x^{4}-\frac {1}{720} n^{6} x^{6}-\frac {1}{240} n^{5} x^{6}+\frac {23}{720} n^{4} x^{6}+\frac {17}{240} n^{3} x^{6}-\frac {47}{360} n^{2} x^{6}-\frac {1}{6} n \,x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{6} n^{2} x^{3}-\frac {1}{6} n \,x^{3}+\frac {1}{3} x^{3}+\frac {1}{120} x^{5} n^{4}+\frac {1}{60} x^{5} n^{3}-\frac {13}{120} x^{5} n^{2}-\frac {7}{60} x^{5} n +\frac {1}{5} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\left (-\frac {1}{2} n^{2}-\frac {1}{2} n \right ) x^{2}+\left (-\frac {5}{24} n^{2}-\frac {1}{4} n +\frac {1}{24} n^{4}+\frac {1}{12} n^{3}\right ) x^{4}\right ) c_{1} +\left (x +\left (-\frac {1}{6} n^{2}-\frac {1}{6} n +\frac {1}{3}\right ) x^{3}+\left (-\frac {13}{120} n^{2}-\frac {7}{60} n +\frac {1}{5}+\frac {1}{120} n^{4}+\frac {1}{60} n^{3}\right ) x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
|
|
Maple solution |
\[ y \left (x \right ) = \left (1-\frac {n \left (n +1\right ) x^{2}}{2}+\frac {n \left (n^{3}+2 n^{2}-5 n -6\right ) x^{4}}{24}\right ) y \left (0\right )+\left (x -\frac {\left (n^{2}+n -2\right ) x^{3}}{6}+\frac {\left (n^{4}+2 n^{3}-13 n^{2}-14 n +24\right ) x^{5}}{120}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \] |
Problem 12071
| ODE |
\[ \boxed {y^{\prime \prime }-y^{\prime } x +y=0} \] With the expansion point for the power series method at \(x = 0\). |
|
program solution |
\[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{24} x^{4}-\frac {1}{240} x^{6}\right ) y \left (0\right )+y^{\prime }\left (0\right ) x +O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{24} x^{4}\right ) c_{1} +c_{2} x +O\left (x^{6}\right ) \] Verified OK.
|
|
Maple solution |
\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}-\frac {1}{24} x^{4}\right ) y \left (0\right )+x D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \] |
Problem 12072
| ODE |
\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+y=0} \] With the expansion point for the power series method at \(x = 0\). |
|
program solution |
\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}-\frac {13}{240} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {7}{120} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}\right ) c_{1} +\left (x -\frac {1}{6} x^{3}+\frac {7}{120} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
|
|
Maple solution |
\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {7}{120} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \] |
Problem 12073
| ODE |
\[ \boxed {2 x y^{\prime \prime }+y^{\prime }-2 y=0} \] With the expansion point for the power series method at \(x = 0\). |
|
program solution |
\[ y = c_{1} \sqrt {x}\, \left (1+\frac {2 x}{3}+\frac {2 x^{2}}{15}+\frac {4 x^{3}}{315}+\frac {2 x^{4}}{2835}+\frac {4 x^{5}}{155925}+O\left (x^{6}\right )\right )+c_{2} \left (1+2 x +\frac {2 x^{2}}{3}+\frac {4 x^{3}}{45}+\frac {2 x^{4}}{315}+\frac {4 x^{5}}{14175}+O\left (x^{6}\right )\right ) \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {2}{3} x +\frac {2}{15} x^{2}+\frac {4}{315} x^{3}+\frac {2}{2835} x^{4}+\frac {4}{155925} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+2 x +\frac {2}{3} x^{2}+\frac {4}{45} x^{3}+\frac {2}{315} x^{4}+\frac {4}{14175} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \] |
Problem 12074
| ODE |
\[ \boxed {y^{\prime \prime }-2 y^{\prime } x -4 y=0} \] With the expansion point for the power series method at \(x = 0\). |
|
program solution |
\[ y = \left (1+2 x^{2}+\frac {4}{3} x^{4}+\frac {8}{15} x^{6}\right ) y \left (0\right )+\left (x +x^{3}+\frac {1}{2} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+2 x^{2}+\frac {4}{3} x^{4}\right ) c_{1} +\left (x +x^{3}+\frac {1}{2} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
|
|
Maple solution |
\[ y \left (x \right ) = \left (1+2 x^{2}+\frac {4}{3} x^{4}\right ) y \left (0\right )+\left (x +x^{3}+\frac {1}{2} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \] |
Problem 12075
| ODE |
\[ \boxed {y^{\prime \prime }-2 y^{\prime } x +4 y=0} \] With the expansion point for the power series method at \(x = 0\). |
|
program solution |
\[ y = \left (-2 x^{2}+1\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}-\frac {1}{30} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (-2 x^{2}+1\right ) c_{1} +\left (x -\frac {1}{3} x^{3}-\frac {1}{30} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
|
|
Maple solution |
\[ y \left (x \right ) = \left (-2 x^{2}+1\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}-\frac {1}{30} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \] |
Problem 12076
| ODE |
\[ \boxed {x \left (1-x \right ) y^{\prime \prime }-3 y^{\prime } x -y=0} \] With the expansion point for the power series method at \(x = 0\). |
|
program solution |
\[ y = c_{1} x \left (1+2 x +3 x^{2}+4 x^{3}+5 x^{4}+6 x^{5}+O\left (x^{6}\right )\right )+c_{2} \left (x \left (1+2 x +3 x^{2}+4 x^{3}+5 x^{4}+6 x^{5}+O\left (x^{6}\right )\right ) \ln \left (x \right )+1-x^{2}-2 x^{3}-3 x^{4}-4 x^{5}+O\left (x^{6}\right )\right ) \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \ln \left (x \right ) \left (x +2 x^{2}+3 x^{3}+4 x^{4}+5 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} +c_{1} x \left (1+2 x +3 x^{2}+4 x^{3}+5 x^{4}+6 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1+3 x +5 x^{2}+7 x^{3}+9 x^{4}+11 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \] |
Problem 12077
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x -x^{2} y=0} \] With the expansion point for the power series method at \(x = 0\). |
|
program solution |
\[ y = c_{1} \left (1+\frac {x^{2}}{4}+\frac {x^{4}}{64}+O\left (x^{6}\right )\right )+c_{2} \left (\left (1+\frac {x^{2}}{4}+\frac {x^{4}}{64}+O\left (x^{6}\right )\right ) \ln \left (x \right )-\frac {x^{2}}{4}-\frac {3 x^{4}}{128}+O\left (x^{6}\right )\right ) \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \left (c_{1} +c_{2} \ln \left (x \right )\right ) \left (1+\frac {1}{4} x^{2}+\frac {1}{64} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {1}{4} x^{2}-\frac {3}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \] |
Problem 12078
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +y \left (x^{2}-1\right )=0} \] With the expansion point for the power series method at \(x = 0\). |
|
program solution |
\[ y = c_{1} x \left (1-\frac {x^{2}}{8}+\frac {x^{4}}{192}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {x \left (1-\frac {x^{2}}{8}+\frac {x^{4}}{192}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{2}+\frac {1-\frac {3 x^{4}}{64}+O\left (x^{6}\right )}{x}\right ) \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {c_{1} x^{2} \left (1-\frac {1}{8} x^{2}+\frac {1}{192} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (x^{2}-\frac {1}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+\frac {3}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \] |
Problem 12079
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (-n^{2}+x^{2}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\). |
|
program solution |
\[ y = c_{1} x^{n} \left (1+\frac {x^{2}}{-4 n -4}+\frac {x^{4}}{32 \left (n +1\right ) \left (n +2\right )}+O\left (x^{6}\right )\right )+c_{2} x^{-n} \left (1+\frac {x^{2}}{4 n -4}+\frac {x^{4}}{32 \left (-2+n \right ) \left (n -1\right )}+O\left (x^{6}\right )\right ) \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = x^{-n} \left (1+\frac {1}{4 n -4} x^{2}+\frac {1}{32} \frac {1}{\left (n -2\right ) \left (n -1\right )} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{1} +c_{2} x^{n} \left (1-\frac {1}{4 n +4} x^{2}+\frac {1}{32} \frac {1}{\left (n +2\right ) \left (n +1\right )} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \] |
Problem 12080
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=4 x \left (t \right )-y \left (t \right )\\ y^{\prime }\left (t \right )&=2 x \left (t \right )+y \left (t \right )+t^{2} \end {align*} With initial conditions \[ [x \left (0\right ) = 0, y \left (0\right ) = 1] \] |
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= -\frac {29 \,{\mathrm e}^{3 t}}{27}+\frac {5 \,{\mathrm e}^{2 t}}{4}-\frac {t^{2}}{6}-\frac {5 t}{18}-\frac {19}{108} \\ y \left (t \right ) &= -\frac {29 \,{\mathrm e}^{3 t}}{27}+\frac {5 \,{\mathrm e}^{2 t}}{2}-\frac {7 t}{9}-\frac {23}{54}-\frac {2 t^{2}}{3} \\ \end{align*} |
Problem 12081
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )-4 y \left (t \right )+2 \cos \left (t \right )^{2}-1\\ y^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 1] \] |
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= \frac {26 \,{\mathrm e}^{t} \cos \left (2 t \right )}{17}-\frac {32 \,{\mathrm e}^{t} \sin \left (2 t \right )}{17}+\frac {2 \sin \left (2 t \right )}{17}-\frac {9 \cos \left (2 t \right )}{17} \\ y \left (t \right ) &= \frac {13 \,{\mathrm e}^{t} \sin \left (2 t \right )}{17}+\frac {16 \,{\mathrm e}^{t} \cos \left (2 t \right )}{17}+\frac {\cos \left (2 t \right )}{17}-\frac {4 \sin \left (2 t \right )}{17} \\ \end{align*} |
Problem 12082
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )+2 y \left (t \right )\\ y^{\prime }\left (t \right )&=6 x \left (t \right )+3 y \left (t \right )+{\mathrm e}^{t} \end {align*} With initial conditions \[ [x \left (0\right ) = 0, y \left (0\right ) = 1] \] |
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= \frac {12 \,{\mathrm e}^{6 t}}{35}-\frac {{\mathrm e}^{-t}}{7}-\frac {{\mathrm e}^{t}}{5} \\ y \left (t \right ) &= \frac {24 \,{\mathrm e}^{6 t}}{35}+\frac {3 \,{\mathrm e}^{-t}}{14}+\frac {{\mathrm e}^{t}}{10} \\ \end{align*} |
Problem 12083
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=5 x \left (t \right )-4 y \left (t \right )+{\mathrm e}^{3 t}\\ y^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = -1] \] |
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= {\mathrm e}^{3 t} \left (t^{2}+7 t +1\right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{3 t} \left (t^{2}+6 t -2\right )}{2} \\ \end{align*} |
Problem 12084
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )+5 y \left (t \right )\\ y^{\prime }\left (t \right )&=-2 x \left (t \right )+4 \cos \left (t \right )^{3}-3 \cos \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 2, y \left (0\right ) = -1] \] |
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= -\frac {16 \,{\mathrm e}^{t} \sin \left (3 t \right )}{111}+\frac {69 \,{\mathrm e}^{t} \cos \left (3 t \right )}{37}+\frac {5 \cos \left (3 t \right )}{37}-\frac {30 \sin \left (3 t \right )}{37} \\ y \left (t \right ) &= -\frac {121 \,{\mathrm e}^{t} \sin \left (3 t \right )}{111}-\frac {17 \,{\mathrm e}^{t} \cos \left (3 t \right )}{37}+\frac {9 \sin \left (3 t \right )}{37}-\frac {20 \cos \left (3 t \right )}{37} \\ \end{align*} |
Problem 12085
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right )+{\mathrm e}^{-t}\\ y^{\prime }\left (t \right )&=4 x \left (t \right )-2 y \left (t \right )+{\mathrm e}^{2 t} \end {align*} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = -1] \] |
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= \frac {62 \,{\mathrm e}^{2 t}}{75}+\frac {17 \,{\mathrm e}^{-3 t}}{50}+\frac {{\mathrm e}^{2 t} t}{5}-\frac {{\mathrm e}^{-t}}{6} \\ y \left (t \right ) &= \frac {77 \,{\mathrm e}^{2 t}}{75}-\frac {34 \,{\mathrm e}^{-3 t}}{25}+\frac {{\mathrm e}^{2 t} t}{5}-\frac {2 \,{\mathrm e}^{-t}}{3} \\ \end{align*} |
Problem 12086
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=8 x \left (t \right )+14 y \left (t \right )\\ y^{\prime }\left (t \right )&=7 x \left (t \right )+y \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 1] \] |
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= -\frac {{\mathrm e}^{-6 t}}{3}+\frac {4 \,{\mathrm e}^{15 t}}{3} \\ y \left (t \right ) &= \frac {{\mathrm e}^{-6 t}}{3}+\frac {2 \,{\mathrm e}^{15 t}}{3} \\ \end{align*} |
Problem 12096
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=8 x \left (t \right )+14 y \left (t \right )\\ y^{\prime }\left (t \right )&=7 x \left (t \right )+y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= {\mathrm e}^{-6 t} c_{1} +c_{2} {\mathrm e}^{15 t} \\ y \left (t \right ) &= -{\mathrm e}^{-6 t} c_{1} +\frac {c_{2} {\mathrm e}^{15 t}}{2} \\ \end{align*} |
Problem 12097
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )\\ y^{\prime }\left (t \right )&=-5 x \left (t \right )-3 y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{2 t} \\ y \left (t \right ) &= -c_{2} {\mathrm e}^{2 t}+c_{1} {\mathrm e}^{-3 t} \\ \end{align*} |
Problem 12098
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=11 x \left (t \right )-2 y \left (t \right )\\ y^{\prime }\left (t \right )&=3 x \left (t \right )+4 y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= {\mathrm e}^{5 t} c_{1} +c_{2} {\mathrm e}^{10 t} \\ y \left (t \right ) &= 3 \,{\mathrm e}^{5 t} c_{1} +\frac {c_{2} {\mathrm e}^{10 t}}{2} \\ \end{align*} |
Problem 12099
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )+20 y \left (t \right )\\ y^{\prime }\left (t \right )&=40 x \left (t \right )-19 y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{21 t}+c_{2} {\mathrm e}^{-39 t} \\ y \left (t \right ) &= c_{1} {\mathrm e}^{21 t}-2 c_{2} {\mathrm e}^{-39 t} \\ \end{align*} |
Problem 12100
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=-2 x \left (t \right )+2 y \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )-y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} +c_{2} {\mathrm e}^{-3 t} \\ y \left (t \right ) &= -\frac {c_{2} {\mathrm e}^{-3 t}}{2}+c_{1} \\ \end{align*} |