2.17.7 Problems 601 to 700

Problem 601

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

Problem 602

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

Problem 603

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

Problem 604

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

Problem 605

ODE	\[ \boxed {y^{\prime \prime }-9 y^{\prime }+9 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {3 \left (\sqrt {5}-3\right ) x}{2}}+\frac {c_{2} \sqrt {5}\, {\mathrm e}^{\frac {3 \left (3+\sqrt {5}\right ) x}{2}}}{15} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{\frac {3 \left (3+\sqrt {5}\right ) x}{2}}+c_{2} {\mathrm e}^{-\frac {3 \left (\sqrt {5}-3\right ) x}{2}} \]

Problem 606

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

Problem 607

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

Problem 608

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

Problem 609

ODE	\[ \boxed {6 y^{\prime \prime }-5 y^{\prime }+y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 4, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = -8 \,{\mathrm e}^{\frac {x}{2}}+12 \,{\mathrm e}^{\frac {x}{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -8 \,{\mathrm e}^{\frac {x}{2}}+12 \,{\mathrm e}^{\frac {x}{3}} \]

Problem 610

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

Problem 611

ODE	\[ \boxed {y^{\prime \prime }+5 y^{\prime }+3 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {\left (13+5 \sqrt {13}\right ) {\mathrm e}^{\frac {\left (-5+\sqrt {13}\right ) x}{2}}}{26}+\frac {\left (13-5 \sqrt {13}\right ) {\mathrm e}^{-\frac {\left (5+\sqrt {13}\right ) x}{2}}}{26} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (13+5 \sqrt {13}\right ) {\mathrm e}^{\frac {\left (-5+\sqrt {13}\right ) x}{2}}}{26}+\frac {\left (13-5 \sqrt {13}\right ) {\mathrm e}^{-\frac {\left (5+\sqrt {13}\right ) x}{2}}}{26} \]

Problem 612

ODE	\[ \boxed {2 y^{\prime \prime }+y^{\prime }-4 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \frac {2 \sqrt {33}\, \left ({\mathrm e}^{\frac {\left (-1+\sqrt {33}\right ) x}{4}}-{\mathrm e}^{-\frac {\left (1+\sqrt {33}\right ) x}{4}}\right )}{33} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {2 \left ({\mathrm e}^{\frac {\left (-1+\sqrt {33}\right ) x}{4}}-{\mathrm e}^{-\frac {\left (1+\sqrt {33}\right ) x}{4}}\right ) \sqrt {33}}{33} \]

Problem 613

ODE	\[ \boxed {y^{\prime \prime }+8 y^{\prime }-9 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1, y^{\prime }\left (1\right ) = 0] \end {align*}
program solution	\[ y = \frac {9 \,{\mathrm e}^{x} {\mathrm e}^{-1}}{10}+\frac {{\mathrm e}^{-9 x} {\mathrm e}^{9}}{10} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{9-9 x}}{10}+\frac {9 \,{\mathrm e}^{x -1}}{10} \]

Problem 614

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

Problem 615

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

Problem 616

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

Problem 617

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

Problem 618

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

Problem 619

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

Problem 620

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

Problem 621

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

Problem 622

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

Problem 623

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

Problem 624

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

Problem 625

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

Problem 626

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

Problem 627

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+13 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-3 x} \cos \left (2 x \right )+\frac {c_{2} {\mathrm e}^{-3 x} \sin \left (2 x \right )}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-3 x} \left (\sin \left (2 x \right ) c_{1} +c_{2} \cos \left (2 x \right )\right ) \]

Problem 628

ODE	\[ \boxed {4 y^{\prime \prime }+9 y=0} \]
program solution	\[ y = c_{1} \cos \left (\frac {3 x}{2}\right )+\frac {2 c_{2} \sin \left (\frac {3 x}{2}\right )}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sin \left (\frac {3 x}{2}\right )+c_{2} \cos \left (\frac {3 x}{2}\right ) \]

Problem 629

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

Problem 630

ODE	\[ \boxed {9 y^{\prime \prime }+9 y^{\prime }-4 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {4 x}{3}}+\frac {3 c_{2} {\mathrm e}^{\frac {x}{3}}}{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} {\mathrm e}^{\frac {5 x}{3}}+c_{1} \right ) {\mathrm e}^{-\frac {4 x}{3}} \]

Problem 631

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

Problem 632

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+\frac {25 y}{4}=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x} \cos \left (\frac {3 x}{2}\right )+\frac {2 c_{2} {\mathrm e}^{-2 x} \sin \left (\frac {3 x}{2}\right )}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-2 x} \left (c_{1} \sin \left (\frac {3 x}{2}\right )+c_{2} \cos \left (\frac {3 x}{2}\right )\right ) \]

Problem 633

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

Problem 634

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

Problem 635

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

Problem 636

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

Problem 637

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }+\frac {5 y}{4}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \frac {{\mathrm e}^{-\frac {x}{2}} \left (6 \cos \left (x \right )+5 \sin \left (x \right )\right )}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{-\frac {x}{2}} \left (5 \sin \left (x \right )+6 \cos \left (x \right )\right )}{2} \]

Problem 638

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

Problem 639

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

Problem 640

ODE	\[ \boxed {5 u^{\prime \prime }+2 u^{\prime }+7 u=0} \] With initial conditions \begin {align*} [u \left (0\right ) = 2, u^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ u = \frac {\left (7 \sqrt {34}\, \sin \left (\frac {\sqrt {34}\, x}{5}\right )+68 \cos \left (\frac {\sqrt {34}\, x}{5}\right )\right ) {\mathrm e}^{-\frac {x}{5}}}{34} \] Verified OK.
Maple solution	\[ u \left (x \right ) = \frac {{\mathrm e}^{-\frac {x}{5}} \left (7 \sqrt {34}\, \sin \left (\frac {\sqrt {34}\, x}{5}\right )+68 \cos \left (\frac {\sqrt {34}\, x}{5}\right )\right )}{34} \]

Problem 641

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+6 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = \alpha ] \end {align*}
program solution	\[ y = \frac {{\mathrm e}^{-x} \left (\sin \left (\sqrt {5}\, x \right ) \sqrt {5}\, \left (\alpha +2\right )+10 \cos \left (\sqrt {5}\, x \right )\right )}{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{-x} \left (\sqrt {5}\, \left (\alpha +2\right ) \sin \left (\sqrt {5}\, x \right )+10 \cos \left (\sqrt {5}\, x \right )\right )}{5} \]

Problem 642

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

Problem 643

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 644

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

Problem 645

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

Problem 646

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

Problem 647

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

Problem 648

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

Problem 649

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

Problem 650

ODE	\[ \boxed {t^{2} y^{\prime \prime }+7 t y^{\prime }+10 y=0} \]
program solution	\[ y = c_{1} t^{-3-i}-\frac {i c_{2} t^{-3+i}}{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {c_{1} \sin \left (\ln \left (t \right )\right )+c_{2} \cos \left (\ln \left (t \right )\right )}{t^{3}} \]

Problem 651

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

Problem 652

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

Problem 653

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

Problem 654

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

Problem 655

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

Problem 656

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

Problem 657

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }+10 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{x} \cos \left (3 x \right )+\frac {c_{2} {\mathrm e}^{x} \sin \left (3 x \right )}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{x} \left (c_{1} \sin \left (3 x \right )+c_{2} \cos \left (3 x \right )\right ) \]

Problem 658

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

Problem 659

ODE	\[ \boxed {4 y^{\prime \prime }+17 y^{\prime }+4 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-4 x}+\frac {4 c_{2} {\mathrm e}^{-\frac {x}{4}}}{15} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{-4 x}+c_{2} {\mathrm e}^{-\frac {x}{4}} \]

Problem 660

ODE	\[ \boxed {16 y^{\prime \prime }+24 y^{\prime }+9 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {3 x}{4}}+c_{2} x \,{\mathrm e}^{-\frac {3 x}{4}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\frac {3 x}{4}} \left (c_{2} x +c_{1} \right ) \]

Problem 661

ODE	\[ \boxed {25 y^{\prime \prime }-20 y^{\prime }+4 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{\frac {2 x}{5}}+c_{2} x \,{\mathrm e}^{\frac {2 x}{5}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{\frac {2 x}{5}} \left (c_{2} x +c_{1} \right ) \]

Problem 662

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

Problem 663

ODE	\[ \boxed {9 y^{\prime \prime }-12 y^{\prime }+4 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = -1] \end {align*}
program solution	\[ y = {\mathrm e}^{\frac {2 t}{3}} \left (2-\frac {7 t}{3}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {{\mathrm e}^{\frac {2 t}{3}} \left (-6+7 t \right )}{3} \]

Problem 664

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

Problem 665

ODE	\[ \boxed {9 y^{\prime \prime }+6 y^{\prime }+82 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = 2] \end {align*}
program solution	\[ y = -\frac {{\mathrm e}^{-\frac {t}{3}} \left (9 \cos \left (3 t \right )-5 \sin \left (3 t \right )\right )}{9} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {{\mathrm e}^{-\frac {t}{3}} \left (5 \sin \left (3 t \right )-9 \cos \left (3 t \right )\right )}{9} \]

Problem 666

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

Problem 667

ODE	\[ \boxed {4 y^{\prime \prime }+12 y^{\prime }+9 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = -4] \end {align*}
program solution	\[ y = {\mathrm e}^{-\frac {3 t}{2}} \left (1-\frac {5 t}{2}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {{\mathrm e}^{-\frac {3 t}{2}} \left (-2+5 t \right )}{2} \]

Problem 668

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

Problem 669

ODE	\[ \boxed {t^{2} y^{\prime \prime }-4 t y^{\prime }+6 y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= t^{2} \end {align*}
program solution	\[ y = c_{2} t^{3}+c_{1} t^{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = t^{2} \left (c_{2} t +c_{1} \right ) \]

Problem 670

ODE	\[ \boxed {t^{2} y^{\prime \prime }+2 t y^{\prime }-2 y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= t \end {align*}
program solution	\[ y = c_{1} t -\frac {c_{2}}{3 t^{2}} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {c_{1} t^{3}+c_{2}}{t^{2}} \]

Problem 671

ODE	\[ \boxed {t^{2} y^{\prime \prime }+3 t y^{\prime }+y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \frac {1}{t} \end {align*}
program solution	\[ y = \frac {c_{1}}{t}+\frac {c_{2} \ln \left (t \right )}{t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {c_{2} \ln \left (t \right )+c_{1}}{t} \]

Problem 672

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

Problem 673

ODE	\[ \boxed {x y^{\prime \prime }-y^{\prime }+4 y x^{3}=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \sin \left (x^{2}\right ) \end {align*}
program solution	\[ y = \sin \left (x^{2}\right ) c_{1} -\frac {c_{2} \sin \left (x^{2}\right ) \cot \left (x^{2}\right )}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sin \left (x^{2}\right )+c_{2} \cos \left (x^{2}\right ) \]

Problem 674

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

Problem 675

ODE	\[ \boxed {x^{2} y^{\prime \prime }-\left (x -\frac {3}{16}\right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= x^{\frac {1}{4}} {\mathrm e}^{2 \sqrt {x}} \end {align*}
program solution	\[ y = x^{\frac {1}{4}} {\mathrm e}^{2 \sqrt {x}} c_{1} -\frac {c_{2} x^{\frac {1}{4}} {\mathrm e}^{2 \sqrt {x}} {\mathrm e}^{-4 \sqrt {x}}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{\frac {1}{4}} \left (c_{1} \sinh \left (2 \sqrt {x}\right )+c_{2} \cosh \left (2 \sqrt {x}\right )\right ) \]

Problem 676

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

Problem 677

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

Problem 678

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

Problem 679

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

Problem 680

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 681

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

Problem 682

ODE	\[ \boxed {t^{2} y^{\prime \prime }+5 t y^{\prime }+13 y=0} \]
program solution	\[ y = c_{1} t^{-2-3 i}-\frac {i c_{2} t^{-2+3 i}}{6} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {c_{1} \sin \left (3 \ln \left (t \right )\right )+c_{2} \cos \left (3 \ln \left (t \right )\right )}{t^{2}} \]

Problem 683

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

Problem 684

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

Problem 685

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

Problem 686

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

Problem 687

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

Problem 688

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

Problem 689

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

Problem 690

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

Problem 691

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

Problem 692

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

Problem 693

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

Problem 694

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

Problem 695

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

Problem 696

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

Problem 697

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

Problem 698

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

Problem 699

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

Problem 700

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (x^{2}-\frac {1}{4}\right ) y=g \left (x \right )} \]
program solution	\[ y = \frac {c_{1} \cos \left (x \right )+c_{2} \sin \left (x \right )}{\sqrt {x}}+\frac {-\left (\int _{0}^{x}\frac {\sin \left (\alpha \right ) g \left (\alpha \right )}{\alpha ^{\frac {3}{2}}}d \alpha \right ) \cos \left (x \right )+\left (\int _{0}^{x}\frac {\cos \left (\alpha \right ) g \left (\alpha \right )}{\alpha ^{\frac {3}{2}}}d \alpha \right ) \sin \left (x \right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\sin \left (x \right ) c_{2} +\cos \left (x \right ) c_{1} +\left (\int \frac {\cos \left (x \right ) g \left (x \right )}{x^{\frac {3}{2}}}d x \right ) \sin \left (x \right )-\left (\int \frac {\sin \left (x \right ) g \left (x \right )}{x^{\frac {3}{2}}}d x \right ) \cos \left (x \right )}{\sqrt {x}} \]