2.17.18 Problems 1701 to 1800

Problem 1701

ODE	\[ \boxed {y^{\prime }-y^{2}={\mathrm e}^{-t^{2}}} \]
program solution	\[ y = -\frac {\frac {d}{d t}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (t \right )+{\mathrm e}^{-t^{2}} \textit {\_Y} \left (t \right )\right \}, \left \{\textit {\_Y} \left (t \right )\right \}\right )}{\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (t \right )+{\mathrm e}^{-t^{2}} \textit {\_Y} \left (t \right )\right \}, \left \{\textit {\_Y} \left (t \right )\right \}\right )} \] Verified OK.
Maple solution	\[ \text {No solution found} \]

Problem 1702

ODE	\[ \boxed {y^{\prime }-y^{2}={\mathrm e}^{-t^{2}}} \]
program solution	\[ y = -\frac {\frac {d}{d t}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (t \right )+{\mathrm e}^{-t^{2}} \textit {\_Y} \left (t \right )\right \}, \left \{\textit {\_Y} \left (t \right )\right \}\right )}{\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (t \right )+{\mathrm e}^{-t^{2}} \textit {\_Y} \left (t \right )\right \}, \left \{\textit {\_Y} \left (t \right )\right \}\right )} \] Verified OK.
Maple solution	\[ \text {No solution found} \]

Problem 1703

ODE	\[ \boxed {y^{\prime }-y-{\mathrm e}^{-y}={\mathrm e}^{-t}} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 1704

ODE	\[ \boxed {y^{\prime }-y^{3}={\mathrm e}^{-5 t}} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 1705

ODE	\[ \boxed {y^{\prime }-{\mathrm e}^{\left (-t +y\right )^{2}}=0} \]
program solution	\[ t = \int _{}^{y}\frac {1}{{\mathrm e}^{\left (-t +\textit {\_a} \right )^{2}}-1}d \textit {\_a} +c_{1} \] Verified OK.
Maple solution	\[ y \left (t \right ) = t +\operatorname {RootOf}\left (-t +\int _{}^{\textit {\_Z}}\frac {1}{-1+{\mathrm e}^{\textit {\_a}^{2}}}d \textit {\_a} +c_{1} \right ) \]

Problem 1706

ODE	\[ \boxed {y^{\prime }-\left (4 y+{\mathrm e}^{-t^{2}}\right ) {\mathrm e}^{2 y}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 1707

ODE	\[ \boxed {y^{\prime }-\ln \left (1+y^{2}\right )={\mathrm e}^{-t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution
Maple solution	\[ \text {No solution found} \]

Problem 1708

ODE	\[ \boxed {y^{\prime }-\frac {\left (1+\cos \left (4 t \right )\right ) y}{4}+\frac {\left (1-\cos \left (4 t \right )\right ) y^{2}}{800}=0} \]
program solution	\[ y = \frac {800 \,{\mathrm e}^{\frac {t}{4}+\frac {\sin \left (4 t \right )}{16}}}{c_{3} +2 \left (\int {\mathrm e}^{\frac {t}{4}+\frac {\sin \left (4 t \right )}{16}} \sin \left (2 t \right )^{2}d t \right )} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {800 \,{\mathrm e}^{\frac {t}{4}+\frac {\sin \left (4 t \right )}{16}}}{\int {\mathrm e}^{\frac {t}{4}+\frac {\sin \left (4 t \right )}{16}} \left (-1+\cos \left (4 t \right )\right )d t -800 c_{1}} \]

Problem 1709

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

Problem 1710

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

Problem 1711

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

Problem 1712

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

Problem 1713

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

Problem 1714

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

Problem 1715

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

Problem 1716

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

Problem 1717

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

Problem 1718

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

Problem 1719

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

Problem 1720

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

Problem 1721

ODE	\[ \boxed {2 y^{\prime \prime }+y^{\prime }-10 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 5, y^{\prime }\left (1\right ) = 2] \end {align*}
program solution	\[ y = \frac {29 \,{\mathrm e}^{2 t} {\mathrm e}^{-2}}{9}+\frac {16 \,{\mathrm e}^{-\frac {5 t}{2}} {\mathrm e}^{\frac {5}{2}}}{9} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {16 \,{\mathrm e}^{\frac {5}{2}-\frac {5 t}{2}}}{9}+\frac {29 \,{\mathrm e}^{2 t -2}}{9} \]

Problem 1722

ODE	\[ \boxed {5 y^{\prime \prime }+5 y^{\prime }-y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = -\frac {\sqrt {5}\, \left (-{\mathrm e}^{\frac {3 \sqrt {5}\, t}{10}-\frac {t}{2}}+{\mathrm e}^{-\frac {3 \sqrt {5}\, t}{10}-\frac {t}{2}}\right )}{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left ({\mathrm e}^{\frac {3 t \sqrt {5}}{10}-\frac {t}{2}}-{\mathrm e}^{-\frac {t}{2}-\frac {3 t \sqrt {5}}{10}}\right ) \sqrt {5}}{3} \]

Problem 1723

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

Problem 1724

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

Problem 1725

ODE	\[ \boxed {t^{2} y^{\prime \prime }+\alpha t y^{\prime }+\beta y=0} \]
program solution	\[ y = c_{1} t^{-\frac {\alpha }{2}+\frac {1}{2}+\frac {\sqrt {\alpha ^{2}-2 \alpha -4 \beta +1}}{2}}-\frac {c_{2} t^{-\frac {\alpha }{2}+\frac {1}{2}-\frac {\sqrt {\alpha ^{2}-2 \alpha -4 \beta +1}}{2}}}{\sqrt {\alpha ^{2}-2 \alpha -4 \beta +1}} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \sqrt {t}\, t^{-\frac {\alpha }{2}} \left (t^{\frac {\sqrt {\alpha ^{2}-2 \alpha -4 \beta +1}}{2}} c_{1} +t^{-\frac {\sqrt {\alpha ^{2}-2 \alpha -4 \beta +1}}{2}} c_{2} \right ) \]

Problem 1726

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

Problem 1727

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

Problem 1728

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

Problem 1729

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

Problem 1730

ODE	\[ \boxed {2 y^{\prime \prime }+3 y^{\prime }+4 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {3 t}{4}} \cos \left (\frac {\sqrt {23}\, t}{4}\right )+\frac {4 c_{2} \sin \left (\frac {\sqrt {23}\, t}{4}\right ) {\mathrm e}^{-\frac {3 t}{4}} \sqrt {23}}{23} \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{-\frac {3 t}{4}} \left (c_{1} \sin \left (\frac {\sqrt {23}\, t}{4}\right )+c_{2} \cos \left (\frac {\sqrt {23}\, t}{4}\right )\right ) \]

Problem 1731

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

Problem 1732

ODE	\[ \boxed {4 y^{\prime \prime }-y^{\prime }+y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{\frac {t}{8}} \cos \left (\frac {\sqrt {15}\, t}{8}\right )+\frac {8 c_{2} \sin \left (\frac {\sqrt {15}\, t}{8}\right ) {\mathrm e}^{\frac {t}{8}} \sqrt {15}}{15} \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{\frac {t}{8}} \left (c_{1} \sin \left (\frac {\sqrt {15}\, t}{8}\right )+c_{2} \cos \left (\frac {\sqrt {15}\, t}{8}\right )\right ) \]

Problem 1733

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

Problem 1734

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

Problem 1735

ODE	\[ \boxed {2 y^{\prime \prime }-y^{\prime }+3 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1, y^{\prime }\left (1\right ) = 1] \end {align*}
program solution	\[ y = -\frac {{\mathrm e}^{\frac {t}{4}-\frac {1}{4}} \left (3 \sqrt {23}\, \cos \left (\frac {\sqrt {23}\, t}{4}\right ) \sin \left (\frac {\sqrt {23}}{4}\right )-3 \sin \left (\frac {\sqrt {23}\, t}{4}\right ) \sqrt {23}\, \cos \left (\frac {\sqrt {23}}{4}\right )-23 \cos \left (\frac {\sqrt {23}\, t}{4}\right ) \cos \left (\frac {\sqrt {23}}{4}\right )-23 \sin \left (\frac {\sqrt {23}\, t}{4}\right ) \sin \left (\frac {\sqrt {23}}{4}\right )\right )}{23} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {{\mathrm e}^{-\frac {1}{4}+\frac {t}{4}} \left (3 \sin \left (\frac {\sqrt {23}}{4}\right ) \sqrt {23}\, \cos \left (\frac {\sqrt {23}\, t}{4}\right )-3 \sqrt {23}\, \cos \left (\frac {\sqrt {23}}{4}\right ) \sin \left (\frac {\sqrt {23}\, t}{4}\right )-23 \sin \left (\frac {\sqrt {23}}{4}\right ) \sin \left (\frac {\sqrt {23}\, t}{4}\right )-23 \cos \left (\frac {\sqrt {23}}{4}\right ) \cos \left (\frac {\sqrt {23}\, t}{4}\right )\right )}{23} \]

Problem 1736

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

Problem 1737

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

Problem 1738

ODE	\[ \boxed {t^{2} y^{\prime \prime }+2 t y^{\prime }+2 y=0} \]
program solution	\[ y = c_{1} t^{-\frac {1}{2}-\frac {i \sqrt {7}}{2}}-\frac {i c_{2} \sqrt {7}\, t^{-\frac {1}{2}+\frac {i \sqrt {7}}{2}}}{7} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {c_{1} \sin \left (\frac {\sqrt {7}\, \ln \left (t \right )}{2}\right )+c_{2} \cos \left (\frac {\sqrt {7}\, \ln \left (t \right )}{2}\right )}{\sqrt {t}} \]

Problem 1739

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

Problem 1740

ODE	\[ \boxed {4 y^{\prime \prime }-12 y^{\prime }+9 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{\frac {3 t}{2}}+c_{2} t \,{\mathrm e}^{\frac {3 t}{2}} \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{\frac {3 t}{2}} \left (c_{2} t +c_{1} \right ) \]

Problem 1741

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

Problem 1742

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

Problem 1743

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

Problem 1744

ODE	\[ \boxed {9 y^{\prime \prime }-12 y^{\prime }+4 y=0} \] With initial conditions \begin {align*} [y \left (\pi \right ) = 0, y^{\prime }\left (\pi \right ) = 2] \end {align*}
program solution	\[ y = -2 \,{\mathrm e}^{\frac {2 t}{3}-\frac {2 \pi }{3}} \left (\pi -t \right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = -2 \,{\mathrm e}^{-\frac {2 \pi }{3}+\frac {2 t}{3}} \left (\pi -t \right ) \]

Problem 1745

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

Problem 1746

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

Problem 1747

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

Problem 1748

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

Problem 1749

ODE	\[ \boxed {\left (-t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+6 y=0} \]
program solution	\[ y = c_{1} \left (t^{2}-\frac {1}{3}\right )+c_{2} \left (\frac {9 \ln \left (-1+t \right ) t^{2}}{8}-\frac {9 \ln \left (t +1\right ) t^{2}}{8}-\frac {3 \ln \left (-1+t \right )}{8}+\frac {3 \ln \left (t +1\right )}{8}+\frac {9 t}{4}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {c_{2} \left (3 t^{2}-1\right ) \ln \left (t -1\right )}{2}+\frac {\left (-3 t^{2}+1\right ) c_{2} \ln \left (t +1\right )}{2}-3 c_{1} t^{2}+3 c_{2} t +c_{1} \]

Problem 1750

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

Problem 1751

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

Problem 1752

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

Problem 1753

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

Problem 1754

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

Problem 1755

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

Problem 1756

ODE	\[ \boxed {2 y^{\prime \prime }-3 y^{\prime }+y=\left (t^{2}+1\right ) {\mathrm e}^{t}} \]
program solution	\[ y = c_{1} {\mathrm e}^{\frac {t}{2}}+2 c_{2} {\mathrm e}^{t}+\frac {{\mathrm e}^{t} \left (t -3\right ) \left (t^{2}-3 t +18\right )}{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{2} {\mathrm e}^{\frac {t}{2}}+\frac {{\mathrm e}^{t} \left (t^{3}-6 t^{2}+6 c_{1} +27 t -54\right )}{3} \]

Problem 1757

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

Problem 1758

ODE	\[ \boxed {3 y^{\prime \prime }+4 y^{\prime }+y={\mathrm e}^{-t} \sin \left (t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {24 \,{\mathrm e}^{-\frac {t}{3}}}{13}+\frac {\left (-13+2 \cos \left (t \right )-3 \sin \left (t \right )\right ) {\mathrm e}^{-t}}{13} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {24 \,{\mathrm e}^{-\frac {t}{3}}}{13}+\frac {\left (-13-3 \sin \left (t \right )+2 \cos \left (t \right )\right ) {\mathrm e}^{-t}}{13} \]

Problem 1759

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+4 y=t^{\frac {5}{2}} {\mathrm e}^{-2 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {4 t^{\frac {9}{2}} {\mathrm e}^{-2 t}}{63} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {4 t^{\frac {9}{2}} {\mathrm e}^{-2 t}}{63} \]

Problem 1760

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

Problem 1761

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

Problem 1762

ODE	\[ \boxed {y^{\prime \prime }+\frac {y t^{2}}{4}=f \cos \left (t \right )} \]
program solution	\[ y = c_{1} \sqrt {t}\, \operatorname {BesselJ}\left (\frac {1}{4}, \frac {t^{2}}{4}\right )+c_{2} \sqrt {t}\, \operatorname {BesselY}\left (\frac {1}{4}, \frac {t^{2}}{4}\right )-\frac {\pi f \sqrt {t}\, \left (\left (\int _{0}^{t}\sqrt {\alpha }\, \operatorname {BesselY}\left (\frac {1}{4}, \frac {\alpha ^{2}}{4}\right ) \cos \left (\alpha \right )d \alpha \right ) \operatorname {BesselJ}\left (\frac {1}{4}, \frac {t^{2}}{4}\right )-\left (\int _{0}^{t}\sqrt {\alpha }\, \operatorname {BesselJ}\left (\frac {1}{4}, \frac {\alpha ^{2}}{4}\right ) \cos \left (\alpha \right )d \alpha \right ) \operatorname {BesselY}\left (\frac {1}{4}, \frac {t^{2}}{4}\right )\right )}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\sqrt {t}\, \left (f \pi \left (\int \sqrt {t}\, \operatorname {BesselJ}\left (\frac {1}{4}, \frac {t^{2}}{4}\right ) \cos \left (t \right )d t \right ) \operatorname {BesselY}\left (\frac {1}{4}, \frac {t^{2}}{4}\right )-f \pi \left (\int \sqrt {t}\, \operatorname {BesselY}\left (\frac {1}{4}, \frac {t^{2}}{4}\right ) \cos \left (t \right )d t \right ) \operatorname {BesselJ}\left (\frac {1}{4}, \frac {t^{2}}{4}\right )+4 \operatorname {BesselY}\left (\frac {1}{4}, \frac {t^{2}}{4}\right ) c_{1} +4 \operatorname {BesselJ}\left (\frac {1}{4}, \frac {t^{2}}{4}\right ) c_{2} \right )}{4} \]

Problem 1763

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

Problem 1764

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

Problem 1765

ODE	\[ \boxed {y^{\prime \prime }+t y^{\prime }+y=0} \] With the expansion point for the power series method at \(t = 0\).
program solution	\[ y = \left (1-\frac {1}{2} t^{2}+\frac {1}{8} t^{4}-\frac {1}{48} t^{6}\right ) y \left (0\right )+\left (t -\frac {1}{3} t^{3}+\frac {1}{15} t^{5}\right ) y^{\prime }\left (0\right )+O\left (t^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} t^{2}+\frac {1}{8} t^{4}\right ) c_{1} +\left (t -\frac {1}{3} t^{3}+\frac {1}{15} t^{5}\right ) c_{2} +O\left (t^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (1-\frac {1}{2} t^{2}+\frac {1}{8} t^{4}\right ) y \left (0\right )+\left (t -\frac {1}{3} t^{3}+\frac {1}{15} t^{5}\right ) D\left (y \right )\left (0\right )+O\left (t^{6}\right ) \]

Problem 1766

ODE	\[ \boxed {y^{\prime \prime }-y t=0} \] With the expansion point for the power series method at \(t = 0\).
program solution	\[ y = \left (1+\frac {1}{6} t^{3}+\frac {1}{180} t^{6}\right ) y \left (0\right )+\left (t +\frac {1}{12} t^{4}\right ) y^{\prime }\left (0\right )+O\left (t^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {t^{3}}{6}\right ) c_{1} +\left (t +\frac {1}{12} t^{4}\right ) c_{2} +O\left (t^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (1+\frac {t^{3}}{6}\right ) y \left (0\right )+\left (t +\frac {1}{12} t^{4}\right ) D\left (y \right )\left (0\right )+O\left (t^{6}\right ) \]

Problem 1767

ODE	\[ \boxed {\left (t^{2}+2\right ) y^{\prime \prime }-t y^{\prime }-3 y=0} \] With the expansion point for the power series method at \(t = 0\).
program solution	\[ y = \left (1+\frac {3}{4} t^{2}+\frac {3}{32} t^{4}-\frac {1}{128} t^{6}\right ) y \left (0\right )+\left (\frac {1}{3} t^{3}+t \right ) y^{\prime }\left (0\right )+O\left (t^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {3}{4} t^{2}+\frac {3}{32} t^{4}\right ) c_{1} +\left (\frac {1}{3} t^{3}+t \right ) c_{2} +O\left (t^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (1+\frac {3}{4} t^{2}+\frac {3}{32} t^{4}\right ) y \left (0\right )+\left (\frac {1}{3} t^{3}+t \right ) D\left (y \right )\left (0\right )+O\left (t^{6}\right ) \]

Problem 1768

ODE	\[ \boxed {y^{\prime \prime }-y t^{3}=0} \] With the expansion point for the power series method at \(t = 0\).
program solution	\[ y = \left (1+\frac {t^{5}}{20}\right ) y \left (0\right )+\left (t +\frac {1}{30} t^{6}\right ) y^{\prime }\left (0\right )+O\left (t^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {t^{5}}{20}\right ) c_{1} +c_{2} t +O\left (t^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (1+\frac {t^{5}}{20}\right ) y \left (0\right )+t D\left (y \right )\left (0\right )+O\left (t^{6}\right ) \]

Problem 1769

ODE	\[ \boxed {t \left (-t +2\right ) y^{\prime \prime }-6 \left (-1+t \right ) y^{\prime }-4 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1, y^{\prime }\left (1\right ) = 0] \end {align*} With the expansion point for the power series method at \(t = 1\).
program solution	\[ y = 4 \left (-1+t \right )^{6}+3 \left (-1+t \right )^{4}+2 \left (-1+t \right )^{2}+1+O\left (\left (-1+t \right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = 1+2 \left (t -1\right )^{2}+3 \left (t -1\right )^{4}+\operatorname {O}\left (\left (t -1\right )^{6}\right ) \]

Problem 1770

ODE	\[ \boxed {y^{\prime \prime }+y t^{2}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = -1] \end {align*} With the expansion point for the power series method at \(t = 0\).
program solution	\[ y = -t +2-\frac {t^{4}}{6}+\frac {t^{5}}{20}+O\left (t^{6}\right ) \] Verified OK. \[ y = -t +2-\frac {t^{4}}{6}+\frac {t^{5}}{20}+O\left (t^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = 2-t -\frac {1}{6} t^{4}+\frac {1}{20} t^{5}+\operatorname {O}\left (t^{6}\right ) \]

Problem 1771

ODE	\[ \boxed {y^{\prime \prime }-y t^{3}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = -2] \end {align*} With the expansion point for the power series method at \(t = 0\).
program solution	\[ y = -2 t -\frac {t^{6}}{15}+O\left (t^{6}\right ) \] Verified OK. \[ y = -2 t +O\left (t^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (-2\right ) t +\operatorname {O}\left (t^{6}\right ) \]

Problem 1772

ODE	\[ \boxed {y^{\prime \prime }+\left (t^{2}+2 t +1\right ) y^{\prime }-\left (4 t +4\right ) y=0} \] With initial conditions \begin {align*} [y \left (-1\right ) = 0, y^{\prime }\left (-1\right ) = 1] \end {align*} With the expansion point for the power series method at \(t = -1\).
program solution	\[ y = \frac {\left (t +1\right )^{4}}{4}+t +1+O\left (\left (t +1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (t +1\right )+\frac {1}{4} \left (t +1\right )^{4}+\operatorname {O}\left (\left (t +1\right )^{6}\right ) \]

Problem 1773

ODE	\[ \boxed {y^{\prime \prime }-2 t y^{\prime }+\lambda y=0} \] With the expansion point for the power series method at \(t = 0\).
program solution	\[ y = \left (1-\frac {1}{2} \lambda \,t^{2}+\frac {1}{24} t^{4} \lambda ^{2}-\frac {1}{6} t^{4} \lambda -\frac {1}{720} t^{6} \lambda ^{3}+\frac {1}{60} t^{6} \lambda ^{2}-\frac {2}{45} t^{6} \lambda \right ) y \left (0\right )+\left (t -\frac {1}{6} t^{3} \lambda +\frac {1}{3} t^{3}+\frac {1}{120} t^{5} \lambda ^{2}-\frac {1}{15} t^{5} \lambda +\frac {1}{10} t^{5}\right ) y^{\prime }\left (0\right )+O\left (t^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {\lambda \,t^{2}}{2}+\left (\frac {1}{24} \lambda ^{2}-\frac {1}{6} \lambda \right ) t^{4}\right ) c_{1} +\left (t +\left (-\frac {\lambda }{6}+\frac {1}{3}\right ) t^{3}+\left (\frac {1}{120} \lambda ^{2}-\frac {1}{15} \lambda +\frac {1}{10}\right ) t^{5}\right ) c_{2} +O\left (t^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (1-\frac {\lambda \,t^{2}}{2}+\frac {\lambda \left (\lambda -4\right ) t^{4}}{24}\right ) y \left (0\right )+\left (t -\frac {\left (\lambda -2\right ) t^{3}}{6}+\frac {\left (\lambda -2\right ) \left (-6+\lambda \right ) t^{5}}{120}\right ) D\left (y \right )\left (0\right )+O\left (t^{6}\right ) \]

Problem 1774

ODE	\[ \boxed {\left (-t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+\alpha \left (\alpha +1\right ) y=0} \] With the expansion point for the power series method at \(t = 0\).
program solution	\[ y = \left (1-\frac {1}{2} t^{2} \alpha ^{2}-\frac {1}{2} t^{2} \alpha +\frac {1}{24} \alpha ^{4} t^{4}+\frac {1}{12} \alpha ^{3} t^{4}-\frac {5}{24} \alpha ^{2} t^{4}-\frac {1}{4} \alpha \,t^{4}-\frac {1}{720} t^{6} \alpha ^{6}-\frac {1}{240} t^{6} \alpha ^{5}+\frac {23}{720} t^{6} \alpha ^{4}+\frac {17}{240} t^{6} \alpha ^{3}-\frac {47}{360} t^{6} \alpha ^{2}-\frac {1}{6} t^{6} \alpha \right ) y \left (0\right )+\left (t -\frac {1}{6} \alpha ^{2} t^{3}-\frac {1}{6} \alpha \,t^{3}+\frac {1}{3} t^{3}+\frac {1}{120} t^{5} \alpha ^{4}+\frac {1}{60} t^{5} \alpha ^{3}-\frac {13}{120} t^{5} \alpha ^{2}-\frac {7}{60} t^{5} \alpha +\frac {1}{5} t^{5}\right ) y^{\prime }\left (0\right )+O\left (t^{6}\right ) \] Verified OK. \[ y = \left (1+\left (-\frac {1}{2} \alpha ^{2}-\frac {1}{2} \alpha \right ) t^{2}+\left (-\frac {5}{24} \alpha ^{2}-\frac {1}{4} \alpha +\frac {1}{24} \alpha ^{4}+\frac {1}{12} \alpha ^{3}\right ) t^{4}\right ) c_{1} +\left (t +\left (-\frac {1}{6} \alpha ^{2}-\frac {1}{6} \alpha +\frac {1}{3}\right ) t^{3}+\left (-\frac {13}{120} \alpha ^{2}-\frac {7}{60} \alpha +\frac {1}{5}+\frac {1}{120} \alpha ^{4}+\frac {1}{60} \alpha ^{3}\right ) t^{5}\right ) c_{2} +O\left (t^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (1-\frac {\alpha \left (1+\alpha \right ) t^{2}}{2}+\frac {\alpha \left (\alpha ^{3}+2 \alpha ^{2}-5 \alpha -6\right ) t^{4}}{24}\right ) y \left (0\right )+\left (t -\frac {\left (\alpha ^{2}+\alpha -2\right ) t^{3}}{6}+\frac {\left (\alpha ^{4}+2 \alpha ^{3}-13 \alpha ^{2}-14 \alpha +24\right ) t^{5}}{120}\right ) D\left (y \right )\left (0\right )+O\left (t^{6}\right ) \]

Problem 1775

ODE	\[ \boxed {\left (-t^{2}+1\right ) y^{\prime \prime }-t y^{\prime }+\alpha ^{2} y=0} \] With the expansion point for the power series method at \(t = 0\).
program solution	\[ y = \left (1-\frac {1}{2} t^{2} \alpha ^{2}+\frac {1}{24} \alpha ^{4} t^{4}-\frac {1}{6} \alpha ^{2} t^{4}-\frac {1}{720} t^{6} \alpha ^{6}+\frac {1}{36} t^{6} \alpha ^{4}-\frac {4}{45} t^{6} \alpha ^{2}\right ) y \left (0\right )+\left (t -\frac {1}{6} \alpha ^{2} t^{3}+\frac {1}{6} t^{3}+\frac {1}{120} t^{5} \alpha ^{4}-\frac {1}{12} t^{5} \alpha ^{2}+\frac {3}{40} t^{5}\right ) y^{\prime }\left (0\right )+O\left (t^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {t^{2} \alpha ^{2}}{2}+\left (\frac {1}{24} \alpha ^{4}-\frac {1}{6} \alpha ^{2}\right ) t^{4}\right ) c_{1} +\left (t +\left (-\frac {\alpha ^{2}}{6}+\frac {1}{6}\right ) t^{3}+\left (\frac {1}{120} \alpha ^{4}-\frac {1}{12} \alpha ^{2}+\frac {3}{40}\right ) t^{5}\right ) c_{2} +O\left (t^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (1-\frac {\alpha ^{2} t^{2}}{2}+\frac {\alpha ^{2} \left (\alpha ^{2}-4\right ) t^{4}}{24}\right ) y \left (0\right )+\left (t -\frac {\left (\alpha ^{2}-1\right ) t^{3}}{6}+\frac {\left (\alpha ^{4}-10 \alpha ^{2}+9\right ) t^{5}}{120}\right ) D\left (y \right )\left (0\right )+O\left (t^{6}\right ) \]

Problem 1776

ODE	\[ \boxed {y^{\prime \prime }+t^{3} y^{\prime }+3 y t^{2}=0} \] With the expansion point for the power series method at \(t = 0\).
program solution	\[ y = \left (1-\frac {t^{4}}{4}\right ) y \left (0\right )+\left (t -\frac {1}{5} t^{5}\right ) y^{\prime }\left (0\right )+O\left (t^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {t^{4}}{4}\right ) c_{1} +\left (t -\frac {1}{5} t^{5}\right ) c_{2} +O\left (t^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (1-\frac {t^{4}}{4}\right ) y \left (0\right )+\left (t -\frac {1}{5} t^{5}\right ) D\left (y \right )\left (0\right )+O\left (t^{6}\right ) \]

Problem 1777

ODE	\[ \boxed {y^{\prime \prime }+t^{3} y^{\prime }+3 y t^{2}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*} With the expansion point for the power series method at \(t = 0\).
program solution	\[ y = O\left (t^{6}\right ) \] Verified OK. \[ y = O\left (t^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = 0 \]

Problem 1778

ODE	\[ \boxed {\left (1-t \right ) y^{\prime \prime }+t y^{\prime }+y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*} With the expansion point for the power series method at \(t = 0\).
program solution	\[ y = 1-\frac {t^{2}}{2}-\frac {t^{3}}{6}+\frac {t^{4}}{24}+\frac {7 t^{5}}{120}+\frac {23 t^{6}}{720}+O\left (t^{6}\right ) \] Verified OK. \[ y = 1-\frac {t^{2}}{2}-\frac {t^{3}}{6}+\frac {t^{4}}{24}+\frac {7 t^{5}}{120}+O\left (t^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = 1-\frac {1}{2} t^{2}-\frac {1}{6} t^{3}+\frac {1}{24} t^{4}+\frac {7}{120} t^{5}+\operatorname {O}\left (t^{6}\right ) \]

Problem 1779

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }+y t=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = 2] \end {align*} With the expansion point for the power series method at \(t = 0\).
program solution	\[ y = -t^{2}+2 t -1+\frac {t^{3}}{2}-\frac {7 t^{4}}{24}+\frac {13 t^{5}}{120}-\frac {5 t^{6}}{144}+O\left (t^{6}\right ) \] Verified OK. \[ y = -1+\frac {t^{3}}{2}-\frac {7 t^{4}}{24}+\frac {13 t^{5}}{120}+2 t -t^{2}+O\left (t^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = -1+2 t -t^{2}+\frac {1}{2} t^{3}-\frac {7}{24} t^{4}+\frac {13}{120} t^{5}+\operatorname {O}\left (t^{6}\right ) \]

Problem 1780

ODE	\[ \boxed {y^{\prime \prime }+t y^{\prime }+y \,{\mathrm e}^{t}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*} With the expansion point for the power series method at \(t = 0\).
program solution	\[ y = 1-\frac {t^{2}}{2}-\frac {t^{3}}{6}+\frac {t^{4}}{12}+\frac {t^{5}}{20}-\frac {t^{6}}{720}+O\left (t^{6}\right ) \] Verified OK. \[ y = 1-\frac {t^{2}}{2}-\frac {t^{3}}{6}+\frac {t^{4}}{12}+\frac {t^{5}}{20}+O\left (t^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = 1-\frac {1}{2} t^{2}-\frac {1}{6} t^{3}+\frac {1}{12} t^{4}+\frac {1}{20} t^{5}+\operatorname {O}\left (t^{6}\right ) \]

Problem 1781

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }+y \,{\mathrm e}^{t}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = -1] \end {align*} With the expansion point for the power series method at \(t = 0\).
program solution	\[ y = -t +\frac {t^{2}}{2}+\frac {t^{4}}{24}-\frac {t^{5}}{120}-\frac {t^{6}}{360}+O\left (t^{6}\right ) \] Verified OK. \[ y = -t +\frac {t^{2}}{2}+\frac {t^{4}}{24}-\frac {t^{5}}{120}+O\left (t^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = -t +\frac {1}{2} t^{2}+\frac {1}{24} t^{4}-\frac {1}{120} t^{5}+\operatorname {O}\left (t^{6}\right ) \]

Problem 1782

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }+{\mathrm e}^{-t} y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = 5] \end {align*} With the expansion point for the power series method at \(t = 0\).
program solution	\[ y = t^{3}-4 t^{2}+5 t +3+\frac {3 t^{4}}{8}-\frac {17 t^{5}}{40}+\frac {131 t^{6}}{720}+O\left (t^{6}\right ) \] Verified OK. \[ y = 3-4 t^{2}+t^{3}+\frac {3 t^{4}}{8}-\frac {17 t^{5}}{40}+5 t +O\left (t^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = 3+5 t -4 t^{2}+t^{3}+\frac {3}{8} t^{4}-\frac {17}{40} t^{5}+\operatorname {O}\left (t^{6}\right ) \]

Problem 1783

ODE	\[ \boxed {t^{2} y^{\prime \prime }-5 t y^{\prime }+9 y=0} \]
program solution	\[ y = c_{1} t^{3}+c_{2} t^{3} \ln \left (t \right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = t^{3} \left (c_{2} \ln \left (t \right )+c_{1} \right ) \]

Problem 1784

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

Problem 1785

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

Problem 1786

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

Problem 1787

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

Problem 1788

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

Problem 1789

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

Problem 1790

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

Problem 1791

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

Problem 1792

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

Problem 1793

ODE	\[ \boxed {t \left (t -2\right )^{2} y^{\prime \prime }+t y^{\prime }+y=0} \] With the expansion point for the power series method at \(t = 0\).
program solution	\[ y = c_{1} t \left (1-\frac {t}{4}-\frac {5 t^{2}}{96}-\frac {13 t^{3}}{1152}-\frac {199 t^{4}}{92160}-\frac {1123 t^{5}}{5529600}+O\left (t^{6}\right )\right )+c_{2} \left (-\frac {t \left (1-\frac {t}{4}-\frac {5 t^{2}}{96}-\frac {13 t^{3}}{1152}-\frac {199 t^{4}}{92160}-\frac {1123 t^{5}}{5529600}+O\left (t^{6}\right )\right ) \ln \left (t \right )}{4}+1-\frac {3 t^{2}}{16}-\frac {25 t^{3}}{2304}+\frac {5 t^{4}}{1728}+\frac {50087 t^{5}}{22118400}+O\left (t^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} t \left (1-\frac {1}{4} t -\frac {5}{96} t^{2}-\frac {13}{1152} t^{3}-\frac {199}{92160} t^{4}-\frac {1123}{5529600} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_{2} \left (\ln \left (t \right ) \left (-\frac {1}{4} t +\frac {1}{16} t^{2}+\frac {5}{384} t^{3}+\frac {13}{4608} t^{4}+\frac {199}{368640} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+\left (1-\frac {1}{4} t -\frac {1}{8} t^{2}+\frac {5}{2304} t^{3}+\frac {79}{13824} t^{4}+\frac {62027}{22118400} t^{5}+\operatorname {O}\left (t^{6}\right )\right )\right ) \]

Problem 1794

ODE	\[ \boxed {t \left (t -2\right )^{2} y^{\prime \prime }+t y^{\prime }+y=0} \] With the expansion point for the power series method at \(t = 2\).
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 1795

ODE	\[ \boxed {\sin \left (t \right ) y^{\prime \prime }+\cos \left (t \right ) y^{\prime }+\frac {y}{t}=0} \] With the expansion point for the power series method at \(t = 0\).
program solution	\[ y = c_{1} t^{i} \left (1+\left (\frac {1}{48}+\frac {i}{16}\right ) t^{2}+\left (\frac {1}{57600}+\frac {217 i}{57600}\right ) t^{4}+O\left (t^{6}\right )\right )+c_{2} t^{-i} \left (1+\left (\frac {1}{48}-\frac {i}{16}\right ) t^{2}+\left (\frac {1}{57600}-\frac {217 i}{57600}\right ) t^{4}+O\left (t^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} t^{-i} \left (1+\left (\frac {1}{48}-\frac {i}{16}\right ) t^{2}+\left (\frac {1}{57600}-\frac {217 i}{57600}\right ) t^{4}+\operatorname {O}\left (t^{6}\right )\right )+c_{2} t^{i} \left (1+\left (\frac {1}{48}+\frac {i}{16}\right ) t^{2}+\left (\frac {1}{57600}+\frac {217 i}{57600}\right ) t^{4}+\operatorname {O}\left (t^{6}\right )\right ) \]

Problem 1796

ODE	\[ \boxed {\left ({\mathrm e}^{t}-1\right ) y^{\prime \prime }+y^{\prime } {\mathrm e}^{t}+y=0} \] With the expansion point for the power series method at \(t = 0\).
program solution	\[ y = c_{1} \left (1-t +\frac {t^{2}}{2}-\frac {t^{3}}{6}+\frac {t^{4}}{24}-\frac {t^{5}}{120}+O\left (t^{6}\right )\right )+c_{2} \left (\left (1-t +\frac {t^{2}}{2}-\frac {t^{3}}{6}+\frac {t^{4}}{24}-\frac {t^{5}}{120}+O\left (t^{6}\right )\right ) \ln \left (t \right )+\frac {3 t}{2}-\frac {23 t^{2}}{24}+\frac {3 t^{3}}{8}-\frac {301 t^{4}}{2880}+\frac {13 t^{5}}{576}+O\left (t^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (c_{2} \ln \left (t \right )+c_{1} \right ) \left (1-t +\frac {1}{2} t^{2}-\frac {1}{6} t^{3}+\frac {1}{24} t^{4}-\frac {1}{120} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+\left (\frac {3}{2} t -\frac {23}{24} t^{2}+\frac {3}{8} t^{3}-\frac {301}{2880} t^{4}+\frac {13}{576} t^{5}+\operatorname {O}\left (t^{6}\right )\right ) c_{2} \]

Problem 1797

ODE	\[ \boxed {\left (-t^{2}+1\right ) y^{\prime \prime }+\frac {y^{\prime }}{\sin \left (t +1\right )}+y=0} \] With the expansion point for the power series method at \(t = -1\).
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 1798

ODE	\[ \boxed {t^{3} y^{\prime \prime }+\sin \left (t^{3}\right ) y^{\prime }+y t=0} \] With the expansion point for the power series method at \(t = 0\).
program solution	\[ y = c_{1} t^{\frac {1}{2}+\frac {i \sqrt {3}}{2}} \left (1-\frac {t}{2}+\frac {\left (i \sqrt {3}+3\right ) t^{2}}{16+8 i \sqrt {3}}+\frac {\left (-i \sqrt {3}-5\right ) t^{3}}{48 i \sqrt {3}+96}+\frac {\left (i \sqrt {3}+5\right ) \left (i \sqrt {3}+7\right ) t^{4}}{384 \left (i \sqrt {3}+4\right ) \left (2+i \sqrt {3}\right )}-\frac {\left (i \sqrt {3}+7\right ) \left (i \sqrt {3}+9\right ) t^{5}}{3840 \left (i \sqrt {3}+4\right ) \left (2+i \sqrt {3}\right )}+\frac {\left (i \sqrt {3}+7\right ) \left (i \sqrt {3}+9\right ) \left (i \sqrt {3}+11\right ) t^{6}}{46080 \left (6+i \sqrt {3}\right ) \left (i \sqrt {3}+4\right ) \left (2+i \sqrt {3}\right )}+O\left (t^{6}\right )\right )+c_{2} t^{\frac {1}{2}-\frac {i \sqrt {3}}{2}} \left (1-\frac {t}{2}+\frac {\left (-i \sqrt {3}+3\right ) t^{2}}{16-8 i \sqrt {3}}+\frac {\left (i \sqrt {3}-5\right ) t^{3}}{-48 i \sqrt {3}+96}+\frac {\left (-i \sqrt {3}+5\right ) \left (-i \sqrt {3}+7\right ) t^{4}}{384 \left (-i \sqrt {3}+4\right ) \left (2-i \sqrt {3}\right )}-\frac {\left (-i \sqrt {3}+7\right ) \left (-i \sqrt {3}+9\right ) t^{5}}{3840 \left (-i \sqrt {3}+4\right ) \left (2-i \sqrt {3}\right )}+\frac {\left (-i \sqrt {3}+7\right ) \left (-i \sqrt {3}+9\right ) \left (-i \sqrt {3}+11\right ) t^{6}}{46080 \left (6-i \sqrt {3}\right ) \left (-i \sqrt {3}+4\right ) \left (2-i \sqrt {3}\right )}+O\left (t^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \sqrt {t}\, \left (c_{2} t^{\frac {i \sqrt {3}}{2}} \left (1-\frac {1}{2} t +\frac {i \sqrt {3}+3}{8 i \sqrt {3}+16} t^{2}+\frac {-i \sqrt {3}-5}{48 i \sqrt {3}+96} t^{3}+\frac {1}{384} \frac {\left (i \sqrt {3}+5\right ) \left (i \sqrt {3}+7\right )}{\left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+2\right )} t^{4}-\frac {1}{3840} \frac {\left (i \sqrt {3}+7\right ) \left (i \sqrt {3}+9\right )}{\left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+2\right )} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_{1} t^{-\frac {i \sqrt {3}}{2}} \left (1-\frac {1}{2} t +\frac {\sqrt {3}+3 i}{8 \sqrt {3}+16 i} t^{2}+\frac {-\sqrt {3}-5 i}{48 \sqrt {3}+96 i} t^{3}+\frac {3 i \sqrt {3}-8}{576 i \sqrt {3}-480} t^{4}-\frac {1}{3840} \frac {\left (\sqrt {3}+7 i\right ) \left (\sqrt {3}+9 i\right )}{\left (\sqrt {3}+4 i\right ) \left (\sqrt {3}+2 i\right )} t^{5}+\operatorname {O}\left (t^{6}\right )\right )\right ) \]

Problem 1799

ODE	\[ \boxed {2 t^{2} y^{\prime \prime }+3 t y^{\prime }-\left (t +1\right ) y=0} \] With the expansion point for the power series method at \(t = 0\).
program solution	\[ y = c_{1} \sqrt {t}\, \left (1+\frac {t}{5}+\frac {t^{2}}{70}+\frac {t^{3}}{1890}+\frac {t^{4}}{83160}+\frac {t^{5}}{5405400}+O\left (t^{6}\right )\right )+\frac {c_{2} \left (1-t -\frac {t^{2}}{2}-\frac {t^{3}}{18}-\frac {t^{4}}{360}-\frac {t^{5}}{12600}+O\left (t^{6}\right )\right )}{t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {c_{2} t^{\frac {3}{2}} \left (1+\frac {1}{5} t +\frac {1}{70} t^{2}+\frac {1}{1890} t^{3}+\frac {1}{83160} t^{4}+\frac {1}{5405400} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_{1} \left (1-t -\frac {1}{2} t^{2}-\frac {1}{18} t^{3}-\frac {1}{360} t^{4}-\frac {1}{12600} t^{5}+\operatorname {O}\left (t^{6}\right )\right )}{t} \]

Problem 1800

ODE	\[ \boxed {2 t y^{\prime \prime }+\left (1-2 t \right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(t = 0\).
program solution	\[ y = c_{1} \sqrt {t}\, \left (1+\frac {2 t}{3}+\frac {4 t^{2}}{15}+\frac {8 t^{3}}{105}+\frac {16 t^{4}}{945}+\frac {32 t^{5}}{10395}+O\left (t^{6}\right )\right )+c_{2} \left (1+t +\frac {t^{2}}{2}+\frac {t^{3}}{6}+\frac {t^{4}}{24}+\frac {t^{5}}{120}+O\left (t^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} \sqrt {t}\, \left (1+\frac {2}{3} t +\frac {4}{15} t^{2}+\frac {8}{105} t^{3}+\frac {16}{945} t^{4}+\frac {32}{10395} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_{2} \left (1+t +\frac {1}{2} t^{2}+\frac {1}{6} t^{3}+\frac {1}{24} t^{4}+\frac {1}{120} t^{5}+\operatorname {O}\left (t^{6}\right )\right ) \]