Problem 1601
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=-2 y_{1} \left (t \right )+2 y_{2} \left (t \right )-6 y_{3} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=2 y_{1} \left (t \right )+6 y_{2} \left (t \right )+2 y_{3} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=-2 y_{1} \left (t \right )-2 y_{2} \left (t \right )+2 y_{3} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= c_{1} {\mathrm e}^{4 t}+c_{2} {\mathrm e}^{-4 t}+c_{3} {\mathrm e}^{6 t} \\ y_{2} \left (t \right ) &= -\frac {c_{2} {\mathrm e}^{-4 t}}{4}+c_{3} {\mathrm e}^{6 t} \\ y_{3} \left (t \right ) &= -c_{1} {\mathrm e}^{4 t}+\frac {c_{2} {\mathrm e}^{-4 t}}{4}-c_{3} {\mathrm e}^{6 t} \\ \end{align*} |
Problem 1602
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=3 y_{1} \left (t \right )+2 y_{2} \left (t \right )-2 y_{3} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=-2 y_{1} \left (t \right )+7 y_{2} \left (t \right )-2 y_{3} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=-10 y_{1} \left (t \right )+10 y_{2} \left (t \right )-5 y_{3} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= c_{2} {\mathrm e}^{5 t}+c_{3} {\mathrm e}^{-5 t} \\ y_{2} \left (t \right ) &= c_{2} {\mathrm e}^{5 t}+c_{3} {\mathrm e}^{-5 t}+{\mathrm e}^{5 t} c_{1} \\ y_{3} \left (t \right ) &= 5 c_{3} {\mathrm e}^{-5 t}+{\mathrm e}^{5 t} c_{1} \\ \end{align*} |
Problem 1603
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=3 y_{1} \left (t \right )+y_{2} \left (t \right )-y_{3} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=3 y_{1} \left (t \right )+5 y_{2} \left (t \right )+y_{3} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=-6 y_{1} \left (t \right )+2 y_{2} \left (t \right )+4 y_{3} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= c_{2} +c_{3} {\mathrm e}^{6 t} \\ y_{2} \left (t \right ) &= -c_{2} -c_{3} {\mathrm e}^{6 t}+{\mathrm e}^{6 t} c_{1} \\ y_{3} \left (t \right ) &= -4 c_{3} {\mathrm e}^{6 t}+2 c_{2} +{\mathrm e}^{6 t} c_{1} \\ \end{align*} |
Problem 1604
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=3 y_{1} \left (t \right )+4 y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=-y_{1} \left (t \right )+7 y_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{5 t} \left (c_{2} t +c_{1} \right ) \\ y_{2} \left (t \right ) &= \frac {{\mathrm e}^{5 t} \left (2 c_{2} t +2 c_{1} +c_{2} \right )}{4} \\ \end{align*} |
Problem 1605
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=-y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=y_{1} \left (t \right )-2 y_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{-t} \left (c_{2} t +c_{1} \right ) \\ y_{2} \left (t \right ) &= {\mathrm e}^{-t} \left (c_{2} t +c_{1} -c_{2} \right ) \\ \end{align*} |
Problem 1606
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=-7 y_{1} \left (t \right )+4 y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=-y_{1} \left (t \right )-11 y_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{-9 t} \left (c_{2} t +c_{1} \right ) \\ y_{2} \left (t \right ) &= -\frac {{\mathrm e}^{-9 t} \left (2 c_{2} t +2 c_{1} -c_{2} \right )}{4} \\ \end{align*} |
Problem 1607
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=3 y_{1} \left (t \right )+y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=-y_{1} \left (t \right )+y_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{2 t} \left (c_{2} t +c_{1} \right ) \\ y_{2} \left (t \right ) &= -{\mathrm e}^{2 t} \left (c_{2} t +c_{1} -c_{2} \right ) \\ \end{align*} |
Problem 1608
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=4 y_{1} \left (t \right )+12 y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=-3 y_{1} \left (t \right )-8 y_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{-2 t} \left (c_{2} t +c_{1} \right ) \\ y_{2} \left (t \right ) &= -\frac {{\mathrm e}^{-2 t} \left (6 c_{2} t +6 c_{1} -c_{2} \right )}{12} \\ \end{align*} |
Problem 1609
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=-10 y_{1} \left (t \right )+9 y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=-4 y_{1} \left (t \right )+2 y_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{-4 t} \left (c_{2} t +c_{1} \right ) \\ y_{2} \left (t \right ) &= \frac {{\mathrm e}^{-4 t} \left (6 c_{2} t +6 c_{1} +c_{2} \right )}{9} \\ \end{align*} |
Problem 1610
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=-13 y_{1} \left (t \right )+16 y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=-9 y_{1} \left (t \right )+11 y_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{-t} \left (c_{2} t +c_{1} \right ) \\ y_{2} \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (12 c_{2} t +12 c_{1} +c_{2} \right )}{16} \\ \end{align*} |
Problem 1611
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=2 y_{2} \left (t \right )+y_{3} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=-4 y_{1} \left (t \right )+6 y_{2} \left (t \right )+y_{3} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=4 y_{2} \left (t \right )+2 y_{3} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= \frac {c_{2} {\mathrm e}^{4 t}}{2}+\frac {c_{3} {\mathrm e}^{4 t} t}{2}-\frac {c_{1}}{2} \\ y_{2} \left (t \right ) &= \frac {c_{2} {\mathrm e}^{4 t}}{2}+\frac {c_{3} {\mathrm e}^{4 t} t}{2}+\frac {c_{3} {\mathrm e}^{4 t}}{4}-\frac {c_{1}}{2} \\ y_{3} \left (t \right ) &= c_{1} +c_{2} {\mathrm e}^{4 t}+c_{3} {\mathrm e}^{4 t} t \\ \end{align*} |
Problem 1612
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=\frac {y_{1} \left (t \right )}{3}+\frac {y_{2} \left (t \right )}{3}-y_{3} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=-\frac {4 y_{1} \left (t \right )}{3}-\frac {4 y_{2} \left (t \right )}{3}+y_{3} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=-\frac {2 y_{1} \left (t \right )}{3}+\frac {y_{2} \left (t \right )}{3} \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= -c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-t}+c_{3} {\mathrm e}^{-t} t \\ y_{2} \left (t \right ) &= c_{1} {\mathrm e}^{t}-c_{2} {\mathrm e}^{-t}-c_{3} {\mathrm e}^{-t} t +3 c_{3} {\mathrm e}^{-t} \\ y_{3} \left (t \right ) &= c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-t}+c_{3} {\mathrm e}^{-t} t \\ \end{align*} |
Problem 1613
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=-y_{1} \left (t \right )+y_{2} \left (t \right )-y_{3} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=-2 y_{1} \left (t \right )+2 y_{3} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=-y_{1} \left (t \right )+3 y_{2} \left (t \right )-y_{3} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= \left (2 c_{2} t +c_{1} \right ) {\mathrm e}^{-2 t} \\ y_{2} \left (t \right ) &= c_{2} {\mathrm e}^{-2 t}+c_{3} {\mathrm e}^{2 t} \\ y_{3} \left (t \right ) &= 2 c_{2} {\mathrm e}^{-2 t} t +c_{3} {\mathrm e}^{2 t}+c_{1} {\mathrm e}^{-2 t}-c_{2} {\mathrm e}^{-2 t} \\ \end{align*} |
Problem 1614
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=4 y_{1} \left (t \right )-2 y_{2} \left (t \right )-2 y_{3} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=-2 y_{1} \left (t \right )+3 y_{2} \left (t \right )-y_{3} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=2 y_{1} \left (t \right )-y_{2} \left (t \right )+3 y_{3} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= c_{2} {\mathrm e}^{4 t}+c_{3} {\mathrm e}^{2 t} \\ y_{2} \left (t \right ) &= -2 c_{2} {\mathrm e}^{4 t} t +c_{1} {\mathrm e}^{4 t}+\frac {3 c_{3} {\mathrm e}^{2 t}}{2} \\ y_{3} \left (t \right ) &= 2 c_{2} {\mathrm e}^{4 t} t -c_{1} {\mathrm e}^{4 t}-\frac {c_{3} {\mathrm e}^{2 t}}{2} \\ \end{align*} |
Problem 1615
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=6 y_{1} \left (t \right )-5 y_{2} \left (t \right )+3 y_{3} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=2 y_{1} \left (t \right )-y_{2} \left (t \right )+3 y_{3} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=2 y_{1} \left (t \right )+y_{2} \left (t \right )+y_{3} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= c_{1} {\mathrm e}^{-2 t}+c_{2} {\mathrm e}^{4 t}+c_{3} {\mathrm e}^{4 t} t \\ y_{2} \left (t \right ) &= c_{1} {\mathrm e}^{-2 t}+c_{2} {\mathrm e}^{4 t}+c_{3} {\mathrm e}^{4 t} t -\frac {c_{3} {\mathrm e}^{4 t}}{2} \\ y_{3} \left (t \right ) &= -c_{1} {\mathrm e}^{-2 t}+c_{2} {\mathrm e}^{4 t}+c_{3} {\mathrm e}^{4 t} t -\frac {c_{3} {\mathrm e}^{4 t}}{2} \\ \end{align*} |
Problem 1616
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=-11 y_{1} \left (t \right )+8 y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=-2 y_{1} \left (t \right )-3 y_{2} \left (t \right ) \end {align*} With initial conditions \[ [y_{1} \left (0\right ) = 6, y_{2} \left (0\right ) = 2] \] |
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{-7 t} \left (-8 t +6\right ) \\ y_{2} \left (t \right ) &= \frac {{\mathrm e}^{-7 t} \left (-32 t +16\right )}{8} \\ \end{align*} |
Problem 1617
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=15 y_{1} \left (t \right )-9 y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=16 y_{1} \left (t \right )-9 y_{2} \left (t \right ) \end {align*} With initial conditions \[ [y_{1} \left (0\right ) = 5, y_{2} \left (0\right ) = 8] \] |
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{3 t} \left (-12 t +5\right ) \\ y_{2} \left (t \right ) &= \frac {{\mathrm e}^{3 t} \left (-144 t +72\right )}{9} \\ \end{align*} |
Problem 1618
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=-3 y_{1} \left (t \right )-4 y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=y_{1} \left (t \right )-7 y_{2} \left (t \right ) \end {align*} With initial conditions \[ [y_{1} \left (0\right ) = 2, y_{2} \left (0\right ) = 3] \] |
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{-5 t} \left (-8 t +2\right ) \\ y_{2} \left (t \right ) &= \frac {{\mathrm e}^{-5 t} \left (-16 t +12\right )}{4} \\ \end{align*} |
Problem 1619
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=-7 y_{1} \left (t \right )+24 y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=-6 y_{1} \left (t \right )+17 y_{2} \left (t \right ) \end {align*} With initial conditions \[ [y_{1} \left (0\right ) = 3, y_{2} \left (0\right ) = 1] \] |
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{5 t} \left (-12 t +3\right ) \\ y_{2} \left (t \right ) &= \frac {{\mathrm e}^{5 t} \left (-144 t +24\right )}{24} \\ \end{align*} |
Problem 1620
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=-7 y_{1} \left (t \right )+3 y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=-3 y_{1} \left (t \right )-y_{2} \left (t \right ) \end {align*} With initial conditions \[ [y_{1} \left (0\right ) = 0, y_{2} \left (0\right ) = 2] \] |
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= 6 \,{\mathrm e}^{-4 t} t \\ y_{2} \left (t \right ) &= \frac {{\mathrm e}^{-4 t} \left (18 t +6\right )}{3} \\ \end{align*} |
Problem 1621
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=-y_{1} \left (t \right )+y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=y_{1} \left (t \right )-y_{2} \left (t \right )-2 y_{3} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=-y_{1} \left (t \right )-y_{2} \left (t \right )-y_{3} \left (t \right ) \end {align*} With initial conditions \[ [y_{1} \left (0\right ) = 6, y_{2} \left (0\right ) = 5, y_{3} \left (0\right ) = -7] \] |
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= 4 \,{\mathrm e}^{t}+2 \,{\mathrm e}^{-2 t}-{\mathrm e}^{-2 t} t \\ y_{2} \left (t \right ) &= 8 \,{\mathrm e}^{t}-3 \,{\mathrm e}^{-2 t}+{\mathrm e}^{-2 t} t \\ y_{3} \left (t \right ) &= -6 \,{\mathrm e}^{t}-{\mathrm e}^{-2 t} \\ \end{align*} |
Problem 1622
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=-2 y_{1} \left (t \right )+2 y_{2} \left (t \right )+y_{3} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=-2 y_{1} \left (t \right )+2 y_{2} \left (t \right )+y_{3} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=-3 y_{1} \left (t \right )+3 y_{2} \left (t \right )+2 y_{3} \left (t \right ) \end {align*} With initial conditions \[ [y_{1} \left (0\right ) = -6, y_{2} \left (0\right ) = -2, y_{3} \left (0\right ) = 0] \] |
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= -9+2 t +3 \,{\mathrm e}^{2 t} \\ y_{2} \left (t \right ) &= 3 \,{\mathrm e}^{2 t}-5+2 t \\ y_{3} \left (t \right ) &= -6+6 \,{\mathrm e}^{2 t} \\ \end{align*} |
Problem 1623
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=-7 y_{1} \left (t \right )-4 y_{2} \left (t \right )+4 y_{3} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=y_{1} \left (t \right )+y_{3} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=-9 y_{1} \left (t \right )-5 y_{2} \left (t \right )+6 y_{3} \left (t \right ) \end {align*} With initial conditions \[ [y_{1} \left (0\right ) = -6, y_{2} \left (0\right ) = 9, y_{3} \left (0\right ) = -1] \] |
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= -\frac {4 \,{\mathrm e}^{-t} \left (\frac {13 \sin \left (2 t \right )}{2}+\frac {39 \cos \left (2 t \right )}{2}\right )}{13} \\ y_{2} \left (t \right ) &= \frac {9 \,{\mathrm e}^{t}}{2}-\frac {7 \,{\mathrm e}^{-t} \sin \left (2 t \right )}{2}+\frac {9 \,{\mathrm e}^{-t} \cos \left (2 t \right )}{2} \\ y_{3} \left (t \right ) &= \frac {9 \,{\mathrm e}^{t}}{2}-\frac {7 \,{\mathrm e}^{-t} \sin \left (2 t \right )}{2}-\frac {11 \,{\mathrm e}^{-t} \cos \left (2 t \right )}{2} \\ \end{align*} |
Problem 1624
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=-y_{1} \left (t \right )-4 y_{2} \left (t \right )-y_{3} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=3 y_{1} \left (t \right )+6 y_{2} \left (t \right )+y_{3} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=-3 y_{1} \left (t \right )-2 y_{2} \left (t \right )+3 y_{3} \left (t \right ) \end {align*} With initial conditions \[ [y_{1} \left (0\right ) = -2, y_{2} \left (0\right ) = 1, y_{3} \left (0\right ) = 3] \] |
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= -2 \,{\mathrm e}^{4 t}+3 \,{\mathrm e}^{2 t} t \\ y_{2} \left (t \right ) &= 2 \,{\mathrm e}^{4 t}-3 \,{\mathrm e}^{2 t} t -{\mathrm e}^{2 t} \\ y_{3} \left (t \right ) &= 2 \,{\mathrm e}^{4 t}+3 \,{\mathrm e}^{2 t} t +{\mathrm e}^{2 t} \\ \end{align*} |
Problem 1625
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=4 y_{1} \left (t \right )-8 y_{2} \left (t \right )-4 y_{3} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=-3 y_{1} \left (t \right )-y_{2} \left (t \right )-4 y_{3} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=y_{1} \left (t \right )-y_{2} \left (t \right )+9 y_{3} \left (t \right ) \end {align*} With initial conditions \[ [y_{1} \left (0\right ) = -4, y_{2} \left (0\right ) = 1, y_{3} \left (0\right ) = -3] \] |
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= -\frac {50 \,{\mathrm e}^{7 t}}{11}-\frac {164 \,{\mathrm e}^{-4 t}}{143}+\frac {22 \,{\mathrm e}^{9 t}}{13} \\ y_{2} \left (t \right ) &= \frac {5 \,{\mathrm e}^{7 t}}{11}-\frac {164 \,{\mathrm e}^{-4 t}}{143}+\frac {22 \,{\mathrm e}^{9 t}}{13} \\ y_{3} \left (t \right ) &= \frac {5 \,{\mathrm e}^{7 t}}{2}-\frac {11 \,{\mathrm e}^{9 t}}{2} \\ \end{align*} |
Problem 1626
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=-5 y_{1} \left (t \right )-y_{2} \left (t \right )+11 y_{3} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=-7 y_{1} \left (t \right )+y_{2} \left (t \right )+13 y_{3} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=-4 y_{1} \left (t \right )+8 y_{3} \left (t \right ) \end {align*} With initial conditions \[ [y_{1} \left (0\right ) = 0, y_{2} \left (0\right ) = 2, y_{3} \left (0\right ) = 2] \] |
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= -3+3 \,{\mathrm e}^{4 t}+8 t \\ y_{2} \left (t \right ) &= 6 \,{\mathrm e}^{4 t}-4+4 t \\ y_{3} \left (t \right ) &= -1+4 t +3 \,{\mathrm e}^{4 t} \\ \end{align*} |
Problem 1627
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=5 y_{1} \left (t \right )-y_{2} \left (t \right )+y_{3} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=-y_{1} \left (t \right )+9 y_{2} \left (t \right )-3 y_{3} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=-2 y_{1} \left (t \right )+2 y_{2} \left (t \right )+4 y_{3} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{6 t} \left (c_{3} t +c_{2} \right ) \\ y_{2} \left (t \right ) &= \left (-2 c_{3} t^{2}-4 c_{2} t -3 c_{3} t +c_{1} \right ) {\mathrm e}^{6 t} \\ y_{3} \left (t \right ) &= {\mathrm e}^{6 t} \left (-2 c_{3} t^{2}-4 c_{2} t -2 c_{3} t +c_{1} +c_{2} +c_{3} \right ) \\ \end{align*} |
Problem 1628
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=y_{1} \left (t \right )+10 y_{2} \left (t \right )-12 y_{3} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=2 y_{1} \left (t \right )+2 y_{2} \left (t \right )+3 y_{3} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=2 y_{1} \left (t \right )-y_{2} \left (t \right )+6 y_{3} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{3 t} \left (c_{3} t^{2}+c_{2} t +c_{1} \right ) \\ y_{2} \left (t \right ) &= -\frac {{\mathrm e}^{3 t} \left (6 c_{3} t^{2}+6 c_{2} t +6 c_{3} t +6 c_{1} +3 c_{2} +4 c_{3} \right )}{6} \\ y_{3} \left (t \right ) &= -\frac {{\mathrm e}^{3 t} \left (18 c_{3} t^{2}+18 c_{2} t +18 c_{3} t +18 c_{1} +9 c_{2} +10 c_{3} \right )}{18} \\ \end{align*} |
Problem 1629
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=-6 y_{1} \left (t \right )-4 y_{2} \left (t \right )-4 y_{3} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=2 y_{1} \left (t \right )-y_{2} \left (t \right )+y_{3} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=2 y_{1} \left (t \right )+3 y_{2} \left (t \right )+y_{3} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{-2 t} \left (c_{3} t +c_{2} \right ) \\ y_{2} \left (t \right ) &= \frac {\left (2 c_{3} t^{2}+4 c_{2} t -c_{3} t +4 c_{1} \right ) {\mathrm e}^{-2 t}}{4} \\ y_{3} \left (t \right ) &= -\frac {{\mathrm e}^{-2 t} \left (2 c_{3} t^{2}+4 c_{2} t +3 c_{3} t +4 c_{1} +4 c_{2} +c_{3} \right )}{4} \\ \end{align*} |
Problem 1630
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=2 y_{2} \left (t \right )-2 y_{3} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=-y_{1} \left (t \right )+5 y_{2} \left (t \right )-3 y_{3} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=y_{1} \left (t \right )+y_{2} \left (t \right )+y_{3} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{2 t} \left (c_{3} t +c_{2} \right ) \\ y_{2} \left (t \right ) &= \frac {\left (2 c_{3} t^{2}+4 c_{2} t +3 c_{3} t +2 c_{1} \right ) {\mathrm e}^{2 t}}{2} \\ y_{3} \left (t \right ) &= \frac {{\mathrm e}^{2 t} \left (2 c_{3} t^{2}+4 c_{2} t +c_{3} t +2 c_{1} -2 c_{2} -c_{3} \right )}{2} \\ \end{align*} |
Problem 1631
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=-2 y_{1} \left (t \right )-12 y_{2} \left (t \right )+10 y_{3} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=2 y_{1} \left (t \right )-24 y_{2} \left (t \right )+11 y_{3} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=2 y_{1} \left (t \right )-24 y_{2} \left (t \right )+8 y_{3} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{-6 t} \left (c_{3} t^{2}+c_{2} t +c_{1} \right ) \\ y_{2} \left (t \right ) &= -\frac {{\mathrm e}^{-6 t} \left (18 c_{3} t^{2}+18 c_{2} t -24 c_{3} t +18 c_{1} -12 c_{2} +5 c_{3} \right )}{36} \\ y_{3} \left (t \right ) &= -\frac {{\mathrm e}^{-6 t} \left (6 c_{3} t^{2}+6 c_{2} t -6 c_{3} t +6 c_{1} -3 c_{2} +c_{3} \right )}{6} \\ \end{align*} |
Problem 1632
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=-y_{1} \left (t \right )-12 y_{2} \left (t \right )+8 y_{3} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=y_{1} \left (t \right )-9 y_{2} \left (t \right )+4 y_{3} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=y_{1} \left (t \right )-6 y_{2} \left (t \right )+y_{3} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{-3 t} \left (c_{3} t +c_{2} \right ) \\ y_{2} \left (t \right ) &= \frac {{\mathrm e}^{-3 t} \left (c_{3} t +2 c_{1} +c_{2} \right )}{2} \\ y_{3} \left (t \right ) &= \frac {{\mathrm e}^{-3 t} \left (4 c_{3} t +12 c_{1} +4 c_{2} +c_{3} \right )}{8} \\ \end{align*} |
Problem 1633
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=-4 y_{1} \left (t \right )-y_{3} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=-y_{1} \left (t \right )-3 y_{2} \left (t \right )-y_{3} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=y_{1} \left (t \right )-2 y_{3} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{-3 t} \left (c_{3} t +c_{2} \right ) \\ y_{2} \left (t \right ) &= {\mathrm e}^{-3 t} \left (c_{3} t +c_{1} +c_{2} \right ) \\ y_{3} \left (t \right ) &= -{\mathrm e}^{-3 t} \left (c_{3} t +c_{2} +c_{3} \right ) \\ \end{align*} |
Problem 1634
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=-3 y_{1} \left (t \right )-3 y_{2} \left (t \right )+4 y_{3} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=4 y_{1} \left (t \right )+5 y_{2} \left (t \right )-8 y_{3} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=2 y_{1} \left (t \right )+3 y_{2} \left (t \right )-5 y_{3} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{-t} \left (c_{3} t +c_{2} \right ) \\ y_{2} \left (t \right ) &= {\mathrm e}^{-t} \left (-2 c_{3} t +c_{1} -2 c_{2} \right ) \\ y_{3} \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (-4 c_{3} t +3 c_{1} -4 c_{2} +c_{3} \right )}{4} \\ \end{align*} |
Problem 1635
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=-3 y_{1} \left (t \right )-y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=y_{1} \left (t \right )-y_{2} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=-y_{1} \left (t \right )-y_{2} \left (t \right )-2 y_{3} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{-2 t} \left (c_{3} t +c_{2} \right ) \\ y_{2} \left (t \right ) &= -{\mathrm e}^{-2 t} \left (c_{3} t +c_{2} +c_{3} \right ) \\ y_{3} \left (t \right ) &= {\mathrm e}^{-2 t} \left (c_{3} t +c_{1} +c_{2} \right ) \\ \end{align*} |
Problem 1636
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=-y_{1} \left (t \right )+2 y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=-5 y_{1} \left (t \right )+5 y_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{2 t} \left (c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right )\right ) \\ y_{2} \left (t \right ) &= \frac {{\mathrm e}^{2 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 )}{2} \\ \end{align*} |
Problem 1637
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=-11 y_{1} \left (t \right )+4 y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=-26 y_{1} \left (t \right )+9 y_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{-t} \left (c_{1} \sin \left (2 t \right )+c_{2} \cos \left (2 t \right )\right ) \\ y_{2} \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (5 c_{1} \sin \left (2 t \right )-c_{2} \sin \left (2 t \right )+c_{1} \cos \left (2 t \right )+5 c_{2} \cos \left (2 t \right )\right )}{2} \\ \end{align*} |
Problem 1638
|
ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=y_{1} \left (t \right )+2 y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=-4 y_{1} \left (t \right )+5 y_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{3 t} \left (c_{1} \sin \left (2 t \right )+c_{2} \cos \left (2 t \right )\right ) \\ y_{2} \left (t \right ) &= {\mathrm e}^{3 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 ) \\ \end{align*} |
Problem 1639
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=5 y_{1} \left (t \right )-6 y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=3 y_{1} \left (t \right )-y_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{2 t} \left (c_{1} \sin \left (3 t \right )+c_{2} \cos \left (3 t \right )\right ) \\ y_{2} \left (t \right ) &= \frac {{\mathrm e}^{2 t} \left (c_{1} \sin \left (3 t \right )+c_{2} \sin \left (3 t \right )-c_{1} \cos \left (3 t \right )+c_{2} \cos \left (3 t \right )\right )}{2} \\ \end{align*} |
Problem 1640
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=-3 y_{1} \left (t \right )-3 y_{2} \left (t \right )+y_{3} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=2 y_{2} \left (t \right )+2 y_{3} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=5 y_{1} \left (t \right )+y_{2} \left (t \right )+y_{3} \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} \\ y_{2} \left (t \right ) &= {\mathrm e}^{\frac {\left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}}{6}+7\right ) t}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \sin \left (\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t \sqrt {3}\, 36^{\frac {1}{3}}}{36 \left (90+\sqrt {6042}\right )^{\frac {1}{3}}}\right ) c_{2} +{\mathrm e}^{\frac {\left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}}{6}+7\right ) t}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \cos \left (\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t \sqrt {3}\, 36^{\frac {1}{3}}}{36 \left (90+\sqrt {6042}\right )^{\frac {1}{3}}}\right ) c_{3} +c_{1} {\mathrm e}^{-\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{3 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \\ y_{3} \left (t \right ) &= \frac {-2 c_{1} \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) {\mathrm e}^{-\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{3 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}}+c_{2} \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) {\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )+c_{2} {\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \left (\sqrt {3}\, \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42 \sqrt {3}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )+c_{3} \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) {\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-c_{3} {\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \left (\sqrt {3}\, \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42 \sqrt {3}\right ) \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-12 c_{1} {\mathrm e}^{-\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{3 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-12 c_{2} {\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-12 c_{3} {\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{12 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}} \\ \end{align*} |
Problem 1641
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=-3 y_{1} \left (t \right )+3 y_{2} \left (t \right )+y_{3} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=y_{1} \left (t \right )-5 y_{2} \left (t \right )-3 y_{3} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=-3 y_{1} \left (t \right )+7 y_{2} \left (t \right )+3 y_{3} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{-t} c_{1} +c_{2} {\mathrm e}^{-2 t} \sin \left (2 t \right )+c_{3} {\mathrm e}^{-2 t} \cos \left (2 t \right ) \\ y_{2} \left (t \right ) &= {\mathrm e}^{-t} c_{1} +c_{2} {\mathrm e}^{-2 t} \cos \left (2 t \right )-c_{3} {\mathrm e}^{-2 t} \sin \left (2 t \right ) \\ y_{3} \left (t \right ) &= -{\mathrm e}^{-t} c_{1} +c_{2} {\mathrm e}^{-2 t} \sin \left (2 t \right )-c_{2} {\mathrm e}^{-2 t} \cos \left (2 t \right )+c_{3} {\mathrm e}^{-2 t} \cos \left (2 t \right )+c_{3} {\mathrm e}^{-2 t} \sin \left (2 t \right ) \\ \end{align*} |
Problem 1642
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=2 y_{1} \left (t \right )+y_{2} \left (t \right )-y_{3} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=y_{2} \left (t \right )+y_{3} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=y_{1} \left (t \right )+y_{3} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= c_{1} {\mathrm e}^{2 t}+c_{2} {\mathrm e}^{t} \cos \left (t \right )-c_{3} {\mathrm e}^{t} \sin \left (t \right ) \\ y_{2} \left (t \right ) &= c_{1} {\mathrm e}^{2 t}-c_{2} {\mathrm e}^{t} \cos \left (t \right )+c_{3} {\mathrm e}^{t} \sin \left (t \right ) \\ y_{3} \left (t \right ) &= c_{1} {\mathrm e}^{2 t}+c_{2} {\mathrm e}^{t} \sin \left (t \right )+c_{3} {\mathrm e}^{t} \cos \left (t \right ) \\ \end{align*} |
Problem 1643
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ODE |
\begin {align*} y_{1}^{\prime }\left (t \right )&=-3 y_{1} \left (t \right )+y_{2} \left (t \right )-3 y_{3} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=4 y_{1} \left (t \right )-y_{2} \left (t \right )+2 y_{3} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=4 y_{1} \left (t \right )-2 y_{2} \left (t \right )+3 y_{3} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (t \right ) &= c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-t} \sin \left (2 t \right )+c_{3} {\mathrm e}^{-t} \cos \left (2 t \right ) \\ y_{2} \left (t \right ) &= c_{1} {\mathrm e}^{t}-c_{2} {\mathrm e}^{-t} \sin \left (2 t \right )-c_{2} {\mathrm e}^{-t} \cos \left (2 t \right )-c_{3} {\mathrm e}^{-t} \cos \left (2 t \right )+c_{3} {\mathrm e}^{-t} \sin \left (2 t \right ) \\ y_{3} \left (t \right ) &= -c_{1} {\mathrm e}^{t}-c_{2} {\mathrm e}^{-t} \sin \left (2 t \right )-c_{2} {\mathrm e}^{-t} \cos \left (2 t \right )-c_{3} {\mathrm e}^{-t} \cos \left (2 t \right )+c_{3} {\mathrm e}^{-t} \sin \left (2 t \right ) \\ \end{align*} |
Problem 1644
| ODE |
\[ \boxed {y^{\prime }+\sin \left (t \right ) y=0} \] With initial conditions \begin {align*} \left [y \left (0\right ) = {\frac {3}{2}}\right ] \end {align*}
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program solution |
\[ y = \frac {3 \,{\mathrm e}^{-1+\cos \left (t \right )}}{2} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {3 \,{\mathrm e}^{\cos \left (t \right )-1}}{2} \] |
Problem 1645
| ODE |
\[ \boxed {y^{\prime }+y \,{\mathrm e}^{t^{2}}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 2] \end {align*}
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program solution |
\[ y = 2 \,{\mathrm e}^{\frac {\sqrt {\pi }\, \left (\operatorname {erfi}\left (1\right )-\operatorname {erfi}\left (t \right )\right )}{2}} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = 2 \,{\mathrm e}^{\frac {\left (\operatorname {erfi}\left (1\right )-\operatorname {erfi}\left (t \right )\right ) \sqrt {\pi }}{2}} \] |
Problem 1646
| ODE |
\[ \boxed {y^{\prime }-2 y t=t} \] |
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program solution |
\[ y = -\frac {1}{2}+c_{1} {\mathrm e}^{t^{2}} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = -\frac {1}{2}+{\mathrm e}^{t^{2}} c_{1} \] |
Problem 1647
| ODE |
\[ \boxed {y^{\prime }+2 y t=t} \] With initial conditions \begin {align*} [y \left (1\right ) = 2] \end {align*}
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program solution |
\[ y = \frac {1}{2}+\frac {3 \,{\mathrm e}^{-\left (-1+t \right ) \left (t +1\right )}}{2} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {1}{2}+\frac {3 \,{\mathrm e}^{-\left (t -1\right ) \left (t +1\right )}}{2} \] |
Problem 1648
| ODE |
\[ \boxed {y^{\prime }+y=\frac {1}{t^{2}+1}} \] With initial conditions \begin {align*} [y \left (2\right ) = 3] \end {align*}
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program solution |
\[ y = \frac {i {\mathrm e}^{-t} {\mathrm e}^{i} \operatorname {expIntegral}_{1}\left (i-t \right )}{2}-\frac {i {\mathrm e}^{-t} {\mathrm e}^{-i} \operatorname {expIntegral}_{1}\left (-t -i\right )}{2}-\frac {i {\mathrm e}^{-t} {\mathrm e}^{i} \operatorname {expIntegral}_{1}\left (-2+i\right )}{2}+\frac {i {\mathrm e}^{-t} {\mathrm e}^{-i} \operatorname {expIntegral}_{1}\left (-2-i\right )}{2}+3 \,{\mathrm e}^{-t +2} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {\left (i {\mathrm e}^{i} \operatorname {expIntegral}_{1}\left (-t +i\right )-i {\mathrm e}^{-i} \operatorname {expIntegral}_{1}\left (-t -i\right )-i {\mathrm e}^{i} \operatorname {expIntegral}_{1}\left (-2+i\right )+i {\mathrm e}^{-i} \operatorname {expIntegral}_{1}\left (-2-i\right )+6 \,{\mathrm e}^{2}\right ) {\mathrm e}^{-t}}{2} \] |
Problem 1649
| ODE |
\[ \boxed {\cos \left (t \right ) y+y^{\prime }=0} \] |
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program solution |
\[ y = {\mathrm e}^{-\sin \left (t \right )-c_{1}} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = c_{1} {\mathrm e}^{-\sin \left (t \right )} \] |
Problem 1650
| ODE |
\[ \boxed {\sqrt {t}\, \sin \left (t \right ) y+y^{\prime }=0} \] |
|
program solution |
\[ y = {\mathrm e}^{-\frac {\sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {t}}{\sqrt {\pi }}\right )}{2}+\cos \left (t \right ) \sqrt {t}-c_{1}} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = c_{1} {\mathrm e}^{\sqrt {t}\, \cos \left (t \right )-\frac {\sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {t}}{\sqrt {\pi }}\right )}{2}} \] |
Problem 1651
| ODE |
\[ \boxed {\frac {2 y t}{t^{2}+1}+y^{\prime }=\frac {1}{t^{2}+1}} \] |
|
program solution |
\[ y = \frac {t +c_{1}}{t^{2}+1} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {t +c_{1}}{t^{2}+1} \] |
Problem 1652
| ODE |
\[ \boxed {y^{\prime }+y=t \,{\mathrm e}^{t}} \] |
|
program solution |
\[ y = \frac {\left (2 \,{\mathrm e}^{2 t} t -{\mathrm e}^{2 t}+4 c_{1} \right ) {\mathrm e}^{-t}}{4} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = {\mathrm e}^{-t} c_{1} +\frac {{\mathrm e}^{t} \left (2 t -1\right )}{4} \] |
Problem 1653
| ODE |
\[ \boxed {y t^{2}+y^{\prime }=1} \] |
|
program solution |
\[ y = -\frac {{\mathrm e}^{-\frac {t^{3}}{3}} \left (-2 \,3^{\frac {5}{6}} t \pi +3 \,3^{\frac {1}{3}} t \Gamma \left (\frac {1}{3}, -\frac {t^{3}}{3}\right ) \Gamma \left (\frac {2}{3}\right )-9 c_{1} \left (-t^{3}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}\right )\right )}{9 \left (-t^{3}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}\right )} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = -\frac {\left (3^{\frac {1}{3}} t \Gamma \left (\frac {1}{3}, -\frac {t^{3}}{3}\right ) \Gamma \left (\frac {2}{3}\right )-\frac {2 \,3^{\frac {5}{6}} t \pi }{3}-3 c_{1} \Gamma \left (\frac {2}{3}\right ) \left (-t^{3}\right )^{\frac {1}{3}}\right ) {\mathrm e}^{-\frac {t^{3}}{3}}}{3 \left (-t^{3}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}\right )} \] |
Problem 1654
| ODE |
\[ \boxed {y t^{2}+y^{\prime }=t^{2}} \] |
|
program solution |
\[ y = {\mathrm e}^{-\frac {t^{3}}{3}-c_{1}}+1 \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = 1+{\mathrm e}^{-\frac {t^{3}}{3}} c_{1} \] |
Problem 1655
| ODE |
\[ \boxed {\frac {y t}{t^{2}+1}+y^{\prime }+\frac {t^{3} y}{t^{4}+1}=1} \] |
|
program solution |
\[ \int _{}^{t}-\frac {\textit {\_a}^{6}-2 \textit {\_a}^{5} y+\textit {\_a}^{4}-\textit {\_a}^{3} y+\textit {\_a}^{2}-\textit {\_a} y+1}{\sqrt {\textit {\_a}^{2}+1}\, \left (\textit {\_a}^{4}+1\right )^{\frac {3}{4}}}d \textit {\_a} = c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {\int \left (t^{4}+1\right )^{\frac {1}{4}} \sqrt {t^{2}+1}d t +c_{1}}{\left (t^{4}+1\right )^{\frac {1}{4}} \sqrt {t^{2}+1}} \] |
Problem 1656
| ODE |
\[ \boxed {\sqrt {t^{2}+1}\, y+y^{\prime }=0} \] With initial conditions \begin {align*} \left [y \left (0\right ) = \sqrt {5}\right ] \end {align*}
|
|
program solution |
\[ y = \frac {{\mathrm e}^{-\frac {\sqrt {t^{2}+1}\, t}{2}} \sqrt {5}}{\sqrt {t +\sqrt {t^{2}+1}}} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \sqrt {5}\, {\mathrm e}^{-\frac {t \sqrt {t^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (t \right )}{2}} \] |
Problem 1657
| ODE |
\[ \boxed {\sqrt {t^{2}+1}\, y \,{\mathrm e}^{-t}+y^{\prime }=0} \] |
|
program solution |
\[ y = {\mathrm e}^{\int _{}^{t}-\sqrt {\textit {\_a}^{2}+1}\, {\mathrm e}^{-\textit {\_a}}d \textit {\_a} -c_{1}} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = c_{1} {\mathrm e}^{-\left (\int \sqrt {t^{2}+1}\, {\mathrm e}^{-t}d t \right )} \] |
Problem 1658
| ODE |
\[ \boxed {y^{\prime }-2 y t=t} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
|
|
program solution |
\[ y = \frac {3 \,{\mathrm e}^{t^{2}}}{2}-\frac {1}{2} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = -\frac {1}{2}+\frac {3 \,{\mathrm e}^{t^{2}}}{2} \] |
Problem 1659
| ODE |
\[ \boxed {y t +y^{\prime }=t +1} \] With initial conditions \begin {align*} \left [y \left (\frac {3}{2}\right ) = 0\right ] \end {align*}
|
|
program solution |
\[ y = {\mathrm e}^{-\frac {t^{2}}{2}} \left (\int _{\frac {3}{2}}^{t}\textit {\_a} \,{\mathrm e}^{\frac {\textit {\_a}^{2}}{2}}d \textit {\_a} \right )+{\mathrm e}^{-\frac {t^{2}}{2}} \left (\int _{\frac {3}{2}}^{t}{\mathrm e}^{\frac {\textit {\_a}^{2}}{2}}d \textit {\_a} \right ) \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = 1-{\mathrm e}^{\frac {9}{8}-\frac {t^{2}}{2}}+\frac {\sqrt {2}\, \sqrt {\pi }\, \left (-i \operatorname {erf}\left (\frac {i \sqrt {2}\, t}{2}\right )-\operatorname {erfi}\left (\frac {3 \sqrt {2}}{4}\right )\right ) {\mathrm e}^{-\frac {t^{2}}{2}}}{2} \] |
Problem 1660
| ODE |
\[ \boxed {y^{\prime }+y=\frac {1}{t^{2}+1}} \] With initial conditions \begin {align*} [y \left (1\right ) = 2] \end {align*}
|
|
program solution |
\[ y = \frac {\int _{1}^{t}\frac {{\mathrm e}^{\textit {\_a}}}{\textit {\_a}^{2}+1}d \textit {\_a} +2 \,{\mathrm e}}{{\mathrm e}^{t}+\int _{1}^{t}\frac {{\mathrm e}^{\textit {\_a}} \textit {\_a}^{2}}{\textit {\_a}^{2}+1}d \textit {\_a} +\int _{1}^{t}\frac {{\mathrm e}^{\textit {\_a}}}{\textit {\_a}^{2}+1}d \textit {\_a} -\left (\int _{1}^{t}{\mathrm e}^{\textit {\_a}}d \textit {\_a} \right )} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = -\frac {\left (i {\mathrm e}^{i} \operatorname {expIntegral}_{1}\left (-1+i\right )-i {\mathrm e}^{i} \operatorname {expIntegral}_{1}\left (-t +i\right )-i {\mathrm e}^{-i} \operatorname {expIntegral}_{1}\left (-1-i\right )+i {\mathrm e}^{-i} \operatorname {expIntegral}_{1}\left (-t -i\right )-4 \,{\mathrm e}\right ) {\mathrm e}^{-t}}{2} \] |
Problem 1661
| ODE |
\[ \boxed {y^{\prime }-2 y t=1} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
|
|
program solution |
\[ y = \frac {{\mathrm e}^{t^{2}} \sqrt {\pi }\, \operatorname {erf}\left (t \right )}{2}+{\mathrm e}^{t^{2}} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {\left (\sqrt {\pi }\, \operatorname {erf}\left (t \right )+2\right ) {\mathrm e}^{t^{2}}}{2} \] |
Problem 1662
| ODE |
\[ \boxed {y t +\left (t^{2}+1\right ) y^{\prime }=\left (t^{2}+1\right )^{\frac {5}{2}}} \] |
|
program solution |
\[ y = \frac {3 t^{5}+10 t^{3}+15 c_{1} +15 t}{15 \sqrt {t^{2}+1}} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {3 t^{5}+10 t^{3}+15 c_{1} +15 t}{15 \sqrt {t^{2}+1}} \] |
Problem 1663
| ODE |
\[ \boxed {4 y t +\left (t^{2}+1\right ) y^{\prime }=t} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
|
|
program solution |
\[ y = \frac {t^{2} \left (t^{2}+2\right )}{4 t^{4}+8 t^{2}+4} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {1}{4}-\frac {1}{4 \left (t^{2}+1\right )^{2}} \] |
Problem 1664
| ODE |
\[ \boxed {y^{\prime }+\frac {y}{t}=\frac {1}{t^{2}}} \] |
|
program solution |
\[ y = \frac {\ln \left (t \right )+c_{1}}{t} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {\ln \left (t \right )+c_{1}}{t} \] |
Problem 1665
| ODE |
\[ \boxed {y^{\prime }+\frac {y}{\sqrt {t}}={\mathrm e}^{\frac {\sqrt {t}}{2}}} \] |
|
program solution |
\[ y = \frac {\left (20 \,{\mathrm e}^{\frac {5 \sqrt {t}}{2}} \sqrt {t}-8 \,{\mathrm e}^{\frac {5 \sqrt {t}}{2}}+25 c_{1} \right ) {\mathrm e}^{-2 \sqrt {t}}}{25} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {\left (20 \,{\mathrm e}^{\frac {5 \sqrt {t}}{2}} \sqrt {t}-8 \,{\mathrm e}^{\frac {5 \sqrt {t}}{2}}+25 c_{1} \right ) {\mathrm e}^{-2 \sqrt {t}}}{25} \] |
Problem 1666
| ODE |
\[ \boxed {y^{\prime }+\frac {y}{t}=\cos \left (t \right )+\frac {\sin \left (t \right )}{t}} \] |
|
program solution |
\[ y = \sin \left (t \right )+\frac {c_{1}}{t} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \sin \left (t \right )+\frac {c_{1}}{t} \] |
Problem 1667
| ODE |
\[ \boxed {\tan \left (t \right ) y+y^{\prime }=\sin \left (t \right ) \cos \left (t \right )} \] |
|
program solution |
\[ y = \cos \left (t \right ) \left (-\cos \left (t \right )+c_{1} \right ) \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \left (-\cos \left (t \right )+c_{1} \right ) \cos \left (t \right ) \] |
Problem 1668
| ODE |
\[ \boxed {\left (t^{2}+1\right ) y^{\prime }-y^{2}=1} \] |
|
program solution |
\[ y = \frac {-c_{3} +t}{c_{3} t +1} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \tan \left (\arctan \left (t \right )+c_{1} \right ) \] |
Problem 1669
| ODE |
\[ \boxed {y^{\prime }-\left (t +1\right ) \left (1+y\right )=0} \] |
|
program solution |
\[ y = {\mathrm e}^{\frac {1}{2} t^{2}+t +c_{1}}-1 \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = -1+{\mathrm e}^{\frac {t \left (2+t \right )}{2}} c_{1} \] |
Problem 1670
| ODE |
\[ \boxed {y^{\prime }-y^{2}+t y^{2}=1-t} \] |
|
program solution |
\[ y = \frac {c_{3} \cos \left (\frac {t \left (t -2\right )}{2}\right )-\sin \left (\frac {t \left (t -2\right )}{2}\right )}{c_{3} \sin \left (\frac {t \left (t -2\right )}{2}\right )+\cos \left (\frac {t \left (t -2\right )}{2}\right )} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = -\tan \left (\frac {1}{2} t^{2}+c_{1} -t \right ) \] |
Problem 1671
| ODE |
\[ \boxed {y^{\prime }-{\mathrm e}^{3+t +y}=0} \] |
|
program solution |
\[ y = -\ln \left (-{\mathrm e}^{t +3}-c_{1} \right ) \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = -3-\ln \left (-{\mathrm e}^{t}-c_{1} \right ) \] |
Problem 1672
| ODE |
\[ \boxed {\cos \left (y\right ) \sin \left (t \right ) y^{\prime }-\cos \left (t \right ) \sin \left (y\right )=0} \] |
|
program solution |
\[ -\ln \left (\sin \left (t \right )\right )+\ln \left (\sin \left (y\right )\right ) = c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \arcsin \left (c_{1} \sin \left (t \right )\right ) \] |
Problem 1673
| ODE |
\[ \boxed {t^{2} \left (1+y^{2}\right )+2 y^{\prime } y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
|
|
program solution |
\[ -\frac {t^{3}}{3}-\ln \left (1+y^{2}\right ) = -\ln \left (2\right ) \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \sqrt {2 \,{\mathrm e}^{-\frac {t^{3}}{3}}-1} \] |
Problem 1674
| ODE |
\[ \boxed {y^{\prime }-\frac {2 t}{y+y t^{2}}=0} \] With initial conditions \begin {align*} [y \left (2\right ) = 3] \end {align*}
|
|
program solution |
\[ -\frac {\ln \left (t^{2}+1\right )}{2}+\frac {y^{2}}{4} = -\frac {\ln \left (5\right )}{2}+\frac {9}{4} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \sqrt {9-2 \ln \left (5\right )+2 \ln \left (t^{2}+1\right )} \] |
Problem 1675
| ODE |
\[ \boxed {\sqrt {t^{2}+1}\, y^{\prime }-\frac {t y^{3}}{\sqrt {t^{2}+1}}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
|
|
program solution |
\[ -\frac {\ln \left (t^{2}+1\right )}{2}-\frac {1}{2 y^{2}} = -{\frac {1}{2}} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {1}{\sqrt {1-\ln \left (t^{2}+1\right )}} \] |
Problem 1676
| ODE |
\[ \boxed {y^{\prime }-\frac {3 t^{2}+4 t +2}{-2+2 y}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -1] \end {align*}
|
|
program solution |
\[ -t^{3}-2 t^{2}+y^{2}-2 t -2 y = 3 \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = -\sqrt {\left (2+t \right ) \left (t^{2}+2\right )}+1 \] |
Problem 1677
| ODE |
\[ \boxed {\cos \left (y\right ) y^{\prime }+\frac {t \sin \left (y\right )}{t^{2}+1}=0} \] With initial conditions \begin {align*} \left [y \left (1\right ) = \frac {\pi }{2}\right ] \end {align*}
|
|
program solution |
\[ -\frac {\ln \left (t^{2}+1\right )}{2}-\ln \left (\sin \left (y\right )\right ) = -\frac {\ln \left (2\right )}{2} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (t \right ) &= \arcsin \left (\frac {\sqrt {2}}{\sqrt {t^{2}+1}}\right ) \\ y \left (t \right ) &= \pi -\arcsin \left (\frac {\sqrt {2}}{\sqrt {t^{2}+1}}\right ) \\ \end{align*} |
Problem 1678
| ODE |
\[ \boxed {y^{\prime }-k \left (a -y\right ) \left (b -y\right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
|
|
program solution |
\[ \frac {\ln \left (-a +y\right )-\ln \left (-b +y\right )}{k \left (-b +a \right )} = t +\frac {\ln \left (-a \right )-\ln \left (-b \right )}{k \left (-b +a \right )} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {a b \left ({\mathrm e}^{t k \left (a -b \right )}-1\right )}{{\mathrm e}^{t k \left (a -b \right )} a -b} \] |
Problem 1679
| ODE |
\[ \boxed {3 t y^{\prime }-\cos \left (t \right ) y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 0] \end {align*}
|
|
program solution |
N/A |
|
Maple solution | \[ y \left (t \right ) = 0 \] |
Problem 1680
| ODE |
\[ \boxed {t y^{\prime }-y-\sqrt {t^{2}+y^{2}}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 0] \end {align*}
|
|
program solution |
\[ y = \frac {t^{2}}{2}-\frac {1}{2} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (t \right ) &= -\frac {t^{2}}{2}+\frac {1}{2} \\ y \left (t \right ) &= \frac {t^{2}}{2}-\frac {1}{2} \\ \end{align*} |
Problem 1681
| ODE |
\[ \boxed {2 t y y^{\prime }-3 y^{2}=-t^{2}} \] |
|
program solution |
\[ \frac {y^{2}}{t^{3}}-\frac {1}{t} = c_{1} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (t \right ) &= \sqrt {c_{1} t +1}\, t \\ y \left (t \right ) &= -\sqrt {c_{1} t +1}\, t \\ \end{align*} |
Problem 1682
| ODE |
\[ \boxed {\left (t -\sqrt {y t}\right ) y^{\prime }-y=0} \] |
|
program solution |
\[ \frac {\ln \left (y\right ) \sqrt {y}+2 \sqrt {t}}{\sqrt {y}} = c_{1} \] Verified OK.
|
|
Maple solution | \[ \ln \left (y \left (t \right )\right )+\frac {2 t}{\sqrt {t y \left (t \right )}}-c_{1} = 0 \] |
Problem 1683
| ODE |
\[ \boxed {y^{\prime }-\frac {t +y}{t -y}=0} \] |
|
program solution |
\[ \frac {\ln \left (t^{2}+y^{2}\right )}{2}+\arctan \left (\frac {t}{y}\right ) = c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \tan \left (\operatorname {RootOf}\left (-2 \textit {\_Z} +\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+2 \ln \left (t \right )+2 c_{1} \right )\right ) t \] |
Problem 1684
| ODE |
\[ \boxed {{\mathrm e}^{\frac {t}{y}} \left (-t +y\right ) y^{\prime }+y \left (1+{\mathrm e}^{\frac {t}{y}}\right )=0} \] |
|
program solution |
\[ y = -\frac {t}{\operatorname {LambertW}\left (-\frac {t}{c_{1} -t}\right )} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = -\frac {t}{\operatorname {LambertW}\left (\frac {c_{1} t}{c_{1} t -1}\right )} \] |
Problem 1685
| ODE |
\[ \boxed {y^{\prime }-\frac {t +y+1}{t -y+3}=0} \] |
|
program solution |
\[ -\frac {\ln \left (y^{2}+t^{2}-2 y+4 t +5\right )}{2}+\arctan \left (\frac {-1+y}{2+t}\right ) = c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = 1+\tan \left (\operatorname {RootOf}\left (2 \textit {\_Z} +\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+2 \ln \left (2+t \right )+2 c_{1} \right )\right ) \left (-2-t \right ) \] |
Problem 1686
| ODE |
\[ \boxed {-2 y+\left (4 t -3 y-6\right ) y^{\prime }=-t -1} \] |
|
program solution |
\[ \frac {5 \ln \left (t +3 y-9\right )}{8}-\frac {\ln \left (-t +y+1\right )}{8} = c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {\left (-t +3\right ) {\operatorname {RootOf}\left (-4+\left (3 c_{1} t^{4}-36 c_{1} t^{3}+162 c_{1} t^{2}-324 c_{1} t +243 c_{1} \right ) \textit {\_Z}^{20}-\textit {\_Z}^{4}\right )}^{4}}{3}-\frac {t}{3}+3 \] |
Problem 1687
| ODE |
\[ \boxed {2 y+\left (2 t +4 y-1\right ) y^{\prime }=-t -3} \] |
|
program solution |
\[ \frac {t \left (t +4 y+6\right )}{2}+2 y^{2}-y = c_{1} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (t \right ) &= -\frac {t}{2}+\frac {1}{4}-\frac {\sqrt {28 c_{1} -28 t +1}}{4} \\ y \left (t \right ) &= -\frac {t}{2}+\frac {1}{4}+\frac {\sqrt {28 c_{1} -28 t +1}}{4} \\ \end{align*} |
Problem 1688
| ODE |
\[ \boxed {2 t \sin \left (y\right )+{\mathrm e}^{t} y^{3}+\left (t^{2} \cos \left (y\right )+3 \,{\mathrm e}^{t} y^{2}\right ) y^{\prime }=0} \] |
|
program solution |
\[ {\mathrm e}^{t} y^{3}+t^{2} \sin \left (y\right ) = c_{1} \] Verified OK.
|
|
Maple solution | \[ {\mathrm e}^{t} y \left (t \right )^{3}+t^{2} \sin \left (y \left (t \right )\right )+c_{1} = 0 \] |
Problem 1689
| ODE |
\[ \boxed {{\mathrm e}^{y t} \left (1+y t \right )+\left (1+{\mathrm e}^{y t} t^{2}\right ) y^{\prime }=-1} \] |
|
program solution |
\[ y = -\frac {-c_{1} t +t^{2}+\operatorname {LambertW}\left (t^{2} {\mathrm e}^{c_{1} t -t^{2}}\right )}{t} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {-c_{1} t -t^{2}-\operatorname {LambertW}\left (t^{2} {\mathrm e}^{-t \left (t +c_{1} \right )}\right )}{t} \] |
Problem 1690
| ODE |
\[ \boxed {\sec \left (t \right )^{2} y+\left (\tan \left (t \right )+2 y\right ) y^{\prime }=-\sec \left (t \right ) \tan \left (t \right )} \] |
|
program solution |
\[ \tan \left (t \right ) y+\sec \left (t \right )+y^{2} = c_{1} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (t \right ) &= -\frac {\tan \left (t \right )}{2}-\frac {\sec \left (t \right ) \sqrt {-4 \cos \left (t \right )^{2} c_{1} +\sin \left (t \right )^{2}-4 \cos \left (t \right )}}{2} \\ y \left (t \right ) &= -\frac {\tan \left (t \right )}{2}+\frac {\sec \left (t \right ) \sqrt {-4 \cos \left (t \right )^{2} c_{1} +\sin \left (t \right )^{2}-4 \cos \left (t \right )}}{2} \\ \end{align*} |
Problem 1691
| ODE |
\[ \boxed {\frac {y^{2}}{2}-2 y \,{\mathrm e}^{t}+\left (-{\mathrm e}^{t}+y\right ) y^{\prime }=0} \] |
|
program solution |
\[ \frac {y \,{\mathrm e}^{t} \left (y-2 \,{\mathrm e}^{t}\right )}{2} = c_{1} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (t \right ) &= \left (1-\sqrt {\left ({\mathrm e}^{3 t}+c_{1} \right ) {\mathrm e}^{-3 t}}\right ) {\mathrm e}^{t} \\ y \left (t \right ) &= \left (1+\sqrt {\left ({\mathrm e}^{3 t}+c_{1} \right ) {\mathrm e}^{-3 t}}\right ) {\mathrm e}^{t} \\ \end{align*} |
Problem 1692
| ODE |
\[ \boxed {2 t y^{3}+3 t^{2} y^{2} y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1] \end {align*}
|
|
program solution |
\[ y = \frac {1}{t^{\frac {2}{3}}} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {1}{t^{\frac {2}{3}}} \] |
Problem 1693
| ODE |
\[ \boxed {2 t \cos \left (y\right )+3 y t^{2}+\left (t^{3}-t^{2} \sin \left (y\right )-y\right ) y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2] \end {align*}
|
|
program solution |
\[ y t^{3}+t^{2} \cos \left (y\right )-\frac {y^{2}}{2} = -2 \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \operatorname {RootOf}\left (-2 \textit {\_Z} \,t^{3}-2 \cos \left (\textit {\_Z} \right ) t^{2}+\textit {\_Z}^{2}-4\right ) \] |
Problem 1694
| ODE |
\[ \boxed {4 y t +\left (2 t^{2}+2 y\right ) y^{\prime }=-3 t^{2}} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
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program solution |
\[ t^{3}+2 y t^{2}+y^{2} = 1 \] Verified OK.
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Maple solution | \[ y \left (t \right ) = -t^{2}+\sqrt {t^{4}-t^{3}+1} \] |
Problem 1695
| ODE |
\[ \boxed {-2 \,{\mathrm e}^{y t} \sin \left (2 t \right )+{\mathrm e}^{y t} \cos \left (2 t \right ) y+\left (-3+{\mathrm e}^{y t} t \cos \left (2 t \right )\right ) y^{\prime }=-2 t} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
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program solution |
\[ y = \frac {t^{3}-3 \operatorname {LambertW}\left (-\frac {{\mathrm e}^{\frac {t \left (-1+t \right ) \left (t +1\right )}{3}} \cos \left (2 t \right ) t}{3}\right )-t}{3 t} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {t^{3}-3 \operatorname {LambertW}\left (-\frac {t \cos \left (2 t \right ) {\mathrm e}^{\frac {t \left (t -1\right ) \left (t +1\right )}{3}}}{3}\right )-t}{3 t} \] |
Problem 1696
| ODE |
\[ \boxed {3 y t +y^{2}+\left (t^{2}+y t \right ) y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (2\right ) = 1] \end {align*}
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program solution |
\[ \frac {t^{2} y \left (y+2 t \right )}{2} = 10 \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {-t^{2}+\sqrt {t^{4}+20}}{t} \] |
Problem 1697
| ODE |
\[ \boxed {y^{\prime }-y^{2}=\cos \left (t^{2}\right )} \] |
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program solution |
\[ y = -\frac {\frac {d}{d t}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (t \right )+\cos \left (t^{2}\right ) \textit {\_Y} \left (t \right )\right \}, \left \{\textit {\_Y} \left (t \right )\right \}\right )}{\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (t \right )+\cos \left (t^{2}\right ) \textit {\_Y} \left (t \right )\right \}, \left \{\textit {\_Y} \left (t \right )\right \}\right )} \] Verified OK.
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Maple solution | \[ \text {No solution found} \] |
Problem 1698
| ODE |
\[ \boxed {y^{\prime }-y-y^{2} \cos \left (t \right )=1} \] |
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program solution |
\[ y = \frac {c_{3} \left (-4+8 \cos \left (\frac {t}{2}\right )^{2}\right ) \operatorname {MathieuC}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )+\left (-4+8 \cos \left (\frac {t}{2}\right )^{2}\right ) \operatorname {MathieuS}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )}{4 \left (c_{3} \operatorname {MathieuCPrime}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )-c_{3} \operatorname {MathieuC}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )-\operatorname {MathieuS}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )+\operatorname {MathieuSPrime}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )\right ) \left (\cos \left (\frac {t}{2}\right )^{2}-\frac {1}{2}\right )} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = -\frac {\operatorname {csgn}\left (\sin \left (\frac {t}{2}\right )\right ) \left (\left (-4 \cos \left (t \right )-\operatorname {csgn}\left (\sin \left (\frac {t}{2}\right )\right )+1\right ) \operatorname {MathieuC}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )-4 c_{1} \left (\cos \left (t \right )+\frac {\operatorname {csgn}\left (\sin \left (\frac {t}{2}\right )\right )}{4}-\frac {1}{4}\right ) \operatorname {MathieuS}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )+\left (-1+\operatorname {csgn}\left (\sin \left (\frac {t}{2}\right )\right )\right ) \left (c_{1} \operatorname {MathieuSPrime}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )+\operatorname {MathieuCPrime}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )\right )\right )}{2 \left (-c_{1} \operatorname {MathieuS}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )+c_{1} \operatorname {MathieuSPrime}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )-\operatorname {MathieuC}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )+\operatorname {MathieuCPrime}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )\right ) \cos \left (t \right )} \] |
Problem 1699
| ODE |
\[ \boxed {y^{\prime }-y^{2}=t} \] |
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program solution |
\[ y = \frac {c_{3} \operatorname {AiryAi}\left (1, -t \right )+\operatorname {AiryBi}\left (1, -t \right )}{c_{3} \operatorname {AiryAi}\left (-t \right )+\operatorname {AiryBi}\left (-t \right )} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {c_{1} \operatorname {AiryAi}\left (1, -t \right )+\operatorname {AiryBi}\left (1, -t \right )}{c_{1} \operatorname {AiryAi}\left (-t \right )+\operatorname {AiryBi}\left (-t \right )} \] |
Problem 1700
| ODE |
\[ \boxed {y^{\prime }-y^{2}={\mathrm e}^{-t^{2}}} \] |
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program solution |
\[ y = -\frac {\frac {d}{d t}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (t \right )+{\mathrm e}^{-t^{2}} \textit {\_Y} \left (t \right )\right \}, \left \{\textit {\_Y} \left (t \right )\right \}\right )}{\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (t \right )+{\mathrm e}^{-t^{2}} \textit {\_Y} \left (t \right )\right \}, \left \{\textit {\_Y} \left (t \right )\right \}\right )} \] Verified OK.
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Maple solution | \[ \text {No solution found} \] |