2.17.146 Problems 14501 to 14600

Problem 14501

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }+y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {\left (\sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{2}\right )+3 \cos \left (\frac {\sqrt {3}\, t}{2}\right )\right ) {\mathrm e}^{-\frac {t}{2}}}{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {{\mathrm e}^{-\frac {t}{2}} \left (\sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{2}\right )+3 \cos \left (\frac {\sqrt {3}\, t}{2}\right )\right )}{3} \]

Problem 14502

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }+y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \frac {2 \sqrt {3}\, {\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right )}{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {2 \sqrt {3}\, {\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right )}{3} \]

Problem 14503

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }-y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \frac {\sqrt {5}\, \left ({\mathrm e}^{\frac {\left (\sqrt {5}+1\right ) t}{2}}-{\mathrm e}^{-\frac {\left (\sqrt {5}-1\right ) t}{2}}\right )}{5} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {\left (-{\mathrm e}^{\frac {\left (\sqrt {5}+1\right ) t}{2}}+{\mathrm e}^{-\frac {\left (\sqrt {5}-1\right ) t}{2}}\right ) \sqrt {5}}{5} \]

Problem 14504

ODE	\[ \boxed {6 y^{\prime \prime }+5 y^{\prime }+y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {t}{2}}+6 c_{2} {\mathrm e}^{-\frac {t}{3}} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} {\mathrm e}^{-\frac {t}{3}}+c_{2} {\mathrm e}^{-\frac {t}{2}} \]

Problem 14505

ODE	\[ \boxed {9 y^{\prime \prime }+6 y^{\prime }+y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {t}{3}}+c_{2} t \,{\mathrm e}^{-\frac {t}{3}} \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{-\frac {t}{3}} \left (c_{2} t +c_{1} \right ) \]

Problem 14506

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+20 y=0} \]
program solution	\[ y = c_{1} \cos \left (4 t \right ) {\mathrm e}^{-2 t}+\frac {c_{2} \sin \left (4 t \right ) {\mathrm e}^{-2 t}}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{-2 t} \left (c_{1} \sin \left (4 t \right )+c_{2} \cos \left (4 t \right )\right ) \]

Problem 14507

ODE	\[ \boxed {3 t^{2} y^{\prime \prime }-2 t y^{\prime }+2 y=0} \]
program solution	\[ y = t^{\frac {2}{3}} c_{1} +3 c_{2} t \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} t^{\frac {2}{3}}+c_{2} t \]

Problem 14508

ODE	\[ \boxed {t^{2} y^{\prime \prime }-t y^{\prime }+y=0} \]
program solution	\[ y = t c_{1} +c_{2} t \ln \left (t \right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = t \left (c_{2} \ln \left (t \right )+c_{1} \right ) \]

Problem 14509

ODE	\[ \boxed {a y^{\prime \prime }+2 b y^{\prime }+c y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{\frac {t \left (\sqrt {\frac {-a c +b^{2}}{a^{2}}}\, a -b \right )}{a}}+\frac {c_{2} a^{2} {\mathrm e}^{-\frac {t \left (\sqrt {\frac {-a c +b^{2}}{a^{2}}}\, a +b \right )}{a}} \sqrt {\frac {-a c +b^{2}}{a^{2}}}}{2 a c -2 b^{2}} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} {\mathrm e}^{\frac {\left (-b +\sqrt {-a c +b^{2}}\right ) t}{a}}+c_{2} {\mathrm e}^{-\frac {\left (b +\sqrt {-a c +b^{2}}\right ) t}{a}} \]

Problem 14510

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+2 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\left (3+\sqrt {7}\right ) t}+\frac {c_{2} {\mathrm e}^{\left (\sqrt {7}-3\right ) t} \sqrt {7}}{14} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} {\mathrm e}^{\left (-3+\sqrt {7}\right ) t}+c_{2} {\mathrm e}^{-\left (3+\sqrt {7}\right ) t} \]

Problem 14511

ODE	\[ \boxed {y^{\prime \prime }-5 y^{\prime }+6 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{2 t}+c_{2} {\mathrm e}^{3 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} {\mathrm e}^{3 t}+c_{2} {\mathrm e}^{2 t} \]

Problem 14512

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }-16 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 t}+\frac {c_{2} {\mathrm e}^{8 t}}{10} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} {\mathrm e}^{8 t}+c_{2} {\mathrm e}^{-2 t} \]

Problem 14513

ODE	\[ \boxed {y^{\prime \prime }-16 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-4 t}+\frac {c_{2} {\mathrm e}^{4 t}}{8} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} {\mathrm e}^{4 t}+c_{2} {\mathrm e}^{-4 t} \]

Problem 14514

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+5 y=0} \] With initial conditions \begin {align*} \left [y \left (0\right ) = 0, y^{\prime }\left (\frac {\pi }{2}\right ) = 0\right ] \end {align*}
program solution	\[ y = 0 \] Verified OK.
Maple solution	\[ y \left (t \right ) = 0 \]

Problem 14515

ODE	\[ \boxed {{y^{\prime \prime }}^{2}-5 y^{\prime } y^{\prime \prime }+4 y^{2}=0} \]
program solution
Maple solution	\begin{align*} y \left (t \right ) &= 0 \\ y \left (t \right ) &= {\mathrm e}^{\int \operatorname {RootOf}\left (t +2 \left (\int _{}^{\textit {\_Z}}\frac {1}{2 \textit {\_f}^{2}+\sqrt {25 \textit {\_f}^{2}-16}-5 \textit {\_f}}d \textit {\_f} \right )+c_{1} \right )d t +c_{2}} \\ y \left (t \right ) &= {\mathrm e}^{\int \operatorname {RootOf}\left (t -2 \left (\int _{}^{\textit {\_Z}}-\frac {1}{2 \textit {\_f}^{2}-\sqrt {25 \textit {\_f}^{2}-16}-5 \textit {\_f}}d \textit {\_f} \right )+c_{1} \right )d t +c_{2}} \\ \end{align*}

Problem 14516

ODE	\[ \boxed {{y^{\prime \prime }}^{2}-2 y^{\prime } y^{\prime \prime }+y^{2}=0} \]
program solution
Maple solution	\begin{align*} y \left (t \right ) &= 0 \\ y \left (t \right ) &= {\mathrm e}^{t} c_{1} \\ y \left (t \right ) &= {\mathrm e}^{\int \operatorname {RootOf}\left (t +\int _{}^{\textit {\_Z}}\frac {1}{\textit {\_f}^{2}+\sqrt {\textit {\_f}^{2}-1}-\textit {\_f}}d \textit {\_f} +c_{1} \right )d t +c_{2}} \\ y \left (t \right ) &= {\mathrm e}^{\int \operatorname {RootOf}\left (t -\left (\int _{}^{\textit {\_Z}}-\frac {1}{\textit {\_f}^{2}-\sqrt {\textit {\_f}^{2}-1}-\textit {\_f}}d \textit {\_f} \right )+c_{1} \right )d t +c_{2}} \\ \end{align*}

Problem 14517

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+3 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = a, y^{\prime }\left (0\right ) = b] \end {align*}
program solution	\[ y = \frac {\left (-a -b \right ) {\mathrm e}^{-3 t}}{2}+\frac {3 \left (a +\frac {b}{3}\right ) {\mathrm e}^{-t}}{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (-a -b \right ) {\mathrm e}^{-3 t}}{2}+\frac {3 \left (a +\frac {b}{3}\right ) {\mathrm e}^{-t}}{2} \]

Problem 14518

ODE	\[ \boxed {y^{\prime \prime }+y=8 \,{\mathrm e}^{2 t}} \]
program solution	\[ y = c_{1} \cos \left (t \right )+c_{2} \sin \left (t \right )+\frac {8 \,{\mathrm e}^{2 t}}{5} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \sin \left (t \right ) c_{2} +\cos \left (t \right ) c_{1} +\frac {8 \,{\mathrm e}^{2 t}}{5} \]

Problem 14519

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+3 y=-{\mathrm e}^{-9 t}} \]
program solution	\[ y = c_{1} {\mathrm e}^{t}+\frac {c_{2} {\mathrm e}^{3 t}}{2}-\frac {{\mathrm e}^{-9 t}}{120} \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{-9 t} \left ({\mathrm e}^{10 t} c_{1} +c_{2} {\mathrm e}^{12 t}-\frac {1}{120}\right ) \]

Problem 14520

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+3 y=2 \,{\mathrm e}^{3 t}} \]
program solution	\[ y = c_{1} {\mathrm e}^{t}+\frac {c_{2} {\mathrm e}^{3 t}}{2}+{\mathrm e}^{3 t} \left (t -\frac {1}{2}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (c_{2} +t \right ) {\mathrm e}^{3 t}+{\mathrm e}^{t} c_{1} \]

Problem 14521

ODE	\[ \boxed {y^{\prime \prime }-y=2 t -4} \]
program solution	\[ y = c_{1} {\mathrm e}^{-t}+\frac {c_{2} {\mathrm e}^{t}}{2}+4-2 t \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{2} {\mathrm e}^{-t}+{\mathrm e}^{t} c_{1} -2 t +4 \]

Problem 14522

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }+y=t^{2}} \]
program solution	\[ y = {\mathrm e}^{t} \left (c_{2} t +c_{1} \right )+t^{2}+4 t +6 \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (c_{1} t +c_{2} \right ) {\mathrm e}^{t}+t^{2}+4 t +6 \]

Problem 14523

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }=3-4 t} \]
program solution	\[ y = -t^{2}+\frac {c_{1}}{2}+\frac {5 t}{2}-\frac {5}{4}+c_{2} {\mathrm e}^{-2 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {{\mathrm e}^{-2 t} c_{1}}{2}-t^{2}+\frac {5 t}{2}+c_{2} \]

Problem 14524

ODE	\[ \boxed {y^{\prime \prime }+y=\cos \left (2 t \right )} \]
program solution	\[ y = c_{1} \cos \left (t \right )+c_{2} \sin \left (t \right )-\frac {\cos \left (2 t \right )}{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \sin \left (t \right ) c_{2} +\cos \left (t \right ) c_{1} -\frac {\cos \left (2 t \right )}{3} \]

Problem 14525

ODE	\[ \boxed {y^{\prime \prime }+4 y=4 \cos \left (t \right )-\sin \left (t \right )} \]
program solution	\[ y = c_{1} \cos \left (2 t \right )+\frac {c_{2} \sin \left (2 t \right )}{2}+\frac {4 \cos \left (t \right )}{3}-\frac {\sin \left (t \right )}{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \sin \left (2 t \right ) c_{2} +\cos \left (2 t \right ) c_{1} +\frac {4 \cos \left (t \right )}{3}-\frac {\sin \left (t \right )}{3} \]

Problem 14526

ODE	\[ \boxed {y^{\prime \prime }+4 y=\cos \left (2 t \right )+t} \]
program solution	\[ y = c_{1} \cos \left (2 t \right )+\frac {c_{2} \sin \left (2 t \right )}{2}+\frac {t}{4}+\frac {\sin \left (2 t \right ) t}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (8 c_{1} +1\right ) \cos \left (2 t \right )}{8}+\frac {\left (8 c_{2} +2 t \right ) \sin \left (2 t \right )}{8}+\frac {t}{4} \]

Problem 14527

ODE	\[ \boxed {y^{\prime \prime }+4 y=3 t \,{\mathrm e}^{-t}} \]
program solution	\[ y = c_{1} \cos \left (2 t \right )+\frac {c_{2} \sin \left (2 t \right )}{2}+\frac {3 t \,{\mathrm e}^{-t}}{5}+\frac {6 \,{\mathrm e}^{-t}}{25} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \sin \left (2 t \right ) c_{2} +\cos \left (2 t \right ) c_{1} +\frac {3 \left (5 t +2\right ) {\mathrm e}^{-t}}{25} \]

Problem 14528

ODE	\[ \boxed {y^{\prime \prime }=3 t^{4}-2 t} \]
program solution	\[ y = \frac {1}{10} t^{6}-\frac {1}{3} t^{3}+t c_{1} +c_{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {1}{10} t^{6}-\frac {1}{3} t^{3}+\frac {5}{18}+c_{1} t +c_{2} \]

Problem 14529

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+13 y=2 t \,{\mathrm e}^{-2 t} \sin \left (3 t \right )} \]
program solution	\[ y = c_{1} \cos \left (3 t \right ) {\mathrm e}^{2 t}+\frac {c_{2} \sin \left (3 t \right ) {\mathrm e}^{2 t}}{3}+\frac {63 \,{\mathrm e}^{-2 t} \cos \left (3 t \right )}{2704}+\frac {{\mathrm e}^{-2 t} \sin \left (3 t \right )}{169}+\frac {3 t \,{\mathrm e}^{-2 t} \cos \left (3 t \right )}{52}+\frac {t \,{\mathrm e}^{-2 t} \sin \left (3 t \right )}{26} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (\left (156 t +63\right ) \cos \left (3 t \right )+\left (104 t +16\right ) \sin \left (3 t \right )\right ) {\mathrm e}^{-2 t}}{2704}+{\mathrm e}^{2 t} \left (c_{1} \cos \left (3 t \right )+c_{2} \sin \left (3 t \right )\right ) \]

Problem 14530

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }-2 y=-1} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 t}+\frac {c_{2} {\mathrm e}^{t}}{3}+\frac {1}{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (2 c_{2} {\mathrm e}^{3 t}+{\mathrm e}^{2 t}+2 c_{1} \right ) {\mathrm e}^{-2 t}}{2} \]

Problem 14531

ODE	\[ \boxed {5 y^{\prime \prime }+y^{\prime }-4 y=-3} \]
program solution	\[ y = c_{1} {\mathrm e}^{-t}+\frac {5 c_{2} {\mathrm e}^{\frac {4 t}{5}}}{9}+\frac {3}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (4 \,{\mathrm e}^{\frac {9 t}{5}} c_{1} +3 \,{\mathrm e}^{t}+4 c_{2} \right ) {\mathrm e}^{-t}}{4} \]

Problem 14532

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }-8 y=32 t} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 t}+\frac {c_{2} {\mathrm e}^{4 t}}{6}+1-4 t \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{4 t} c_{2} +{\mathrm e}^{-2 t} c_{1} -4 t +1 \]

Problem 14533

ODE	\[ \boxed {16 y^{\prime \prime }-8 y^{\prime }-15 y=75 t} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {3 t}{4}}+\frac {c_{2} {\mathrm e}^{\frac {5 t}{4}}}{2}-5 t +\frac {8}{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (3 c_{2} {\mathrm e}^{2 t}-15 t \,{\mathrm e}^{\frac {3 t}{4}}+8 \,{\mathrm e}^{\frac {3 t}{4}}+3 c_{1} \right ) {\mathrm e}^{-\frac {3 t}{4}}}{3} \]

Problem 14534

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+26 y=-338 t} \]
program solution	\[ y = c_{1} {\mathrm e}^{-t} \cos \left (5 t \right )+\frac {c_{2} {\mathrm e}^{-t} \sin \left (5 t \right )}{5}-13 t +1 \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{-t} \sin \left (5 t \right ) c_{2} +{\mathrm e}^{-t} \cos \left (5 t \right ) c_{1} -13 t +1 \]

Problem 14535

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }-4 y=-32 t^{2}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-4 t}+\frac {c_{2} {\mathrm e}^{t}}{5}+8 t^{2}+12 t +13 \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (8 t^{2}+12 t +13\right ) {\mathrm e}^{-4 t} {\mathrm e}^{4 t}+\left (c_{2} {\mathrm e}^{5 t}+c_{1} \right ) {\mathrm e}^{-4 t} \]

Problem 14536

ODE	\[ \boxed {8 y^{\prime \prime }+6 y^{\prime }+y=5 t^{2}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {t}{2}}+4 \,{\mathrm e}^{-\frac {t}{4}} c_{2} +5 t^{2}-60 t +280 \] Verified OK.
Maple solution	\[ y \left (t \right ) = -4 \,{\mathrm e}^{-\frac {t}{2}} c_{1} +{\mathrm e}^{-\frac {t}{4}} c_{2} +5 t^{2}-60 t +280 \]

Problem 14537

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+8 y=-256 t^{3}} \]
program solution	\[ y = c_{1} {\mathrm e}^{2 t}+\frac {c_{2} {\mathrm e}^{4 t}}{2}-32 t^{3}-72 t^{2}-84 t -45 \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {c_{1} {\mathrm e}^{4 t}}{2}-45-84 t -72 t^{2}-32 t^{3}+c_{2} {\mathrm e}^{2 t} \]

Problem 14538

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }=52 \sin \left (3 t \right )} \]
program solution	\[ y = -\frac {c_{1}}{2}+c_{2} {\mathrm e}^{2 t}+\frac {8 \cos \left (3 t \right )}{3}-4 \sin \left (3 t \right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {c_{1} {\mathrm e}^{2 t}}{2}-4 \sin \left (3 t \right )+\frac {8 \cos \left (3 t \right )}{3}+c_{2} \]

Problem 14539

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+13 y=25 \sin \left (2 t \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{3 t} \cos \left (2 t \right )+\frac {c_{2} {\mathrm e}^{3 t} \sin \left (2 t \right )}{2}+\frac {4 \cos \left (2 t \right )}{3}+\sin \left (2 t \right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (3 c_{2} {\mathrm e}^{3 t}+3\right ) \sin \left (2 t \right )}{3}+\cos \left (2 t \right ) {\mathrm e}^{3 t} c_{1} +\frac {4 \cos \left (2 t \right )}{3} \]

Problem 14540

ODE	\[ \boxed {y^{\prime \prime }-9 y=54 \sin \left (2 t \right ) t} \]
program solution	\[ y = c_{1} {\mathrm e}^{-3 t}+\frac {c_{2} {\mathrm e}^{3 t}}{6}-\frac {54 \sin \left (2 t \right ) t}{13}-\frac {216 \cos \left (2 t \right )}{169} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{2} {\mathrm e}^{-3 t}+c_{1} {\mathrm e}^{3 t}-\frac {216 \cos \left (2 t \right )}{169}-\frac {54 t \sin \left (2 t \right )}{13} \]

Problem 14541

ODE	\[ \boxed {y^{\prime \prime }-5 y^{\prime }+6 y=-78 \cos \left (3 t \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{2 t}+c_{2} {\mathrm e}^{3 t}+\cos \left (3 t \right )+5 \sin \left (3 t \right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{2} {\mathrm e}^{3 t}+c_{1} {\mathrm e}^{2 t}+5 \sin \left (3 t \right )+\cos \left (3 t \right ) \]

Problem 14542

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+4 y=-32 t^{2} \cos \left (2 t \right )} \]
program solution	\[ y = {\mathrm e}^{-2 t} \left (c_{2} t +c_{1} \right )-4 t^{2} \sin \left (2 t \right )-4 \cos \left (2 t \right ) t +4 \sin \left (2 t \right ) t +3 \cos \left (2 t \right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (3-4 t \right ) \cos \left (2 t \right )+\left (c_{1} t +c_{2} \right ) {\mathrm e}^{-2 t}-4 t \sin \left (2 t \right ) \left (t -1\right ) \]

Problem 14543

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }-20 y=-2 \,{\mathrm e}^{t}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-4 t}+\frac {c_{2} {\mathrm e}^{5 t}}{9}+\frac {{\mathrm e}^{t}}{10} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (10 c_{2} {\mathrm e}^{9 t}+{\mathrm e}^{5 t}+10 c_{1} \right ) {\mathrm e}^{-4 t}}{10} \]

Problem 14544

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }-5 y=-648 t^{2} {\mathrm e}^{5 t}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-t}+\frac {c_{2} {\mathrm e}^{5 t}}{6}+{\mathrm e}^{5 t} \left (-36 t^{3}+18 t^{2}-6 t +1\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (-36 t^{3}+18 t^{2}+c_{2} -6 t \right ) {\mathrm e}^{5 t}+{\mathrm e}^{-t} c_{1} \]

Problem 14545

ODE	\[ \boxed {y^{\prime \prime }-7 y^{\prime }+12 y=-2 t^{3} {\mathrm e}^{4 t}} \]
program solution	\[ y = c_{1} {\mathrm e}^{3 t}+c_{2} {\mathrm e}^{4 t}+12 t \,{\mathrm e}^{4 t}-6 t^{2} {\mathrm e}^{4 t}+2 t^{3} {\mathrm e}^{4 t}-\frac {t^{4} {\mathrm e}^{4 t}}{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (-t^{4}+4 t^{3}-12 t^{2}+2 c_{2} +24 t \right ) {\mathrm e}^{4 t}}{2}+c_{1} {\mathrm e}^{3 t} \]

Problem 14546

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }=8 \,{\mathrm e}^{4 t}-4 \,{\mathrm e}^{-4 t}} \]
program solution	\[ y = \left (t +c_{2} \right ) {\mathrm e}^{-4 t}+\frac {c_{1}}{4}+\frac {{\mathrm e}^{4 t}}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (4 t -c_{1} +1\right ) {\mathrm e}^{-4 t}}{4}+c_{2} +\frac {{\mathrm e}^{4 t}}{4} \]

Problem 14547

ODE	\[ \boxed {y^{\prime \prime }-3 y^{\prime }=t^{2}-{\mathrm e}^{3 t}} \]
program solution	\[ y = -\frac {2}{81}+\frac {\left (-t +3 c_{2} \right ) {\mathrm e}^{3 t}}{3}-\frac {t^{3}}{9}-\frac {t^{2}}{9}-\frac {2 t}{27}-\frac {c_{1}}{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (1-3 t +3 c_{1} \right ) {\mathrm e}^{3 t}}{9}-\frac {t^{3}}{9}-\frac {t^{2}}{9}-\frac {2 t}{27}+c_{2} \]

Problem 14548

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }=-24 t -6-4 t \,{\mathrm e}^{-4 t}+{\mathrm e}^{-4 t}} \]
program solution	\[ y = \frac {\left (2 t^{2}+4 c_{2} \right ) {\mathrm e}^{-4 t}}{4}-3 t^{2}+\frac {c_{1}}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (2 t^{2}-c_{1} \right ) {\mathrm e}^{-4 t}}{4}-3 t^{2}+c_{2} \]

Problem 14549

ODE	\[ \boxed {y^{\prime \prime }-3 y^{\prime }=t^{2}-{\mathrm e}^{3 t}} \]
program solution	\[ y = -\frac {2}{81}+\frac {\left (-t +3 c_{2} \right ) {\mathrm e}^{3 t}}{3}-\frac {t^{3}}{9}-\frac {t^{2}}{9}-\frac {2 t}{27}-\frac {c_{1}}{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (1-3 t +3 c_{1} \right ) {\mathrm e}^{3 t}}{9}-\frac {t^{3}}{9}-\frac {t^{2}}{9}-\frac {2 t}{27}+c_{2} \]

Problem 14550

ODE	\[ \boxed {y^{\prime \prime }=t^{2}+{\mathrm e}^{t}+\sin \left (t \right )} \]
program solution	\[ y = \frac {t^{4}}{12}-\sin \left (t \right )+{\mathrm e}^{t}+t c_{1} +c_{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {t^{4}}{12}-\sin \left (t \right )+{\mathrm e}^{t}+c_{1} t +c_{2} \]

Problem 14551

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }=18} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 3] \end {align*}
program solution	\[ y = -1+{\mathrm e}^{-3 t}+6 t \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{-3 t}+6 t -1 \]

Problem 14552

ODE	\[ \boxed {y^{\prime \prime }-y=4} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = -4+2 \,{\mathrm e}^{t}+2 \,{\mathrm e}^{-t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = 2 \,{\mathrm e}^{-t}+2 \,{\mathrm e}^{t}-4 \]

Problem 14553

ODE	\[ \boxed {y^{\prime \prime }-4 y=32 t} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 6] \end {align*}
program solution	\[ y = \frac {7 \,{\mathrm e}^{2 t}}{2}-\frac {7 \,{\mathrm e}^{-2 t}}{2}-8 t \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {7 \,{\mathrm e}^{2 t}}{2}-\frac {7 \,{\mathrm e}^{-2 t}}{2}-8 t \]

Problem 14554

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }-3 y=-2} \] With initial conditions \begin {align*} \left [y \left (0\right ) = {\frac {2}{3}}, y^{\prime }\left (0\right ) = 8\right ] \end {align*}
program solution	\[ y = \frac {2}{3}+2 \,{\mathrm e}^{t}-2 \,{\mathrm e}^{-3 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {2 \left (3 \,{\mathrm e}^{4 t}+{\mathrm e}^{3 t}-3\right ) {\mathrm e}^{-3 t}}{3} \]

Problem 14555

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }-6 y=3 t} \] With initial conditions \begin {align*} \left [y \left (0\right ) = {\frac {23}{12}}, y^{\prime }\left (0\right ) = -{\frac {3}{2}}\right ] \end {align*}
program solution	\[ y = -\frac {1}{12}+{\mathrm e}^{2 t}+{\mathrm e}^{-3 t}-\frac {t}{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {\left (\left (t +\frac {1}{6}\right ) {\mathrm e}^{3 t}-2 \,{\mathrm e}^{5 t}-2\right ) {\mathrm e}^{-3 t}}{2} \]

Problem 14556

ODE	\[ \boxed {y^{\prime \prime }+8 y^{\prime }+16 y=4} \] With initial conditions \begin {align*} \left [y \left (0\right ) = {\frac {5}{4}}, y^{\prime }\left (0\right ) = 0\right ] \end {align*}
program solution	\[ y = \frac {1}{4}+4 t \,{\mathrm e}^{-4 t}+{\mathrm e}^{-4 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{-4 t}+4 t \,{\mathrm e}^{-4 t}+\frac {1}{4} \]

Problem 14557

ODE	\[ \boxed {y^{\prime \prime }+7 y^{\prime }+10 y=t \,{\mathrm e}^{-t}} \] With initial conditions \begin {align*} \left [y \left (0\right ) = -{\frac {5}{16}}, y^{\prime }\left (0\right ) = {\frac {9}{16}}\right ] \end {align*}
program solution	\[ y = \frac {{\mathrm e}^{-t} \left (4 t -5\right )}{16} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (4 t -5\right ) {\mathrm e}^{-t}}{16} \]

Problem 14558

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+25 y=-1} \] With initial conditions \begin {align*} \left [y \left (0\right ) = -{\frac {1}{25}}, y^{\prime }\left (0\right ) = 7\right ] \end {align*}
program solution	\[ y = -\frac {1}{25}+\frac {7 \,{\mathrm e}^{-3 t} \sin \left (4 t \right )}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {7 \,{\mathrm e}^{-3 t} \sin \left (4 t \right )}{4}-\frac {1}{25} \]

Problem 14559

ODE	\[ \boxed {y^{\prime \prime }-3 y^{\prime }=-{\mathrm e}^{3 t}-2 t} \] With initial conditions \begin {align*} \left [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = {\frac {8}{9}}\right ] \end {align*}
program solution	\[ y = \frac {\left (-3 t +3\right ) {\mathrm e}^{3 t}}{9}+\frac {t^{2}}{3}+\frac {2 t}{9}-\frac {1}{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (-3 t +3\right ) {\mathrm e}^{3 t}}{9}+\frac {t^{2}}{3}+\frac {2 t}{9}-\frac {1}{3} \]

Problem 14560

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }=-3 t -4 t^{2} {\mathrm e}^{2 t}} \] With initial conditions \begin {align*} \left [y \left (0\right ) = -{\frac {7}{2}}, y^{\prime }\left (0\right ) = 0\right ] \end {align*}
program solution	\[ y = -\frac {3}{2}+\left (-2 t^{2}+6 t -7\right ) {\mathrm e}^{2 t}+\frac {3 t^{2}}{2}+3 t +5 \,{\mathrm e}^{t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {3}{2}+\left (-2 t^{2}+6 t -7\right ) {\mathrm e}^{2 t}+\frac {3 t^{2}}{2}+3 t +5 \,{\mathrm e}^{t} \]

Problem 14561

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }=2 t^{2}} \] With initial conditions \begin {align*} \left [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = {\frac {3}{2}}\right ] \end {align*}
program solution	\[ y = {\mathrm e}^{2 t}+2-\frac {t^{3}}{3}-\frac {t^{2}}{2}-\frac {t}{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{2 t}-\frac {t^{2}}{2}-\frac {t^{3}}{3}-\frac {t}{2}+2 \]

Problem 14562

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }=-24 t -6-4 t \,{\mathrm e}^{-4 t}+{\mathrm e}^{-4 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = t^{2} \left (-3+\frac {{\mathrm e}^{-4 t}}{2}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {t^{2} \left (-6+{\mathrm e}^{-4 t}\right )}{2} \]

Problem 14563

ODE	\[ \boxed {y^{\prime \prime }-3 y^{\prime }={\mathrm e}^{-3 t}-{\mathrm e}^{3 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \frac {4}{9}+\frac {\left (3-2 t \right ) {\mathrm e}^{3 t}}{6}+\frac {{\mathrm e}^{-3 t}}{18} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {4}{9}+\frac {\left (3-2 t \right ) {\mathrm e}^{3 t}}{6}+\frac {{\mathrm e}^{-3 t}}{18} \]

Problem 14564

ODE	\[ \boxed {y^{\prime \prime }+9 y=\left \{\begin {array}{cc} 2 t & 0\le t <\pi \\ 0 & \pi \le t \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = -\frac {2 \left (\left \{\begin {array}{cc} 0 & t \le 0 \\ \sin \left (3 t \right )-3 t & t \le \pi \\ 3 \pi \cos \left (3 t \right )+2 \sin \left (3 t \right ) & \pi
Maple solution	\[ y \left (t \right ) = \frac {2 \left (\left \{\begin {array}{cc} 0 & t <0 \\ 3 t -\sin \left (3 t \right ) & t <\pi \\ -3 \cos \left (3 t \right ) \pi -2 \sin \left (3 t \right ) & \pi \le t \end {array}\right .\right )}{27} \]

Problem 14565

ODE	\[ \boxed {y^{\prime \prime }+9 \pi ^{2} y=\left \{\begin {array}{cc} 2 t & 0\le t <\pi \\ 2 t -2 \pi & \pi \le t <2 \pi \\ 0 & 2 \pi \le t \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	N/A
Maple solution	\[ y \left (t \right ) = \frac {2 \left (\left \{\begin {array}{cc} 0 & t <0 \\ 3 \pi t -\sin \left (3 \pi t \right ) & t <\pi \\ 3 \cos \left (3 \pi ^{2}-3 \pi t \right ) \pi ^{2}-3 \pi ^{2}+3 \pi t -\sin \left (3 \pi t \right ) & t <2 \pi \\ 3 \left (\cos \left (3 \pi ^{2}-3 \pi t \right )+\cos \left (6 \pi ^{2}-3 \pi t \right )\right ) \pi ^{2}-\sin \left (3 \pi t \right )-\sin \left (6 \pi ^{2}-3 \pi t \right ) & 2 \pi \le t \end {array}\right .\right )}{27 \pi ^{3}} \]

Problem 14566

ODE	\[ \boxed {y^{\prime \prime }+4 y=\left \{\begin {array}{cc} 0 & 0\le t <\pi \\ 10 & \pi \le t <2 \pi \\ 0 & 2 \pi \le t \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 2] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} \sin \left (2 t \right ) & t \le \pi \\ \sin \left (2 t \right )-\frac {5 \cos \left (2 t \right )}{2}+\frac {5}{2} & t \le 2 \pi \\ \sin \left (2 t \right ) & \operatorname {otherwise} \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (t \right ) = \sin \left (2 t \right )+\left (\left \{\begin {array}{cc} 0 & t <\pi \\ \frac {5}{2}-\frac {5 \cos \left (2 t \right )}{2} & t <2 \pi \\ 0 & 2 \pi \le t \end {array}\right .\right ) \]

Problem 14567

ODE	\[ \boxed {y^{\prime }-4 y=t^{2}} \]
program solution	\[ y = -\frac {\left (8 t^{2} {\mathrm e}^{-4 t}+4 t \,{\mathrm e}^{-4 t}+{\mathrm e}^{-4 t}-32 c_{1} \right ) {\mathrm e}^{4 t}}{32} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {t^{2}}{4}-\frac {t}{8}-\frac {1}{32}+c_{1} {\mathrm e}^{4 t} \]

Problem 14568

ODE	\[ \boxed {y^{\prime }+y=\cos \left (2 t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ y = -\frac {{\mathrm e}^{-t}}{5}+\frac {2 \sin \left (2 t \right )}{5}+\frac {\cos \left (2 t \right )}{5} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\cos \left (2 t \right )}{5}+\frac {2 \sin \left (2 t \right )}{5}-\frac {{\mathrm e}^{-t}}{5} \]

Problem 14569

ODE	\[ \boxed {y^{\prime }-y={\mathrm e}^{4 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {{\mathrm e}^{4 t}}{3}-\frac {{\mathrm e}^{t}}{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left ({\mathrm e}^{3 t}-1\right ) {\mathrm e}^{t}}{3} \]

Problem 14570

ODE	\[ \boxed {4 y+y^{\prime }={\mathrm e}^{-4 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
program solution	\[ y = t \,{\mathrm e}^{-4 t}+{\mathrm e}^{-4 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (t +1\right ) {\mathrm e}^{-4 t} \]

Problem 14571

ODE	\[ \boxed {4 y+y^{\prime }=t \,{\mathrm e}^{-4 t}} \]
program solution	\[ y = \frac {{\mathrm e}^{-4 t} \left (t^{2}+2 c_{1} \right )}{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (t^{2}+2 c_{1} \right ) {\mathrm e}^{-4 t}}{2} \]

Problem 14572

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }-2 y=f \left (t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = a] \end {align*}
program solution	\[ y = \frac {{\mathrm e}^{-2 t} \left (-\left (\int _{0}^{t}f \left (\alpha \right ) {\mathrm e}^{2 \alpha }d \alpha \right )+\left (\int _{0}^{t}f \left (\alpha \right ) {\mathrm e}^{-\alpha }d \alpha \right ) {\mathrm e}^{3 t}+a \left ({\mathrm e}^{3 t}-1\right )\right )}{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (-\left (\int _{0}^{t}f \left (\textit {\_z1} \right ) {\mathrm e}^{2 \textit {\_z1}}d \textit {\_z1} \right )+\left (\int _{0}^{t}f \left (\textit {\_z1} \right ) {\mathrm e}^{-\textit {\_z1}}d \textit {\_z1} \right ) {\mathrm e}^{3 t}+a \left ({\mathrm e}^{3 t}-1\right )\right ) {\mathrm e}^{-2 t}}{3} \]

Problem 14573

ODE	\[ \boxed {x^{\prime \prime }+9 x=\sin \left (3 t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ x = \frac {\sin \left (3 t \right )}{18}-\frac {\cos \left (3 t \right ) t}{6} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {\sin \left (3 t \right )}{18}-\frac {\cos \left (3 t \right ) t}{6} \]

Problem 14574

ODE	\[ \boxed {4 y^{\prime \prime }+4 y^{\prime }+37 y=\cos \left (3 t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = a, y^{\prime }\left (\pi \right ) = a] \end {align*}
program solution	\[ y = \frac {\left (-290 a -72\right ) \sin \left (3 t \right ) {\mathrm e}^{-\frac {t}{2}+\frac {\pi }{2}}}{870}+\left (a -\frac {1}{145}\right ) \left (\cos \left (3 t \right )+\frac {\sin \left (3 t \right )}{6}\right ) {\mathrm e}^{-\frac {t}{2}}+\frac {\cos \left (3 t \right )}{145}+\frac {12 \sin \left (3 t \right )}{145} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (-290 a -72\right ) \sin \left (3 t \right ) {\mathrm e}^{-\frac {t}{2}+\frac {\pi }{2}}}{870}+\left (\frac {\sin \left (3 t \right )}{6}+\cos \left (3 t \right )\right ) \left (a -\frac {1}{145}\right ) {\mathrm e}^{-\frac {t}{2}}+\frac {\cos \left (3 t \right )}{145}+\frac {12 \sin \left (3 t \right )}{145} \]

Problem 14575

ODE	\[ \boxed {y^{\prime \prime }+4 y=1} \]
program solution	\[ y = c_{1} \cos \left (2 t \right )+\frac {c_{2} \sin \left (2 t \right )}{2}+\frac {1}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {1}{4}+\sin \left (2 t \right ) c_{2} +\cos \left (2 t \right ) c_{1} \]

Problem 14576

ODE	\[ \boxed {y^{\prime \prime }+16 y^{\prime }=t} \]
program solution	\[ y = \frac {t^{2}}{32}+\frac {c_{1}}{16}-\frac {t}{256}+\frac {1}{4096}+c_{2} {\mathrm e}^{-16 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {t^{2}}{32}-\frac {{\mathrm e}^{-16 t} c_{1}}{16}-\frac {t}{256}+c_{2} \]

Problem 14577

ODE	\[ \boxed {y^{\prime \prime }-7 y^{\prime }+10 y={\mathrm e}^{3 t}} \]
program solution	\[ y = c_{1} {\mathrm e}^{2 t}+\frac {c_{2} {\mathrm e}^{5 t}}{3}-\frac {{\mathrm e}^{3 t}}{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{2} {\mathrm e}^{5 t}+c_{1} {\mathrm e}^{2 t}-\frac {{\mathrm e}^{3 t}}{2} \]

Problem 14578

ODE	\[ \boxed {y^{\prime \prime }+16 y=2 \cos \left (4 t \right )} \]
program solution	\[ y = c_{1} \cos \left (4 t \right )+\frac {c_{2} \sin \left (4 t \right )}{4}+\frac {t \sin \left (4 t \right )}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (16 c_{1} +1\right ) \cos \left (4 t \right )}{16}+\frac {\sin \left (4 t \right ) \left (t +4 c_{2} \right )}{4} \]

Problem 14579

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+20 y=2 t \,{\mathrm e}^{-2 t}} \]
program solution	\[ y = c_{1} \cos \left (4 t \right ) {\mathrm e}^{-2 t}+\frac {c_{2} \sin \left (4 t \right ) {\mathrm e}^{-2 t}}{4}+\frac {t \,{\mathrm e}^{-2 t}}{8} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {{\mathrm e}^{-2 t} \left (8 c_{1} \cos \left (4 t \right )+8 c_{2} \sin \left (4 t \right )+t \right )}{8} \]

Problem 14580

ODE	\[ \boxed {y^{\prime \prime }+\frac {y}{4}=\sec \left (\frac {t}{2}\right )+\csc \left (\frac {t}{2}\right )} \]
program solution	\[ y = c_{1} \cos \left (\frac {t}{2}\right )+2 c_{2} \sin \left (\frac {t}{2}\right )+\left (-2 t -2 \ln \left (\sec \left (\frac {t}{2}\right )^{2}\right )\right ) \cos \left (\frac {t}{2}\right )+2 \left (t -\ln \left (\csc \left (\frac {t}{2}\right )^{2}\right )\right ) \sin \left (\frac {t}{2}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = 4 \cos \left (\frac {t}{2}\right ) \ln \left (\cos \left (\frac {t}{2}\right )\right )+4 \sin \left (\frac {t}{2}\right ) \ln \left (\sin \left (\frac {t}{2}\right )\right )+\left (c_{1} -2 t \right ) \cos \left (\frac {t}{2}\right )+2 \left (t +\frac {c_{2}}{2}\right ) \sin \left (\frac {t}{2}\right ) \]

Problem 14581

ODE	\[ \boxed {y^{\prime \prime }+16 y=\csc \left (4 t \right )} \]
program solution	\[ y = c_{1} \cos \left (4 t \right )+\frac {c_{2} \sin \left (4 t \right )}{4}-\frac {t \cos \left (4 t \right )}{4}-\frac {\ln \left (\csc \left (4 t \right )^{2}\right ) \sin \left (4 t \right )}{32} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {\ln \left (\csc \left (4 t \right )\right ) \sin \left (4 t \right )}{16}+\frac {\left (-t +4 c_{1} \right ) \cos \left (4 t \right )}{4}+c_{2} \sin \left (4 t \right ) \]

Problem 14582

ODE	\[ \boxed {y^{\prime \prime }+16 y=\cot \left (4 t \right )} \]
program solution	\[ y = c_{1} \cos \left (4 t \right )+\frac {c_{2} \sin \left (4 t \right )}{4}+\frac {\sin \left (4 t \right ) \ln \left (-\cot \left (4 t \right )+\csc \left (4 t \right )\right )}{16} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{2} \sin \left (4 t \right )+c_{1} \cos \left (4 t \right )+\frac {\sin \left (4 t \right ) \ln \left (\csc \left (4 t \right )-\cot \left (4 t \right )\right )}{16} \]

Problem 14583

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+50 y={\mathrm e}^{-t} \csc \left (7 t \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-t} \cos \left (7 t \right )+\frac {c_{2} {\mathrm e}^{-t} \sin \left (7 t \right )}{7}-\frac {{\mathrm e}^{-t} \left (14 t \cos \left (7 t \right )+\ln \left (\csc \left (7 t \right )^{2}\right ) \sin \left (7 t \right )\right )}{98} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {\left (\frac {\sin \left (7 t \right ) \ln \left (\csc \left (7 t \right )\right )}{7}+\left (t -7 c_{1} \right ) \cos \left (7 t \right )-7 \sin \left (7 t \right ) c_{2} \right ) {\mathrm e}^{-t}}{7} \]

Problem 14584

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+25 y={\mathrm e}^{-3 t} \left (\sec \left (4 t \right )+\csc \left (4 t \right )\right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-3 t} \cos \left (4 t \right )+\frac {c_{2} {\mathrm e}^{-3 t} \sin \left (4 t \right )}{4}-\frac {\left (\frac {\ln \left (\csc \left (4 t \right )^{2}\right ) \sin \left (4 t \right )}{8}+\frac {\cos \left (4 t \right ) \ln \left (\sec \left (4 t \right )^{2}\right )}{8}+t \left (\cos \left (4 t \right )-\sin \left (4 t \right )\right )\right ) {\mathrm e}^{-3 t}}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {\left (-\frac {\ln \left (\cos \left (4 t \right )\right ) \cos \left (4 t \right )}{4}-\frac {\ln \left (\sin \left (4 t \right )\right ) \sin \left (4 t \right )}{4}+\left (-4 c_{1} +t \right ) \cos \left (4 t \right )-\sin \left (4 t \right ) \left (t +4 c_{2} \right )\right ) {\mathrm e}^{-3 t}}{4} \]

Problem 14585

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }+26 y={\mathrm e}^{t} \left (\sec \left (5 t \right )+\csc \left (5 t \right )\right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{t} \cos \left (5 t \right )+\frac {c_{2} {\mathrm e}^{t} \sin \left (5 t \right )}{5}-\frac {{\mathrm e}^{t} \left (\frac {\sin \left (5 t \right ) \ln \left (\csc \left (5 t \right )^{2}\right )}{10}+\frac {\cos \left (5 t \right ) \ln \left (\sec \left (5 t \right )^{2}\right )}{10}+t \left (\cos \left (5 t \right )-\sin \left (5 t \right )\right )\right )}{5} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {{\mathrm e}^{t} \left (-\frac {\ln \left (\cos \left (5 t \right )\right ) \cos \left (5 t \right )}{5}-\frac {\ln \left (\sin \left (5 t \right )\right ) \sin \left (5 t \right )}{5}+\left (t -5 c_{1} \right ) \cos \left (5 t \right )-\sin \left (5 t \right ) \left (t +5 c_{2} \right )\right )}{5} \]

Problem 14586

ODE	\[ \boxed {y^{\prime \prime }+12 y^{\prime }+37 y={\mathrm e}^{-6 t} \csc \left (t \right )} \]
program solution	\[ y = {\mathrm e}^{-6 t} \left (c_{1} \cos \left (t \right )+c_{2} \sin \left (t \right )\right )+{\mathrm e}^{-6 t} \left (-t \cos \left (t \right )+\ln \left (\sin \left (t \right )\right ) \sin \left (t \right )\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\left (\sin \left (t \right ) \ln \left (\csc \left (t \right )\right )+\left (-c_{1} +t \right ) \cos \left (t \right )-\sin \left (t \right ) c_{2} \right ) {\mathrm e}^{-6 t} \]

Problem 14587

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+34 y={\mathrm e}^{3 t} \tan \left (5 t \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{3 t} \cos \left (5 t \right )+\frac {c_{2} {\mathrm e}^{3 t} \sin \left (5 t \right )}{5}-\frac {{\mathrm e}^{3 t} \cos \left (5 t \right ) \ln \left (\sec \left (5 t \right )+\tan \left (5 t \right )\right )}{25} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {{\mathrm e}^{3 t} \left (\ln \left (\sec \left (5 t \right )+\tan \left (5 t \right )\right ) \cos \left (5 t \right )-25 \cos \left (5 t \right ) c_{1} -25 \sin \left (5 t \right ) c_{2} \right )}{25} \]

Problem 14588

ODE	\[ \boxed {y^{\prime \prime }-10 y^{\prime }+34 y={\mathrm e}^{5 t} \cot \left (3 t \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{5 t} \cos \left (3 t \right )+\frac {c_{2} \sin \left (3 t \right ) {\mathrm e}^{5 t}}{3}+\frac {\sin \left (3 t \right ) {\mathrm e}^{5 t} \ln \left (-\cot \left (3 t \right )+\csc \left (3 t \right )\right )}{9} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {{\mathrm e}^{5 t} \left (\ln \left (\csc \left (3 t \right )-\cot \left (3 t \right )\right ) \sin \left (3 t \right )+9 c_{2} \sin \left (3 t \right )+9 c_{1} \cos \left (3 t \right )\right )}{9} \]

Problem 14589

ODE	\[ \boxed {y^{\prime \prime }-12 y^{\prime }+37 y={\mathrm e}^{6 t} \sec \left (t \right )} \]
program solution	\[ y = {\mathrm e}^{6 t} \left (c_{1} \cos \left (t \right )+c_{2} \sin \left (t \right )\right )+{\mathrm e}^{6 t} \left (\cos \left (t \right ) \ln \left (\cos \left (t \right )\right )+t \sin \left (t \right )\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (-\cos \left (t \right ) \ln \left (\sec \left (t \right )\right )+\cos \left (t \right ) c_{1} +\sin \left (t \right ) \left (c_{2} +t \right )\right ) {\mathrm e}^{6 t} \]

Problem 14590

ODE	\[ \boxed {y^{\prime \prime }-8 y^{\prime }+17 y={\mathrm e}^{4 t} \sec \left (t \right )} \]
program solution	\[ y = {\mathrm e}^{4 t} \left (c_{1} \cos \left (t \right )+c_{2} \sin \left (t \right )\right )+{\mathrm e}^{4 t} \left (\cos \left (t \right ) \ln \left (\cos \left (t \right )\right )+t \sin \left (t \right )\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (-\cos \left (t \right ) \ln \left (\sec \left (t \right )\right )+\cos \left (t \right ) c_{1} +\sin \left (t \right ) \left (c_{2} +t \right )\right ) {\mathrm e}^{4 t} \]

Problem 14591

ODE	\[ \boxed {y^{\prime \prime }-9 y=\frac {1}{1+{\mathrm e}^{3 t}}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-3 t}+\frac {c_{2} {\mathrm e}^{3 t}}{6}-\frac {\left (3 \,{\mathrm e}^{6 t} \ln \left ({\mathrm e}^{t}\right )-{\mathrm e}^{6 t} \ln \left ({\mathrm e}^{t}+1\right )-{\mathrm e}^{6 t} \ln \left ({\mathrm e}^{2 t}-{\mathrm e}^{t}+1\right )+{\mathrm e}^{3 t}+\ln \left (1+{\mathrm e}^{3 t}\right )\right ) {\mathrm e}^{-3 t}}{18} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {{\mathrm e}^{-3 t} \left (\ln \left ({\mathrm e}^{2 t}-{\mathrm e}^{t}+1\right ) {\mathrm e}^{6 t}-\ln \left (1+{\mathrm e}^{3 t}\right )+\ln \left (1+{\mathrm e}^{t}\right ) {\mathrm e}^{6 t}+\left (18 c_{1} -3 \ln \left ({\mathrm e}^{t}\right )\right ) {\mathrm e}^{6 t}+18 c_{2} -{\mathrm e}^{3 t}\right )}{18} \]

Problem 14592

ODE	\[ \boxed {y^{\prime \prime }-25 y=\frac {1}{1-{\mathrm e}^{5 t}}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-5 t}+\frac {c_{2} {\mathrm e}^{5 t}}{10}-\frac {\left (\ln \left ({\mathrm e}^{t}-1\right ) {\mathrm e}^{10 t}-5 \ln \left ({\mathrm e}^{t}\right ) {\mathrm e}^{10 t}+{\mathrm e}^{10 t} \ln \left ({\mathrm e}^{4 t}+{\mathrm e}^{3 t}+{\mathrm e}^{2 t}+{\mathrm e}^{t}+1\right )+{\mathrm e}^{5 t}-\ln \left ({\mathrm e}^{5 t}-1\right )\right ) {\mathrm e}^{-5 t}}{50} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {\left (\ln \left ({\mathrm e}^{4 t}+{\mathrm e}^{3 t}+{\mathrm e}^{2 t}+{\mathrm e}^{t}+1\right ) {\mathrm e}^{10 t}+\ln \left ({\mathrm e}^{t}-1\right ) {\mathrm e}^{10 t}-5 \ln \left ({\mathrm e}^{t}\right ) {\mathrm e}^{10 t}-50 c_{2} {\mathrm e}^{10 t}+{\mathrm e}^{5 t}-\ln \left (1-{\mathrm e}^{5 t}\right )-50 c_{1} \right ) {\mathrm e}^{-5 t}}{50} \]

Problem 14593

ODE	\[ \boxed {y^{\prime \prime }-y=2 \sinh \left (t \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-t}+\frac {c_{2} {\mathrm e}^{t}}{2}+\frac {\left (2 t -1-\sinh \left (2 t \right )-\cosh \left (2 t \right )\right ) {\mathrm e}^{-t}}{4}+\frac {\left (t -\frac {\sinh \left (2 t \right )}{2}+\frac {\cosh \left (2 t \right )}{2}\right ) {\mathrm e}^{t}}{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (2 t +4 c_{2} \right ) {\mathrm e}^{-t}}{4}+\frac {\left (t +2 c_{1} -\frac {1}{2}\right ) {\mathrm e}^{t}}{2} \]

Problem 14594

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }+y=\frac {{\mathrm e}^{t}}{t}} \]
program solution	\[ y = {\mathrm e}^{t} \left (c_{2} t +c_{1} \right )+{\mathrm e}^{t} t \left (-1+\ln \left (t \right )\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (t \ln \left (t \right )+t \left (c_{1} -1\right )+c_{2} \right ) {\mathrm e}^{t} \]

Problem 14595

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+4 y=\frac {{\mathrm e}^{2 t}}{t^{2}}} \]
program solution	\[ y = {\mathrm e}^{2 t} \left (c_{2} t +c_{1} \right )+{\mathrm e}^{2 t} \left (-\ln \left (t \right )-1\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{2 t} \left (-1+c_{1} t -\ln \left (t \right )+c_{2} \right ) \]

Problem 14596

ODE	\[ \boxed {y^{\prime \prime }+8 y^{\prime }+16 y=\frac {{\mathrm e}^{-4 t}}{t^{4}}} \]
program solution	\[ y = {\mathrm e}^{-4 t} \left (c_{2} t +c_{1} \right )+\frac {{\mathrm e}^{-4 t}}{6 t^{2}} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {{\mathrm e}^{-4 t} \left (6 c_{1} t^{3}+6 c_{2} t^{2}+1\right )}{6 t^{2}} \]

Problem 14597

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+9 y=\frac {{\mathrm e}^{-3 t}}{t}} \]
program solution	\[ y = {\mathrm e}^{-3 t} \left (c_{2} t +c_{1} \right )+{\mathrm e}^{-3 t} t \left (-1+\ln \left (t \right )\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (t \ln \left (t \right )+t \left (c_{1} -1\right )+c_{2} \right ) {\mathrm e}^{-3 t} \]

Problem 14598

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+9 y={\mathrm e}^{-3 t} \ln \left (t \right )} \]
program solution	\[ y = {\mathrm e}^{-3 t} \left (c_{2} t +c_{1} \right )+\frac {t^{2} {\mathrm e}^{-3 t} \left (-3+2 \ln \left (t \right )\right )}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {{\mathrm e}^{-3 t} \left (2 \ln \left (t \right ) t^{2}+4 c_{1} t -3 t^{2}+4 c_{2} \right )}{4} \]

Problem 14599

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }+2 y=\cos \left ({\mathrm e}^{t}\right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 t}+c_{2} {\mathrm e}^{-t}-\left (\cos \left ({\mathrm e}^{t}\right )+1\right ) {\mathrm e}^{-2 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\left (-c_{2} {\mathrm e}^{t}+\cos \left ({\mathrm e}^{t}\right )+c_{1} +1\right ) {\mathrm e}^{-2 t} \]

Problem 14600

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+4 y={\mathrm e}^{-2 t} \sqrt {-t^{2}+1}} \]
program solution	\[ y = {\mathrm e}^{-2 t} \left (c_{2} t +c_{1} \right )+\frac {{\mathrm e}^{-2 t} \left (t^{2} \sqrt {-t^{2}+1}+3 t \arcsin \left (t \right )+2 \sqrt {-t^{2}+1}\right )}{6} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (\left (t^{2}+2\right ) \sqrt {-t^{2}+1}+6 c_{1} t +3 \arcsin \left (t \right ) t +6 c_{2} \right ) {\mathrm e}^{-2 t}}{6} \]