Problem 15501
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=\frac {x_{1} \left (t \right )^{2}}{x_{2} \left (t \right )}\\ x_{2}^{\prime }\left (t \right )&=x_{2} \left (t \right )-x_{1} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} \\ \left [\left \{x_{1} \left (t \right ) &= \frac {1}{\sqrt {2 \,{\mathrm e}^{-t} c_{1} -2 c_{2}}}, x_{1} \left (t \right ) &= -\frac {1}{\sqrt {2 \,{\mathrm e}^{-t} c_{1} -2 c_{2}}}\right \}, \left \{x_{2} \left (t \right ) &= \frac {x_{1} \left (t \right )^{2}}{\frac {d}{d t}x_{1} \left (t \right )}\right \}\right ] \\ \end{align*} |
Problem 15502
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=\frac {{\mathrm e}^{-x \left (t \right )}}{t}\\ y^{\prime }\left (t \right )&=\frac {x \left (t \right ) {\mathrm e}^{-y \left (t \right )}}{t} \end {align*}
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program solution |
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Maple solution |
\begin{align*} \{x \left (t \right ) &= \ln \left (\ln \left (t \right )+c_{2} \right )\} \\ \left \{y \left (t \right ) &= \ln \left (\int \frac {x \left (t \right )}{t}d t +c_{1} \right )\right \} \\ \end{align*} |
Problem 15503
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=\frac {y \left (t \right )}{x \left (t \right )+y \left (t \right )}+\frac {t}{x \left (t \right )+y \left (t \right )}\\ y^{\prime }\left (t \right )&=\frac {x \left (t \right )}{x \left (t \right )+y \left (t \right )}-\frac {t}{x \left (t \right )+y \left (t \right )} \end {align*}
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program solution |
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Maple solution |
\begin{align*} \\ \left [\left \{x \left (t \right ) &= \frac {c_{1} t^{2}-c_{2} t +1}{c_{1} t -c_{2}}\right \}, \left \{y \left (t \right ) &= \frac {-x \left (t \right ) \left (\frac {d}{d t}x \left (t \right )\right )+t}{\frac {d}{d t}x \left (t \right )-1}\right \}\right ] \\ \end{align*} |
Problem 15504
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=\frac {t}{y \left (t \right )-x \left (t \right )}-\frac {y \left (t \right )}{y \left (t \right )-x \left (t \right )}\\ y^{\prime }\left (t \right )&=\frac {x \left (t \right )}{y \left (t \right )-x \left (t \right )}-\frac {t}{y \left (t \right )-x \left (t \right )} \end {align*}
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program solution |
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Maple solution |
\begin{align*} \left \{x \left (t \right ) &= t +\operatorname {RootOf}\left (-t +\int _{}^{\textit {\_Z}}-\frac {2 \left ({\mathrm e}^{c_{1}} \textit {\_f}^{2}-1\right )}{-4+3 \,{\mathrm e}^{c_{1}} \textit {\_f}^{2}-\sqrt {-3 \,{\mathrm e}^{c_{1}} \textit {\_f}^{2}+4}\, {\mathrm e}^{\frac {c_{1}}{2}} \textit {\_f}}d \textit {\_f} +c_{2} \right ), x \left (t \right ) &= t +\operatorname {RootOf}\left (-t +\int _{}^{\textit {\_Z}}-\frac {2 \left ({\mathrm e}^{c_{1}} \textit {\_f}^{2}-1\right )}{3 \,{\mathrm e}^{c_{1}} \textit {\_f}^{2}+\sqrt {-3 \,{\mathrm e}^{c_{1}} \textit {\_f}^{2}+4}\, {\mathrm e}^{\frac {c_{1}}{2}} \textit {\_f} -4}d \textit {\_f} +c_{2} \right )\right \} \\ \left \{y \left (t \right ) &= \frac {x \left (t \right ) \left (\frac {d}{d t}x \left (t \right )\right )+t}{\frac {d}{d t}x \left (t \right )+1}\right \} \\ \end{align*} |
Problem 15505
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=\frac {y \left (t \right )}{x \left (t \right )+y \left (t \right )}+\frac {t}{x \left (t \right )+y \left (t \right )}\\ y^{\prime }\left (t \right )&=\frac {t}{x \left (t \right )+y \left (t \right )}+\frac {x \left (t \right )}{x \left (t \right )+y \left (t \right )} \end {align*}
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program solution |
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Maple solution |
\begin{align*} \text {Expression too large to display} \\ \text {Expression too large to display} \\ \end{align*} |
Problem 15506
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=-9 y \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right ) \end {align*}
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program solution |
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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 {c_{1} \cos \left (3 t \right )}{3}+\frac {c_{2} \sin \left (3 t \right )}{3} \\ \end{align*} |
Problem 15507
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=y \left (t \right )+t\\ y^{\prime }\left (t \right )&=x \left (t \right )-t \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{t}+{\mathrm e}^{-t} c_{1} +t -1 \\ y \left (t \right ) &= c_{2} {\mathrm e}^{t}-{\mathrm e}^{-t} c_{1} +1-t \\ \end{align*} |
Problem 15508
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=-3 x \left (t \right )-4 y \left (t \right )\\ y^{\prime }\left (t \right )&=-2 x \left (t \right )-5 y \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 4] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= 3 \,{\mathrm e}^{-7 t}-2 \,{\mathrm e}^{-t} \\ y \left (t \right ) &= 3 \,{\mathrm e}^{-7 t}+{\mathrm e}^{-t} \\ \end{align*} |
Problem 15509
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )+5 y \left (t \right )\\ y^{\prime }\left (t \right )&=-x \left (t \right )-3 y \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = -2, y \left (0\right ) = 1] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} \left (\sin \left (t \right )-2 \cos \left (t \right )\right ) \\ y \left (t \right ) &= {\mathrm e}^{-t} \cos \left (t \right ) \\ \end{align*} |
Problem 15510
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=-y \left (t \right )+\cos \left (t \right )\\ y^{\prime }\left (t \right )&=-4 y \left (t \right )+4 \cos \left (t \right )+3 x \left (t \right )-\sin \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= \frac {c_{2} {\mathrm e}^{-3 t}}{3}+{\mathrm e}^{-t} c_{1} \\ y \left (t \right ) &= c_{2} {\mathrm e}^{-3 t}+{\mathrm e}^{-t} c_{1} +\cos \left (t \right ) \\ \end{align*} |
Problem 15511
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=-y \left (t \right )+z \left (t \right )\\ y^{\prime }\left (t \right )&=z \left (t \right )\\ z^{\prime }\left (t \right )&=-x \left (t \right )+z \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{2} \sin \left (t \right )+c_{3} \sin \left (t \right )-c_{2} \cos \left (t \right )+c_{3} \cos \left (t \right ) \\ y \left (t \right ) &= c_{1} {\mathrm e}^{t}-c_{2} \cos \left (t \right )+c_{3} \sin \left (t \right ) \\ z \left (t \right ) &= c_{1} {\mathrm e}^{t}+c_{2} \sin \left (t \right )+c_{3} \cos \left (t \right ) \\ \end{align*} |
Problem 15512
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=y \left (t \right )+z \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )+z \left (t \right )\\ z^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{-t} \\ y \left (t \right ) &= c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{-t}+{\mathrm e}^{-t} c_{1} \\ z \left (t \right ) &= c_{2} {\mathrm e}^{2 t}-2 c_{3} {\mathrm e}^{-t}-{\mathrm e}^{-t} c_{1} \\ \end{align*} |
Problem 15513
| ODE |
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program solution |
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Maple solution | \begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-t}-c_{3} \sin \left (t \right )-c_{4} \cos \left (t \right ) \\ y \left (t \right ) &= c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-t}+c_{3} \sin \left (t \right )+c_{4} \cos \left (t \right ) \\ \end{align*} |
Problem 15514
| ODE |
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program solution |
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Maple solution | \begin{align*} x \left (t \right ) &= c_{1} -\frac {1}{2} t^{2} c_{1} -c_{2} t -c_{3} \\ y \left (t \right ) &= \frac {1}{6} t^{3} c_{1} +\frac {1}{2} c_{2} t^{2}+c_{3} t +c_{4} \\ \end{align*} |
Problem 15515
| ODE |
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program solution |
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Maple solution | \begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{-2 t}-\frac {c_{2} {\mathrm e}^{t}}{2}-\frac {c_{3} {\mathrm e}^{t} t}{2}-\frac {c_{3} {\mathrm e}^{t}}{2} \\ y \left (t \right ) &= c_{1} {\mathrm e}^{-2 t}+c_{2} {\mathrm e}^{t}+c_{3} {\mathrm e}^{t} t \\ \end{align*} |
Problem 15516
| ODE |
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program solution |
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Maple solution | \begin{align*} x \left (t \right ) &= {\mathrm e}^{t} \\ y \left (t \right ) &= -{\mathrm e}^{2 t}+{\mathrm e}^{t} \\ \end{align*} |
Problem 15517
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )^{2}+y \left (t \right )^{2}\\ y^{\prime }\left (t \right )&=2 x \left (t \right ) y \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} \left [\{y \left (t \right ) = 0\}, \left \{x \left (t \right ) &= \frac {1}{-t +c_{1}}\right \}\right ] \\ \left [\left \{y \left (t \right ) &= \frac {4 c_{1}}{c_{1}^{2} c_{2}^{2}+2 c_{1}^{2} c_{2} t +c_{1}^{2} t^{2}-16}\right \}, \left \{x \left (t \right ) &= \frac {\frac {d}{d t}y \left (t \right )}{2 y \left (t \right )}\right \}\right ] \\ \end{align*} |
Problem 15518
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=-\frac {1}{y \left (t \right )}\\ y^{\prime }\left (t \right )&=\frac {1}{x \left (t \right )} \end {align*}
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program solution |
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Maple solution |
\begin{align*} \{x \left (t \right ) &= {\mathrm e}^{c_{1} t} c_{2}\} \\ \left \{y \left (t \right ) &= -\frac {1}{\frac {d}{d t}x \left (t \right )}\right \} \\ \end{align*} |
Problem 15519
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=\frac {x \left (t \right )}{y \left (t \right )}\\ y^{\prime }\left (t \right )&=\frac {y \left (t \right )}{x \left (t \right )} \end {align*}
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program solution |
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Maple solution |
\begin{align*} \left \{x \left (t \right ) &= \frac {-1+{\mathrm e}^{c_{2} c_{1}} {\mathrm e}^{c_{1} t}}{c_{1}}\right \} \\ \left \{y \left (t \right ) &= \frac {x \left (t \right )}{\frac {d}{d t}x \left (t \right )}\right \} \\ \end{align*} |
Problem 15520
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=\frac {y \left (t \right )}{x \left (t \right )-y \left (t \right )}\\ y^{\prime }\left (t \right )&=\frac {x \left (t \right )}{x \left (t \right )-y \left (t \right )} \end {align*}
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program solution |
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Maple solution |
\begin{align*} \left \{x \left (t \right ) &= \frac {-c_{1} t^{2}-2 c_{2} t -2}{2 c_{1} t +2 c_{2}}\right \} \\ \left \{y \left (t \right ) &= \frac {x \left (t \right ) \left (\frac {d}{d t}x \left (t \right )\right )}{\frac {d}{d t}x \left (t \right )+1}\right \} \\ \end{align*} |
Problem 15521
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=\sin \left (x \left (t \right )\right ) \cos \left (y \left (t \right )\right )\\ y^{\prime }\left (t \right )&=\cos \left (x \left (t \right )\right ) \sin \left (y \left (t \right )\right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} \left \{y \left (t \right ) &= \operatorname {arccot}\left (\frac {\left (c_{1} {\mathrm e}^{2 t}-c_{2} \right ) {\mathrm e}^{-t}}{2}\right )\right \} \\ \left \{x \left (t \right ) &= \arccos \left (\frac {\frac {d}{d t}y \left (t \right )}{\sin \left (y \left (t \right )\right )}\right )\right \} \\ \end{align*} |
Problem 15522
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=\frac {{\mathrm e}^{-t}}{y \left (t \right )}\\ y^{\prime }\left (t \right )&=\frac {{\mathrm e}^{-t}}{x \left (t \right )} \end {align*}
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program solution |
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Maple solution |
\begin{align*} \left \{x \left (t \right ) &= \sqrt {-2 \,{\mathrm e}^{-t} c_{1} +2 c_{2}}, x \left (t \right ) &= -\sqrt {-2 \,{\mathrm e}^{-t} c_{1} +2 c_{2}}\right \} \\ \left \{y \left (t \right ) &= \frac {{\mathrm e}^{-t}}{\frac {d}{d t}x \left (t \right )}\right \} \\ \end{align*} |
Problem 15523
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=\cos \left (x \left (t \right )\right )^{2} \cos \left (y \left (t \right )\right )^{2}+\sin \left (x \left (t \right )\right )^{2} \cos \left (y \left (t \right )\right )^{2}\\ y^{\prime }\left (t \right )&=-2 \sin \left (x \left (t \right )\right ) \cos \left (x \left (t \right )\right ) \sin \left (y \left (t \right )\right ) \cos \left (y \left (t \right )\right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 0, y \left (0\right ) = 0] \] |
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program solution |
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Maple solution |
\[ \text {No solution found} \] |
Problem 15524
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=8 y \left (t \right )-x \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{3 t}+c_{2} {\mathrm e}^{-3 t} \\ y \left (t \right ) &= \frac {c_{1} {\mathrm e}^{3 t}}{2}-\frac {c_{2} {\mathrm e}^{-3 t}}{4} \\ \end{align*} |
Problem 15525
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )-y \left (t \right )\\ y^{\prime }\left (t \right )&=y \left (t \right )-x \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} +c_{2} {\mathrm e}^{2 t} \\ y \left (t \right ) &= -c_{2} {\mathrm e}^{2 t}+c_{1} \\ \end{align*} |
Problem 15526
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )+y \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )-3 y \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 0, y \left (0\right ) = 0] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= 0 \\ y \left (t \right ) &= 0 \\ \end{align*} |
Problem 15527
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right )\\ y^{\prime }\left (t \right )&=4 y \left (t \right )-2 x \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 0, y \left (0\right ) = -1] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= -{\mathrm e}^{3 t}+{\mathrm e}^{2 t} \\ y \left (t \right ) &= -2 \,{\mathrm e}^{3 t}+{\mathrm e}^{2 t} \\ \end{align*} |
Problem 15528
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=4 x \left (t \right )-5 y \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 0, y \left (0\right ) = 1] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= -5 \,{\mathrm e}^{2 t} \sin \left (t \right ) \\ y \left (t \right ) &= {\mathrm e}^{2 t} \left (-2 \sin \left (t \right )+\cos \left (t \right )\right ) \\ \end{align*} |
Problem 15529
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right )+y \left (t \right )+z \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )-y \left (t \right )+z \left (t \right )\\ z^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right )-z \left (t \right ) \end {align*}
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program solution |
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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}-2 c_{3} {\mathrm e}^{-2 t}-c_{1} {\mathrm e}^{-2 t} \\ \end{align*} |
Problem 15530
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )-y \left (t \right )+z \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )+2 y \left (t \right )-z \left (t \right )\\ z^{\prime }\left (t \right )&=x \left (t \right )-y \left (t \right )+2 z \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{3 t}+c_{3} {\mathrm e}^{2 t} \\ y \left (t \right ) &= c_{3} {\mathrm e}^{2 t}+c_{1} {\mathrm e}^{t} \\ z \left (t \right ) &= c_{3} {\mathrm e}^{2 t}+c_{2} {\mathrm e}^{3 t}+c_{1} {\mathrm e}^{t} \\ \end{align*} |
Problem 15531
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )-y \left (t \right )+z \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )+z \left (t \right )\\ z^{\prime }\left (t \right )&=y \left (t \right )-2 z \left (t \right )-3 x \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 0, y \left (0\right ) = 0, z \left (0\right ) = 1] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= 1-{\mathrm e}^{-t} \\ y \left (t \right ) &= 1-{\mathrm e}^{-t} \\ z \left (t \right ) &= 2 \,{\mathrm e}^{-t}-1 \\ \end{align*} |
Problem 15532
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=-2 x \left (t \right )+y \left (t \right )-{\mathrm e}^{2 t}\\ y^{\prime }\left (t \right )&=-3 x \left (t \right )+2 y \left (t \right )+6 \,{\mathrm e}^{2 t} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{t}+{\mathrm e}^{-t} c_{1} +2 \,{\mathrm e}^{2 t} \\ y \left (t \right ) &= 3 c_{2} {\mathrm e}^{t}+{\mathrm e}^{-t} c_{1} +9 \,{\mathrm e}^{2 t} \\ \end{align*} |
Problem 15533
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right )-\cos \left (t \right )\\ y^{\prime }\left (t \right )&=-y \left (t \right )-2 x \left (t \right )+\cos \left (t \right )+\sin \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = -2] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= -\sin \left (t \right )+\cos \left (t \right )-\cos \left (t \right ) t \\ y \left (t \right ) &= -2 \cos \left (t \right )+\sin \left (t \right ) t +\cos \left (t \right ) t \\ \end{align*} |
Problem 15534
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=y \left (t \right )+\tan \left (t \right )^{2}-1\\ y^{\prime }\left (t \right )&=\tan \left (t \right )-x \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{2} \sin \left (t \right )+c_{1} \cos \left (t \right )+\tan \left (t \right ) \\ y \left (t \right ) &= c_{2} \cos \left (t \right )-c_{1} \sin \left (t \right )+2 \\ \end{align*} |
Problem 15535
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=-\frac {4 \,{\mathrm e}^{t} x \left (t \right )}{{\mathrm e}^{t}-1}-\frac {2 \,{\mathrm e}^{t} y \left (t \right )}{{\mathrm e}^{t}-1}+\frac {4 x \left (t \right )}{{\mathrm e}^{t}-1}+\frac {2 y \left (t \right )}{{\mathrm e}^{t}-1}+\frac {2}{{\mathrm e}^{t}-1}\\ y^{\prime }\left (t \right )&=\frac {6 \,{\mathrm e}^{t} x \left (t \right )}{{\mathrm e}^{t}-1}+\frac {3 \,{\mathrm e}^{t} y \left (t \right )}{{\mathrm e}^{t}-1}-\frac {6 x \left (t \right )}{{\mathrm e}^{t}-1}-\frac {3 y \left (t \right )}{{\mathrm e}^{t}-1}-\frac {3}{{\mathrm e}^{t}-1} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= 2 \,{\mathrm e}^{-t} \ln \left ({\mathrm e}^{t}-1\right )-{\mathrm e}^{-t} c_{1} +2 \,{\mathrm e}^{-t}+c_{2} \\ y \left (t \right ) &= \frac {6 \,{\mathrm e}^{-t} \ln \left ({\mathrm e}^{t}-1\right )-4 c_{2} {\mathrm e}^{t}-3 \,{\mathrm e}^{-t} c_{1} -6 \ln \left ({\mathrm e}^{t}-1\right )+6 \,{\mathrm e}^{-t}+3 c_{1} +4 c_{2} -6}{2 \,{\mathrm e}^{t}-2} \\ \end{align*} |
Problem 15536
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=y \left (t \right )\\ y^{\prime }\left (t \right )&=-x \left (t \right )+\frac {1}{\cos \left (t \right )} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{2} \sin \left (t \right )+c_{1} \cos \left (t \right )+\sin \left (t \right ) t +\cos \left (t \right ) \ln \left (\cos \left (t \right )\right ) \\ y \left (t \right ) &= c_{2} \cos \left (t \right )-c_{1} \sin \left (t \right )+\cos \left (t \right ) t -\sin \left (t \right ) \ln \left (\cos \left (t \right )\right ) \\ \end{align*} |
Problem 15537
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=y \left (t \right )\\ y^{\prime }\left (t \right )&=1-x \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{2} \sin \left (t \right )+c_{1} \cos \left (t \right )+1 \\ y \left (t \right ) &= c_{2} \cos \left (t \right )-c_{1} \sin \left (t \right ) \\ \end{align*} |
Problem 15538
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=3-2 y \left (t \right )\\ y^{\prime }\left (t \right )&=2 x \left (t \right )-2 t \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{2} \sin \left (2 t \right )+c_{1} \cos \left (2 t \right )+t \\ y \left (t \right ) &= -c_{2} \cos \left (2 t \right )+c_{1} \sin \left (2 t \right )+1 \\ \end{align*} |
Problem 15539
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=-y \left (t \right )+\sin \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )+\cos \left (t \right ) \end {align*}
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program solution |
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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 )+\sin \left (t \right ) \\ \end{align*} |
Problem 15540
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right )+{\mathrm e}^{t}\\ y^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right )-{\mathrm e}^{t} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= \frac {c_{1} {\mathrm e}^{2 t}}{2}+{\mathrm e}^{t}+c_{2} \\ y \left (t \right ) &= \frac {c_{1} {\mathrm e}^{2 t}}{2}-{\mathrm e}^{t}-c_{2} \\ \end{align*} |
Problem 15541
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=4 x \left (t \right )-5 y \left (t \right )+4 t -1\\ y^{\prime }\left (t \right )&=x \left (t \right )-2 y \left (t \right )+t \end {align*} With initial conditions \[ [x \left (0\right ) = 0, y \left (0\right ) = 0] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= -t \\ y \left (t \right ) &= 0 \\ \end{align*} |
Problem 15542
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=y \left (t \right )-x \left (t \right )+{\mathrm e}^{t}\\ y^{\prime }\left (t \right )&=x \left (t \right )-y \left (t \right )+{\mathrm e}^{t} \end {align*} With initial conditions \[ [x \left (0\right ) = 0, y \left (0\right ) = 1] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= -\frac {{\mathrm e}^{-2 t}}{2}+{\mathrm e}^{t}-\frac {1}{2} \\ y \left (t \right ) &= \frac {{\mathrm e}^{-2 t}}{2}+{\mathrm e}^{t}-\frac {1}{2} \\ \end{align*} |
Problem 15543
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=t^{2}-y \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )+t \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{2} \sin \left (t \right )+c_{1} \cos \left (t \right )+t \\ y \left (t \right ) &= t^{2}-c_{2} \cos \left (t \right )+c_{1} \sin \left (t \right )-1 \\ \end{align*} |
Problem 15544
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=\sin \left (t \right )-{\mathrm e}^{-t}-y \left (t \right )\\ y^{\prime }\left (t \right )&=-\sin \left (t \right )+2 \,{\mathrm e}^{-t} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= -\sin \left (t \right )-{\mathrm e}^{-t}-\cos \left (t \right )+c_{1} t +c_{2} \\ y \left (t \right ) &= \cos \left (t \right )-2 \,{\mathrm e}^{-t}-c_{1} \\ \end{align*} |
Problem 15545
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )+y \left (t \right )-2 z \left (t \right )+2-t\\ y^{\prime }\left (t \right )&=1-x \left (t \right )\\ z^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right )-z \left (t \right )+1-t \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} \sin \left (t \right )-c_{2} {\mathrm e}^{t}-c_{3} \cos \left (t \right ) \\ y \left (t \right ) &= t +c_{1} \cos \left (t \right )+c_{2} {\mathrm e}^{t}+c_{3} \sin \left (t \right ) \\ z \left (t \right ) &= 1+c_{1} \sin \left (t \right )-c_{3} \cos \left (t \right ) \\ \end{align*} |
Problem 15546
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right )-2 y \left (t \right )+2 \,{\mathrm e}^{-t}\\ y^{\prime }\left (t \right )&=-y \left (t \right )-z \left (t \right )+1\\ z^{\prime }\left (t \right )&=-z \left (t \right )+1 \end {align*} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 1, z \left (0\right ) = 1] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} \\ y \left (t \right ) &= {\mathrm e}^{-t} \\ z \left (t \right ) &= 1 \\ \end{align*} |
Problem 15547
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=5 x \left (t \right )+4 y \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )+2 y \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{6 t} \\ y \left (t \right ) &= -c_{1} {\mathrm e}^{t}+\frac {c_{2} {\mathrm e}^{6 t}}{4} \\ \end{align*} |
Problem 15548
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=6 x \left (t \right )+y \left (t \right )\\ y^{\prime }\left (t \right )&=4 x \left (t \right )+3 y \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{7 t}+c_{2} {\mathrm e}^{2 t} \\ y \left (t \right ) &= c_{1} {\mathrm e}^{7 t}-4 c_{2} {\mathrm e}^{2 t} \\ \end{align*} |
Problem 15549
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )-4 y \left (t \right )+1\\ y^{\prime }\left (t \right )&=-x \left (t \right )+5 y \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{t}+{\mathrm e}^{6 t} c_{1} -\frac {5}{6} \\ y \left (t \right ) &= \frac {c_{2} {\mathrm e}^{t}}{4}-{\mathrm e}^{6 t} c_{1} -\frac {1}{6} \\ \end{align*} |
Problem 15550
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=3 x \left (t \right )+y \left (t \right )+{\mathrm e}^{t}\\ y^{\prime }\left (t \right )&=x \left (t \right )+3 y \left (t \right )-{\mathrm e}^{t} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= \frac {c_{1} {\mathrm e}^{4 t}}{2}-{\mathrm e}^{t}+c_{2} {\mathrm e}^{2 t} \\ y \left (t \right ) &= \frac {c_{1} {\mathrm e}^{4 t}}{2}+{\mathrm e}^{t}-c_{2} {\mathrm e}^{2 t} \\ \end{align*} |
Problem 15551
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )+4 y \left (t \right )+\cos \left (t \right )\\ y^{\prime }\left (t \right )&=-x \left (t \right )-2 y \left (t \right )+\sin \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= -2 \cos \left (t \right )-3 \sin \left (t \right )+c_{1} t +c_{2} \\ y \left (t \right ) &= 2 \sin \left (t \right )+\frac {c_{1}}{4}-\frac {c_{1} t}{2}-\frac {c_{2}}{2} \\ \end{align*} |
Problem 15552
| ODE |
\[ \boxed {x^{\prime }+3 x={\mathrm e}^{-2 t}} \] With initial conditions \begin {align*} [x \left (0\right ) = 0] \end {align*}
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program solution |
\[ x = {\mathrm e}^{-2 t}-{\mathrm e}^{-3 t} \] Verified OK.
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Maple solution | \[ x \left (t \right ) = {\mathrm e}^{-2 t}-{\mathrm e}^{-3 t} \] |
Problem 15553
| ODE |
\[ \boxed {x^{\prime }-3 x=3 t^{3}+3 t^{2}+2 t +1} \] With initial conditions \begin {align*} [x \left (0\right ) = -1] \end {align*}
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program solution |
\[ x = -\left (t +1\right ) \left (t^{2}+t +1\right ) \] Verified OK.
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Maple solution | \[ x \left (t \right ) = -\left (t +1\right ) \left (t^{2}+t +1\right ) \] |
Problem 15554
| ODE |
\[ \boxed {x^{\prime }-x=\cos \left (t \right )-\sin \left (t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0] \end {align*}
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program solution |
\[ x = \sin \left (t \right ) \] Verified OK.
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Maple solution | \[ x \left (t \right ) = \sin \left (t \right ) \] |
Problem 15555
| ODE |
\[ \boxed {2 x^{\prime }+6 x={\mathrm e}^{-3 t} t} \] With initial conditions \begin {align*} \left [x \left (0\right ) = -{\frac {1}{2}}\right ] \end {align*}
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program solution |
\[ x = \frac {{\mathrm e}^{-3 t} \left (t^{2}-2\right )}{4} \] Verified OK.
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Maple solution | \[ x \left (t \right ) = \frac {{\mathrm e}^{-3 t} \left (t^{2}-2\right )}{4} \] |
Problem 15556
| ODE |
\[ \boxed {x^{\prime }+x=2 \sin \left (t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0] \end {align*}
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program solution |
\[ x = \sin \left (t \right )-\cos \left (t \right )+{\mathrm e}^{-t} \] Verified OK.
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Maple solution | \[ x \left (t \right ) = -\cos \left (t \right )+\sin \left (t \right )+{\mathrm e}^{-t} \] |
Problem 15557
| ODE |
\[ \boxed {x^{\prime \prime }=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ x = 0 \] Verified OK.
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Maple solution | \[ x \left (t \right ) = 0 \] |
Problem 15558
| ODE |
\[ \boxed {x^{\prime \prime }=1} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ x = \frac {t^{2}}{2} \] Verified OK.
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Maple solution | \[ x \left (t \right ) = \frac {t^{2}}{2} \] |
Problem 15559
| ODE |
\[ \boxed {x^{\prime \prime }=\cos \left (t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ x = 1-\cos \left (t \right ) \] Verified OK.
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Maple solution | \[ x \left (t \right ) = 1-\cos \left (t \right ) \] |
Problem 15560
| ODE |
\[ \boxed {x^{\prime \prime }+x^{\prime }=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ x = 0 \] Verified OK.
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Maple solution | \[ x \left (t \right ) = 0 \] |
Problem 15561
| ODE |
\[ \boxed {x^{\prime \prime }+x^{\prime }=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 1, x^{\prime }\left (0\right ) = -1] \end {align*}
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program solution |
\[ x = {\mathrm e}^{-t} \] Verified OK.
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Maple solution | \[ x \left (t \right ) = {\mathrm e}^{-t} \] |
Problem 15562
| ODE |
\[ \boxed {x^{\prime \prime }-x^{\prime }=1} \] With initial conditions \begin {align*} [x \left (0\right ) = -1, x^{\prime }\left (0\right ) = -1] \end {align*}
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program solution |
\[ x = -t -1 \] Verified OK.
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Maple solution | \[ x \left (t \right ) = -t -1 \] |
Problem 15563
| ODE |
\[ \boxed {x^{\prime \prime }+x=t} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 1] \end {align*}
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program solution |
\[ x = t \] Verified OK.
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Maple solution | \[ x \left (t \right ) = t \] |
Problem 15564
| ODE |
\[ \boxed {x^{\prime \prime }+6 x^{\prime }=12 t +2} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ x = t^{2} \] Verified OK.
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Maple solution | \[ x = t^{2} \] |
Problem 15565
| ODE |
\[ \boxed {x^{\prime \prime }-2 x^{\prime }+2 x=2} \] With initial conditions \begin {align*} [x \left (0\right ) = 1, x^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ x = 1 \] Verified OK.
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Maple solution | \[ x = 1 \] |
Problem 15566
| ODE |
\[ \boxed {x^{\prime \prime }+4 x^{\prime }+4 x=4} \] With initial conditions \begin {align*} [x \left (0\right ) = 1, x^{\prime }\left (0\right ) = -4] \end {align*}
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program solution |
\[ x = -4 t \,{\mathrm e}^{-2 t}+1 \] Verified OK.
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Maple solution | \[ x = 1-4 t \,{\mathrm e}^{-2 t} \] |
Problem 15567
| ODE |
\[ \boxed {2 x^{\prime \prime }-2 x^{\prime }={\mathrm e}^{t} \left (t +1\right )} \] With initial conditions \begin {align*} \left [x \left (0\right ) = {\frac {1}{2}}, x^{\prime }\left (0\right ) = {\frac {1}{2}}\right ] \end {align*}
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program solution |
\[ x = \frac {{\mathrm e}^{t} \left (t^{2}+2\right )}{4} \] Verified OK.
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Maple solution | \[ x = \frac {{\mathrm e}^{t} \left (t^{2}+2\right )}{4} \] |
Problem 15568
| ODE |
\[ \boxed {x^{\prime \prime }+x=2 \cos \left (t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = -1, x^{\prime }\left (0\right ) = 1] \end {align*}
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program solution |
\[ x = -\cos \left (t \right )+\left (t +1\right ) \sin \left (t \right ) \] Verified OK.
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Maple solution | \[ x = -\cos \left (t \right )+\sin \left (t \right ) \left (1+t \right ) \] |