Problems 13101 to 13200

Problem 13101


ODE	\begin {align} x^{\prime }\left (t \right )&=2 x \left (t \right )-6 y\\ y^{\prime }&=2 x \left (t \right )+y \end {align} With initial conditions \[ [x \left (0\right ) = 2, y \left (0\right ) = 1] \]

program solution

Maple solution	\begin{align} x \left (t \right ) &= {\mathrm e}^{\frac {3 t}{2}} \left (-\frac {10 \sqrt {47}\, \sin \left (\frac {\sqrt {47}\, t}{2}\right )}{47}+2 \cos \left (\frac {\sqrt {47}\, t}{2}\right )\right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{\frac {3 t}{2}} \left (\frac {84 \sqrt {47}\, \sin \left (\frac {\sqrt {47}\, t}{2}\right )}{47}+12 \cos \left (\frac {\sqrt {47}\, t}{2}\right )\right )}{12} \\ \end{align}

Problem 13102


ODE	\begin {align} x^{\prime }\left (t \right )&=x \left (t \right )+4 y\\ y^{\prime }&=-3 x \left (t \right )+2 y \end {align} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = -1] \]

program solution

Maple solution	\begin{align} x \left (t \right ) &= {\mathrm e}^{\frac {3 t}{2}} \left (-\frac {9 \sqrt {47}\, \sin \left (\frac {\sqrt {47}\, t}{2}\right )}{47}+\cos \left (\frac {\sqrt {47}\, t}{2}\right )\right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{\frac {3 t}{2}} \left (\frac {56 \sqrt {47}\, \sin \left (\frac {\sqrt {47}\, t}{2}\right )}{47}+8 \cos \left (\frac {\sqrt {47}\, t}{2}\right )\right )}{8} \\ \end{align}

Problem 13103


ODE	\begin {align} x^{\prime }\left (t \right )&=2 y\\ y^{\prime }&=-2 x \left (t \right ) \end {align} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 0] \]

program solution

Maple solution	\begin{align} x \left (t \right ) &= \cos \left (2 t \right ) \\ y \left (t \right ) &= -\sin \left (2 t \right ) \\ \end{align}

Problem 13104


ODE	\begin {align} x^{\prime }\left (t \right )&=2 x \left (t \right )+2 y\\ y^{\prime }&=-4 x \left (t \right )+6 y \end {align} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 1] \]

program solution

Maple solution	\begin{align} x \left (t \right ) &= {\mathrm e}^{4 t} \cos \left (2 t \right ) \\ y \left (t \right ) &= {\mathrm e}^{4 t} \left (\cos \left (2 t \right )-\sin \left (2 t \right )\right ) \\ \end{align}

Problem 13105


ODE	\begin {align} x^{\prime }\left (t \right )&=-3 x \left (t \right )-5 y\\ y^{\prime }&=3 x \left (t \right )+y \end {align} With initial conditions \[ [x \left (0\right ) = 4, y \left (0\right ) = 0] \]

program solution

Maple solution	\begin{align} x \left (t \right ) &= {\mathrm e}^{-t} \left (-\frac {8 \sqrt {11}\, \sin \left (\sqrt {11}\, t \right )}{11}+4 \cos \left (\sqrt {11}\, t \right )\right ) \\ y \left (t \right ) &= \frac {12 \,{\mathrm e}^{-t} \sqrt {11}\, \sin \left (\sqrt {11}\, t \right )}{11} \\ \end{align}

Problem 13106


ODE	\begin {align} x^{\prime }\left (t \right )&=2 y\\ y^{\prime }&=-2 x \left (t \right )-y \end {align} With initial conditions \[ [x \left (0\right ) = -1, y \left (0\right ) = 1] \]

program solution

Maple solution	\begin{align} x \left (t \right ) &= {\mathrm e}^{-\frac {t}{2}} \left (\frac {\sqrt {15}\, \sin \left (\frac {t \sqrt {15}}{2}\right )}{5}-\cos \left (\frac {t \sqrt {15}}{2}\right )\right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{-\frac {t}{2}} \left (-\frac {4 \sqrt {15}\, \sin \left (\frac {t \sqrt {15}}{2}\right )}{5}-4 \cos \left (\frac {t \sqrt {15}}{2}\right )\right )}{4} \\ \end{align}

Problem 13107


ODE	\begin {align} x^{\prime }\left (t \right )&=2 x \left (t \right )-6 y\\ y^{\prime }&=2 x \left (t \right )+y \end {align} With initial conditions \[ [x \left (0\right ) = 2, y \left (0\right ) = 1] \]

program solution

Maple solution	\begin{align} x \left (t \right ) &= {\mathrm e}^{\frac {3 t}{2}} \left (-\frac {10 \sqrt {47}\, \sin \left (\frac {\sqrt {47}\, t}{2}\right )}{47}+2 \cos \left (\frac {\sqrt {47}\, t}{2}\right )\right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{\frac {3 t}{2}} \left (\frac {84 \sqrt {47}\, \sin \left (\frac {\sqrt {47}\, t}{2}\right )}{47}+12 \cos \left (\frac {\sqrt {47}\, t}{2}\right )\right )}{12} \\ \end{align}

Problem 13108


ODE	\begin {align} x^{\prime }\left (t \right )&=x \left (t \right )+4 y\\ y^{\prime }&=-3 x \left (t \right )+2 y \end {align} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = -1] \]

program solution

Maple solution	\begin{align} x \left (t \right ) &= {\mathrm e}^{\frac {3 t}{2}} \left (-\frac {9 \sqrt {47}\, \sin \left (\frac {\sqrt {47}\, t}{2}\right )}{47}+\cos \left (\frac {\sqrt {47}\, t}{2}\right )\right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{\frac {3 t}{2}} \left (\frac {56 \sqrt {47}\, \sin \left (\frac {\sqrt {47}\, t}{2}\right )}{47}+8 \cos \left (\frac {\sqrt {47}\, t}{2}\right )\right )}{8} \\ \end{align}

Problem 13109


ODE	\begin {align} x^{\prime }\left (t \right )&=-\frac {9 x \left (t \right )}{10}-2 y\\ y^{\prime }&=x \left (t \right )+\frac {11 y}{10} \end {align} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 1] \]

program solution

Maple solution	\begin{align} x \left (t \right ) &= {\mathrm e}^{\frac {t}{10}} \left (-3 \sin \left (t \right )+\cos \left (t \right )\right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{\frac {t}{10}} \left (-4 \sin \left (t \right )-2 \cos \left (t \right )\right )}{2} \\ \end{align}

Problem 13110


ODE	\begin {align} x^{\prime }\left (t \right )&=-3 x \left (t \right )+10 y\\ y^{\prime }&=-x \left (t \right )+3 y \end {align}

program solution

Maple solution	\begin{align} x \left (t \right ) &= c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right ) \\ y \left (t \right ) &= \frac {c_{1} \cos \left (t \right )}{10}-\frac {c_{2} \sin \left (t \right )}{10}+\frac {3 c_{1} \sin \left (t \right )}{10}+\frac {3 c_{2} \cos \left (t \right )}{10} \\ \end{align}

Problem 13111


ODE	\begin {align} x^{\prime }\left (t \right )&=-3 x \left (t \right )\\ y^{\prime }&=x \left (t \right )-3 y \end {align} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 0] \]

program solution

Maple solution	\begin{align} x \left (t \right ) &= {\mathrm e}^{-3 t} \\ y \left (t \right ) &= t \,{\mathrm e}^{-3 t} \\ \end{align}

Problem 13112


ODE	\begin {align} x^{\prime }\left (t \right )&=2 x \left (t \right )+y\\ y^{\prime }&=-x \left (t \right )-2 y \end {align} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 0] \]

program solution

Maple solution	\begin{align} x \left (t \right ) &= \left (\frac {1}{2}+\frac {\sqrt {3}}{3}\right ) {\mathrm e}^{\sqrt {3}\, t}+\left (\frac {1}{2}-\frac {\sqrt {3}}{3}\right ) {\mathrm e}^{-\sqrt {3}\, t} \\ y \left (t \right ) &= \left (\frac {1}{2}+\frac {\sqrt {3}}{3}\right ) \sqrt {3}\, {\mathrm e}^{\sqrt {3}\, t}-\left (\frac {1}{2}-\frac {\sqrt {3}}{3}\right ) \sqrt {3}\, {\mathrm e}^{-\sqrt {3}\, t}-2 \left (\frac {1}{2}+\frac {\sqrt {3}}{3}\right ) {\mathrm e}^{\sqrt {3}\, t}-2 \left (\frac {1}{2}-\frac {\sqrt {3}}{3}\right ) {\mathrm e}^{-\sqrt {3}\, t} \\ \end{align}

Problem 13113


ODE	\begin {align} x^{\prime }\left (t \right )&=-2 x \left (t \right )-y\\ y^{\prime }&=x \left (t \right )-4 y \end {align} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 0] \]

program solution

Maple solution	\begin{align} x \left (t \right ) &= \left (t +1\right ) {\mathrm e}^{-3 t} \\ y \left (t \right ) &= t \,{\mathrm e}^{-3 t} \\ \end{align}

Problem 13114


ODE	\begin {align} x^{\prime }\left (t \right )&=y\\ y^{\prime }&=-x \left (t \right )-2 y \end {align} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 0] \]

program solution

Maple solution	\begin{align} x \left (t \right ) &= {\mathrm e}^{-t} \left (t +1\right ) \\ y \left (t \right ) &= -t \,{\mathrm e}^{-t} \\ \end{align}

Problem 13115


ODE	\begin {align} x^{\prime }\left (t \right )&=-3 x \left (t \right )\\ y^{\prime }&=x \left (t \right )-3 y \end {align} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 0] \]

program solution

Maple solution	\begin{align} x \left (t \right ) &= {\mathrm e}^{-3 t} \\ y \left (t \right ) &= t \,{\mathrm e}^{-3 t} \\ \end{align}

Problem 13116


ODE	\begin {align} x^{\prime }\left (t \right )&=2 x \left (t \right )+y\\ y^{\prime }&=-x \left (t \right )+4 y \end {align} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 0] \]

program solution

Maple solution	\begin{align} x \left (t \right ) &= {\mathrm e}^{3 t} \left (-t +1\right ) \\ y \left (t \right ) &= -{\mathrm e}^{3 t} t \\ \end{align}

Problem 13117


ODE	\begin {align} x^{\prime }\left (t \right )&=-2 x \left (t \right )-y\\ y^{\prime }&=x \left (t \right )-4 y \end {align} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 0] \]

program solution

Maple solution	\begin{align} x \left (t \right ) &= \left (t +1\right ) {\mathrm e}^{-3 t} \\ y \left (t \right ) &= t \,{\mathrm e}^{-3 t} \\ \end{align}

Problem 13118


ODE	\begin {align} x^{\prime }\left (t \right )&=y\\ y^{\prime }&=-x \left (t \right )-2 y \end {align} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 0] \]

program solution

Maple solution	\begin{align} x \left (t \right ) &= {\mathrm e}^{-t} \left (t +1\right ) \\ y \left (t \right ) &= -t \,{\mathrm e}^{-t} \\ \end{align}

Problem 13119


ODE	\begin {align} x^{\prime }\left (t \right )&=2 y\\ y^{\prime }&=-y \end {align} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 0] \]

program solution

Maple solution	\begin{align} x \left (t \right ) &= 1 \\ y \left (t \right ) &= 0 \\ \end{align}

Problem 13120


ODE	\begin {align} x^{\prime }\left (t \right )&=2 x \left (t \right )+4 y\\ y^{\prime }&=3 x \left (t \right )+6 y \end {align} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 0] \]

program solution

Maple solution	\begin{align} x \left (t \right ) &= \frac {3}{4}+\frac {{\mathrm e}^{8 t}}{4} \\ y \left (t \right ) &= \frac {3 \,{\mathrm e}^{8 t}}{8}-\frac {3}{8} \\ \end{align}

Problem 13121


ODE	\begin {align} x^{\prime }\left (t \right )&=4 x \left (t \right )+2 y\\ y^{\prime }&=2 x \left (t \right )+y \end {align} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 0] \]

program solution

Maple solution	\begin{align} x \left (t \right ) &= \frac {1}{5}+\frac {4 \,{\mathrm e}^{5 t}}{5} \\ y \left (t \right ) &= \frac {2 \,{\mathrm e}^{5 t}}{5}-\frac {2}{5} \\ \end{align}

Problem 13122


ODE	\begin {align} x^{\prime }\left (t \right )&=2 y\\ y^{\prime }&=0 \end {align}

program solution

Maple solution	\begin{align} x \left (t \right ) &= 2 c_{2} t +c_{1} \\ y \left (t \right ) &= c_{2} \\ \end{align}

Problem 13123


ODE	\begin {align} x^{\prime }\left (t \right )&=-2 y\\ y^{\prime }&=0 \end {align}

program solution

Maple solution	\begin{align} x \left (t \right ) &= -2 c_{2} t +c_{1} \\ y \left (t \right ) &= c_{2} \\ \end{align}

Problem 13124


ODE	\begin {align} x^{\prime }\left (t \right )&=-3 x \left (t \right )-y\\ y^{\prime }&=4 x \left (t \right )+y \end {align} With initial conditions \[ [x \left (0\right ) = -1, y \left (0\right ) = 2] \]

program solution

Maple solution	\begin{align} x \left (t \right ) &= -{\mathrm e}^{-t} \\ y \left (t \right ) &= 2 \,{\mathrm e}^{-t} \\ \end{align}

Problem 13125


ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }-7 y=0} \]

program solution	\[ y = c_{1} {\mathrm e}^{-t}+\frac {c_{2} {\mathrm e}^{7 t}}{8} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = c_{1} {\mathrm e}^{7 t}+c_{2} {\mathrm e}^{-t} \]

Problem 13126


ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }-12 y=0} \]

program solution	\[ y = c_{1} {\mathrm e}^{-3 t}+\frac {c_{2} {\mathrm e}^{4 t}}{7} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = \left ({\mathrm e}^{7 t} c_{2} +c_{1} \right ) {\mathrm e}^{-3 t} \]

Problem 13127


ODE	\begin {align} x^{\prime }\left (t \right )&=\frac {y}{10}\\ y^{\prime }&=\frac {z \left (t \right )}{5}\\ z^{\prime }\left (t \right )&=\frac {2 x \left (t \right )}{5} \end {align}

program solution

Maple solution	\begin{align} x \left (t \right ) &= \frac {{\mathrm e}^{\frac {t}{5}} c_{1}}{2}-\frac {c_{2} {\mathrm e}^{-\frac {t}{10}} \sin \left (\frac {\sqrt {3}\, t}{10}\right )}{4}+\frac {c_{2} {\mathrm e}^{-\frac {t}{10}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, t}{10}\right )}{4}-\frac {c_{3} {\mathrm e}^{-\frac {t}{10}} \cos \left (\frac {\sqrt {3}\, t}{10}\right )}{4}-\frac {c_{3} {\mathrm e}^{-\frac {t}{10}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{10}\right )}{4} \\ y \left (t \right ) &= {\mathrm e}^{\frac {t}{5}} c_{1} -\frac {c_{2} {\mathrm e}^{-\frac {t}{10}} \sin \left (\frac {\sqrt {3}\, t}{10}\right )}{2}-\frac {c_{2} {\mathrm e}^{-\frac {t}{10}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, t}{10}\right )}{2}-\frac {c_{3} {\mathrm e}^{-\frac {t}{10}} \cos \left (\frac {\sqrt {3}\, t}{10}\right )}{2}+\frac {c_{3} {\mathrm e}^{-\frac {t}{10}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{10}\right )}{2} \\ z \left (t \right ) &= {\mathrm e}^{\frac {t}{5}} c_{1} +c_{2} {\mathrm e}^{-\frac {t}{10}} \sin \left (\frac {\sqrt {3}\, t}{10}\right )+c_{3} {\mathrm e}^{-\frac {t}{10}} \cos \left (\frac {\sqrt {3}\, t}{10}\right ) \\ \end{align}

Problem 13128


ODE	\begin {align} x^{\prime }\left (t \right )&=y\\ y^{\prime }&=-x \left (t \right )\\ z^{\prime }\left (t \right )&=2 z \left (t \right ) \end {align}

program solution

Maple solution	\begin{align} x \left (t \right ) &= c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right ) \\ y \left (t \right ) &= c_{1} \cos \left (t \right )-c_{2} \sin \left (t \right ) \\ z \left (t \right ) &= c_{3} {\mathrm e}^{2 t} \\ \end{align}

Problem 13129


ODE	\begin {align} x^{\prime }\left (t \right )&=-2 x \left (t \right )+3 y\\ y^{\prime }&=3 x \left (t \right )-2 y\\ z^{\prime }\left (t \right )&=-z \left (t \right ) \end {align}

program solution

Maple solution	\begin{align} x \left (t \right ) &= c_{1} {\mathrm e}^{-5 t}+c_{2} {\mathrm e}^{t} \\ y \left (t \right ) &= -c_{1} {\mathrm e}^{-5 t}+c_{2} {\mathrm e}^{t} \\ z \left (t \right ) &= c_{3} {\mathrm e}^{-t} \\ \end{align}

Problem 13130


ODE	\begin {align} x^{\prime }\left (t \right )&=x \left (t \right )+3 z \left (t \right )\\ y^{\prime }&=-y\\ z^{\prime }\left (t \right )&=-3 x \left (t \right )+z \left (t \right ) \end {align}

program solution

Maple solution	\begin{align} x \left (t \right ) &= {\mathrm e}^{t} \left (c_{1} \sin \left (3 t \right )+c_{2} \cos \left (3 t \right )\right ) \\ y \left (t \right ) &= c_{3} {\mathrm e}^{-t} \\ z \left (t \right ) &= {\mathrm e}^{t} \left (c_{1} \cos \left (3 t \right )-c_{2} \sin \left (3 t \right )\right ) \\ \end{align}

Problem 13131


ODE	\begin {align} x^{\prime }\left (t \right )&=x \left (t \right )\\ y^{\prime }&=2 y-z \left (t \right )\\ z^{\prime }\left (t \right )&=-y+2 z \left (t \right ) \end {align}

program solution

Maple solution	\begin{align} x \left (t \right ) &= c_{3} {\mathrm e}^{t} \\ y \left (t \right ) &= c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{3 t} \\ z \left (t \right ) &= c_{1} {\mathrm e}^{t}-c_{2} {\mathrm e}^{3 t} \\ \end{align}

Problem 13132


ODE	\begin {align} x^{\prime }\left (t \right )&=-2 x \left (t \right )+y\\ y^{\prime }&=-2 y\\ z^{\prime }\left (t \right )&=-z \left (t \right ) \end {align}

program solution

Maple solution	\begin{align} x \left (t \right ) &= \left (c_{2} t +c_{1} \right ) {\mathrm e}^{-2 t} \\ y \left (t \right ) &= c_{2} {\mathrm e}^{-2 t} \\ z \left (t \right ) &= c_{3} {\mathrm e}^{-t} \\ \end{align}

Problem 13133


ODE	\begin {align} x^{\prime }\left (t \right )&=-2 x \left (t \right )+y\\ y^{\prime }&=-2 y\\ z^{\prime }\left (t \right )&=z \left (t \right ) \end {align}

program solution

Maple solution	\begin{align} x \left (t \right ) &= \left (c_{2} t +c_{1} \right ) {\mathrm e}^{-2 t} \\ y \left (t \right ) &= c_{2} {\mathrm e}^{-2 t} \\ z \left (t \right ) &= c_{3} {\mathrm e}^{t} \\ \end{align}

Problem 13134


ODE	\begin {align} x^{\prime }\left (t \right )&=-x \left (t \right )+2 y\\ y^{\prime }&=2 x \left (t \right )-4 y\\ z^{\prime }\left (t \right )&=-z \left (t \right ) \end {align}

program solution

Maple solution	\begin{align} x \left (t \right ) &= c_{1} +c_{2} {\mathrm e}^{-5 t} \\ y \left (t \right ) &= -2 c_{2} {\mathrm e}^{-5 t}+\frac {c_{1}}{2} \\ z \left (t \right ) &= c_{3} {\mathrm e}^{-t} \\ \end{align}

Problem 13135


ODE	\begin {align} x^{\prime }\left (t \right )&=-x \left (t \right )+2 y\\ y^{\prime }&=2 x \left (t \right )-4 y\\ z^{\prime }\left (t \right )&=0 \end {align}

program solution

Maple solution	\begin{align} x \left (t \right ) &= c_{1} +c_{2} {\mathrm e}^{-5 t} \\ y \left (t \right ) &= -2 c_{2} {\mathrm e}^{-5 t}+\frac {c_{1}}{2} \\ z \left (t \right ) &= c_{3} \\ \end{align}

Problem 13136


ODE	\begin {align} x^{\prime }\left (t \right )&=-2 x \left (t \right )+y\\ y^{\prime }&=-2 y+z \left (t \right )\\ z^{\prime }\left (t \right )&=-2 z \left (t \right ) \end {align}

program solution

Maple solution	\begin{align} x \left (t \right ) &= \frac {\left (c_{3} t^{2}+2 c_{2} t +2 c_{1} \right ) {\mathrm e}^{-2 t}}{2} \\ y \left (t \right ) &= \left (c_{3} t +c_{2} \right ) {\mathrm e}^{-2 t} \\ z \left (t \right ) &= c_{3} {\mathrm e}^{-2 t} \\ \end{align}

Problem 13137


ODE	\begin {align} x^{\prime }\left (t \right )&=y\\ y^{\prime }&=z \left (t \right )\\ z^{\prime }\left (t \right )&=0 \end {align}

program solution

Maple solution	\begin{align} x \left (t \right ) &= \frac {1}{2} c_{3} t^{2}+c_{2} t +c_{1} \\ y \left (t \right ) &= c_{3} t +c_{2} \\ z \left (t \right ) &= c_{3} \\ \end{align}

Problem 13138


ODE	\begin {align} x^{\prime }\left (t \right )&=2 x \left (t \right )-y\\ y^{\prime }&=-2 y+3 z \left (t \right )\\ z^{\prime }\left (t \right )&=-x \left (t \right )+3 y-z \left (t \right ) \end {align}

program solution

Maple solution	\begin{align} x \left (t \right ) &= -c_{2} {\mathrm e}^{\left (-1+2 \sqrt {3}\right ) t}-c_{3} {\mathrm e}^{-\left (1+2 \sqrt {3}\right ) t}-\frac {2 c_{2} {\mathrm e}^{\left (-1+2 \sqrt {3}\right ) t} \sqrt {3}}{3}+\frac {2 c_{3} {\mathrm e}^{-\left (1+2 \sqrt {3}\right ) t} \sqrt {3}}{3}+c_{1} {\mathrm e}^{t} \\ y \left (t \right ) &= c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{\left (-1+2 \sqrt {3}\right ) t}+c_{3} {\mathrm e}^{-\left (1+2 \sqrt {3}\right ) t} \\ z \left (t \right ) &= \frac {2 c_{2} {\mathrm e}^{\left (-1+2 \sqrt {3}\right ) t} \sqrt {3}}{3}-\frac {2 c_{3} {\mathrm e}^{-\left (1+2 \sqrt {3}\right ) t} \sqrt {3}}{3}+\frac {c_{2} {\mathrm e}^{\left (-1+2 \sqrt {3}\right ) t}}{3}+\frac {c_{3} {\mathrm e}^{-\left (1+2 \sqrt {3}\right ) t}}{3}+c_{1} {\mathrm e}^{t} \\ \end{align}

Problem 13139


ODE	\begin {align} x^{\prime }\left (t \right )&=-4 x \left (t \right )+3 y\\ y^{\prime }&=-y+z \left (t \right )\\ z^{\prime }\left (t \right )&=5 x \left (t \right )-5 y \end {align}

program solution

Maple solution	\begin{align} x \left (t \right ) &= {\mathrm e}^{-t} c_{1} +\frac {6 c_{2} {\mathrm e}^{-2 t} \sin \left (t \right )}{5}-\frac {3 c_{2} {\mathrm e}^{-2 t} \cos \left (t \right )}{5}+\frac {6 \,{\mathrm e}^{-2 t} \cos \left (t \right ) c_{3}}{5}+\frac {3 \,{\mathrm e}^{-2 t} \sin \left (t \right ) c_{3}}{5} \\ y \left (t \right ) &= {\mathrm e}^{-t} c_{1} +c_{2} {\mathrm e}^{-2 t} \sin \left (t \right )+{\mathrm e}^{-2 t} \cos \left (t \right ) c_{3} \\ z \left (t \right ) &= -{\mathrm e}^{-2 t} \left (c_{2} \sin \left (t \right )+c_{3} \sin \left (t \right )-c_{2} \cos \left (t \right )+c_{3} \cos \left (t \right )\right ) \\ \end{align}

Problem 13140


ODE	\begin {align} x^{\prime }\left (t \right )&=-10 x \left (t \right )+10 y\\ y^{\prime }&=28 x \left (t \right )-y\\ z^{\prime }\left (t \right )&=-\frac {8 z \left (t \right )}{3} \end {align}

program solution

Maple solution	\begin{align} x \left (t \right ) &= c_{1} {\mathrm e}^{\frac {\left (-11+\sqrt {1201}\right ) t}{2}}+c_{2} {\mathrm e}^{-\frac {\left (11+\sqrt {1201}\right ) t}{2}} \\ y \left (t \right ) &= \frac {c_{1} {\mathrm e}^{\frac {\left (-11+\sqrt {1201}\right ) t}{2}} \sqrt {1201}}{20}-\frac {c_{2} {\mathrm e}^{-\frac {\left (11+\sqrt {1201}\right ) t}{2}} \sqrt {1201}}{20}+\frac {9 c_{1} {\mathrm e}^{\frac {\left (-11+\sqrt {1201}\right ) t}{2}}}{20}+\frac {9 c_{2} {\mathrm e}^{-\frac {\left (11+\sqrt {1201}\right ) t}{2}}}{20} \\ z \left (t \right ) &= c_{3} {\mathrm e}^{-\frac {8 t}{3}} \\ \end{align}

Problem 13141


ODE	\begin {align} x^{\prime }\left (t \right )&=-y+z \left (t \right )\\ y^{\prime }&=-x \left (t \right )+z \left (t \right )\\ z^{\prime }\left (t \right )&=z \left (t \right ) \end {align}

program solution

Maple solution	\begin{align} x \left (t \right ) &= c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-t} \\ y \left (t \right ) &= -c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-t}+c_{3} {\mathrm e}^{t} \\ z \left (t \right ) &= c_{3} {\mathrm e}^{t} \\ \end{align}

Problem 13144


ODE	\begin {align} x^{\prime }\left (t \right )&=3 x \left (t \right )\\ y^{\prime }&=-2 y \end {align}

program solution

Maple solution	\begin{align} x \left (t \right ) &= c_{2} {\mathrm e}^{3 t} \\ y \left (t \right ) &= c_{1} {\mathrm e}^{-2 t} \\ \end{align}

Problem 13146


ODE	\begin {align} x^{\prime }\left (t \right )&=0\\ y^{\prime }&=-y+x \left (t \right ) \end {align}

program solution

Maple solution	\begin{align} x \left (t \right ) &= c_{2} \\ y \left (t \right ) &= c_{2} +{\mathrm e}^{-t} c_{1} \\ \end{align}

Problem 13147


ODE	\begin {align} x^{\prime }\left (t \right )&=\pi ^{2} x \left (t \right )+\frac {187 y}{5}\\ y^{\prime }&=\sqrt {555}\, x \left (t \right )+\frac {400617 y}{5000} \end {align} With initial conditions \[ [x \left (0\right ) = 0, y \left (0\right ) = 0] \]

program solution

Maple solution	\begin{align} x \left (t \right ) &= 0 \\ y \left (t \right ) &= 0 \\ \end{align}

Problem 13148


ODE	\begin {align} x^{\prime }\left (t \right )&=x \left (t \right )+y\\ y^{\prime }&=-2 x \left (t \right )-y \end {align}

program solution

Maple solution	\begin{align} x \left (t \right ) &= c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right ) \\ y \left (t \right ) &= c_{1} \cos \left (t \right )-c_{2} \sin \left (t \right )-c_{1} \sin \left (t \right )-c_{2} \cos \left (t \right ) \\ \end{align}

Problem 13149


ODE	\begin {align} x^{\prime }\left (t \right )&=-3 x \left (t \right )+y\\ y^{\prime }&=y-x \left (t \right ) \end {align}

program solution

Maple solution	\begin{align} x \left (t \right ) &= c_{1} {\mathrm e}^{\left (\sqrt {3}-1\right ) t}+c_{2} {\mathrm e}^{-\left (1+\sqrt {3}\right ) t} \\ y \left (t \right ) &= c_{1} {\mathrm e}^{\left (\sqrt {3}-1\right ) t} \sqrt {3}-c_{2} {\mathrm e}^{-\left (1+\sqrt {3}\right ) t} \sqrt {3}+2 c_{1} {\mathrm e}^{\left (\sqrt {3}-1\right ) t}+2 c_{2} {\mathrm e}^{-\left (1+\sqrt {3}\right ) t} \\ \end{align}

Problem 13150


ODE	\begin {align} x^{\prime }\left (t \right )&=-3 x \left (t \right )+y\\ y^{\prime }&=-x \left (t \right ) \end {align}

program solution

Maple solution	\begin{align} x \left (t \right ) &= \left (-\frac {\sqrt {5}}{2}+\frac {3}{2}\right ) c_{1} {\mathrm e}^{\frac {\left (\sqrt {5}-3\right ) t}{2}}+\left (\frac {3}{2}+\frac {\sqrt {5}}{2}\right ) c_{2} {\mathrm e}^{-\frac {\left (3+\sqrt {5}\right ) t}{2}} \\ y \left (t \right ) &= c_{1} {\mathrm e}^{\frac {\left (\sqrt {5}-3\right ) t}{2}}+c_{2} {\mathrm e}^{-\frac {\left (3+\sqrt {5}\right ) t}{2}} \\ \end{align}

Problem 13151


ODE	\begin {align} x^{\prime }\left (t \right )&=y-x \left (t \right )\\ y^{\prime }&=-2 x \left (t \right )+y \end {align}

program solution

Maple solution	\begin{align} x \left (t \right ) &= c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right ) \\ y \left (t \right ) &= c_{1} \sin \left (t \right )-c_{2} \sin \left (t \right )+c_{1} \cos \left (t \right )+c_{2} \cos \left (t \right ) \\ \end{align}

Problem 13152


ODE	\begin {align} x^{\prime }\left (t \right )&=2 x \left (t \right )\\ y^{\prime }&=-y+x \left (t \right ) \end {align}

program solution

Maple solution	\begin{align} x \left (t \right ) &= c_{2} {\mathrm e}^{2 t} \\ y \left (t \right ) &= \frac {c_{2} {\mathrm e}^{2 t}}{3}+{\mathrm e}^{-t} c_{1} \\ \end{align}

Problem 13153


ODE	\begin {align} x^{\prime }\left (t \right )&=3 x \left (t \right )+y\\ y^{\prime }&=-x \left (t \right ) \end {align}

program solution

Maple solution	\begin{align} x \left (t \right ) &= \left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) c_{2} {\mathrm e}^{-\frac {\left (\sqrt {5}-3\right ) t}{2}}+\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) c_{1} {\mathrm e}^{\frac {\left (3+\sqrt {5}\right ) t}{2}} \\ 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}} \\ \end{align}

Problem 13154


ODE	\begin {align} x^{\prime }\left (t \right )&=y\\ y^{\prime }&=-4 x \left (t \right )-4 y \end {align}

program solution

Maple solution	\begin{align} x \left (t \right ) &= \left (c_{2} t +c_{1} \right ) {\mathrm e}^{-2 t} \\ y \left (t \right ) &= -{\mathrm e}^{-2 t} \left (2 c_{2} t +2 c_{1} -c_{2} \right ) \\ \end{align}

Problem 13155


ODE	\begin {align} x^{\prime }\left (t \right )&=-3 x \left (t \right )-3 y\\ y^{\prime }&=2 x \left (t \right )+y \end {align}

program solution

Maple solution	\begin{align} x \left (t \right ) &= {\mathrm e}^{-t} \left (c_{1} \sin \left (\sqrt {2}\, t \right )+c_{2} \cos \left (\sqrt {2}\, t \right )\right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (\sin \left (\sqrt {2}\, t \right ) \sqrt {2}\, c_{2} -\cos \left (\sqrt {2}\, t \right ) \sqrt {2}\, c_{1} -2 c_{1} \sin \left (\sqrt {2}\, t \right )-2 c_{2} \cos \left (\sqrt {2}\, t \right )\right )}{3} \\ \end{align}

Problem 13156


ODE	\[ \boxed {y^{\prime \prime }+5 y^{\prime }+6 y=0} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 2] \end {align}

program solution	\[ y = 2 \,{\mathrm e}^{-2 t}-2 \,{\mathrm e}^{-3 t} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = -2 \,{\mathrm e}^{-3 t}+2 \,{\mathrm e}^{-2 t} \]

Problem 13157


ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+5 y=0} \] With initial conditions \begin {align} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = -1] \end {align}

program solution	\[ y = \left (3 \cos \left (2 t \right )+\sin \left (2 t \right )\right ) {\mathrm e}^{-t} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = {\mathrm e}^{-t} \left (\sin \left (2 t \right )+3 \cos \left (2 t \right )\right ) \]

Problem 13158


ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+y=0} \] With initial conditions \begin {align} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 1] \end {align}

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

Maple solution	\[ y \left (t \right ) = {\mathrm e}^{-t} \left (2 t +1\right ) \]

Problem 13159


ODE	\[ \boxed {y^{\prime \prime }+2 y=0} \] With initial conditions \begin {align} \left [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = -\sqrt {2}\right ] \end {align}

program solution	\[ y = 3 \cos \left (\sqrt {2}\, t \right )-\sin \left (\sqrt {2}\, t \right ) \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = -\sin \left (\sqrt {2}\, t \right )+3 \cos \left (\sqrt {2}\, t \right ) \]

Problem 13160


ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }-6 y={\mathrm e}^{4 t}} \]

program solution	\[ y = c_{1} {\mathrm e}^{-2 t}+\frac {c_{2} {\mathrm e}^{3 t}}{5}+\frac {{\mathrm e}^{4 t}}{6} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = \frac {\left ({\mathrm e}^{6 t}+6 c_{2} {\mathrm e}^{5 t}+6 c_{1} \right ) {\mathrm e}^{-2 t}}{6} \]

Problem 13161


ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+8 y=2 \,{\mathrm e}^{-3 t}} \]

program solution	\[ y = c_{1} {\mathrm e}^{-4 t}+\frac {c_{2} {\mathrm e}^{-2 t}}{2}-2 \,{\mathrm e}^{-3 t} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = -\frac {\left ({\mathrm e}^{-2 t} c_{1} +4 \,{\mathrm e}^{-t}-2 c_{2} \right ) {\mathrm e}^{-2 t}}{2} \]

Problem 13162


ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }-2 y=5 \,{\mathrm e}^{3 t}} \]

program solution	\[ y = c_{1} {\mathrm e}^{-t}+\frac {c_{2} {\mathrm e}^{2 t}}{3}+\frac {5 \,{\mathrm e}^{3 t}}{4} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = c_{2} {\mathrm e}^{-t}+c_{1} {\mathrm e}^{2 t}+\frac {5 \,{\mathrm e}^{3 t}}{4} \]

Problem 13163


ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+13 y={\mathrm e}^{-t}} \]

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

Maple solution	\[ y \left (t \right ) = c_{2} {\mathrm e}^{-2 t} \sin \left (3 t \right )+c_{1} {\mathrm e}^{-2 t} \cos \left (3 t \right )+\frac {{\mathrm e}^{-t}}{10} \]

Problem 13164


ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+13 y=-3 \,{\mathrm e}^{-2 t}} \]

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

Maple solution	\[ y \left (t \right ) = \frac {{\mathrm e}^{-2 t} \left (3 c_{1} \cos \left (3 t \right )+3 c_{2} \sin \left (3 t \right )-1\right )}{3} \]

Problem 13165


ODE	\[ \boxed {y^{\prime \prime }+7 y^{\prime }+10 y={\mathrm e}^{-2 t}} \]

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

Maple solution	\[ y \left (t \right ) = \frac {\left (t +3 c_{1} \right ) {\mathrm e}^{-2 t}}{3}+c_{2} {\mathrm e}^{-5 t} \]

Problem 13166


ODE	\[ \boxed {y^{\prime \prime }-5 y^{\prime }+4 y={\mathrm e}^{4 t}} \]

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

Maple solution	\[ y \left (t \right ) = \frac {\left (t +3 c_{2} \right ) {\mathrm e}^{4 t}}{3}+c_{1} {\mathrm e}^{t} \]

Problem 13167


ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }-6 y=4 \,{\mathrm e}^{-3 t}} \]

program solution	\[ y = c_{1} {\mathrm e}^{-3 t}+\frac {c_{2} {\mathrm e}^{2 t}}{5}-\frac {4 \,{\mathrm e}^{-3 t} t}{5} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = \frac {\left (5 c_{1} {\mathrm e}^{5 t}+5 c_{2} -4 t \right ) {\mathrm e}^{-3 t}}{5} \]

Problem 13168


ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+8 y={\mathrm e}^{-t}} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = -\frac {{\mathrm e}^{-2 t}}{2}+\frac {{\mathrm e}^{-4 t}}{6}+\frac {{\mathrm e}^{-t}}{3} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = \frac {\left (2 \,{\mathrm e}^{3 t}-3 \,{\mathrm e}^{2 t}+1\right ) {\mathrm e}^{-4 t}}{6} \]

Problem 13169


ODE	\[ \boxed {y^{\prime \prime }+7 y^{\prime }+12 y=3 \,{\mathrm e}^{-t}} \] With initial conditions \begin {align} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 1] \end {align}

program solution	\[ y = \frac {15 \,{\mathrm e}^{-3 t}}{2}-6 \,{\mathrm e}^{-4 t}+\frac {{\mathrm e}^{-t}}{2} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = \frac {15 \,{\mathrm e}^{-3 t}}{2}-6 \,{\mathrm e}^{-4 t}+\frac {{\mathrm e}^{-t}}{2} \]

Problem 13170


ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+13 y=-3 \,{\mathrm e}^{-2 t}} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align}

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

Maple solution	\[ y \left (t \right ) = \frac {{\mathrm e}^{-2 t} \left (\cos \left (3 t \right )-1\right )}{3} \]

Problem 13171


ODE	\[ \boxed {y^{\prime \prime }+7 y^{\prime }+10 y={\mathrm e}^{-2 t}} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align}

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

Maple solution	\[ y \left (t \right ) = \frac {\left (3 t -1\right ) {\mathrm e}^{-2 t}}{9}+\frac {{\mathrm e}^{-5 t}}{9} \]

Problem 13172


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

program solution	\[ y = -{\mathrm e}^{-t}+\frac {{\mathrm e}^{-3 t}}{5}+\frac {4 \,{\mathrm e}^{-\frac {t}{2}}}{5} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = \frac {{\mathrm e}^{-3 t}}{5}-{\mathrm e}^{-t}+\frac {4 \,{\mathrm e}^{-\frac {t}{2}}}{5} \]

Problem 13173


ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+3 y={\mathrm e}^{-2 t}} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = \frac {{\mathrm e}^{-t}}{2}+\frac {{\mathrm e}^{-3 t}}{2}-{\mathrm e}^{-2 t} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = \frac {{\mathrm e}^{-3 t}}{2}+\frac {{\mathrm e}^{-t}}{2}-{\mathrm e}^{-2 t} \]

Problem 13174


ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+3 y={\mathrm e}^{-4 t}} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = \frac {{\mathrm e}^{-t}}{6}-\frac {{\mathrm e}^{-3 t}}{2}+\frac {{\mathrm e}^{-4 t}}{3} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = -\frac {{\mathrm e}^{-3 t}}{2}+\frac {{\mathrm e}^{-t}}{6}+\frac {{\mathrm e}^{-4 t}}{3} \]

Problem 13175


ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+20 y={\mathrm e}^{-\frac {t}{2}}} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = \frac {4 \,{\mathrm e}^{-\frac {t}{2}}}{73}+\frac {\left (-8 \cos \left (4 t \right )-3 \sin \left (4 t \right )\right ) {\mathrm e}^{-2 t}}{146} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = \frac {4 \,{\mathrm e}^{-\frac {t}{2}}}{73}+\frac {\left (-3 \sin \left (4 t \right )-8 \cos \left (4 t \right )\right ) {\mathrm e}^{-2 t}}{146} \]

Problem 13176


ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+20 y={\mathrm e}^{-2 t}} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = -\frac {{\mathrm e}^{-2 t} \left (-1+\cos \left (4 t \right )\right )}{16} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = -\frac {{\mathrm e}^{-2 t} \left (-1+\cos \left (4 t \right )\right )}{16} \]

Problem 13177


ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+20 y={\mathrm e}^{-4 t}} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = \frac {\left (-2 \cos \left (4 t \right )+\sin \left (4 t \right )\right ) {\mathrm e}^{-2 t}}{40}+\frac {{\mathrm e}^{-4 t}}{20} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = \frac {\left (\sin \left (4 t \right )-2 \cos \left (4 t \right )\right ) {\mathrm e}^{-2 t}}{40}+\frac {{\mathrm e}^{-4 t}}{20} \]

Problem 13178


ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+y={\mathrm e}^{-t}} \]

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

Maple solution	\[ y \left (t \right ) = {\mathrm e}^{-t} \left (\frac {1}{2} t^{2}+c_{1} t +c_{2} \right ) \]

Problem 13179


ODE	\[ \boxed {y^{\prime \prime }-5 y^{\prime }+4 y=5} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = \frac {5}{4}+\frac {5 \,{\mathrm e}^{4 t}}{12}-\frac {5 \,{\mathrm e}^{t}}{3} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = \frac {5 \,{\mathrm e}^{4 t}}{12}-\frac {5 \,{\mathrm e}^{t}}{3}+\frac {5}{4} \]

Problem 13180


ODE	\[ \boxed {y^{\prime \prime }+5 y^{\prime }+6 y=2} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = \frac {1}{3}-{\mathrm e}^{-2 t}+\frac {2 \,{\mathrm e}^{-3 t}}{3} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = \frac {2 \,{\mathrm e}^{-3 t}}{3}-{\mathrm e}^{-2 t}+\frac {1}{3} \]

Problem 13181


ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+10 y=10} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = 1+\frac {\left (-3 \cos \left (3 t \right )-\sin \left (3 t \right )\right ) {\mathrm e}^{-t}}{3} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = 1+\frac {\left (-3 \cos \left (3 t \right )-\sin \left (3 t \right )\right ) {\mathrm e}^{-t}}{3} \]

Problem 13182


ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+6 y=-8} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = \frac {4 \,{\mathrm e}^{-2 t} \cos \left (\sqrt {2}\, t \right )}{3}+\frac {4 \,{\mathrm e}^{-2 t} \sin \left (\sqrt {2}\, t \right ) \sqrt {2}}{3}-\frac {4}{3} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = \frac {4 \,{\mathrm e}^{-2 t} \sin \left (\sqrt {2}\, t \right ) \sqrt {2}}{3}+\frac {4 \,{\mathrm e}^{-2 t} \cos \left (\sqrt {2}\, t \right )}{3}-\frac {4}{3} \]

Problem 13183


ODE	\[ \boxed {y^{\prime \prime }+9 y={\mathrm e}^{-t}} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = -\frac {\cos \left (3 t \right )}{10}+\frac {\sin \left (3 t \right )}{30}+\frac {{\mathrm e}^{-t}}{10} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = \frac {\sin \left (3 t \right )}{30}-\frac {\cos \left (3 t \right )}{10}+\frac {{\mathrm e}^{-t}}{10} \]

Problem 13184


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

program solution	\[ y = -\frac {\cos \left (2 t \right )}{4}+\frac {\sin \left (2 t \right )}{4}+\frac {{\mathrm e}^{-2 t}}{4} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = \frac {\sin \left (2 t \right )}{4}-\frac {\cos \left (2 t \right )}{4}+\frac {{\mathrm e}^{-2 t}}{4} \]

Problem 13185


ODE	\[ \boxed {y^{\prime \prime }+2 y=-3} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = \frac {3 \cos \left (\sqrt {2}\, t \right )}{2}-\frac {3}{2} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = -\frac {3}{2}+\frac {3 \cos \left (\sqrt {2}\, t \right )}{2} \]

Problem 13186


ODE	\[ \boxed {y^{\prime \prime }+4 y={\mathrm e}^{t}} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = \frac {{\mathrm e}^{t}}{5}-\frac {\cos \left (2 t \right )}{5}-\frac {\sin \left (2 t \right )}{10} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = -\frac {\sin \left (2 t \right )}{10}-\frac {\cos \left (2 t \right )}{5}+\frac {{\mathrm e}^{t}}{5} \]

Problem 13187


ODE	\[ \boxed {y^{\prime \prime }+9 y=6} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = \frac {2}{3}-\frac {2 \cos \left (3 t \right )}{3} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = \frac {2}{3}-\frac {2 \cos \left (3 t \right )}{3} \]

Problem 13188


ODE	\[ \boxed {y^{\prime \prime }+2 y=-{\mathrm e}^{t}} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = \frac {\cos \left (\sqrt {2}\, t \right )}{3}+\frac {\sqrt {2}\, \sin \left (\sqrt {2}\, t \right )}{6}-\frac {{\mathrm e}^{t}}{3} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = \frac {\sqrt {2}\, \sin \left (\sqrt {2}\, t \right )}{6}+\frac {\cos \left (\sqrt {2}\, t \right )}{3}-\frac {{\mathrm e}^{t}}{3} \]

Problem 13189


ODE	\[ \boxed {y^{\prime \prime }+4 y=-3 t^{2}+2 t +3} \] With initial conditions \begin {align} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = \frac {9}{8}+\frac {7 \cos \left (2 t \right )}{8}-\frac {\sin \left (2 t \right )}{4}-\frac {3 t^{2}}{4}+\frac {t}{2} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = -\frac {\sin \left (2 t \right )}{4}+\frac {7 \cos \left (2 t \right )}{8}-\frac {3 t^{2}}{4}+\frac {t}{2}+\frac {9}{8} \]

Problem 13190


ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }=3 t +2} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = -\frac {1}{8}+\frac {{\mathrm e}^{-2 t}}{8}+\frac {3 t^{2}}{4}+\frac {t}{4} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = \frac {3 t^{2}}{4}+\frac {{\mathrm e}^{-2 t}}{8}+\frac {t}{4}-\frac {1}{8} \]

Problem 13191


ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }=3 t +2} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = -\frac {5}{64}+\frac {5 \,{\mathrm e}^{-4 t}}{64}+\frac {3 t^{2}}{8}+\frac {5 t}{16} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = \frac {3 t^{2}}{8}+\frac {5 \,{\mathrm e}^{-4 t}}{64}+\frac {5 t}{16}-\frac {5}{64} \]

Problem 13192


ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }+2 y=t^{2}} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = \frac {7}{4}-2 \,{\mathrm e}^{-t}+\frac {{\mathrm e}^{-2 t}}{4}+\frac {t^{2}}{2}-\frac {3 t}{2} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = \frac {7}{4}-\frac {3 t}{2}+\frac {t^{2}}{2}+\frac {{\mathrm e}^{-2 t}}{4}-2 \,{\mathrm e}^{-t} \]

Problem 13193


ODE	\[ \boxed {y^{\prime \prime }+4 y=t -\frac {1}{20} t^{2}} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = \frac {1}{160}-\frac {\cos \left (2 t \right )}{160}-\frac {\sin \left (2 t \right )}{8}-\frac {t^{2}}{80}+\frac {t}{4} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = -\frac {\sin \left (2 t \right )}{8}-\frac {\cos \left (2 t \right )}{160}-\frac {t^{2}}{80}+\frac {t}{4}+\frac {1}{160} \]

Problem 13194


ODE	\[ \boxed {y^{\prime \prime }+5 y^{\prime }+6 y=4+{\mathrm e}^{-t}} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = \frac {2}{3}-3 \,{\mathrm e}^{-2 t}+\frac {11 \,{\mathrm e}^{-3 t}}{6}+\frac {{\mathrm e}^{-t}}{2} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = \frac {11 \,{\mathrm e}^{-3 t}}{6}-3 \,{\mathrm e}^{-2 t}+\frac {{\mathrm e}^{-t}}{2}+\frac {2}{3} \]

Problem 13195


ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }+2 y={\mathrm e}^{-t}-4} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align}

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

Maple solution	\[ y \left (t \right ) = -\left (2 \,{\mathrm e}^{2 t}+\ln \left ({\mathrm e}^{-t}\right ) {\mathrm e}^{t}-3 \,{\mathrm e}^{t}+1\right ) {\mathrm e}^{-2 t} \]

Problem 13196


ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+8 y=2 t +{\mathrm e}^{-t}} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = -\frac {3}{16}-\frac {{\mathrm e}^{-2 t}}{4}+\frac {5 \,{\mathrm e}^{-4 t}}{48}+\frac {{\mathrm e}^{-t}}{3}+\frac {t}{4} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = \frac {5 \,{\mathrm e}^{-4 t}}{48}-\frac {3}{16}+\frac {t}{4}+\frac {{\mathrm e}^{-t}}{3}-\frac {{\mathrm e}^{-2 t}}{4} \]

Problem 13197


ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+8 y=2 t +{\mathrm e}^{t}} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = -\frac {3}{16}+\frac {{\mathrm e}^{-2 t}}{12}+\frac {3 \,{\mathrm e}^{-4 t}}{80}+\frac {{\mathrm e}^{t}}{15}+\frac {t}{4} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = \frac {\left (16 \,{\mathrm e}^{5 t}+60 t \,{\mathrm e}^{4 t}-45 \,{\mathrm e}^{4 t}+20 \,{\mathrm e}^{2 t}+9\right ) {\mathrm e}^{-4 t}}{240} \]

Problem 13198


ODE	\[ \boxed {y^{\prime \prime }+4 y=t +{\mathrm e}^{-t}} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = -\frac {\cos \left (2 t \right )}{5}-\frac {\sin \left (2 t \right )}{40}+\frac {{\mathrm e}^{-t}}{5}+\frac {t}{4} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = -\frac {\sin \left (2 t \right )}{40}-\frac {\cos \left (2 t \right )}{5}+\frac {t}{4}+\frac {{\mathrm e}^{-t}}{5} \]

Problem 13199


ODE	\[ \boxed {y^{\prime \prime }+4 y=6+t^{2}+{\mathrm e}^{t}} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = \frac {11}{8}-\frac {63 \cos \left (2 t \right )}{40}-\frac {\sin \left (2 t \right )}{10}+\frac {{\mathrm e}^{t}}{5}+\frac {t^{2}}{4} \] Veriﬁed OK.

Maple solution	\[ y \left (t \right ) = -\frac {\sin \left (2 t \right )}{10}-\frac {63 \cos \left (2 t \right )}{40}+\frac {11}{8}+\frac {t^{2}}{4}+\frac {{\mathrm e}^{t}}{5} \]

Problem 13200


ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }+2 y=\cos \left (t \right )} \]

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

Maple solution	\[ y \left (t \right ) = -{\mathrm e}^{-2 t} c_{1} +\frac {\cos \left (t \right )}{10}+\frac {3 \sin \left (t \right )}{10}+c_{2} {\mathrm e}^{-t} \]

2.17.132 Problems 13101 to 13200