2.17.68 Problems 6701 to 6800

Problem 6701

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

Problem 6702

ODE	\[ \boxed {y^{\prime \prime }+y=\delta \left (t -2 \pi \right )+\delta \left (t -4 \pi \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \cos \left (t \right )+\sin \left (t \right ) \left (\left \{\begin {array}{cc} 0 & t <2 \pi \\ 1 & t <4 \pi \\ 2 & 4 \pi \le t \end {array}\right .\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \sin \left (t \right ) \operatorname {Heaviside}\left (t -2 \pi \right )+\sin \left (t \right ) \operatorname {Heaviside}\left (t -4 \pi \right )+\cos \left (t \right ) \]

Problem 6703

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

Problem 6704

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }=1+\delta \left (-2+t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = -\frac {\left (\left \{\begin {array}{cc} 3-3 \,{\mathrm e}^{2 t}+2 t & t <2 \\ 9-3 \,{\mathrm e}^{4} & t =2 \\ 5-3 \,{\mathrm e}^{2 t}+2 t -2 \,{\mathrm e}^{2 t -4} & 2
Maple solution	\[ y \left (t \right ) = -\frac {3}{4}-\frac {\operatorname {Heaviside}\left (t -2\right )}{2}+\frac {3 \,{\mathrm e}^{2 t}}{4}-\frac {t}{2}+\frac {\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{2 t -4}}{2} \]

Problem 6705

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

Problem 6706

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

Problem 6707

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+13 y=\delta \left (t -\pi \right )+\delta \left (t -3 \pi \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {\left (\left \{\begin {array}{cc} {\mathrm e}^{-2 t} \left (3 \cos \left (3 t \right )+2 \sin \left (3 t \right )\right ) & t <\pi \\ -\sin \left (3 t \right ) {\mathrm e}^{2 \pi -2 t}+3 \,{\mathrm e}^{-2 t} \left (\cos \left (3 t \right )+\frac {2 \sin \left (3 t \right )}{3}\right ) & t <3 \pi \\ -\sin \left (3 t \right ) {\mathrm e}^{2 \pi -2 t}-\sin \left (3 t \right ) {\mathrm e}^{-2 t +6 \pi }+3 \,{\mathrm e}^{-2 t} \left (\cos \left (3 t \right )+\frac {2 \sin \left (3 t \right )}{3}\right ) & 3 \pi \le t \end {array}\right .\right )}{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (\cos \left (3 t \right )+\frac {2 \sin \left (3 t \right )}{3}\right ) {\mathrm e}^{-2 t}-\frac {\sin \left (3 t \right ) {\mathrm e}^{-2 t +6 \pi } \operatorname {Heaviside}\left (t -3 \pi \right )}{3}-\frac {\sin \left (3 t \right ) {\mathrm e}^{-2 t +2 \pi } \operatorname {Heaviside}\left (t -\pi \right )}{3} \]

Problem 6708

ODE	\[ \boxed {y^{\prime \prime }-7 y^{\prime }+6 y={\mathrm e}^{t}+\delta \left (-2+t \right )+\delta \left (t -4\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {\left (\left \{\begin {array}{cc} 0 & t \le 2 \\ {\mathrm e}^{-12+6 t}-{\mathrm e}^{-2+t} & t \le 4 \\ {\mathrm e}^{-12+6 t}-{\mathrm e}^{-2+t}-{\mathrm e}^{t -4}+{\mathrm e}^{6 t -24} & 4
Maple solution	\[ y \left (t \right ) = \frac {{\mathrm e}^{-24+6 t} \operatorname {Heaviside}\left (t -4\right )}{5}+\frac {{\mathrm e}^{-12+6 t} \operatorname {Heaviside}\left (t -2\right )}{5}-\frac {{\mathrm e}^{t -4} \operatorname {Heaviside}\left (t -4\right )}{5}-\frac {{\mathrm e}^{t -2} \operatorname {Heaviside}\left (t -2\right )}{5}+\frac {{\mathrm e}^{6 t}}{25}+\frac {\left (-5 t -1\right ) {\mathrm e}^{t}}{25} \]

Problem 6709

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

Problem 6710

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

Problem 6711

ODE	\begin {align*} x^{\prime }\left (t \right )&=3 x \left (t \right )-5 y\\ y^{\prime }&=4 x \left (t \right )+8 y \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{\frac {11 t}{2}} \left (\sin \left (\frac {\sqrt {55}\, t}{2}\right ) c_{1} +\cos \left (\frac {\sqrt {55}\, t}{2}\right ) c_{2} \right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{\frac {11 t}{2}} \left (\sin \left (\frac {\sqrt {55}\, t}{2}\right ) \sqrt {55}\, c_{2} -\cos \left (\frac {\sqrt {55}\, t}{2}\right ) \sqrt {55}\, c_{1} -5 \sin \left (\frac {\sqrt {55}\, t}{2}\right ) c_{1} -5 \cos \left (\frac {\sqrt {55}\, t}{2}\right ) c_{2} \right )}{10} \\ \end{align*}

Problem 6712

ODE	\begin {align*} x^{\prime }\left (t \right )&=4 x \left (t \right )-7 y\\ y^{\prime }&=5 x \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= -\frac {{\mathrm e}^{2 t} \left (\sqrt {31}\, \sin \left (\sqrt {31}\, t \right ) c_{2} -\sqrt {31}\, \cos \left (\sqrt {31}\, t \right ) c_{1} -2 \sin \left (\sqrt {31}\, t \right ) c_{1} -2 \cos \left (\sqrt {31}\, t \right ) c_{2} \right )}{5} \\ y \left (t \right ) &= {\mathrm e}^{2 t} \left (\sin \left (\sqrt {31}\, t \right ) c_{1} +\cos \left (\sqrt {31}\, t \right ) c_{2} \right ) \\ \end{align*}

Problem 6713

ODE	\begin {align*} x^{\prime }\left (t \right )&=-3 x \left (t \right )+4 y-9 z \left (t \right )\\ y^{\prime }&=6 x \left (t \right )-y\\ z^{\prime }\left (t \right )&=10 x \left (t \right )+4 y+3 z \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} \text {Expression too large to display} \\ y \left (t \right ) &= \cos \left (\frac {\left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t \sqrt {3}\, 1156^{\frac {1}{3}}}{204 \left (139+9 \sqrt {291}\right )^{\frac {1}{3}}}\right ) {\mathrm e}^{\frac {\left (-170+\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-2 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}} c_{3} +{\mathrm e}^{\frac {\left (-170+\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-2 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}} \sin \left (\frac {\left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t \sqrt {3}\, 1156^{\frac {1}{3}}}{204 \left (139+9 \sqrt {291}\right )^{\frac {1}{3}}}\right ) c_{2} +c_{1} {\mathrm e}^{-\frac {\left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170\right ) t}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}} \\ \text {Expression too large to display} \\ \end{align*}

Problem 6714

ODE	\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )-y\\ y^{\prime }&=x \left (t \right )+2 z \left (t \right )\\ z^{\prime }\left (t \right )&=-x \left (t \right )+z \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{-\frac {\left (-8+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-8 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}} \cos \left (\frac {\left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t \sqrt {3}\, 2^{\frac {1}{3}}}{24 \left (61+3 \sqrt {417}\right )^{\frac {1}{3}}}\right ) c_{3} -{\mathrm e}^{-\frac {\left (-8+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-8 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}} \sin \left (\frac {\left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t \sqrt {3}\, 2^{\frac {1}{3}}}{24 \left (61+3 \sqrt {417}\right )^{\frac {1}{3}}}\right ) c_{2} +c_{1} {\mathrm e}^{\frac {\left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+4 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-8\right ) t}{6 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}} \\ \text {Expression too large to display} \\ \text {Expression too large to display} \\ \end{align*}

Problem 6715

ODE	\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )-y+z \left (t \right )+t -1\\ y^{\prime }&=2 x \left (t \right )+y-z \left (t \right )-3 t^{2}\\ z^{\prime }\left (t \right )&=x \left (t \right )+y+z \left (t \right )+t^{2}-t +2 \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= t^{2}-\frac {1}{6}+c_{1} {\mathrm e}^{2 t}+c_{2} {\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {11}\, t}{2}\right )+c_{3} {\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {11}\, t}{2}\right ) \\ y \left (t \right ) &= -\frac {t^{2}}{2}-\frac {7}{4}+\frac {c_{1} {\mathrm e}^{2 t}}{2}-\frac {c_{2} {\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {11}\, t}{2}\right )}{2}+\frac {c_{2} {\mathrm e}^{\frac {t}{2}} \sqrt {11}\, \sin \left (\frac {\sqrt {11}\, t}{2}\right )}{2}-\frac {c_{3} {\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {11}\, t}{2}\right )}{2}-\frac {c_{3} {\mathrm e}^{\frac {t}{2}} \sqrt {11}\, \cos \left (\frac {\sqrt {11}\, t}{2}\right )}{2}-\frac {3 t}{2} \\ z \left (t \right ) &= -\frac {t}{2}+\frac {3 c_{1} {\mathrm e}^{2 t}}{2}-c_{2} {\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {11}\, t}{2}\right )-c_{3} {\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {11}\, t}{2}\right )-\frac {3 t^{2}}{2}-\frac {7}{12} \\ \end{align*}

Problem 6716

ODE	\begin {align*} x^{\prime }\left (t \right )&=-3 x \left (t \right )+4 y+2 \,{\mathrm e}^{-t} \sin \left (t \right ) \cos \left (t \right )\\ y^{\prime }&=5 x \left (t \right )+9 z \left (t \right )+8 \,{\mathrm e}^{-t} \cos \left (t \right )^{2}-4 \,{\mathrm e}^{-t}\\ z^{\prime }\left (t \right )&=y+6 z \left (t \right )-{\mathrm e}^{-t} \end {align*}
program solution
Maple solution	\[ \text {No solution found} \]

Problem 6717

ODE	\begin {align*} x^{\prime }\left (t \right )&=4 x \left (t \right )+2 y+{\mathrm e}^{t}\\ y^{\prime }&=-x \left (t \right )+3 y-{\mathrm e}^{t} \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{\frac {7 t}{2}} \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_{2} +{\mathrm e}^{\frac {7 t}{2}} \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_{1} -\frac {{\mathrm e}^{t}}{2} \\ y \left (t \right ) &= -\frac {{\mathrm e}^{\frac {7 t}{2}} \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_{2}}{4}+\frac {{\mathrm e}^{\frac {7 t}{2}} \sqrt {7}\, \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_{2}}{4}-\frac {{\mathrm e}^{\frac {7 t}{2}} \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_{1}}{4}-\frac {{\mathrm e}^{\frac {7 t}{2}} \sqrt {7}\, \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_{1}}{4}+\frac {{\mathrm e}^{t}}{4} \\ \end{align*}

Problem 6718

ODE	\begin {align*} x^{\prime }\left (t \right )&=7 x \left (t \right )+5 y-9 z \left (t \right )-8 \,{\mathrm e}^{-2 t}\\ y^{\prime }&=4 x \left (t \right )+y+z \left (t \right )+2 \,{\mathrm e}^{5 t}\\ z^{\prime }\left (t \right )&=-2 y+3 z \left (t \right )+{\mathrm e}^{5 t}-3 \,{\mathrm e}^{-2 t} \end {align*}
program solution
Maple solution	\begin{align*} \text {Expression too large to display} \\ \text {Expression too large to display} \\ \text {Expression too large to display} \\ \end{align*}

Problem 6719

ODE	\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )-y \left (t \right )+2 z \left (t \right )+{\mathrm e}^{-t}-3 t\\ y^{\prime }\left (t \right )&=3 x \left (t \right )-4 y \left (t \right )+z \left (t \right )+2 \,{\mathrm e}^{-t}+t\\ z^{\prime }\left (t \right )&=-2 x \left (t \right )+5 y \left (t \right )+6 z \left (t \right )+2 \,{\mathrm e}^{-t}-t \end {align*}
program solution
Maple solution	\[ \text {No solution found} \]

Problem 6720

ODE	\begin {align*} x^{\prime }\left (t \right )&=t \,{\mathrm e}^{4 t}+3 x \left (t \right )-7 y+4 \sin \left (t \right )-4 \,{\mathrm e}^{4 t}\\ y^{\prime }&=2 t \,{\mathrm e}^{4 t}+{\mathrm e}^{4 t}+x \left (t \right )+y+8 \sin \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{2 t} \sin \left (\sqrt {6}\, t \right ) c_{2} +{\mathrm e}^{2 t} \cos \left (\sqrt {6}\, t \right ) c_{1} -\frac {11 \,{\mathrm e}^{4 t} t}{10}-\frac {34 \,{\mathrm e}^{4 t}}{25}-\frac {204 \cos \left (t \right )}{97}-\frac {556 \sin \left (t \right )}{97} \\ y \left (t \right ) &= \frac {3 \,{\mathrm e}^{4 t} t}{10}+\frac {{\mathrm e}^{2 t} \sqrt {6}\, \sin \left (\sqrt {6}\, t \right ) c_{1}}{7}-\frac {{\mathrm e}^{2 t} \sqrt {6}\, \cos \left (\sqrt {6}\, t \right ) c_{2}}{7}-\frac {11 \,{\mathrm e}^{4 t}}{50}+\frac {{\mathrm e}^{2 t} \sin \left (\sqrt {6}\, t \right ) c_{2}}{7}+\frac {{\mathrm e}^{2 t} \cos \left (\sqrt {6}\, t \right ) c_{1}}{7}-\frac {8 \cos \left (t \right )}{97}-\frac {212 \sin \left (t \right )}{97} \\ \end{align*}

Problem 6721

ODE	\begin {align*} x^{\prime }\left (t \right )&=3 x \left (t \right )-4 y\\ y^{\prime }&=4 x \left (t \right )-7 y \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 ) &= 2 c_{1} {\mathrm e}^{-5 t}+\frac {c_{2} {\mathrm e}^{t}}{2} \\ \end{align*}

Problem 6722

ODE	\begin {align*} x^{\prime }\left (t \right )&=-2 x \left (t \right )+5 y\\ y^{\prime }&=-2 x \left (t \right )+4 y \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{t} \left (c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right )\right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{t} \left (3 c_{1} \sin \left (t \right )-c_{2} \sin \left (t \right )+c_{1} \cos \left (t \right )+3 c_{2} \cos \left (t \right )\right )}{5} \\ \end{align*}

Problem 6723

ODE	\begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right )+\frac {y}{4}\\ y^{\prime }&=x \left (t \right )-y \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{-\frac {t}{2}}+c_{2} {\mathrm e}^{-\frac {3 t}{2}} \\ y \left (t \right ) &= 2 c_{1} {\mathrm e}^{-\frac {t}{2}}-2 c_{2} {\mathrm e}^{-\frac {3 t}{2}} \\ \end{align*}

Problem 6724

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

Problem 6725

ODE	\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )+2 y+z \left (t \right )\\ y^{\prime }&=6 x \left (t \right )-y\\ z^{\prime }\left (t \right )&=-x \left (t \right )-2 y-z \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= -\frac {c_{2} {\mathrm e}^{-4 t}}{2}+\frac {2 c_{3} {\mathrm e}^{3 t}}{3}+\frac {c_{1}}{6} \\ y \left (t \right ) &= c_{1} +c_{2} {\mathrm e}^{-4 t}+c_{3} {\mathrm e}^{3 t} \\ z \left (t \right ) &= \frac {c_{2} {\mathrm e}^{-4 t}}{2}-\frac {2 c_{3} {\mathrm e}^{3 t}}{3}-\frac {13 c_{1}}{6} \\ \end{align*}

Problem 6726

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

Problem 6727

ODE	\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )+2 y\\ y^{\prime }&=4 x \left (t \right )+3 y \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} c_{1} +c_{2} {\mathrm e}^{5 t} \\ y \left (t \right ) &= -{\mathrm e}^{-t} c_{1} +2 c_{2} {\mathrm e}^{5 t} \\ \end{align*}

Problem 6728

ODE	\begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )+2 y\\ y^{\prime }&=x \left (t \right )+3 y \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{4 t} \\ y \left (t \right ) &= -\frac {c_{1} {\mathrm e}^{t}}{2}+c_{2} {\mathrm e}^{4 t} \\ \end{align*}

Problem 6729

ODE	\begin {align*} x^{\prime }\left (t \right )&=-4 x \left (t \right )+2 y\\ y^{\prime }&=-\frac {5 x \left (t \right )}{2}+2 y \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-3 t} \\ y \left (t \right ) &= \frac {5 c_{1} {\mathrm e}^{t}}{2}+\frac {c_{2} {\mathrm e}^{-3 t}}{2} \\ \end{align*}

Problem 6730

ODE	\begin {align*} x^{\prime }\left (t \right )&=-\frac {5 x \left (t \right )}{2}+2 y\\ y^{\prime }&=\frac {3 x \left (t \right )}{4}-2 y \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{-\frac {7 t}{2}} c_{1} +c_{2} {\mathrm e}^{-t} \\ y \left (t \right ) &= -\frac {{\mathrm e}^{-\frac {7 t}{2}} c_{1}}{2}+\frac {3 c_{2} {\mathrm e}^{-t}}{4} \\ \end{align*}

Problem 6731

ODE	\begin {align*} x^{\prime }\left (t \right )&=10 x \left (t \right )-5 y\\ y^{\prime }&=8 x \left (t \right )-12 y \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{8 t}+c_{2} {\mathrm e}^{-10 t} \\ y \left (t \right ) &= \frac {2 c_{1} {\mathrm e}^{8 t}}{5}+4 c_{2} {\mathrm e}^{-10 t} \\ \end{align*}

Problem 6732

ODE	\begin {align*} x^{\prime }\left (t \right )&=-6 x \left (t \right )+2 y\\ y^{\prime }&=-3 x \left (t \right )+y \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} +c_{2} {\mathrm e}^{-5 t} \\ y \left (t \right ) &= \frac {c_{2} {\mathrm e}^{-5 t}}{2}+3 c_{1} \\ \end{align*}

Problem 6733

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

Problem 6734

ODE	\begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )-7 y\\ y^{\prime }&=5 x \left (t \right )+10 y+4 z \left (t \right )\\ z^{\prime }\left (t \right )&=5 y+2 z \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= -\frac {7 c_{3} {\mathrm e}^{7 t}}{5}-\frac {7 c_{2} {\mathrm e}^{5 t}}{3}+c_{1} {\mathrm e}^{2 t} \\ y \left (t \right ) &= c_{2} {\mathrm e}^{5 t}+c_{3} {\mathrm e}^{7 t} \\ z \left (t \right ) &= \frac {5 c_{2} {\mathrm e}^{5 t}}{3}+c_{3} {\mathrm e}^{7 t}-\frac {5 c_{1} {\mathrm e}^{2 t}}{4} \\ \end{align*}

Problem 6735

ODE	\begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right )+y\\ y^{\prime }&=x \left (t \right )+2 y+z \left (t \right )\\ z^{\prime }\left (t \right )&=3 y-z \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= \frac {c_{2} {\mathrm e}^{3 t}}{4}+{\mathrm e}^{-t} c_{1} -c_{3} {\mathrm e}^{-2 t} \\ y \left (t \right ) &= c_{2} {\mathrm e}^{3 t}+c_{3} {\mathrm e}^{-2 t} \\ z \left (t \right ) &= \frac {3 c_{2} {\mathrm e}^{3 t}}{4}-{\mathrm e}^{-t} c_{1} -3 c_{3} {\mathrm e}^{-2 t} \\ \end{align*}

Problem 6736

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

Problem 6737

ODE	\begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right )-y\\ y^{\prime }&=\frac {3 x \left (t \right )}{4}-\frac {3 y}{2}+3 z \left (t \right )\\ z^{\prime }\left (t \right )&=\frac {x \left (t \right )}{8}+\frac {y}{4}-\frac {z \left (t \right )}{2} \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{-\frac {3 t}{2}}+c_{2} {\mathrm e}^{-t}+c_{3} {\mathrm e}^{-\frac {t}{2}} \\ y \left (t \right ) &= \frac {c_{1} {\mathrm e}^{-\frac {3 t}{2}}}{2}-\frac {c_{3} {\mathrm e}^{-\frac {t}{2}}}{2} \\ z \left (t \right ) &= -\frac {c_{1} {\mathrm e}^{-\frac {3 t}{2}}}{4}-\frac {c_{2} {\mathrm e}^{-t}}{4}-\frac {5 c_{3} {\mathrm e}^{-\frac {t}{2}}}{12} \\ \end{align*}

Problem 6738

ODE	\begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right )-y\\ y^{\prime }&=\frac {3 x \left (t \right )}{4}-\frac {3 y}{2}+3 z \left (t \right )\\ z^{\prime }\left (t \right )&=\frac {x \left (t \right )}{8}+\frac {y}{4}-\frac {z \left (t \right )}{2} \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{-\frac {3 t}{2}}+c_{2} {\mathrm e}^{-t}+c_{3} {\mathrm e}^{-\frac {t}{2}} \\ y \left (t \right ) &= \frac {c_{1} {\mathrm e}^{-\frac {3 t}{2}}}{2}-\frac {c_{3} {\mathrm e}^{-\frac {t}{2}}}{2} \\ z \left (t \right ) &= -\frac {c_{1} {\mathrm e}^{-\frac {3 t}{2}}}{4}-\frac {c_{2} {\mathrm e}^{-t}}{4}-\frac {5 c_{3} {\mathrm e}^{-\frac {t}{2}}}{12} \\ \end{align*}

Problem 6739

ODE	\begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right )+4 y+2 z \left (t \right )\\ y^{\prime }&=4 x \left (t \right )-y-2 z \left (t \right )\\ z^{\prime }\left (t \right )&=6 z \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{3 t}+c_{1} {\mathrm e}^{-5 t}+\frac {2 c_{3} {\mathrm e}^{6 t}}{11} \\ y \left (t \right ) &= -\frac {2 c_{3} {\mathrm e}^{6 t}}{11}-c_{1} {\mathrm e}^{-5 t}+c_{2} {\mathrm e}^{3 t} \\ z \left (t \right ) &= c_{3} {\mathrm e}^{6 t} \\ \end{align*}

Problem 6740

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

Problem 6741

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

Problem 6742

ODE	\begin {align*} x^{\prime }\left (t \right )&=\frac {9 x \left (t \right )}{10}+\frac {21 y}{10}+\frac {16 z \left (t \right )}{5}\\ y^{\prime }&=\frac {7 x \left (t \right )}{10}+\frac {13 y}{2}+\frac {21 z \left (t \right )}{5}\\ z^{\prime }\left (t \right )&=\frac {11 x \left (t \right )}{10}+\frac {17 y}{10}+\frac {17 z \left (t \right )}{5} \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{-\frac {\left (i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6000 i \sqrt {3}+\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-216 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right ) t}{60 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}}+c_{2} {\mathrm e}^{\frac {\left (i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6000 i \sqrt {3}-\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+216 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-6000\right ) t}{60 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}}+c_{3} {\mathrm e}^{\frac {\left (\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+108 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right ) t}{30 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}} \\ y \left (t \right ) &= \frac {\left (8 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}-8 \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}+207000 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-299382930 i \sqrt {3}+20757 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-2070 i \sqrt {29760999}+6210 \sqrt {9920333}-207000 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-299382930\right ) c_{1} {\mathrm e}^{-\frac {\left (i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6000 i \sqrt {3}+\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-216 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right ) t}{60 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}}}{88731 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}}-\frac {\left (8 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}+8 \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}+207000 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-299382930 i \sqrt {3}-20757 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+2070 i \sqrt {29760999}+207000 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6210 \sqrt {9920333}[phone]\right ) c_{2} {\mathrm e}^{\frac {\left (i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6000 i \sqrt {3}-\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+216 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-6000\right ) t}{60 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}}}{88731 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}}+\frac {\left (16 \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}+20757 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+4140 i \sqrt {29760999}+414000 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}[phone]\right ) c_{3} {\mathrm e}^{\frac {\left (\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+108 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right ) t}{30 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}}}{88731 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}} \\ z \left (t \right ) &= -\frac {\left (7 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}-7 \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}-3516000 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-58655160 i \sqrt {3}-81660 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+35160 i \sqrt {29760999}+3516000 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-105480 \sqrt {9920333}-58655160\right ) c_{1} {\mathrm e}^{-\frac {\left (i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6000 i \sqrt {3}+\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-216 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right ) t}{60 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}}}{118308 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}}+\frac {\left (7 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}+7 \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}-3516000 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-58655160 i \sqrt {3}+81660 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-35160 i \sqrt {29760999}-3516000 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-105480 \sqrt {9920333}[phone]\right ) c_{2} {\mathrm e}^{\frac {\left (i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6000 i \sqrt {3}-\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+216 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-6000\right ) t}{60 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}}}{118308 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}}-\frac {\left (7 \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}-40830 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-35160 i \sqrt {29760999}-3516000 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}[phone]\right ) c_{3} {\mathrm e}^{\frac {\left (\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+108 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right ) t}{30 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}}}{59154 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}} \\ \end{align*}

Problem 6743

ODE	\begin {align*} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )+2 x_{3} \left (t \right )-\frac {9 x_{4} \left (t \right )}{5}\\ x_{2}^{\prime }\left (t \right )&=\frac {51 x_{2} \left (t \right )}{10}-x_{4} \left (t \right )+3 x_{5} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=x_{1} \left (t \right )+2 x_{2} \left (t \right )-3 x_{3} \left (t \right )\\ x_{4}^{\prime }\left (t \right )&=x_{2} \left (t \right )-\frac {31 x_{3} \left (t \right )}{10}+4 x_{4} \left (t \right )\\ x_{5}^{\prime }\left (t \right )&=-\frac {14 x_{1} \left (t \right )}{5}+\frac {3 x_{4} \left (t \right )}{2}-x_{5} \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x_{1} \left (t \right ) &= \frac {1334393 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{1052220}+\frac {3565 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right )^{4} {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{105222}-\frac {29663 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right )^{3} {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{210444}-\frac {49145 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right )^{2} {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{105222}+\frac {20174057 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}{\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{10522200} \\ x_{2} \left (t \right ) &= -\frac {16027 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right )^{3} {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{420888}+\frac {10385 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right )^{4} {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{210444}-\frac {210025 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right )^{2} {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{210444}-\frac {371003 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{2104440}+\frac {92973853 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}{\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{21044400} \\ x_{3} \left (t \right ) &= -\frac {33875 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right )^{2} {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{105222}-\frac {2585 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right )^{3} {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{210444}-\frac {79853 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{210444}+\frac {1675 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right )^{4} {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{105222}+\frac {5714563 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}{\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{2104440} \\ x_{4} \left (t \right ) &= \moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}{\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}} \\ x_{5} \left (t \right ) &= \frac {527 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right )^{4} {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{140296}-\frac {511531 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right )^{3} {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{4208880}+\frac {203195 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right )^{2} {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{420888}+\frac {7637447 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{7014800}-\frac {943531891 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}{\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{210444000} \\ \end{align*}

Problem 6744

ODE	\begin {align*} x^{\prime }\left (t \right )&=3 x \left (t \right )-y\\ y^{\prime }&=9 x \left (t \right )-3 y \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} t +c_{2} \\ y \left (t \right ) &= 3 c_{1} t -c_{1} +3 c_{2} \\ \end{align*}

Problem 6745

ODE	\begin {align*} x^{\prime }\left (t \right )&=-6 x \left (t \right )+5 y\\ y^{\prime }&=-5 x \left (t \right )+4 y \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} \left (c_{2} t +c_{1} \right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (5 c_{2} t +5 c_{1} +c_{2} \right )}{5} \\ \end{align*}

Problem 6746

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

Problem 6747

ODE	\begin {align*} x^{\prime }\left (t \right )&=12 x \left (t \right )-9 y\\ y^{\prime }&=4 x \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{6 t} \left (6 c_{2} t +6 c_{1} +c_{2} \right )}{4} \\ y \left (t \right ) &= {\mathrm e}^{6 t} \left (c_{2} t +c_{1} \right ) \\ \end{align*}

Problem 6748

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

Problem 6749

ODE	\begin {align*} x^{\prime }\left (t \right )&=3 x \left (t \right )+2 y+4 z \left (t \right )\\ y^{\prime }&=2 x \left (t \right )+2 z \left (t \right )\\ z^{\prime }\left (t \right )&=4 x \left (t \right )+2 y+3 z \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= 2 c_{2} {\mathrm e}^{8 t}+2 c_{3} {\mathrm e}^{-t}+{\mathrm e}^{-t} c_{1} \\ y \left (t \right ) &= c_{2} {\mathrm e}^{8 t}+c_{3} {\mathrm e}^{-t} \\ z \left (t \right ) &= 2 c_{2} {\mathrm e}^{8 t}-\frac {5 c_{3} {\mathrm e}^{-t}}{2}-{\mathrm e}^{-t} c_{1} \\ \end{align*}

Problem 6750

ODE	\begin {align*} x^{\prime }\left (t \right )&=5 x \left (t \right )-4 y\\ y^{\prime }&=x \left (t \right )+2 z \left (t \right )\\ z^{\prime }\left (t \right )&=2 y+5 z \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= -4 \,{\mathrm e}^{5 t} c_{3} t +{\mathrm e}^{5 t} c_{1} +\frac {4 c_{2}}{5} \\ y \left (t \right ) &= c_{2} +c_{3} {\mathrm e}^{5 t} \\ z \left (t \right ) &= 2 \,{\mathrm e}^{5 t} c_{3} t -\frac {{\mathrm e}^{5 t} c_{1}}{2}+\frac {5 c_{3} {\mathrm e}^{5 t}}{2}-\frac {2 c_{2}}{5} \\ \end{align*}

Problem 6751

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

Problem 6752

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

Problem 6753

ODE	\begin {align*} x^{\prime }\left (t \right )&=4 x \left (t \right )+y\\ y^{\prime }&=4 y+z \left (t \right )\\ z^{\prime }\left (t \right )&=4 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}^{4 t}}{2} \\ y \left (t \right ) &= \left (c_{3} t +c_{2} \right ) {\mathrm e}^{4 t} \\ z \left (t \right ) &= c_{3} {\mathrm e}^{4 t} \\ \end{align*}

Problem 6754

ODE	\begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )+4 y\\ y^{\prime }&=-x \left (t \right )+6 y \end {align*} With initial conditions \[ [x \left (0\right ) = -1, y \left (0\right ) = 6] \]
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{4 t} \left (26 t -1\right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{4 t} \left (52 t +24\right )}{4} \\ \end{align*}

Problem 6755

ODE	\begin {align*} x^{\prime }\left (t \right )&=z \left (t \right )\\ y^{\prime }&=y\\ z^{\prime }\left (t \right )&=x \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 2, z \left (0\right ) = 5] \]
program solution
Maple solution	\begin{align*} x \left (t \right ) &= 3 \,{\mathrm e}^{t}-2 \,{\mathrm e}^{-t} \\ y \left (t \right ) &= 2 \,{\mathrm e}^{t} \\ z \left (t \right ) &= 3 \,{\mathrm e}^{t}+2 \,{\mathrm e}^{-t} \\ \end{align*}

Problem 6756

ODE	\begin {align*} x^{\prime }\left (t \right )&=6 x \left (t \right )-y\\ y^{\prime }&=5 x \left (t \right )+2 y \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{4 t} \left (c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right )\right ) \\ y \left (t \right ) &= {\mathrm e}^{4 t} \left (-c_{1} \cos \left (t \right )+c_{2} \sin \left (t \right )+2 c_{1} \sin \left (t \right )+2 c_{2} \cos \left (t \right )\right ) \\ \end{align*}

Problem 6757

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 6758

ODE	\begin {align*} x^{\prime }\left (t \right )&=5 x \left (t \right )+y\\ y^{\prime }&=-2 x \left (t \right )+3 y \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{4 t} \left (c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right )\right ) \\ y \left (t \right ) &= -{\mathrm e}^{4 t} \left (c_{1} \sin \left (t \right )+c_{2} \sin \left (t \right )-c_{1} \cos \left (t \right )+c_{2} \cos \left (t \right )\right ) \\ \end{align*}

Problem 6759

ODE	\begin {align*} x^{\prime }\left (t \right )&=4 x \left (t \right )+5 y\\ y^{\prime }&=-2 x \left (t \right )+6 y \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{5 t} \left (c_{1} \sin \left (3 t \right )+c_{2} \cos \left (3 t \right )\right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{5 t} \left (c_{1} \sin \left (3 t \right )-3 c_{2} \sin \left (3 t \right )+3 c_{1} \cos \left (3 t \right )+c_{2} \cos \left (3 t \right )\right )}{5} \\ \end{align*}

Problem 6760

ODE	\begin {align*} x^{\prime }\left (t \right )&=4 x \left (t \right )-5 y\\ y^{\prime }&=5 x \left (t \right )-4 y \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} \sin \left (3 t \right )+c_{2} \cos \left (3 t \right ) \\ y \left (t \right ) &= -\frac {3 c_{1} \cos \left (3 t \right )}{5}+\frac {3 c_{2} \sin \left (3 t \right )}{5}+\frac {4 c_{1} \sin \left (3 t \right )}{5}+\frac {4 c_{2} \cos \left (3 t \right )}{5} \\ \end{align*}

Problem 6761

ODE	\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )-8 y\\ y^{\prime }&=x \left (t \right )-3 y \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} \left (c_{1} \sin \left (2 t \right )+c_{2} \cos \left (2 t \right )\right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (c_{1} \sin \left (2 t \right )+c_{2} \sin \left (2 t \right )-c_{1} \cos \left (2 t \right )+c_{2} \cos \left (2 t \right )\right )}{4} \\ \end{align*}

Problem 6762

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

Problem 6763

ODE	\begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )+y+2 z \left (t \right )\\ y^{\prime }&=3 x \left (t \right )+6 z \left (t \right )\\ z^{\prime }\left (t \right )&=-4 x \left (t \right )-3 z \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= -\frac {{\mathrm e}^{t} \left (2 c_{2} \sin \left (2 t \right )-c_{3} \sin \left (2 t \right )+c_{2} \cos \left (2 t \right )+2 c_{3} \cos \left (2 t \right )\right )}{2} \\ y \left (t \right ) &= -2 c_{1} {\mathrm e}^{-3 t}-\frac {3 c_{2} {\mathrm e}^{t} \cos \left (2 t \right )}{2}+\frac {3 \sin \left (2 t \right ) {\mathrm e}^{t} c_{3}}{2} \\ z \left (t \right ) &= c_{1} {\mathrm e}^{-3 t}+c_{2} {\mathrm e}^{t} \sin \left (2 t \right )+{\mathrm e}^{t} c_{3} \cos \left (2 t \right ) \\ \end{align*}

Problem 6764

ODE	\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )-12 y-14 z \left (t \right )\\ y^{\prime }&=x \left (t \right )+2 y-3 z \left (t \right )\\ z^{\prime }\left (t \right )&=x \left (t \right )+y-2 z \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 4, y \left (0\right ) = 6, z \left (0\right ) = -7] \]
program solution
Maple solution	\begin{align*} x \left (t \right ) &= -25 \,{\mathrm e}^{t}+11 \sin \left (5 t \right )+29 \cos \left (5 t \right ) \\ y \left (t \right ) &= 7 \,{\mathrm e}^{t}+6 \sin \left (5 t \right )-\cos \left (5 t \right ) \\ z \left (t \right ) &= -6 \,{\mathrm e}^{t}-\cos \left (5 t \right )+6 \sin \left (5 t \right ) \\ \end{align*}

Problem 6765

ODE	\begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )+3 y-7\\ y^{\prime }&=-x \left (t \right )-2 y+5 \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{t}+{\mathrm e}^{-t} c_{1} -1 \\ y \left (t \right ) &= -\frac {c_{2} {\mathrm e}^{t}}{3}-{\mathrm e}^{-t} c_{1} +3 \\ \end{align*}

Problem 6766

ODE	\begin {align*} x^{\prime }\left (t \right )&=5 x \left (t \right )+9 y+2\\ y^{\prime }&=-x \left (t \right )+11 y+6 \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= \frac {1}{2}+{\mathrm e}^{8 t} \left (c_{1} t +c_{2} \right ) \\ y \left (t \right ) &= -\frac {1}{2}+\frac {{\mathrm e}^{8 t} \left (3 c_{1} t +c_{1} +3 c_{2} \right )}{9} \\ \end{align*}

Problem 6767

ODE	\[ \boxed {x^{2} {y^{\prime }}^{2}-y^{2}=0} \]
program solution	\[ y = \frac {c_{2}}{x} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= c_{1} x \\ y \left (x \right ) &= \frac {c_{1}}{x} \\ \end{align*}

Problem 6768

ODE	\[ \boxed {x {y^{\prime }}^{2}-\left (2 x +3 y\right ) y^{\prime }+6 y=0} \]
program solution	\[ y = c_{2} x^{3} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= c_{1} x^{3} \\ y \left (x \right ) &= c_{1} +2 x \\ \end{align*}

Problem 6769

ODE	\[ \boxed {x^{2} {y^{\prime }}^{2}-5 y^{\prime } x y+6 y^{2}=0} \]
program solution	\[ y = c_{2} x^{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= c_{1} x^{3} \\ y \left (x \right ) &= c_{1} x^{2} \\ \end{align*}

Problem 6770

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

Problem 6771

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

Problem 6772

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

Problem 6773

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

Problem 6774

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

Problem 6775

ODE	\[ \boxed {\left (x +y\right )^{2} {y^{\prime }}^{2}-y^{2}=0} \]
program solution	\[ -\frac {x}{y}+\ln \left (\frac {y}{x}\right )+\ln \left (x \right )-c_{5} = 0 \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {x}{\operatorname {LambertW}\left (x \,{\mathrm e}^{c_{1}}\right )} \\ y \left (x \right ) &= -x -\sqrt {x^{2}+2 c_{1}} \\ y \left (x \right ) &= -x +\sqrt {x^{2}+2 c_{1}} \\ \end{align*}

Problem 6776

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

Problem 6777

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

Problem 6778

ODE	\[ \boxed {\left (4 x -y\right ) {y^{\prime }}^{2}+6 \left (x -y\right ) y^{\prime }-5 y=-2 x} \]
program solution	\[ -\frac {\left (y+2 x \right )^{2}}{x -y} = c_{4} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -x +c_{1} \\ y \left (x \right ) &= \frac {-4 c_{1} x +\sqrt {-12 c_{1} x +1}+1}{2 c_{1}} \\ y \left (x \right ) &= \frac {-4 c_{1} x -\sqrt {-12 c_{1} x +1}+1}{2 c_{1}} \\ \end{align*}

Problem 6779

ODE	\[ \boxed {\left (x -y\right )^{2} {y^{\prime }}^{2}-y^{2}=0} \]
program solution	\[ \frac {x}{y}+\ln \left (\frac {y}{x}\right )+\ln \left (x \right )-c_{5} = 0 \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= x -\sqrt {x^{2}-2 c_{1}} \\ y \left (x \right ) &= x +\sqrt {x^{2}-2 c_{1}} \\ y \left (x \right ) &= -\frac {x}{\operatorname {LambertW}\left (-x \,{\mathrm e}^{-c_{1}}\right )} \\ \end{align*}

Problem 6780

ODE	\[ \boxed {y {y^{\prime }}^{2} x +\left (x y^{2}-1\right ) y^{\prime }-y=0} \]
program solution	\[ y = \sqrt {2 \ln \left (x \right )+2 c_{2}} \] Verified OK. \[ y = -\sqrt {2 \ln \left (x \right )+2 c_{2}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \sqrt {2 \ln \left (x \right )+c_{1}} \\ y \left (x \right ) &= -\sqrt {2 \ln \left (x \right )+c_{1}} \\ y \left (x \right ) &= c_{1} {\mathrm e}^{-x} \\ \end{align*}

Problem 6781

ODE	\[ \boxed {\left (x^{2}+y^{2}\right )^{2} {y^{\prime }}^{2}-4 x^{2} y^{2}=0} \]
program solution	\[ -\frac {\left (x -y\right ) \left (x +y\right )}{y} = c_{6} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {1-\sqrt {4 x^{2} c_{1}^{2}+1}}{2 c_{1}} \\ y \left (x \right ) &= \frac {1+\sqrt {4 x^{2} c_{1}^{2}+1}}{2 c_{1}} \\ y \left (x \right ) &= -\frac {2 \left (c_{1} x^{2}-\frac {\left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{\frac {2}{3}}}{4}\right )}{\sqrt {c_{1}}\, \left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{\frac {1}{3}}}{4 \sqrt {c_{1}}}-\frac {\left (i \sqrt {3}-1\right ) x^{2} \sqrt {c_{1}}}{\left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {4 i \sqrt {3}\, c_{1} x^{2}+i \sqrt {3}\, \left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{\frac {2}{3}}+4 c_{1} x^{2}-\left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{\frac {2}{3}}}{4 \left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{\frac {1}{3}} \sqrt {c_{1}}} \\ \end{align*}

Problem 6782

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

Problem 6783

ODE	\[ \boxed {x y \left (x^{2}+y^{2}\right ) \left ({y^{\prime }}^{2}-1\right )-y^{\prime } \left (x^{4}+x^{2} y^{2}+y^{4}\right )=0} \]
program solution	\[ \left (\frac {y^{2}+2 x^{2}}{x^{2}}\right )^{\frac {1}{4}} \sqrt {\frac {y}{x}} = \frac {c_{5}}{x} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\sqrt {x^{2} c_{1} \left (c_{1} x^{2}-\sqrt {c_{1}^{2} x^{4}+1}\right )}}{x \left (c_{1} x^{2}-\sqrt {c_{1}^{2} x^{4}+1}\right ) c_{1}} \\ y \left (x \right ) &= \frac {\sqrt {x^{2} c_{1} \left (c_{1} x^{2}+\sqrt {c_{1}^{2} x^{4}+1}\right )}}{x \left (c_{1} x^{2}+\sqrt {c_{1}^{2} x^{4}+1}\right ) c_{1}} \\ y \left (x \right ) &= \frac {\sqrt {x^{2} c_{1} \left (c_{1} x^{2}-\sqrt {c_{1}^{2} x^{4}+1}\right )}}{x \left (-c_{1} x^{2}+\sqrt {c_{1}^{2} x^{4}+1}\right ) c_{1}} \\ y \left (x \right ) &= -\frac {\sqrt {x^{2} c_{1} \left (c_{1} x^{2}+\sqrt {c_{1}^{2} x^{4}+1}\right )}}{x \left (c_{1} x^{2}+\sqrt {c_{1}^{2} x^{4}+1}\right ) c_{1}} \\ y \left (x \right ) &= \sqrt {2 \ln \left (x \right )+c_{1}}\, x \\ y \left (x \right ) &= -\sqrt {2 \ln \left (x \right )+c_{1}}\, x \\ \end{align*}

Problem 6784

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

Problem 6785

ODE	\[ \boxed {y {y^{\prime }}^{2} x +\left (x +y\right ) y^{\prime }=-1} \]
program solution	\[ y = \sqrt {-2 c_{2} -2 x} \] Verified OK. \[ y = -\sqrt {-2 c_{2} -2 x} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\ln \left (x \right )+c_{1} \\ y \left (x \right ) &= \sqrt {c_{1} -2 x} \\ y \left (x \right ) &= -\sqrt {c_{1} -2 x} \\ \end{align*}

Problem 6786

ODE	\[ \boxed {x {y^{\prime }}^{2}-2 y^{\prime } y=-4 x} \]
program solution	\[ y = -2 x \] Verified OK. \[ y = 2 x \] Verified OK. \[ y = \frac {c_{1}^{2} x^{2}+4}{2 c_{1}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -2 x \\ y \left (x \right ) &= 2 x \\ y \left (x \right ) &= \frac {4 c_{1}^{2}+x^{2}}{2 c_{1}} \\ \end{align*}

Problem 6787

ODE	\[ \boxed {3 x^{4} {y^{\prime }}^{2}-y^{\prime } x -y=0} \]
program solution	\[ -\frac {\ln \left (y\right )}{2}-\operatorname {arctanh}\left (\sqrt {1+12 y x^{2}}\right ) = c_{1} \] Verified OK. \[ -\frac {\ln \left (y\right )}{2}+\operatorname {arctanh}\left (\sqrt {1+12 y x^{2}}\right ) = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {1}{12 x^{2}} \\ y \left (x \right ) &= \frac {-i \sqrt {3}\, c_{1} -3 x}{3 c_{1}^{2} x} \\ y \left (x \right ) &= \frac {i \sqrt {3}\, c_{1} -3 x}{3 x \,c_{1}^{2}} \\ y \left (x \right ) &= \frac {i \sqrt {3}\, c_{1} -3 x}{3 x \,c_{1}^{2}} \\ y \left (x \right ) &= \frac {-i \sqrt {3}\, c_{1} -3 x}{3 c_{1}^{2} x} \\ \end{align*}

Problem 6788

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

Problem 6789

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

Problem 6790

ODE	\[ \boxed {{y^{\prime }}^{2}+4 x^{5} y^{\prime }-12 x^{4} y=0} \]
program solution	\[ \frac {\ln \left (y\right )}{6}+\frac {\ln \left (-x^{3}+\sqrt {x^{6}+3 y}\right )}{6}-\frac {\ln \left (x^{3}+\sqrt {x^{6}+3 y}\right )}{6} = c_{1} \] Verified OK. \[ \frac {\ln \left (y\right )}{6}-\frac {\ln \left (-x^{3}+\sqrt {x^{6}+3 y}\right )}{6}+\frac {\ln \left (x^{3}+\sqrt {x^{6}+3 y}\right )}{6} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {x^{6}}{3} \\ y \left (x \right ) &= c_{1} x^{3}+\frac {3}{4} c_{1}^{2} \\ \end{align*}

Problem 6791

ODE	\[ \boxed {4 y^{3} {y^{\prime }}^{2}-4 y^{\prime } x +y=0} \]
program solution	\[ \frac {\ln \left (2\right )}{2}+\frac {\ln \left (x \right )}{2}+\frac {\ln \left (x +\sqrt {x^{2}-y^{4}}\right )}{2} = \frac {\ln \left (x \right )}{2}+c_{1} \] Verified OK. \[ y = {\mathrm e}^{\frac {\ln \left (2\right )}{4}+\frac {\ln \left (-2 \,{\mathrm e}^{-2 c_{1}} {\mathrm e}^{4 c_{1}}+2 \,{\mathrm e}^{-2 c_{1}} {\mathrm e}^{2 c_{1}} x \right )}{4}+\frac {c_{1}}{2}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \sqrt {-x} \\ y \left (x \right ) &= -\sqrt {-x} \\ y \left (x \right ) &= \sqrt {x} \\ y \left (x \right ) &= -\sqrt {x} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )+2 \left (\int _{}^{\textit {\_Z}}-\frac {\textit {\_a}^{4}-\sqrt {-\textit {\_a}^{4}+1}-1}{\textit {\_a} \left (\textit {\_a}^{4}-1\right )}d \textit {\_a} \right )+c_{1} \right ) \sqrt {x} \\ \end{align*}

Problem 6792

ODE	\[ \boxed {4 y^{3} {y^{\prime }}^{2}+4 y^{\prime } x +y=0} \]
program solution	\[ \frac {\ln \left (y^{4}+3 x^{2}\right )}{6}+\frac {2 \ln \left (y\right )}{3}+\frac {\ln \left (2\right )}{6}+\frac {\ln \left (x \right )}{6}-\frac {\ln \left (x +\sqrt {x^{2}-y^{4}}\right )}{6}+\frac {\ln \left (i x +\sqrt {3}\, y^{2}+2 i \sqrt {x^{2}-y^{4}}\right )}{6}-\frac {\ln \left (x \sqrt {3}+i y^{2}\right )}{6}+\frac {\ln \left (-i x +\sqrt {3}\, y^{2}-2 i \sqrt {x^{2}-y^{4}}\right )}{6}-\frac {\ln \left (-i y^{2}+x \sqrt {3}\right )}{6} = \frac {\ln \left (x \right )}{6}+c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ \frac {\left (\int _{\textit {\_b}}^{x}\frac {-2 \textit {\_a} +\sqrt {-y \left (x \right )^{4}+\textit {\_a}^{2}}}{y \left (x \right )^{4}+3 \textit {\_a}^{2}}d \textit {\_a} \right )}{2}-\left (\int _{}^{y \left (x \right )}\frac {\left (1+\left (\textit {\_f}^{4}-\sqrt {-\textit {\_f}^{4}+x^{2}}\, x +x^{2}\right ) \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_f}^{4}+4 \sqrt {-\textit {\_f}^{4}+\textit {\_a}^{2}}\, \textit {\_a} -5 \textit {\_a}^{2}}{\sqrt {-\textit {\_f}^{4}+\textit {\_a}^{2}}\, \left (\textit {\_f}^{4}+3 \textit {\_a}^{2}\right )^{2}}d \textit {\_a} \right )\right ) \textit {\_f}^{3}}{\textit {\_f}^{4}-\sqrt {-\textit {\_f}^{4}+x^{2}}\, x +x^{2}}d \textit {\_f} \right )+c_{1} &= 0 \\ -\frac {\left (\int _{\textit {\_b}}^{x}\frac {2 \textit {\_a} +\sqrt {-y \left (x \right )^{4}+\textit {\_a}^{2}}}{y \left (x \right )^{4}+3 \textit {\_a}^{2}}d \textit {\_a} \right )}{2}-\left (\int _{}^{y \left (x \right )}\frac {\left (1+\left (\textit {\_f}^{4}+\sqrt {-\textit {\_f}^{4}+x^{2}}\, x +x^{2}\right ) \left (\int _{\textit {\_b}}^{x}\frac {-\textit {\_f}^{4}+5 \textit {\_a}^{2}+4 \sqrt {-\textit {\_f}^{4}+\textit {\_a}^{2}}\, \textit {\_a}}{\sqrt {-\textit {\_f}^{4}+\textit {\_a}^{2}}\, \left (\textit {\_f}^{4}+3 \textit {\_a}^{2}\right )^{2}}d \textit {\_a} \right )\right ) \textit {\_f}^{3}}{\textit {\_f}^{4}+\sqrt {-\textit {\_f}^{4}+x^{2}}\, x +x^{2}}d \textit {\_f} \right )+c_{1} &= 0 \\ \end{align*}

Problem 6793

ODE	\[ \boxed {{y^{\prime }}^{3}+x {y^{\prime }}^{2}-y=0} \]
program solution	\[ y = 0 \] Verified OK. \[ y = 1+x \] Verified OK. \[ x = \frac {24 \left (x^{3}+\frac {3 x^{2}}{2}-3 y+3 c_{1} \right ) \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}}{2}-\frac {27 y}{2}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}+96 \left (x +\frac {3}{2}\right ) \left (\left (\sqrt {3}\, \left (x^{3}-\frac {27 y}{4}\right ) \sqrt {27 y^{2}-4 x^{3} y}-\frac {x^{6}}{2}+\frac {27 x^{3} y}{2}-\frac {243 y^{2}}{4}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}+\left (-\frac {5 \sqrt {3}\, \left (x^{3}-\frac {54 y}{5}\right ) \sqrt {27 y^{2}-4 x^{3} y}}{2}+x^{6}-\frac {81 x^{3} y}{2}+243 y^{2}\right ) x \right )}{\left (2 x^{3}-3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}-27 y\right ) {\left (\left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}-2 x \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}+4 x^{2}-6 \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}\right )}^{2}} \] Warning, solution could not be verified \[ x = \frac {96 \left (x^{3}+\frac {3 x^{2}}{2}-3 y+3 c_{1} \right ) \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}}{2}-\frac {27 y}{2}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}+192 \left (x +\frac {3}{2}\right ) \left (\left (-3 \left (i+\frac {\sqrt {3}}{3}\right ) \left (x^{3}-\frac {27 y}{4}\right ) \sqrt {27 y^{2}-4 x^{3} y}+\frac {\left (x^{6}-27 x^{3} y+\frac {243 y^{2}}{2}\right ) \left (1+i \sqrt {3}\right )}{2}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}+x \left (-\frac {15 \left (i-\frac {\sqrt {3}}{3}\right ) \left (x^{3}-\frac {54 y}{5}\right ) \sqrt {27 y^{2}-4 x^{3} y}}{2}+\left (i \sqrt {3}-1\right ) \left (x^{6}-\frac {81 x^{3} y}{2}+243 y^{2}\right )\right )\right )}{\left (2 x^{3}-3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}-27 y\right ) \left (4 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}+4 x^{2}+4 x \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}+\left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}+12 \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}\right )^{2}} \] Warning, solution could not be verified \[ x = \frac {96 \left (x^{3}+\frac {3 x^{2}}{2}-3 y+3 c_{1} \right ) \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}}{2}-\frac {27 y}{2}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}-192 \left (\left (-3 \left (x^{3}-\frac {27 y}{4}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {27 y^{2}-4 x^{3} y}+\frac {\left (x^{6}-27 x^{3} y+\frac {243 y^{2}}{2}\right ) \left (i \sqrt {3}-1\right )}{2}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}+\left (-\frac {15 \left (i+\frac {\sqrt {3}}{3}\right ) \left (x^{3}-\frac {54 y}{5}\right ) \sqrt {27 y^{2}-4 x^{3} y}}{2}+\left (1+i \sqrt {3}\right ) \left (x^{6}-\frac {81 x^{3} y}{2}+243 y^{2}\right )\right ) x \right ) \left (x +\frac {3}{2}\right )}{\left (2 x^{3}-3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}-27 y\right ) \left (4 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}-4 x^{2}-4 x \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}-\left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}-12 \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}\right )^{2}} \] Warning, solution could not be verified
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\left (4 x^{2}-2 x \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+12 x +3 \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {2}{3}}+9\right )^{2} \left (4 x^{2}+4 x \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+12 x +3 \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {2}{3}}+9\right )}{-1728 x^{3}-7776 x^{2}-11664 x +23328 c_{1} +1296 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}+5832} \\ y \left (x \right ) &= \frac {\left (\frac {\left (-i \sqrt {3}-1\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {2}{3}}}{4}+\left (2 x +\frac {3}{2}\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+\left (x +\frac {3}{2}\right )^{2} \left (i \sqrt {3}-1\right )\right ) {\left (\frac {\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {2}{3}} \left (i-\sqrt {3}\right )}{4}-i \left (-x +\frac {3}{2}\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+\left (x +\frac {3}{2}\right )^{2} \left (\sqrt {3}+i\right )\right )}^{2}}{216 x^{3}+972 x^{2}+1458 x -2916 c_{1} -162 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}-729} \\ y \left (x \right ) &= \frac {\left (\frac {\left (i \sqrt {3}-1\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {2}{3}}}{4}-\left (-2 x -\frac {3}{2}\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+\left (-i \sqrt {3}-1\right ) \left (x +\frac {3}{2}\right )^{2}\right ) {\left (\frac {\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )}{4}+i \left (x -\frac {3}{2}\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+\left (x +\frac {3}{2}\right )^{2} \left (i-\sqrt {3}\right )\right )}^{2}}{216 x^{3}+972 x^{2}+1458 x -2916 c_{1} -162 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}-729} \\ \end{align*}

Problem 6794

ODE	\[ \boxed {y^{4} {y^{\prime }}^{3}-6 y^{\prime } x +2 y=0} \]
program solution	\[ \frac {\ln \left (x \right )}{2} = \int _{}^{\frac {y}{\sqrt {x}}}\frac {\textit {\_a}^{2} \left (-\textit {\_a}^{3}+\sqrt {\textit {\_a}^{6}-8}\right )^{\frac {1}{3}}}{-\left (-\textit {\_a}^{3}+\sqrt {\textit {\_a}^{6}-8}\right )^{\frac {1}{3}} \textit {\_a}^{3}+2 \left (-\textit {\_a}^{3}+\sqrt {\textit {\_a}^{6}-8}\right )^{\frac {2}{3}}+4}d \textit {\_a} +c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \sqrt {x \left (-i \sqrt {3}-1\right )} \\ y \left (x \right ) &= \sqrt {\left (i \sqrt {3}-1\right ) x} \\ y \left (x \right ) &= -\sqrt {-\left (1+i \sqrt {3}\right ) x} \\ y \left (x \right ) &= -\sqrt {\left (i \sqrt {3}-1\right ) x} \\ y \left (x \right ) &= \sqrt {x}\, \sqrt {2} \\ y \left (x \right ) &= -\sqrt {x}\, \sqrt {2} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {2^{\frac {2}{3}} \left (-c_{1}^{3}+6 c_{1} x \right )^{\frac {1}{3}}}{2} \\ y \left (x \right ) &= -\frac {2^{\frac {2}{3}} \left (-c_{1}^{3}+6 c_{1} x \right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{4} \\ y \left (x \right ) &= \frac {2^{\frac {2}{3}} \left (-c_{1}^{3}+6 c_{1} x \right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{4} \\ \end{align*}

Problem 6795

ODE	\[ \boxed {{y^{\prime }}^{2}+x^{3} y^{\prime }-2 y x^{2}=0} \]
program solution	\[ \frac {\ln \left (y\right )}{4}-\frac {\ln \left (x^{2}+\sqrt {x^{4}+8 y}\right )}{4}+\frac {\ln \left (-x^{2}+\sqrt {x^{4}+8 y}\right )}{4} = c_{1} \] Verified OK. \[ \frac {\ln \left (y\right )}{4}+\frac {\ln \left (x^{2}+\sqrt {x^{4}+8 y}\right )}{4}-\frac {\ln \left (-x^{2}+\sqrt {x^{4}+8 y}\right )}{4} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {x^{4}}{8} \\ y \left (x \right ) &= c_{1} \left (x^{2}+2 c_{1} \right ) \\ \end{align*}

Problem 6796

ODE	\[ \boxed {{y^{\prime }}^{2}+4 x^{5} y^{\prime }-12 x^{4} y=0} \]
program solution	\[ \frac {\ln \left (y\right )}{6}+\frac {\ln \left (-x^{3}+\sqrt {x^{6}+3 y}\right )}{6}-\frac {\ln \left (x^{3}+\sqrt {x^{6}+3 y}\right )}{6} = c_{1} \] Verified OK. \[ \frac {\ln \left (y\right )}{6}-\frac {\ln \left (-x^{3}+\sqrt {x^{6}+3 y}\right )}{6}+\frac {\ln \left (x^{3}+\sqrt {x^{6}+3 y}\right )}{6} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {x^{6}}{3} \\ y \left (x \right ) &= c_{1} x^{3}+\frac {3}{4} c_{1}^{2} \\ \end{align*}

Problem 6797

ODE	\[ \boxed {2 x {y^{\prime }}^{3}-6 y {y^{\prime }}^{2}=-x^{4}} \]
program solution	\[ \ln \left (x \right )-2 \left (\int _{}^{\frac {y}{x^{2}}}\frac {\left (-2+2 i \sqrt {2 \textit {\_a} -1}\, \sqrt {4 \textit {\_a}^{2}+2 \textit {\_a} +1}+8 \textit {\_a}^{3}\right )^{\frac {1}{3}}}{\left (-2+2 i \sqrt {2 \textit {\_a} -1}\, \sqrt {4 \textit {\_a}^{2}+2 \textit {\_a} +1}+8 \textit {\_a}^{3}\right )^{\frac {2}{3}}-2 \textit {\_a} \left (-2+2 i \sqrt {2 \textit {\_a} -1}\, \sqrt {4 \textit {\_a}^{2}+2 \textit {\_a} +1}+8 \textit {\_a}^{3}\right )^{\frac {1}{3}}+4 \textit {\_a}^{2}}d \textit {\_a} \right )-c_{1} = 0 \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) x^{2}}{4} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) x^{2}}{4} \\ y \left (x \right ) &= \frac {x^{2}}{2} \\ y \left (x \right ) &= \frac {1}{6 c_{1}^{2}}+\frac {c_{1} x^{3}}{3} \\ \end{align*}

Problem 6798

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

Problem 6799

ODE	\[ \boxed {y-y^{\prime } x -k {y^{\prime }}^{2}=0} \]
program solution	\[ y = c_{1}^{2} k +c_{1} x \] Verified OK. \[ y = -\frac {x^{2}}{4 k} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {x^{2}}{4 k} \\ y \left (x \right ) &= c_{1} \left (c_{1} k +x \right ) \\ \end{align*}

Problem 6800

ODE	\[ \boxed {x^{8} {y^{\prime }}^{2}+3 y^{\prime } x +9 y=0} \]
program solution	\[ -\frac {\ln \left (y\right )}{6}+\frac {\operatorname {arctanh}\left (\sqrt {1-4 y x^{6}}\right )}{3} = c_{1} \] Verified OK. \[ -\frac {\ln \left (y\right )}{6}-\frac {\operatorname {arctanh}\left (\sqrt {1-4 y x^{6}}\right )}{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {1}{4 x^{6}} \\ y \left (x \right ) &= \frac {-x^{3}+c_{1}}{x^{3} c_{1}^{2}} \\ y \left (x \right ) &= \frac {-x^{3}-c_{1}}{x^{3} c_{1}^{2}} \\ \end{align*}