2.17.147 Problems 14601 to 14700

Problem 14601

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

Problem 14602

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

Problem 14603

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

Problem 14604

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

Problem 14605

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

Problem 14606

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

Problem 14607

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

Problem 14608

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

Problem 14609

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

Problem 14610

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

Problem 14611

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

Problem 14612

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

Problem 14613

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

Problem 14614

ODE	\[ \boxed {y^{\prime \prime }+9 y=\frac {\csc \left (3 t \right )}{2}} \] With initial conditions \begin {align*} \left [y \left (\frac {\pi }{4}\right ) = \sqrt {2}, y^{\prime }\left (\frac {\pi }{4}\right ) = 0\right ] \end {align*}
program solution	\[ y = -\frac {\ln \left (\csc \left (3 t \right )^{2}\right ) \sin \left (3 t \right )}{36}+\frac {\left (-24-4 t +\pi \right ) \cos \left (3 t \right )}{24}+\frac {\sin \left (3 t \right ) \left (\ln \left (2\right )+36\right )}{36} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {\ln \left (\csc \left (3 t \right )\right ) \sin \left (3 t \right )}{18}+\frac {\left (-24-4 t +\pi \right ) \cos \left (3 t \right )}{24}+\frac {\left (\ln \left (2\right )+36\right ) \sin \left (3 t \right )}{36} \]

Problem 14615

ODE	\[ \boxed {y^{\prime \prime }+4 y=\sec \left (2 t \right )^{2}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \frac {\cos \left (2 t \right )}{4}+\frac {\sin \left (2 t \right )}{2}-\frac {1}{4}+\frac {\ln \left (\sec \left (2 t \right )+\tan \left (2 t \right )\right ) \sin \left (2 t \right )}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\sin \left (2 t \right )}{2}+\frac {\cos \left (2 t \right )}{4}-\frac {1}{4}+\frac {\ln \left (\sec \left (2 t \right )+\tan \left (2 t \right )\right ) \sin \left (2 t \right )}{4} \]

Problem 14616

ODE	\[ \boxed {y^{\prime \prime }-16 y=16 t \,{\mathrm e}^{-4 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {\left (-32 t^{2}-8 t -1\right ) {\mathrm e}^{-4 t}}{32}+\frac {{\mathrm e}^{4 t}}{32} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (-32 t^{2}-8 t -1\right ) {\mathrm e}^{-4 t}}{32}+\frac {{\mathrm e}^{4 t}}{32} \]

Problem 14617

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

Problem 14618

ODE	\[ \boxed {y^{\prime \prime }+4 y=\sec \left (2 t \right )+\tan \left (2 t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = -\frac {i \cos \left (2 t \right ) \pi }{4}+\cos \left (2 t \right )+\frac {\sin \left (2 t \right )}{4}+\frac {\sin \left (2 t \right ) t}{2}+\frac {\cos \left (2 t \right ) \ln \left (\sin \left (2 t \right )-1\right )}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\sin \left (2 t \right )}{4}-\frac {i \cos \left (2 t \right ) \pi }{4}+\cos \left (2 t \right )+\frac {t \sin \left (2 t \right )}{2}+\frac {\cos \left (2 t \right ) \ln \left (\sin \left (2 t \right )-1\right )}{4} \]

Problem 14619

ODE	\[ \boxed {y^{\prime \prime }+9 y=\csc \left (3 t \right )} \] With initial conditions \begin {align*} \left [y \left (\frac {\pi }{12}\right ) = 0, y^{\prime }\left (\frac {\pi }{12}\right ) = 1\right ] \end {align*}
program solution	\[ y = -\frac {\ln \left (\csc \left (3 t \right )^{2}\right ) \sin \left (3 t \right )}{18}+\frac {\left (-12 t +\pi -6 \sqrt {2}\right ) \cos \left (3 t \right )}{36}+\frac {\sin \left (3 t \right ) \left (\ln \left (2\right )+3 \sqrt {2}\right )}{18} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {\ln \left (\csc \left (3 t \right )\right ) \sin \left (3 t \right )}{9}+\frac {\left (-12 t +\pi -6 \sqrt {2}\right ) \cos \left (3 t \right )}{36}+\frac {\sin \left (3 t \right ) \left (3 \sqrt {2}+\ln \left (2\right )\right )}{18} \]

Problem 14620

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+3 y=65 \cos \left (2 t \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-3 t}+\frac {c_{2} {\mathrm e}^{-t}}{2}-\cos \left (2 t \right )+8 \sin \left (2 t \right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{2} {\mathrm e}^{-3 t}+{\mathrm e}^{-t} c_{1} -\cos \left (2 t \right )+8 \sin \left (2 t \right ) \]

Problem 14621

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

Problem 14622

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

Problem 14623

ODE	\[ \boxed {t^{2} y^{\prime \prime }-4 t y^{\prime }-6 y=2 \ln \left (t \right )} \]
program solution	\[ y = \frac {126 c_{2} t^{7}-42 \ln \left (t \right ) t -18 c_{1} +35 t}{126 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = t^{6} c_{2} -\frac {\ln \left (t \right )}{3}+\frac {5}{18}+\frac {c_{1}}{t} \]

Problem 14624

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

Problem 14625

ODE	\[ \boxed {-2 t \left (1+t \right ) y=-{\mathrm e}^{-2 t} \left (y y^{\prime \prime }-{y^{\prime }}^{2}\right )} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 14626

ODE	\[ \boxed {y^{\prime \prime }+4 y=f \left (t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 2] \end {align*}
program solution	\[ y = \sin \left (2 t \right )-\frac {\left (\int _{0}^{t}\sin \left (2 \alpha \right ) f \left (\alpha \right )d \alpha \right ) \cos \left (2 t \right )}{2}+\frac {\left (\int _{0}^{t}\cos \left (2 \alpha \right ) f \left (\alpha \right )d \alpha \right ) \sin \left (2 t \right )}{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \sin \left (2 t \right )+\frac {\left (\int _{0}^{t}\cos \left (2 \textit {\_z1} \right ) f \left (\textit {\_z1} \right )d \textit {\_z1} \right ) \sin \left (2 t \right )}{2}-\frac {\left (\int _{0}^{t}\sin \left (2 \textit {\_z1} \right ) f \left (\textit {\_z1} \right )d \textit {\_z1} \right ) \cos \left (2 t \right )}{2} \]

Problem 14627

ODE	\[ \boxed {t^{2} y^{\prime \prime }-4 t y^{\prime }+\left (t^{2}+6\right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \cos \left (t \right ) t^{2} \end {align*}
program solution	\[ y = \cos \left (t \right ) t^{2} c_{1} +c_{2} \cos \left (t \right ) t^{2} \tan \left (t \right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = t^{2} \left (c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right )\right ) \]

Problem 14628

ODE	\[ \boxed {t^{2} y^{\prime \prime }-4 t y^{\prime }+\left (t^{2}+6\right ) y=t^{3}+2 t} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	N/A
Maple solution	\[ y \left (t \right ) = t \left (\sin \left (t \right ) t c_{2} +t \cos \left (t \right ) c_{1} +1\right ) \]

Problem 14629

ODE	\[ \boxed {t y^{\prime \prime }+2 y^{\prime }+y t=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \frac {\cos \left (t \right )}{t} \end {align*}
program solution	\[ y = \frac {\cos \left (t \right ) c_{1}}{t}+\frac {c_{2} \cos \left (t \right ) \tan \left (t \right )}{t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right )}{t} \]

Problem 14630

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

Problem 14631

ODE	\[ \boxed {4 t^{2} y^{\prime \prime }+4 t y^{\prime }+\left (16 t^{2}-1\right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \frac {\sin \left (2 t \right )}{\sqrt {t}} \end {align*}
program solution	\[ y = \frac {\sin \left (2 t \right ) c_{1}}{\sqrt {t}}-\frac {c_{2} \sin \left (2 t \right ) \cot \left (2 t \right )}{2 \sqrt {t}} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\sin \left (2 t \right ) c_{1} +c_{2} \cos \left (2 t \right )}{\sqrt {t}} \]

Problem 14632

ODE	\[ \boxed {4 t^{2} y^{\prime \prime }+4 t y^{\prime }+\left (16 t^{2}-1\right ) y=16 t^{\frac {3}{2}}} \] With initial conditions \begin {align*} \left [y \left (\pi \right ) = 0, y \left (\frac {3 \pi }{2}\right ) = 0\right ] \end {align*}
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 14633

ODE	\[ \boxed {4 t^{2} y^{\prime \prime }+4 t y^{\prime }+\left (16 t^{2}-1\right ) y=16 t^{\frac {3}{2}}} \] With initial conditions \begin {align*} [y \left (\pi \right ) = 0, y^{\prime }\left (2 \pi \right ) = 0] \end {align*}
program solution	\[ y = -\frac {-1+\cos \left (2 t \right )}{\sqrt {t}} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {-1+\cos \left (2 t \right )}{\sqrt {t}} \]

Problem 14634

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

Problem 14635

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

Problem 14636

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

Problem 14637

ODE	\[ \boxed {y^{\prime \prime \prime }-10 y^{\prime \prime }+25 y^{\prime }=0} \]
program solution	\[ y = c_{1} +c_{2} {\mathrm e}^{5 t}+t \,{\mathrm e}^{5 t} c_{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (c_{3} t +c_{2} \right ) {\mathrm e}^{5 t}+c_{1} \]

Problem 14638

ODE	\[ \boxed {8 y^{\prime \prime \prime }+y^{\prime \prime }=0} \]
program solution	\[ y = c_{2} t +c_{1} +{\mathrm e}^{-\frac {t}{8}} c_{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} +t c_{2} +c_{3} {\mathrm e}^{-\frac {t}{8}} \]

Problem 14639

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+16 y^{\prime \prime }=0} \]
program solution	\[ y = c_{2} t +c_{1} +{\mathrm e}^{-4 i t} c_{3} +{\mathrm e}^{4 i t} c_{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} +t c_{2} +c_{3} \sin \left (4 t \right )+c_{4} \cos \left (4 t \right ) \]

Problem 14640

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

Problem 14641

ODE	\[ \boxed {3 y^{\prime \prime \prime }-4 y^{\prime \prime }-5 y^{\prime }+2 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-t}+c_{2} {\mathrm e}^{2 t}+{\mathrm e}^{\frac {t}{3}} c_{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (c_{3} {\mathrm e}^{3 t}+{\mathrm e}^{\frac {4 t}{3}} c_{1} +c_{2} \right ) {\mathrm e}^{-t} \]

Problem 14642

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

Problem 14643

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

Problem 14644

ODE	\[ \boxed {5 y^{\prime \prime \prime }-15 y^{\prime }+11 y=0} \]
program solution	\[ y = {\mathrm e}^{\left (\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} c_{1} +{\mathrm e}^{\left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}-\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right ) t} c_{2} +{\mathrm e}^{\left (\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{20}+\frac {5}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}{10}+\frac {10}{\left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} c_{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (c_{2} {\mathrm e}^{\frac {3 \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}+100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {3}\, \sqrt {7}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {3}\, \sqrt {7}\right )^{\frac {1}{3}}}\right )+c_{3} {\mathrm e}^{\frac {3 \left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}+100\right ) t}{20 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (1100+100 \sqrt {3}\, \sqrt {7}\right )^{\frac {2}{3}}-100\right ) t}{20 \left (1100+100 \sqrt {3}\, \sqrt {7}\right )^{\frac {1}{3}}}\right )+c_{1} \right ) {\mathrm e}^{-\frac {\left (\left (1100+100 \sqrt {21}\right )^{\frac {2}{3}}+100\right ) t}{10 \left (1100+100 \sqrt {21}\right )^{\frac {1}{3}}}} \]

Problem 14645

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+y^{\prime \prime \prime }=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-t}+c_{2} +c_{3} t +t^{2} c_{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} +t c_{2} +c_{3} t^{2}+c_{4} {\mathrm e}^{-t} \]

Problem 14646

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-9 y^{\prime \prime }=0} \]
program solution	\[ y = c_{2} t +c_{1} +{\mathrm e}^{-3 t} c_{3} +{\mathrm e}^{3 t} c_{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} +t c_{2} +c_{3} {\mathrm e}^{-3 t}+c_{4} {\mathrm e}^{3 t} \]

Problem 14647

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-16 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 t}+c_{2} {\mathrm e}^{2 t}+{\mathrm e}^{-2 i t} c_{3} +{\mathrm e}^{2 i t} c_{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} {\mathrm e}^{2 t}+c_{2} {\mathrm e}^{-2 t}+c_{3} \sin \left (2 t \right )+c_{4} \cos \left (2 t \right ) \]

Problem 14648

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-6 y^{\prime \prime \prime }-y^{\prime \prime }+54 y^{\prime }-72 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-3 t}+c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{3 t}+{\mathrm e}^{4 t} c_{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left ({\mathrm e}^{7 t} c_{2} +c_{3} {\mathrm e}^{6 t}+c_{4} {\mathrm e}^{5 t}+c_{1} \right ) {\mathrm e}^{-3 t} \]

Problem 14649

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+7 y^{\prime \prime \prime }+6 y^{\prime \prime }-32 y^{\prime }-32 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-t}+c_{2} {\mathrm e}^{2 t}+{\mathrm e}^{-4 t} c_{3} +t \,{\mathrm e}^{-4 t} c_{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (c_{2} {\mathrm e}^{6 t}+c_{1} {\mathrm e}^{3 t}+t c_{4} +c_{3} \right ) {\mathrm e}^{-4 t} \]

Problem 14650

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }-2 y^{\prime \prime }+8 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 t}+c_{2} {\mathrm e}^{-2 t} t +{\mathrm e}^{\left (1+i\right ) t} c_{3} +{\mathrm e}^{\left (1-i\right ) t} c_{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (c_{3} {\mathrm e}^{3 t} \sin \left (t \right )+c_{4} {\mathrm e}^{3 t} \cos \left (t \right )+t c_{2} +c_{1} \right ) {\mathrm e}^{-2 t} \]

Problem 14651

ODE	\[ \boxed {y^{\left (5\right )}+4 y^{\prime \prime \prime \prime }=0} \]
program solution	\[ y = t^{3} c_{4} +t^{2} c_{3} +c_{2} t +c_{1} +{\mathrm e}^{-4 t} c_{5} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} +t c_{2} +c_{3} t^{2}+c_{4} t^{3}+c_{5} {\mathrm e}^{-4 t} \]

Problem 14652

ODE	\[ \boxed {y^{\left (5\right )}+4 y^{\prime \prime \prime }=0} \]
program solution	\[ y = t^{2} c_{3} +c_{2} t +c_{1} +{\mathrm e}^{-2 i t} c_{4} +{\mathrm e}^{2 i t} c_{5} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} +t c_{2} +c_{3} t^{2}+c_{4} \sin \left (2 t \right )+c_{5} \cos \left (2 t \right ) \]

Problem 14653

ODE	\[ \boxed {y^{\left (5\right )}+3 y^{\prime \prime \prime \prime }+3 y^{\prime \prime \prime }+y^{\prime \prime }=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-t}+c_{2} t \,{\mathrm e}^{-t}+t^{2} {\mathrm e}^{-t} c_{3} +c_{4} +t c_{5} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (c_{5} t^{2}+t c_{4} +c_{3} \right ) {\mathrm e}^{-t}+t c_{2} +c_{1} \]

Problem 14654

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+2 y^{\prime \prime }+y=0} \]
program solution	\[ y = {\mathrm e}^{i t} c_{1} +t \,{\mathrm e}^{i t} c_{2} +{\mathrm e}^{-i t} c_{3} +t \,{\mathrm e}^{-i t} c_{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (t c_{4} +c_{2} \right ) \cos \left (t \right )+\sin \left (t \right ) \left (c_{3} t +c_{1} \right ) \]

Problem 14655

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+8 y^{\prime \prime }+16 y=0} \]
program solution	\[ y = {\mathrm e}^{-2 i t} c_{1} +t \,{\mathrm e}^{-2 i t} c_{2} +{\mathrm e}^{2 i t} c_{3} +t \,{\mathrm e}^{2 i t} c_{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (t c_{4} +c_{2} \right ) \cos \left (2 t \right )+\sin \left (2 t \right ) \left (c_{3} t +c_{1} \right ) \]

Problem 14656

ODE	\[ \boxed {y^{\left (6\right )}+3 y^{\prime \prime \prime \prime }+3 y^{\prime \prime }+y=0} \]
program solution	\[ y = {\mathrm e}^{i t} c_{1} +t \,{\mathrm e}^{i t} c_{2} +t^{2} {\mathrm e}^{i t} c_{3} +{\mathrm e}^{-i t} c_{4} +t \,{\mathrm e}^{-i t} c_{5} +t^{2} {\mathrm e}^{-i t} c_{6} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (c_{6} t^{2}+t c_{4} +c_{2} \right ) \cos \left (t \right )+\sin \left (t \right ) \left (c_{5} t^{2}+c_{3} t +c_{1} \right ) \]

Problem 14657

ODE	\[ \boxed {y^{\left (6\right )}+12 y^{\prime \prime \prime \prime }+48 y^{\prime \prime }+64 y=0} \]
program solution	\[ y = {\mathrm e}^{-2 i t} c_{1} +t \,{\mathrm e}^{-2 i t} c_{2} +t^{2} {\mathrm e}^{-2 i t} c_{3} +{\mathrm e}^{2 i t} c_{4} +t \,{\mathrm e}^{2 i t} c_{5} +t^{2} {\mathrm e}^{2 i t} c_{6} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (c_{6} t^{2}+t c_{4} +c_{2} \right ) \cos \left (2 t \right )+\sin \left (2 t \right ) \left (c_{5} t^{2}+c_{3} t +c_{1} \right ) \]

Problem 14658

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

Problem 14659

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

Problem 14660

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+16 y^{\prime \prime \prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1, y^{\prime \prime }\left (0\right ) = 0, y^{\prime \prime \prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \frac {1}{4096}+\frac {t^{2}}{32}+\frac {255 t}{256}-\frac {{\mathrm e}^{-16 t}}{4096} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {1}{4096}+\frac {255 t}{256}+\frac {t^{2}}{32}-\frac {{\mathrm e}^{-16 t}}{4096} \]

Problem 14661

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-8 y^{\prime \prime }+16 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = 8, y^{\prime \prime \prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = -t \,{\mathrm e}^{-2 t}+t \,{\mathrm e}^{2 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -t \left (-{\mathrm e}^{2 t}+{\mathrm e}^{-2 t}\right ) \]

Problem 14662

ODE	\[ \boxed {24 y^{\prime \prime \prime }-26 y^{\prime \prime }+9 y^{\prime }-y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1, y^{\prime \prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = 54 \,{\mathrm e}^{\frac {t}{3}}-40 \,{\mathrm e}^{\frac {t}{4}}-14 \,{\mathrm e}^{\frac {t}{2}} \] Verified OK.
Maple solution	\[ y \left (t \right ) = 54 \,{\mathrm e}^{\frac {t}{3}}-14 \,{\mathrm e}^{\frac {t}{2}}-40 \,{\mathrm e}^{\frac {t}{4}} \]

Problem 14663

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-5 y^{\prime \prime }+4 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = 3, y^{\prime \prime }\left (0\right ) = -7, y^{\prime \prime \prime }\left (0\right ) = 15] \end {align*}
program solution	\[ y = {\mathrm e}^{-t}-2 \,{\mathrm e}^{-2 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{-t}-2 \,{\mathrm e}^{-2 t} \]

Problem 14664

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-16 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 2, y^{\prime \prime }\left (0\right ) = 4, y^{\prime \prime \prime }\left (0\right ) = -24] \end {align*}
program solution	\[ y = {\mathrm e}^{-2 t}+2 \sin \left (2 t \right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{-2 t}+2 \sin \left (2 t \right ) \]

Problem 14665

ODE	\[ \boxed {8 y^{\left (5\right )}+4 y^{\prime \prime \prime \prime }+66 y^{\prime \prime \prime }-41 y^{\prime \prime }-37 y^{\prime }=0} \] With initial conditions \begin {align*} \left [y \left (0\right ) = 4, y^{\prime }\left (0\right ) = -14, y^{\prime \prime }\left (0\right ) = -14, y^{\prime \prime \prime }\left (0\right ) = 139, y^{\prime \prime \prime \prime }\left (0\right ) = -{\frac {29}{4}}\right ] \end {align*}
program solution	\[ y = \left (\frac {3}{2}-2 i\right ) {\mathrm e}^{\left (-\frac {1}{2}-3 i\right ) t}+\left (\frac {3}{2}+2 i\right ) {\mathrm e}^{\left (-\frac {1}{2}+3 i\right ) t}+{\mathrm e}^{-\frac {t}{2}} \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{-\frac {t}{2}} \left (1-4 \sin \left (3 t \right )+3 \cos \left (3 t \right )\right ) \]

Problem 14666

ODE	\[ \boxed {2 y^{\left (5\right )}+7 y^{\prime \prime \prime \prime }+17 y^{\prime \prime \prime }+17 y^{\prime \prime }+5 y^{\prime }=0} \] With initial conditions \begin {align*} \left [y \left (0\right ) = -3, y^{\prime }\left (0\right ) = {\frac {15}{2}}, y^{\prime \prime }\left (0\right ) = {\frac {17}{4}}, y^{\prime \prime \prime }\left (0\right ) = -{\frac {385}{8}}, y^{\prime \prime \prime \prime }\left (0\right ) = {\frac {1217}{16}}\right ] \end {align*}
program solution	\[ y = \left (-2+i\right ) {\mathrm e}^{\left (-1-2 i\right ) t}+\left (-2-i\right ) {\mathrm e}^{\left (-1+2 i\right ) t}+{\mathrm e}^{-\frac {t}{2}} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (2 \sin \left (2 t \right )-4 \cos \left (2 t \right )\right ) {\mathrm e}^{-t}+{\mathrm e}^{-\frac {t}{2}} \]

Problem 14667

ODE	\[ \boxed {y^{\left (5\right )}+8 y^{\prime \prime \prime \prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 8, y^{\prime }\left (0\right ) = 4, y^{\prime \prime }\left (0\right ) = 0, y^{\prime \prime \prime }\left (0\right ) = 48, y^{\prime \prime \prime \prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = 8 t^{3}+4 t +8 \] Verified OK.
Maple solution	\[ y \left (t \right ) = 8 t^{3}+4 t +8 \]

Problem 14668

ODE	\[ \boxed {y^{\left (6\right )}-3 y^{\prime \prime \prime \prime }+3 y^{\prime \prime }-y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 16, y^{\prime }\left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = 0, y^{\prime \prime \prime }\left (0\right ) = 0, y^{\prime \prime \prime \prime }\left (0\right ) = 0, y^{\left (5\right )}\left (0\right ) = 0] \end {align*}
program solution	\[ y = \left (t^{2}+5 t +8\right ) {\mathrm e}^{-t}+{\mathrm e}^{t} \left (t^{2}-5 t +8\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (t^{2}+5 t +8\right ) {\mathrm e}^{-t}+{\mathrm e}^{t} \left (t^{2}-5 t +8\right ) \]

Problem 14669

ODE	\[ \boxed {y^{\prime \prime \prime }+9 y^{\prime \prime }+16 y^{\prime }-26 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{t}+{\mathrm e}^{\left (-5-i\right ) t} c_{2} +{\mathrm e}^{\left (-5+i\right ) t} c_{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left ({\mathrm e}^{6 t} c_{1} +\sin \left (t \right ) c_{2} +\cos \left (t \right ) c_{3} \right ) {\mathrm e}^{-5 t} \]

Problem 14670

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+12 y^{\prime \prime \prime }+60 y^{\prime \prime }+124 y^{\prime }+75 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-t}+c_{2} {\mathrm e}^{-3 t}+{\mathrm e}^{\left (-4-3 i\right ) t} c_{3} +{\mathrm e}^{\left (-4+3 i\right ) t} c_{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} {\mathrm e}^{-3 t}+c_{2} {\mathrm e}^{-t}+c_{3} {\mathrm e}^{-4 t} \sin \left (3 t \right )+c_{4} {\mathrm e}^{-4 t} \cos \left (3 t \right ) \]

Problem 14671

ODE	\[ \boxed {y^{\prime \prime \prime }+3 y^{\prime \prime }+2 y^{\prime }+6 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1, y^{\prime \prime }\left (0\right ) = -1] \end {align*}
program solution	\[ y = -\frac {{\mathrm e}^{-3 t}}{11}+\frac {\cos \left (\sqrt {2}\, t \right )}{11}+\frac {4 \sqrt {2}\, \sin \left (\sqrt {2}\, t \right )}{11} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {{\mathrm e}^{-3 t}}{11}+\frac {4 \sqrt {2}\, \sin \left (\sqrt {2}\, t \right )}{11}+\frac {\cos \left (\sqrt {2}\, t \right )}{11} \]

Problem 14672

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-8 y^{\prime \prime \prime }+30 y^{\prime \prime }-56 y^{\prime }+49 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 2, y^{\prime \prime }\left (0\right ) = -1, y^{\prime \prime \prime }\left (0\right ) = -1] \end {align*}
program solution	\[ y = \frac {{\mathrm e}^{-\left (-2+i \sqrt {3}\right ) t} \left (6+i \left (7-2 t \right ) \sqrt {3}-21 t \right )}{12}+\frac {{\mathrm e}^{\left (i \sqrt {3}+2\right ) t} \left (3+i \left (-\frac {7}{2}+t \right ) \sqrt {3}-\frac {21 t}{2}\right )}{6} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {\left (\left (\frac {21 t}{2}-3\right ) \cos \left (\sqrt {3}\, t \right )+\sqrt {3}\, \sin \left (\sqrt {3}\, t \right ) \left (t -\frac {7}{2}\right )\right ) {\mathrm e}^{2 t}}{3} \]

Problem 14673

ODE	\[ \boxed {\frac {31 y^{\prime \prime \prime }}{100}+\frac {56 y^{\prime \prime }}{5}-\frac {49 y^{\prime }}{5}+\frac {53 y}{10}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = -1, y^{\prime \prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {\left (\left (19974365614660 \sqrt {3}-71021195271894 i-462753660 \sqrt {19889065283}-163214002 i \sqrt {59667195849}\right ) \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {2}{3}}+\left (379649387003 i \sqrt {59667195849}+6730996696529841 i-55067075370969947 \sqrt {3}-49087884123 \sqrt {19889065283}\right ) \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}-164845850396080 i \sqrt {59667195849}+88745628993688032240 i+34559521450794478600 \sqrt {3}[phone] \sqrt {19889065283}\right ) {\mathrm e}^{\frac {\left (i \sqrt {3}\, \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {2}{3}}-1345540 i \sqrt {3}+\left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {2}{3}}-2240 \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {1}{3}}+1345540\right ) t}{186 \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {1}{3}}}}+\left (\left (163214002 i \sqrt {59667195849}[phone] \sqrt {3}-462753660 \sqrt {19889065283}[phone] i\right ) \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {2}{3}}+\left (-6730996696529841 i-55067075370969947 \sqrt {3}-49087884123 \sqrt {19889065283}-379649387003 i \sqrt {59667195849}\right ) \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}[phone] i \sqrt {59667195849}-88745628993688032240 i+34559521450794478600 \sqrt {3}[phone] \sqrt {19889065283}\right ) {\mathrm e}^{-\frac {t \left (\left (i \sqrt {3}-1\right ) \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {2}{3}}-1345540 i \sqrt {3}+2240 \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {1}{3}}-1345540\right )}{186 \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {1}{3}}}}+\left (\left (-51046829657234 \sqrt {3}-952395666 \sqrt {19889065283}\right ) \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {2}{3}}+\left (110134150741939894 \sqrt {3}[phone] \sqrt {19889065283}\right ) \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}-54186107542893553640 \sqrt {3}+1171956228719640 \sqrt {19889065283}\right ) {\mathrm e}^{-\frac {\left (\left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {2}{3}}+1120 \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {1}{3}}+1345540\right ) t}{93 \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {1}{3}}}}}{6 \left (\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {2}{3}}-1345540\right ) \left (1849683071319 \sqrt {3}[phone] \sqrt {19889065283}\right )} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (\left (\left (-1228360 \sqrt {3}\, \sqrt {19889065283}[phone]\right ) \left (1564919155+465 \sqrt {3}\, \sqrt {19889065283}\right )^{\frac {1}{3}}+\left (-1213 \sqrt {3}\, \sqrt {19889065283}-527826711\right ) \left (1564919155+465 \sqrt {3}\, \sqrt {19889065283}\right )^{\frac {2}{3}}-6259676620 \sqrt {3}\, \sqrt {19889065283}-110980984279140\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (1564919155+465 \sqrt {3}\, \sqrt {19889065283}\right )^{\frac {2}{3}}-1345540\right ) t}{186 \left (1564919155+465 \sqrt {3}\, \sqrt {19889065283}\right )^{\frac {1}{3}}}\right )-527826711 \sin \left (\frac {\sqrt {3}\, \left (\left (1564919155+465 \sqrt {3}\, \sqrt {19889065283}\right )^{\frac {2}{3}}-1345540\right ) t}{186 \left (1564919155+465 \sqrt {3}\, \sqrt {19889065283}\right )^{\frac {1}{3}}}\right ) \left (\left (\frac {196290745000 \sqrt {3}}{175942237}-\frac {1228360 \sqrt {19889065283}}{175942237}\right ) \left (1564919155+465 \sqrt {3}\, \sqrt {19889065283}\right )^{\frac {1}{3}}+\left (1564919155+465 \sqrt {3}\, \sqrt {19889065283}\right )^{\frac {2}{3}} \left (\sqrt {3}+\frac {1213 \sqrt {19889065283}}{175942237}\right )\right )\right ) {\mathrm e}^{\frac {\left (1345540+\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {2}{3}}-2240 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}\right ) t}{186 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}}+1213 \,{\mathrm e}^{-\frac {\left (\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {2}{3}}+1120 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}+1345540\right ) t}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}} \left (\left (\frac {1228360 \sqrt {3}\, \sqrt {19889065283}}{1213}-\frac {588872235000}{1213}\right ) \left (1564919155+465 \sqrt {3}\, \sqrt {19889065283}\right )^{\frac {1}{3}}+\left (\sqrt {3}\, \sqrt {19889065283}+\frac {527826711}{1213}\right ) \left (1564919155+465 \sqrt {3}\, \sqrt {19889065283}\right )^{\frac {2}{3}}-\frac {3129838310 \sqrt {3}\, \sqrt {19889065283}}{1213}-\frac {55490492139570}{1213}\right )}{9389514930 \sqrt {3}\, \sqrt {19889065283}[phone]} \]

Problem 14674

ODE	\[ \boxed {2 y y^{\prime \prime }+y^{2}-{y^{\prime }}^{2}=0} \]
program solution	\[ y = \frac {c_{3}^{2} {\mathrm e}^{2 c_{2}}}{2}+\frac {\tan \left (t +c_{4} \right ) \sqrt {\frac {c_{3}^{4} {\mathrm e}^{4 c_{2}}}{\tan \left (t +c_{4} \right )^{2}+1}}}{2} \] Verified OK. \[ y = \frac {c_{3}^{2} {\mathrm e}^{2 c_{2}}}{2}-\frac {\tan \left (t +c_{5} \right ) \sqrt {\frac {c_{3}^{4} {\mathrm e}^{4 c_{2}}}{\tan \left (t +c_{5} \right )^{2}+1}}}{2} \] Verified OK.
Maple solution	\begin{align*} y \left (t \right ) &= 0 \\ y \left (t \right ) &= \sqrt {c_{1}^{2}+c_{2}^{2}}+c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right ) \\ \end{align*}

Problem 14675

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

Problem 14676

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

Problem 14677

ODE	\[ \boxed {y^{\left (5\right )}-y^{\prime \prime \prime \prime }=1} \]
program solution	\[ y = t^{3} c_{4} +t^{2} c_{3} +c_{2} t +c_{1} +{\mathrm e}^{t} c_{5} -\frac {t^{4}}{24} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {t^{4}}{24}+\frac {c_{3} t^{2}}{2}+\frac {c_{2} t^{3}}{6}+{\mathrm e}^{t} c_{1} +t c_{4} +c_{5} \]

Problem 14678

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+9 y^{\prime \prime }=1} \]
program solution	\[ y = c_{2} t +c_{1} +{\mathrm e}^{-3 i t} c_{3} +{\mathrm e}^{3 i t} c_{4} +\frac {t^{2}}{18} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {t^{2}}{18}-\frac {c_{1} \cos \left (3 t \right )}{9}-\frac {c_{2} \sin \left (3 t \right )}{9}+c_{3} t +c_{4} \]

Problem 14679

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+9 y^{\prime \prime }=9 \,{\mathrm e}^{3 t}} \]
program solution	\[ y = c_{2} t +c_{1} +{\mathrm e}^{-3 i t} c_{3} +{\mathrm e}^{3 i t} c_{4} +\frac {{\mathrm e}^{3 t}}{18} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {c_{1} \cos \left (3 t \right )}{9}-\frac {c_{2} \sin \left (3 t \right )}{9}+\frac {{\mathrm e}^{3 t}}{18}+c_{3} t +c_{4} \]

Problem 14680

ODE	\[ \boxed {y^{\prime \prime \prime }+10 y^{\prime \prime }+34 y^{\prime }+40 y=t \,{\mathrm e}^{-4 t}+2 \,{\mathrm e}^{-3 t} \cos \left (t \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-4 t}+{\mathrm e}^{\left (-3+i\right ) t} c_{2} +{\mathrm e}^{\left (-3-i\right ) t} c_{3} +\frac {\left (\left (5+i-4 t \right ) \cos \left (t \right )-\sin \left (t \right ) \left (-4 t -1+i\right )\right ) {\mathrm e}^{-3 t}}{8}+\frac {{\mathrm e}^{-4 t} \left (1+t \right )^{2}}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (\left (-2 t +4 c_{2} +4\right ) \cos \left (t \right )+2 \sin \left (t \right ) \left (t +2 c_{3} +1\right )\right ) {\mathrm e}^{-3 t}}{4}+\frac {{\mathrm e}^{-4 t} \left (t^{2}+4 c_{1} +2 t +1\right )}{4} \]

Problem 14681

ODE	\[ \boxed {y^{\prime \prime \prime }+6 y^{\prime \prime }+11 y^{\prime }+6 y=2 \,{\mathrm e}^{-3 t}-t \,{\mathrm e}^{-t}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-t}+c_{2} {\mathrm e}^{-2 t}+{\mathrm e}^{-3 t} c_{3} +{\mathrm e}^{-3 t} t +\frac {3 t \,{\mathrm e}^{-t}}{4}-\frac {t^{2} {\mathrm e}^{-t}}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (-2 t^{2}+8 c_{3} +6 t -7\right ) {\mathrm e}^{-t}}{8}+\frac {\left (3+2 t +2 c_{1} \right ) {\mathrm e}^{-3 t}}{2}+c_{2} {\mathrm e}^{-2 t} \]

Problem 14682

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-6 y^{\prime \prime \prime }+13 y^{\prime \prime }-24 y^{\prime }+36 y=108 t} \]
program solution	\[ y = {\mathrm e}^{-2 i t} c_{3} +{\mathrm e}^{2 i t} c_{4} +{\mathrm e}^{3 t} \left (c_{2} t +c_{1} \right )+3 t +2 \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (t c_{4} +c_{2} \right ) {\mathrm e}^{3 t}+\cos \left (2 t \right ) c_{1} +c_{3} \sin \left (2 t \right )+3 t +2 \]

Problem 14683

ODE	\[ \boxed {y^{\prime \prime \prime }+6 y^{\prime \prime }-14 y^{\prime }-104 y=-111 \,{\mathrm e}^{t}} \]
program solution	\[ y = c_{1} {\mathrm e}^{4 t}+{\mathrm e}^{\left (-5-i\right ) t} c_{2} +{\mathrm e}^{\left (-5+i\right ) t} c_{3} +{\mathrm e}^{t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left ({\mathrm e}^{9 t} c_{1} +{\mathrm e}^{6 t}+c_{3} \sin \left (t \right )+c_{2} \cos \left (t \right )\right ) {\mathrm e}^{-5 t} \]

Problem 14684

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-10 y^{\prime \prime \prime }+38 y^{\prime \prime }-64 y^{\prime }+40 y=153 \,{\mathrm e}^{-t}} \]
program solution	\[ y = {\mathrm e}^{\left (3-i\right ) t} c_{3} +{\mathrm e}^{\left (3+i\right ) t} c_{4} +{\mathrm e}^{2 t} \left (c_{2} t +c_{1} \right )+{\mathrm e}^{-t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (t c_{2} +c_{1} \right ) {\mathrm e}^{2 t}+\left (\cos \left (t \right ) c_{3} +\sin \left (t \right ) c_{4} \right ) {\mathrm e}^{3 t}+{\mathrm e}^{-t} \]

Problem 14685

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

Problem 14686

ODE	\[ \boxed {y^{\prime \prime \prime }+4 y^{\prime }=\sec \left (2 t \right ) \tan \left (2 t \right )} \]
program solution	\[ y = c_{1} +{\mathrm e}^{-2 i t} c_{2} +{\mathrm e}^{2 i t} c_{3} +\frac {i \arctan \left (\frac {\sin \left (4 t \right )}{1+\cos \left (4 t \right )}\right ) \cos \left (2 t \right )}{8}+\frac {\cos \left (2 t \right ) \ln \left (1+\cos \left (4 t \right )\right )}{16}+\frac {\left (-4 i \arctan \left (\sin \left (t \right ), \cos \left (t \right )\right )+\ln \left (2\right )+2\right ) \cos \left (2 t \right )}{16}-\frac {\sin \left (2 t \right ) \left (i-2 \arctan \left (\sin \left (t \right ), \cos \left (t \right )\right )\right )}{8} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {\ln \left (\sec \left (2 t \right )\right ) \cos \left (2 t \right )}{8}+\frac {\left (1-4 c_{2} \right ) \cos \left (2 t \right )}{8}+\frac {\left (2 t +4 c_{1} \right ) \sin \left (2 t \right )}{8}+c_{3} \]

Problem 14687

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+4 y^{\prime \prime }=\sec \left (2 t \right )^{2}} \]
program solution	\[ y = c_{2} t +c_{1} +{\mathrm e}^{-2 i t} c_{3} +{\mathrm e}^{2 i t} c_{4} +\frac {i \ln \left ({\mathrm e}^{2 i t}-i\right ) {\mathrm e}^{-2 i t}}{32}-\frac {i \ln \left ({\mathrm e}^{2 i t}+i\right ) {\mathrm e}^{-2 i t}}{32}+\frac {{\mathrm e}^{2 i t} \arctan \left ({\mathrm e}^{2 i t}\right )}{16}+\frac {1}{16}-\frac {\ln \left (\cos \left (2 t \right )\right )}{16} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {\pi \,{\mathrm e}^{-2 i t} \left (\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{-4 i t}+1}\right ) \operatorname {csgn}\left (2 i {\mathrm e}^{-2 i t}-{\mathrm e}^{-4 i t}+1\right )+1\right ) \operatorname {csgn}\left (\frac {2 i {\mathrm e}^{-2 i t}-{\mathrm e}^{-4 i t}+1}{{\mathrm e}^{-4 i t}+1}\right )}{64}+\frac {{\mathrm e}^{2 i t} \pi \left (\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{4 i t}+1}\right ) \operatorname {csgn}\left ({\mathrm e}^{4 i t}+2 i {\mathrm e}^{2 i t}-1\right )+1\right ) \operatorname {csgn}\left (\frac {{\mathrm e}^{4 i t}+2 i {\mathrm e}^{2 i t}-1}{{\mathrm e}^{4 i t}+1}\right )}{64}+\frac {\pi \,\operatorname {csgn}\left (2 i {\mathrm e}^{-2 i t}-{\mathrm e}^{-4 i t}+1\right ) {\mathrm e}^{-2 i t}}{64}+\frac {\pi \,\operatorname {csgn}\left ({\mathrm e}^{4 i t}+2 i {\mathrm e}^{2 i t}-1\right ) {\mathrm e}^{2 i t}}{64}+\frac {i {\mathrm e}^{2 i t} \ln \left (i \left ({\mathrm e}^{2 i t}+i\right )^{2}\right )}{32}-\frac {i \ln \left ({\mathrm e}^{i t}\right ) t}{4}+\frac {\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{-4 i t}+1}\right ) {\mathrm e}^{-2 i t}}{64}-\frac {\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{4 i t}+1}\right ) {\mathrm e}^{2 i t}}{64}+\frac {\left (-1+i {\mathrm e}^{-2 i t}\right ) \ln \left ({\mathrm e}^{-4 i t}+1\right )}{32}-\frac {i {\mathrm e}^{-2 i t} \ln \left (i \left ({\mathrm e}^{-2 i t}-i\right )^{2}\right )}{32}-\frac {{\mathrm e}^{-2 i t} \left (c_{2} i+c_{1} \right )}{8}+\frac {\left (-i {\mathrm e}^{2 i t}-1\right ) \ln \left ({\mathrm e}^{4 i t}+1\right )}{32}+\frac {\left (c_{2} i-c_{1} \right ) {\mathrm e}^{2 i t}}{8}-\frac {t^{2}}{4}+c_{3} t +c_{4} \]

Problem 14688

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+4 y^{\prime \prime }=\tan \left (2 t \right )^{2}} \]
program solution	\[ y = c_{2} t +c_{1} +{\mathrm e}^{-2 i t} c_{3} +{\mathrm e}^{2 i t} c_{4} -\frac {i \ln \left ({\mathrm e}^{2 i t}+i\right ) {\mathrm e}^{-2 i t}}{32}+\frac {i \ln \left ({\mathrm e}^{2 i t}-i\right ) {\mathrm e}^{-2 i t}}{32}+\frac {1}{8}+\frac {{\mathrm e}^{2 i t} \arctan \left ({\mathrm e}^{2 i t}\right )}{16}-\frac {t^{2}}{8}-\frac {\ln \left (\cos \left (2 t \right )\right )}{16} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {\pi \,{\mathrm e}^{-2 i t} \left (\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{-4 i t}+1}\right ) \operatorname {csgn}\left (2 i {\mathrm e}^{-2 i t}-{\mathrm e}^{-4 i t}+1\right )+1\right ) \operatorname {csgn}\left (\frac {2 i {\mathrm e}^{-2 i t}-{\mathrm e}^{-4 i t}+1}{{\mathrm e}^{-4 i t}+1}\right )}{64}+\frac {{\mathrm e}^{2 i t} \pi \left (\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{4 i t}+1}\right ) \operatorname {csgn}\left ({\mathrm e}^{4 i t}+2 i {\mathrm e}^{2 i t}-1\right )+1\right ) \operatorname {csgn}\left (\frac {{\mathrm e}^{4 i t}+2 i {\mathrm e}^{2 i t}-1}{{\mathrm e}^{4 i t}+1}\right )}{64}+\frac {\pi \,\operatorname {csgn}\left (2 i {\mathrm e}^{-2 i t}-{\mathrm e}^{-4 i t}+1\right ) {\mathrm e}^{-2 i t}}{64}+\frac {\pi \,\operatorname {csgn}\left ({\mathrm e}^{4 i t}+2 i {\mathrm e}^{2 i t}-1\right ) {\mathrm e}^{2 i t}}{64}+\frac {i {\mathrm e}^{2 i t} \ln \left (i \left ({\mathrm e}^{2 i t}+i\right )^{2}\right )}{32}-\frac {i \ln \left ({\mathrm e}^{i t}\right ) t}{4}+\frac {\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{-4 i t}+1}\right ) {\mathrm e}^{-2 i t}}{64}-\frac {\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{4 i t}+1}\right ) {\mathrm e}^{2 i t}}{64}+\frac {\left (-1+i {\mathrm e}^{-2 i t}\right ) \ln \left ({\mathrm e}^{-4 i t}+1\right )}{32}-\frac {i {\mathrm e}^{-2 i t} \ln \left (i \left ({\mathrm e}^{-2 i t}-i\right )^{2}\right )}{32}-\frac {{\mathrm e}^{-2 i t} \left (c_{2} i+c_{1} \right )}{8}+\frac {\left (-i {\mathrm e}^{2 i t}-1\right ) \ln \left ({\mathrm e}^{4 i t}+1\right )}{32}+\frac {\left (c_{2} i-c_{1} \right ) {\mathrm e}^{2 i t}}{8}-\frac {3 t^{2}}{8}+c_{3} t +c_{4} \]

Problem 14689

ODE	\[ \boxed {y^{\prime \prime \prime }+9 y^{\prime }=\sec \left (3 t \right )} \]
program solution	\[ y = c_{1} +{\mathrm e}^{-3 i t} c_{2} +{\mathrm e}^{3 i t} c_{3} +\frac {i \ln \left ({\mathrm e}^{6 i t}+1\right ) {\mathrm e}^{-3 i t}}{54}-\frac {i {\mathrm e}^{3 i t} \ln \left ({\mathrm e}^{4 i t}-{\mathrm e}^{2 i t}+1\right )}{54}-\frac {i {\mathrm e}^{3 i t} \ln \left ({\mathrm e}^{2 i t}+1\right )}{54}+\frac {i {\mathrm e}^{3 i t} \ln \left ({\mathrm e}^{i t}\right )}{9}+\frac {\ln \left (\sec \left (3 t \right )+\tan \left (3 t \right )\right )}{27} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {i \left ({\mathrm e}^{3 i t}-{\mathrm e}^{-3 i t}\right ) \ln \left (\frac {{\mathrm e}^{3 i t}}{{\mathrm e}^{6 i t}+1}\right )}{54}-\frac {i \arctan \left (2 \,{\mathrm e}^{i t}-\sqrt {3}\right )}{27}-\frac {i \arctan \left (2 \,{\mathrm e}^{i t}+\sqrt {3}\right )}{27}-\frac {i \arctan \left ({\mathrm e}^{3 i t}\right )}{27}-\frac {i {\mathrm e}^{-3 i t}}{54}+\frac {i {\mathrm e}^{3 i t}}{54}+\frac {i \arctan \left ({\mathrm e}^{i t}\right )}{27}+\frac {\left (1+9 c_{1} -\ln \left (2\right )\right ) \sin \left (3 t \right )}{27}+\frac {\left (-t -3 c_{2} \right ) \cos \left (3 t \right )}{9}+c_{3} \]

Problem 14690

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

Problem 14691

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

Problem 14692

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

Problem 14693

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

Problem 14694

ODE	\[ \boxed {y^{\prime \prime \prime }-4 y^{\prime \prime }-11 y^{\prime }+30 y={\mathrm e}^{4 t}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-3 t}+c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{5 t}-\frac {{\mathrm e}^{4 t}}{14} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {\left (-14 c_{3} {\mathrm e}^{8 t}+{\mathrm e}^{7 t}-14 c_{2} {\mathrm e}^{5 t}-14 c_{1} \right ) {\mathrm e}^{-3 t}}{14} \]

Problem 14695

ODE	\[ \boxed {y^{\prime \prime \prime }+3 y^{\prime \prime }-10 y^{\prime }-24 y={\mathrm e}^{-3 t}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 t}+c_{2} {\mathrm e}^{-4 t}+c_{3} {\mathrm e}^{3 t}+\frac {{\mathrm e}^{-3 t}}{6} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (6 c_{3} {\mathrm e}^{7 t}+6 c_{2} {\mathrm e}^{2 t}+{\mathrm e}^{t}+6 c_{1} \right ) {\mathrm e}^{-4 t}}{6} \]

Problem 14696

ODE	\[ \boxed {y^{\prime \prime \prime }-13 y^{\prime }+12 y=\cos \left (t \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-4 t}+c_{3} {\mathrm e}^{3 t}+\frac {3 \cos \left (t \right )}{85}-\frac {7 \sin \left (t \right )}{170} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (\left (\frac {3 \cos \left (t \right )}{85}-\frac {7 \sin \left (t \right )}{170}\right ) {\mathrm e}^{4 t}+c_{3} {\mathrm e}^{7 t}+c_{1} {\mathrm e}^{5 t}+c_{2} \right ) {\mathrm e}^{-4 t} \]

Problem 14697

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

Problem 14698

ODE	\[ \boxed {y^{\left (6\right )}+y^{\prime \prime \prime \prime }=-24} \]
program solution	\[ y = t^{3} c_{4} +t^{2} c_{3} +c_{2} t +c_{1} +{\mathrm e}^{i t} c_{5} +{\mathrm e}^{-i t} c_{6} -t^{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {c_{3} t^{3}}{6}-t^{4}+\frac {c_{4} t^{2}}{2}+\cos \left (t \right ) c_{1} +\sin \left (t \right ) c_{2} +c_{5} t +c_{6} \]

Problem 14699

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

Problem 14700

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