Problem 10201
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=-\frac {2 x \left (t \right )}{t}+\frac {2 y \left (t \right )}{t}+1\\ y^{\prime }\left (t \right )&=t -\frac {x \left (t \right )}{t}-\frac {5 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 ) &= \frac {2 t^{6}+9 t^{5}+30 c_{1} t +30 c_{2}}{30 t^{4}} \\ y \left (t \right ) &= -\frac {-8 t^{6}+3 t^{5}+30 c_{1} t +60 c_{2}}{60 t^{4}} \\ \end{align*} |
Problem 10202
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=\frac {2 x \left (t \right ) \sin \left (t \right )}{t \left (\sin \left (t \right )-1\right )}-\frac {y \left (t \right )}{\sin \left (t \right )-1}-\frac {x \left (t \right )}{t \left (\sin \left (t \right )-1\right )}\\ y^{\prime }\left (t \right )&=\frac {y \left (t \right ) \cos \left (t \right )}{\sin \left (t \right )-1}-\frac {x \left (t \right ) \cos \left (t \right )}{t \left (\sin \left (t \right )-1\right )}+\frac {x \left (t \right ) \sin \left (t \right )}{t^{2} \left (\sin \left (t \right )-1\right )}-\frac {y \left (t \right )}{t \left (\sin \left (t \right )-1\right )} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= t \left (c_{2} t +c_{1} \right ) \\ y \left (t \right ) &= c_{1} \sin \left (t \right )+c_{2} t \\ \end{align*} |
Problem 10203
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ODE |
\begin {align*} x^{\prime }\left (t \right )+y^{\prime }\left (t \right )+y \left (t \right )&=f \left (t \right )\\ x^{\prime \prime }\left (t \right )+y^{\prime \prime }\left (t \right )+y^{\prime }\left (t \right )+x \left (t \right )+y \left (t \right )&=g \left (t \right ) \end {align*}
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program solution |
N/A |
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Maple solution |
\begin{align*} x \left (t \right ) &= -\frac {d}{d t}f \left (t \right )-f \left (t \right )-\frac {d^{2}}{d t^{2}}f \left (t \right )+\frac {d}{d t}g \left (t \right )+g \left (t \right ) \\ y \left (t \right ) &= f \left (t \right )+\frac {d^{2}}{d t^{2}}f \left (t \right )-\frac {d}{d t}g \left (t \right ) \\ \end{align*} |
Problem 10204
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ODE |
\begin {align*} x^{\prime \prime }\left (t \right )+y^{\prime }\left (t \right )-2 y \left (t \right )&={\mathrm e}^{2 t}\\ 2 x^{\prime }\left (t \right )+y^{\prime }\left (t \right )-3 x \left (t \right )&=0 \end {align*}
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program solution |
N/A |
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Maple solution |
\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{2 t}}{4}+c_{1} {\mathrm e}^{t}-\frac {7 c_{2} {\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {23}\, t}{2}\right )}{18}-\frac {c_{2} {\mathrm e}^{\frac {t}{2}} \sqrt {23}\, \sin \left (\frac {\sqrt {23}\, t}{2}\right )}{18}-\frac {7 c_{3} {\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {23}\, t}{2}\right )}{18}+\frac {c_{3} {\mathrm e}^{\frac {t}{2}} \sqrt {23}\, \cos \left (\frac {\sqrt {23}\, t}{2}\right )}{18} \\ y \left (t \right ) &= -\frac {{\mathrm e}^{2 t}}{8}+c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {23}\, t}{2}\right )+c_{3} {\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {23}\, t}{2}\right ) \\ \end{align*} |
Problem 10205
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ODE |
\begin {align*} x^{\prime }\left (t \right )-y^{\prime }\left (t \right )+x \left (t \right )&=2 t\\ x^{\prime \prime }\left (t \right )+y^{\prime }\left (t \right )-9 x \left (t \right )+3 y \left (t \right )&=\sin \left (2 t \right ) \end {align*}
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program solution |
N/A |
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Maple solution |
\begin{align*} x \left (t \right ) &= -\frac {36 \sin \left (2 t \right )}{325}-\frac {2 \cos \left (2 t \right )}{325}+\frac {c_{1} {\mathrm e}^{t}}{2}+\frac {3 c_{2} {\mathrm e}^{-3 t}}{2}+\frac {c_{3} {\mathrm e}^{t}}{4}+\frac {c_{3} t \,{\mathrm e}^{t}}{2}+2 t +4 \\ y \left (t \right ) &= 6 t +10-\frac {37 \sin \left (2 t \right )}{325}+\frac {16 \cos \left (2 t \right )}{325}+c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-3 t}+c_{3} t \,{\mathrm e}^{t} \\ \end{align*} |
Problem 10206
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )-2 y \left (t \right )\\ y^{\prime }\left (t \right )&=\frac {x \left (t \right )}{4}-\frac {y \left (t \right )}{2}-\frac {t}{2}+\frac {\cos \left (t \right )^{2}}{2}-\frac {1}{4} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= -t^{2}+8 \,{\mathrm e}^{\frac {t}{2}} c_{1} +\frac {\sin \left (2 t \right )}{34}+\frac {2 \cos \left (2 t \right )}{17}-4 t +2 c_{2} -4 \\ y \left (t \right ) &= -\frac {t^{2}}{2}+2 \,{\mathrm e}^{\frac {t}{2}} c_{1} +\frac {9 \sin \left (2 t \right )}{68}+\frac {\cos \left (2 t \right )}{34}-t +c_{2} \\ \end{align*} |
Problem 10207
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ODE |
\begin {align*} t x^{\prime \prime }\left (t \right )+2 x^{\prime }\left (t \right )+x \left (t \right ) t&=0\\ t x^{\prime }\left (t \right )-y^{\prime }\left (t \right ) t -2 y \left (t \right )&=0 \end {align*}
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program solution |
N/A |
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Maple solution |
\begin{align*} x \left (t \right ) &= -\frac {c_{3} \cos \left (t \right )-\sin \left (t \right ) c_{2}}{t} \\ y \left (t \right ) &= \frac {-\cos \left (t \right ) c_{3} t +\sin \left (t \right ) c_{2} t +2 c_{2} \cos \left (t \right )+2 \sin \left (t \right ) c_{3} +c_{1}}{t^{2}} \\ \end{align*} |
Problem 10208
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ODE |
\begin {align*} x^{\prime \prime }\left (t \right )+a y \left (t \right )&=0\\ y^{\prime \prime }\left (t \right )-a^{2} y \left (t \right )&=0 \end {align*}
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program solution |
N/A |
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Maple solution |
\begin{align*} x \left (t \right ) &= -\frac {-c_{1} t a +c_{3} {\mathrm e}^{-a t}+c_{4} {\mathrm e}^{a t}-c_{2} a}{a} \\ y \left (t \right ) &= c_{3} {\mathrm e}^{-a t}+c_{4} {\mathrm e}^{a t} \\ \end{align*} |
Problem 10209
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ODE |
\begin {align*} x^{\prime \prime }\left (t \right )&=a x \left (t \right )+b y \left (t \right )\\ y^{\prime \prime }\left (t \right )&=c x \left (t \right )+d y \left (t \right ) \end {align*}
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program solution |
N/A |
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Maple solution |
\begin{align*} x \left (t \right ) &= \left (-\frac {d}{2 c}+\frac {\frac {\sqrt {a^{2}-2 a d +4 b c +d^{2}}}{2}+\frac {a}{2}}{c}\right ) c_{4} {\mathrm e}^{\frac {\sqrt {2 \sqrt {a^{2}-2 a d +4 b c +d^{2}}+2 a +2 d}\, t}{2}}+\left (-\frac {d}{2 c}+\frac {\frac {\sqrt {a^{2}-2 a d +4 b c +d^{2}}}{2}+\frac {a}{2}}{c}\right ) c_{3} {\mathrm e}^{-\frac {\sqrt {2 \sqrt {a^{2}-2 a d +4 b c +d^{2}}+2 a +2 d}\, t}{2}}+\left (-\frac {d}{2 c}+\frac {-\frac {\sqrt {a^{2}-2 a d +4 b c +d^{2}}}{2}+\frac {a}{2}}{c}\right ) c_{2} {\mathrm e}^{\frac {\sqrt {-2 \sqrt {a^{2}-2 a d +4 b c +d^{2}}+2 a +2 d}\, t}{2}}+\left (-\frac {d}{2 c}+\frac {-\frac {\sqrt {a^{2}-2 a d +4 b c +d^{2}}}{2}+\frac {a}{2}}{c}\right ) c_{1} {\mathrm e}^{-\frac {\sqrt {-2 \sqrt {a^{2}-2 a d +4 b c +d^{2}}+2 a +2 d}\, t}{2}} \\ y \left (t \right ) &= c_{1} {\mathrm e}^{-\frac {\sqrt {-2 \sqrt {a^{2}-2 a d +4 b c +d^{2}}+2 a +2 d}\, t}{2}}+c_{2} {\mathrm e}^{\frac {\sqrt {-2 \sqrt {a^{2}-2 a d +4 b c +d^{2}}+2 a +2 d}\, t}{2}}+c_{3} {\mathrm e}^{-\frac {\sqrt {2 \sqrt {a^{2}-2 a d +4 b c +d^{2}}+2 a +2 d}\, t}{2}}+c_{4} {\mathrm e}^{\frac {\sqrt {2 \sqrt {a^{2}-2 a d +4 b c +d^{2}}+2 a +2 d}\, t}{2}} \\ \end{align*} |
Problem 10210
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ODE |
\begin {align*} x^{\prime \prime }\left (t \right )&=a_{1} x \left (t \right )+b_{1} y \left (t \right )+c_{1}\\ y^{\prime \prime }\left (t \right )&=a_{2} x \left (t \right )+b_{2} y \left (t \right )+c_{2} \end {align*}
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program solution |
N/A |
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Maple solution |
\begin{align*} x \left (t \right ) &= -\frac {\left (c_{6} a_{1} b_{2}^{2}+\left (-\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\, c_{6} a_{1} -c_{6} a_{1}^{2}-c_{6} a_{2} b_{1} \right ) b_{2} +\left (\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\, c_{6} +c_{6} a_{1} \right ) a_{2} b_{1} \right ) {\mathrm e}^{\frac {\sqrt {2 \sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}+2 a_{1} +2 b_{2}}\, t}{2}}}{2 a_{2} \left (a_{1} b_{2} -a_{2} b_{1} \right )}-\frac {\left (c_{5} a_{1} b_{2}^{2}+\left (-\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\, c_{5} a_{1} -c_{5} a_{1}^{2}-c_{5} a_{2} b_{1} \right ) b_{2} +\left (\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\, c_{5} +c_{5} a_{1} \right ) a_{2} b_{1} \right ) {\mathrm e}^{-\frac {\sqrt {2 \sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}+2 a_{1} +2 b_{2}}\, t}{2}}}{2 a_{2} \left (a_{1} b_{2} -a_{2} b_{1} \right )}-\frac {\left (c_{4} a_{1} b_{2}^{2}+\left (\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\, c_{4} a_{1} -c_{4} a_{1}^{2}-c_{4} a_{2} b_{1} \right ) b_{2} +\left (-\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\, c_{4} +c_{4} a_{1} \right ) a_{2} b_{1} \right ) {\mathrm e}^{\frac {\sqrt {2 a_{1} +2 b_{2} -2 \sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}}\, t}{2}}}{2 a_{2} \left (a_{1} b_{2} -a_{2} b_{1} \right )}-\frac {\left (c_{3} a_{1} b_{2}^{2}+\left (\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\, c_{3} a_{1} -c_{3} a_{1}^{2}-c_{3} a_{2} b_{1} \right ) b_{2} +\left (-\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\, c_{3} +c_{3} a_{1} \right ) a_{2} b_{1} \right ) {\mathrm e}^{-\frac {\sqrt {2 a_{1} +2 b_{2} -2 \sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}}\, t}{2}}}{2 a_{2} \left (a_{1} b_{2} -a_{2} b_{1} \right )}-\frac {-2 a_{2} b_{1} c_{2} +2 a_{2} b_{2} c_{1}}{2 a_{2} \left (a_{1} b_{2} -a_{2} b_{1} \right )} \\ y \left (t \right ) &= \frac {-a_{1} c_{2} +a_{2} c_{1}}{a_{1} b_{2} -a_{2} b_{1}}+c_{3} {\mathrm e}^{-\frac {\sqrt {2 a_{1} +2 b_{2} -2 \sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}}\, t}{2}}+c_{4} {\mathrm e}^{\frac {\sqrt {2 a_{1} +2 b_{2} -2 \sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}}\, t}{2}}+c_{5} {\mathrm e}^{-\frac {\sqrt {2 \sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}+2 a_{1} +2 b_{2}}\, t}{2}}+c_{6} {\mathrm e}^{\frac {\sqrt {2 \sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}+2 a_{1} +2 b_{2}}\, t}{2}} \\ \end{align*} |
Problem 10211
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ODE |
\begin {align*} x^{\prime \prime }\left (t \right )+x \left (t \right )+y \left (t \right )&=-5\\ y^{\prime \prime }\left (t \right )-4 x \left (t \right )-3 y \left (t \right )&=-3 \end {align*}
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program solution |
N/A |
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Maple solution |
\begin{align*} x \left (t \right ) &= -\frac {c_{1} {\mathrm e}^{t}}{2}-\frac {c_{2} {\mathrm e}^{-t}}{2}+\frac {c_{3} {\mathrm e}^{t}}{2}-\frac {c_{3} t \,{\mathrm e}^{t}}{2}-\frac {c_{4} {\mathrm e}^{-t} t}{2}-\frac {c_{4} {\mathrm e}^{-t}}{2}+18 \\ y \left (t \right ) &= -23+c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-t}+c_{3} t \,{\mathrm e}^{t}+c_{4} {\mathrm e}^{-t} t \\ \end{align*} |
Problem 10212
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ODE |
\begin {align*} x^{\prime \prime }\left (t \right )&=\left (3 \left (\cos ^{2}\left (a t +b \right )\right )-1\right ) c^{2} x \left (t \right )+\frac {3 c^{2} y \left (t \right ) \sin \left (2 a t b \right )}{2}\\ y^{\prime \prime }\left (t \right )&=\left (3 \left (\sin ^{2}\left (a t +b \right )\right )-1\right ) c^{2} y \left (t \right )+\frac {3 c^{2} x \left (t \right ) \sin \left (2 a t b \right )}{2} \end {align*}
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program solution |
N/A |
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Maple solution |
\[ \text {No solution found} \] |
Problem 10213
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ODE |
\begin {align*} x^{\prime \prime }\left (t \right )+6 x \left (t \right )+7 y \left (t \right )&=0\\ y^{\prime \prime }\left (t \right )+3 x \left (t \right )+2 y \left (t \right )&=2 t \end {align*}
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program solution |
N/A |
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Maple solution |
\begin{align*} x \left (t \right ) &= -c_{1} {\mathrm e}^{t}+\frac {7 c_{2} \cos \left (3 t \right )}{3}-c_{3} {\mathrm e}^{-t}+\frac {7 c_{4} \sin \left (3 t \right )}{3}+\frac {14 t}{9} \\ y \left (t \right ) &= -\frac {4 t}{3}+c_{1} {\mathrm e}^{t}+c_{2} \cos \left (3 t \right )+c_{3} {\mathrm e}^{-t}+c_{4} \sin \left (3 t \right ) \\ \end{align*} |
Problem 10214
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ODE |
\begin {align*} x^{\prime \prime }\left (t \right )-a y^{\prime }\left (t \right )+b x \left (t \right )&=0\\ y^{\prime \prime }\left (t \right )+a x^{\prime }\left (t \right )+b y \left (t \right )&=0 \end {align*}
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program solution |
N/A |
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Maple solution |
\begin{align*} x \left (t \right ) &= -\frac {c_{1} \left (-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b \right )^{\frac {3}{2}} {\mathrm e}^{-\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}}+4 \,{\mathrm e}^{-\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} \sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, c_{1} a^{2}-c_{2} \left (-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b \right )^{\frac {3}{2}} {\mathrm e}^{\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}}-4 \,{\mathrm e}^{\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} \sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, c_{2} a^{2}+c_{3} \left (-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b \right )^{\frac {3}{2}} {\mathrm e}^{-\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}}+4 \,{\mathrm e}^{-\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} \sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, c_{3} a^{2}-c_{4} \left (-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b \right )^{\frac {3}{2}} {\mathrm e}^{\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}}-4 \,{\mathrm e}^{\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} \sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, c_{4} a^{2}+4 \,{\mathrm e}^{-\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} \sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, c_{1} b -4 \,{\mathrm e}^{\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} \sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, c_{2} b +4 \,{\mathrm e}^{-\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} \sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, c_{3} b -4 \,{\mathrm e}^{\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} \sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, c_{4} b}{8 a b} \\ y \left (t \right ) &= c_{1} {\mathrm e}^{-\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}}+c_{2} {\mathrm e}^{\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}}+c_{3} {\mathrm e}^{-\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}}+c_{4} {\mathrm e}^{\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} \\ \end{align*} |
Problem 10215
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ODE |
\begin {align*} a_{1} x^{\prime \prime }\left (t \right )+b_{1} x^{\prime }\left (t \right )+c_{1} x \left (t \right )-A y^{\prime }\left (t \right )&=B \,{\mathrm e}^{i \omega t}\\ a_{2} y^{\prime \prime }\left (t \right )+b_{2} y^{\prime }\left (t \right )+c_{2} y \left (t \right )+A x^{\prime }\left (t \right )&=0 \end {align*}
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program solution |
N/A |
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Maple solution |
\begin{align*} \text {Expression too large to display} \\ y \left (t \right ) &= \frac {i {\mathrm e}^{i \omega t} \omega A B}{-a_{1} a_{2} \omega ^{4}+i a_{1} b_{2} \omega ^{3}+i a_{2} b_{1} \omega ^{3}+A^{2} \omega ^{2}+a_{1} c_{2} \omega ^{2}+a_{2} c_{1} \omega ^{2}+b_{1} b_{2} \omega ^{2}-b_{1} c_{2} \omega i-b_{2} c_{1} \omega i-c_{2} c_{1}}+c_{3} {\mathrm e}^{\operatorname {RootOf}\left (a_{1} a_{2} \textit {\_Z}^{4}+\left (a_{1} b_{2} +a_{2} b_{1} \right ) \textit {\_Z}^{3}+\left (A^{2}+c_{2} a_{1} +a_{2} c_{1} +b_{1} b_{2} \right ) \textit {\_Z}^{2}+\left (b_{1} c_{2} +b_{2} c_{1} \right ) \textit {\_Z} +c_{2} c_{1} , \operatorname {index} &=1\right ) t}+c_{4} {\mathrm e}^{\operatorname {RootOf}\left (a_{1} a_{2} \textit {\_Z}^{4}+\left (a_{1} b_{2} +a_{2} b_{1} \right ) \textit {\_Z}^{3}+\left (A^{2}+c_{2} a_{1} +a_{2} c_{1} +b_{1} b_{2} \right ) \textit {\_Z}^{2}+\left (b_{1} c_{2} +b_{2} c_{1} \right ) \textit {\_Z} +c_{2} c_{1} , \operatorname {index} &=2\right ) t}+c_{5} {\mathrm e}^{\operatorname {RootOf}\left (a_{1} a_{2} \textit {\_Z}^{4}+\left (a_{1} b_{2} +a_{2} b_{1} \right ) \textit {\_Z}^{3}+\left (A^{2}+c_{2} a_{1} +a_{2} c_{1} +b_{1} b_{2} \right ) \textit {\_Z}^{2}+\left (b_{1} c_{2} +b_{2} c_{1} \right ) \textit {\_Z} +c_{2} c_{1} , \operatorname {index} &=3\right ) t}+c_{6} {\mathrm e}^{\operatorname {RootOf}\left (a_{1} a_{2} \textit {\_Z}^{4}+\left (a_{1} b_{2} +a_{2} b_{1} \right ) \textit {\_Z}^{3}+\left (A^{2}+c_{2} a_{1} +a_{2} c_{1} +b_{1} b_{2} \right ) \textit {\_Z}^{2}+\left (b_{1} c_{2} +b_{2} c_{1} \right ) \textit {\_Z} +c_{2} c_{1} , \operatorname {index} &=4\right ) t} \\ \end{align*} |
Problem 10216
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ODE |
\begin {align*} x^{\prime \prime }\left (t \right )+a \left (x^{\prime }\left (t \right )-y^{\prime }\left (t \right )\right )+b_{1} x \left (t \right )&=c_{1} {\mathrm e}^{i \omega t}\\ y^{\prime \prime }\left (t \right )+a \left (y^{\prime }\left (t \right )-x^{\prime }\left (t \right )\right )+b_{2} y \left (t \right )&=c_{2} {\mathrm e}^{i \omega t} \end {align*}
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program solution |
N/A |
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Maple solution |
\begin{align*} \text {Expression too large to display} \\ y \left (t \right ) &= \frac {i {\mathrm e}^{i \omega t} c_{1} a \omega +i {\mathrm e}^{i \omega t} c_{2} a \omega -{\mathrm e}^{i \omega t} \omega ^{2} c_{2} +{\mathrm e}^{i \omega t} b_{1} c_{2}}{-2 i a \,\omega ^{3}+i a b_{1} \omega +i a b_{2} \omega +\omega ^{4}-b_{1} \omega ^{2}-b_{2} \omega ^{2}+b_{1} b_{2}}+c_{3} {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 a \,\textit {\_Z}^{3}+\left (b_{1} +b_{2} \right ) \textit {\_Z}^{2}+\left (b_{1} a +b_{2} a \right ) \textit {\_Z} +b_{1} b_{2} , \operatorname {index} &=1\right ) t}+c_{4} {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 a \,\textit {\_Z}^{3}+\left (b_{1} +b_{2} \right ) \textit {\_Z}^{2}+\left (b_{1} a +b_{2} a \right ) \textit {\_Z} +b_{1} b_{2} , \operatorname {index} &=2\right ) t}+c_{5} {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 a \,\textit {\_Z}^{3}+\left (b_{1} +b_{2} \right ) \textit {\_Z}^{2}+\left (b_{1} a +b_{2} a \right ) \textit {\_Z} +b_{1} b_{2} , \operatorname {index} &=3\right ) t}+c_{6} {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 a \,\textit {\_Z}^{3}+\left (b_{1} +b_{2} \right ) \textit {\_Z}^{2}+\left (b_{1} a +b_{2} a \right ) \textit {\_Z} +b_{1} b_{2} , \operatorname {index} &=4\right ) t} \\ \end{align*} |
Problem 10217
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ODE |
\begin {align*} \mathit {a11} x^{\prime \prime }\left (t \right )+\mathit {b11} x^{\prime }\left (t \right )+\mathit {c11} x \left (t \right )+\mathit {a12} y^{\prime \prime }\left (t \right )+\mathit {b12} y^{\prime }\left (t \right )+\mathit {c12} y \left (t \right )&=0\\ \mathit {a21} x^{\prime \prime }\left (t \right )+\mathit {b21} x^{\prime }\left (t \right )+\mathit {c21} x \left (t \right )+\mathit {a22} y^{\prime \prime }\left (t \right )+\mathit {b22} y^{\prime }\left (t \right )+\mathit {c22} y \left (t \right )&=0 \end {align*}
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program solution |
N/A |
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Maple solution |
\begin{align*} \text {Expression too large to display} \\ y \left (t \right ) &= \moverset {4}{\munderset {\textit {\_a} &=1}{\sum }}{\mathrm e}^{\operatorname {RootOf}\left (\left (\operatorname {a11} \operatorname {a22} -\operatorname {a12} \operatorname {a21} \right ) \textit {\_Z}^{4}+\left (\operatorname {a11} \operatorname {b22} -\operatorname {a12} \operatorname {b21} -\operatorname {a21} \operatorname {b12} +\operatorname {a22} \operatorname {b11} \right ) \textit {\_Z}^{3}+\left (\operatorname {a11} \operatorname {c22} -\operatorname {a12} \operatorname {c21} -\operatorname {a21} \operatorname {c12} +\operatorname {a22} \operatorname {c11} +\operatorname {b11} \operatorname {b22} -\operatorname {b12} \operatorname {b21} \right ) \textit {\_Z}^{2}+\left (\operatorname {b11} \operatorname {c22} -\operatorname {b12} \operatorname {c21} -\operatorname {b21} \operatorname {c12} +\operatorname {b22} \operatorname {c11} \right ) \textit {\_Z} +\operatorname {c11} \operatorname {c22} -\operatorname {c12} \operatorname {c21} , \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}} \\ \end{align*} |
Problem 10218
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ODE |
\begin {align*} y^{\prime \prime \prime }\left (t \right )-y^{\prime \prime }\left (t \right )+2 x^{\prime }\left (t \right )-x \left (t \right )&=t\\ x^{\prime \prime }\left (t \right )-2 x^{\prime }\left (t \right )-y^{\prime }\left (t \right )+y \left (t \right )&=0 \end {align*}
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program solution |
N/A |
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Maple solution |
\begin{align*} x \left (t \right ) &= -2-6 c_{5} {\mathrm e}^{t}-t -\frac {2 c_{2} {\mathrm e}^{-t}}{3}-2 c_{4} {\mathrm e}^{t} t -3 c_{5} {\mathrm e}^{t} t^{2}-c_{3} {\mathrm e}^{t} \\ y \left (t \right ) &= -2+c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-t}+c_{3} t \,{\mathrm e}^{t}+c_{4} {\mathrm e}^{t} t^{2}+c_{5} {\mathrm e}^{t} t^{3} \\ \end{align*} |
Problem 10219
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ODE |
\begin {align*} 2 x^{\prime \prime }\left (t \right )+y^{\prime \prime }\left (t \right )&=2 t\\ x^{\prime \prime }\left (t \right )+y^{\prime \prime }\left (t \right )+y^{\prime }\left (t \right )&=\sinh \left (2 t \right ) \end {align*}
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program solution |
N/A |
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Maple solution |
\begin{align*} x \left (t \right ) &= \frac {t^{2}}{4}+c_{4} t +\frac {t^{3}}{6}+\frac {t \sinh \left (2 t \right )}{4}-\frac {\cosh \left (2 t \right )}{8}-\frac {t \cosh \left (2 t \right )}{4}+\frac {\cosh \left (2 t \right ) c_{3}}{4}-\frac {c_{3} \sinh \left (2 t \right )}{4}+c_{1} t +c_{2} \\ y \left (t \right ) &= \frac {t}{2}+\frac {c_{3} \sinh \left (2 t \right )}{2}-\frac {t \sinh \left (2 t \right )}{2}+\frac {\cosh \left (2 t \right )}{4}-\frac {\cosh \left (2 t \right ) c_{3}}{2}+\frac {t \cosh \left (2 t \right )}{2}-\frac {t^{2}}{2}+c_{4} \\ \end{align*} |
Problem 10220
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ODE |
\begin {align*} x^{\prime \prime }\left (t \right )+y^{\prime \prime }\left (t \right )-x \left (t \right )&=0\\ x^{\prime \prime }\left (t \right )-x^{\prime }\left (t \right )+y^{\prime }\left (t \right )&=0 \end {align*}
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program solution |
N/A |
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Maple solution |
\begin{align*} x \left (t \right ) &= \left (-\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) c_{3} {\mathrm e}^{\frac {\left (\sqrt {5}+1\right ) t}{2}}+\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) c_{4} {\mathrm e}^{-\frac {\left (\sqrt {5}-1\right ) t}{2}}+c_{1} {\mathrm e}^{t} \\ y \left (t \right ) &= c_{2} +c_{3} {\mathrm e}^{\frac {\left (\sqrt {5}+1\right ) t}{2}}+c_{4} {\mathrm e}^{-\frac {\left (\sqrt {5}-1\right ) t}{2}} \\ \end{align*} |
Problem 10221
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )\\ y^{\prime }\left (t \right )&=3 x \left (t \right )-2 y \left (t \right )\\ z^{\prime }\left (t \right )&=2 y \left (t \right )+3 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_{3} {\mathrm e}^{2 t} \\ y \left (t \right ) &= \frac {3 c_{3} {\mathrm e}^{2 t}}{4}+c_{2} {\mathrm e}^{-2 t} \\ z \left (t \right ) &= c_{1} {\mathrm e}^{3 t}-\frac {3 c_{3} {\mathrm e}^{2 t}}{2}-\frac {2 c_{2} {\mathrm e}^{-2 t}}{5} \\ \end{align*} |
Problem 10222
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=4 x \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )-2 y \left (t \right )\\ z^{\prime }\left (t \right )&=x \left (t \right )-4 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_{3} {\mathrm e}^{4 t} \\ y \left (t \right ) &= \frac {c_{3} {\mathrm e}^{4 t}}{6}+c_{2} {\mathrm e}^{-2 t} \\ z \left (t \right ) &= \frac {c_{3} {\mathrm e}^{4 t}}{9}+c_{1} {\mathrm e}^{t}+\frac {4 c_{2} {\mathrm e}^{-2 t}}{3} \\ \end{align*} |
Problem 10223
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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 )+y \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} +c_{3} {\mathrm e}^{t} \\ y \left (t \right ) &= c_{3} {\mathrm e}^{t} t +c_{1} {\mathrm e}^{t}-c_{2} \\ z \left (t \right ) &= c_{3} {\mathrm e}^{t} t +c_{1} {\mathrm e}^{t}-c_{3} {\mathrm e}^{t}-c_{2} \\ \end{align*} |
Problem 10224
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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 )+y \left (t \right )+t\\ z^{\prime }\left (t \right )&=x \left (t \right )+z \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} +c_{3} {\mathrm e}^{t} \\ y \left (t \right ) &= c_{3} {\mathrm e}^{t} t +c_{1} {\mathrm e}^{t}-c_{2} -t -1 \\ z \left (t \right ) &= c_{3} {\mathrm e}^{t} t +c_{1} {\mathrm e}^{t}-c_{3} {\mathrm e}^{t}-c_{2} -t -1 \\ \end{align*} |
Problem 10225
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=\frac {b c y \left (t \right )}{a}-\frac {b c z \left (t \right )}{a}\\ y^{\prime }\left (t \right )&=-\frac {c a x \left (t \right )}{b}+\frac {c a z \left (t \right )}{b}\\ z^{\prime }\left (t \right )&=\frac {a b x \left (t \right )}{c}-\frac {a b y \left (t \right )}{c} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} +c_{2} \sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right )+c_{3} \cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) \\ y \left (t \right ) &= -\frac {\sqrt {a^{2}+b^{2}+c^{2}}\, \sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{3} a c -\sqrt {a^{2}+b^{2}+c^{2}}\, \cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{2} a c +\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{2} a^{2} b +\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{3} a^{2} b -c_{1} b^{3}-c_{1} b \,c^{2}}{b \left (b^{2}+c^{2}\right )} \\ z \left (t \right ) &= \frac {\sqrt {a^{2}+b^{2}+c^{2}}\, \sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{3} a b -\sqrt {a^{2}+b^{2}+c^{2}}\, \cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{2} a b -\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{2} a^{2} c -\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{3} a^{2} c +c_{1} b^{2} c +c_{1} c^{3}}{\left (b^{2}+c^{2}\right ) c} \\ \end{align*} |
Problem 10226
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=c y \left (t \right )-b z \left (t \right )\\ y^{\prime }\left (t \right )&=a z \left (t \right )-c x \left (t \right )\\ z^{\prime }\left (t \right )&=b x \left (t \right )-a 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} +c_{2} \sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right )+c_{3} \cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) \\ y \left (t \right ) &= -\frac {\sqrt {a^{2}+b^{2}+c^{2}}\, \sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{3} a c -\sqrt {a^{2}+b^{2}+c^{2}}\, \cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{2} a c +\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{2} a^{2} b +\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{3} a^{2} b -c_{1} b^{3}-c_{1} b \,c^{2}}{a \left (b^{2}+c^{2}\right )} \\ z \left (t \right ) &= \frac {\sqrt {a^{2}+b^{2}+c^{2}}\, \sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{3} a b -\sqrt {a^{2}+b^{2}+c^{2}}\, \cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{2} a b -\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{2} a^{2} c -\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{3} a^{2} c +c_{1} b^{2} c +c_{1} c^{3}}{a \left (b^{2}+c^{2}\right )} \\ \end{align*} |
Problem 10227
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=h \left (t \right ) y \left (t \right )-g \left (t \right ) z \left (t \right )\\ y^{\prime }\left (t \right )&=f \left (t \right ) z \left (t \right )-h \left (t \right ) x \left (t \right )\\ z^{\prime }\left (t \right )&=x \left (t \right ) g \left (t \right )-y \left (t \right ) f \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\[ \text {No solution found} \] |
Problem 10228
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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 )&=y \left (t \right )+z \left (t \right )-x \left (t \right )\\ z^{\prime }\left (t \right )&=z \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 ) &= {\mathrm e}^{t} \left (c_{1} +\sin \left (\sqrt {3}\, t \right ) c_{2} +\cos \left (\sqrt {3}\, t \right ) c_{3} \right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{t} \left (\sin \left (\sqrt {3}\, t \right ) \sqrt {3}\, c_{3} -\cos \left (\sqrt {3}\, t \right ) \sqrt {3}\, c_{2} +\sin \left (\sqrt {3}\, t \right ) c_{2} +\cos \left (\sqrt {3}\, t \right ) c_{3} -2 c_{1} \right )}{2} \\ z \left (t \right ) &= \frac {{\mathrm e}^{t} \left (\sin \left (\sqrt {3}\, t \right ) \sqrt {3}\, c_{3} -\cos \left (\sqrt {3}\, t \right ) \sqrt {3}\, c_{2} -\sin \left (\sqrt {3}\, t \right ) c_{2} -\cos \left (\sqrt {3}\, t \right ) c_{3} +2 c_{1} \right )}{2} \\ \end{align*} |
Problem 10229
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=-3 x \left (t \right )+48 y \left (t \right )-28 z \left (t \right )\\ y^{\prime }\left (t \right )&=-4 x \left (t \right )+40 y \left (t \right )-22 z \left (t \right )\\ z^{\prime }\left (t \right )&=-6 x \left (t \right )+57 y \left (t \right )-31 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_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{3 t} \\ y \left (t \right ) &= \frac {2 c_{1} {\mathrm e}^{t}}{3}+\frac {c_{2} {\mathrm e}^{2 t}}{4}+c_{3} {\mathrm e}^{3 t} \\ z \left (t \right ) &= c_{1} {\mathrm e}^{t}+\frac {c_{2} {\mathrm e}^{2 t}}{4}+\frac {3 c_{3} {\mathrm e}^{3 t}}{2} \\ \end{align*} |
Problem 10230
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=6 x \left (t \right )-72 y \left (t \right )+44 z \left (t \right )\\ y^{\prime }\left (t \right )&=4 x \left (t \right )-4 y \left (t \right )+26 z \left (t \right )\\ z^{\prime }\left (t \right )&=6 x \left (t \right )-63 y \left (t \right )+38 z \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= \cos \left (\frac {\left (\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+3542\right ) t \sqrt {3}\, 4^{\frac {1}{3}}}{12 \left (131737+9 \sqrt {351406311}\right )^{\frac {1}{3}}}\right ) {\mathrm e}^{\frac {\left (-3542+\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+80 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}\right ) t}{6 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}} c_{3} +\sin \left (\frac {\left (\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+3542\right ) t \sqrt {3}\, 4^{\frac {1}{3}}}{12 \left (131737+9 \sqrt {351406311}\right )^{\frac {1}{3}}}\right ) {\mathrm e}^{\frac {\left (-3542+\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+80 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}\right ) t}{6 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}} c_{2} +c_{1} {\mathrm e}^{-\frac {\left (\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-40 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-3542\right ) t}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}} \\ \text {Expression too large to display} \\ \text {Expression too large to display} \\ \end{align*} |
Problem 10231
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=a x \left (t \right )+g y \left (t \right )+\beta z \left (t \right )\\ y^{\prime }\left (t \right )&=g x \left (t \right )+b y \left (t \right )+\alpha z \left (t \right )\\ z^{\prime }\left (t \right )&=\beta x \left (t \right )+\alpha y \left (t \right )+c z \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} \\ \text {Expression too large to display} \\ \end{align*} |
Problem 10232
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=\frac {2 x \left (t \right )}{t}-1\\ y^{\prime }\left (t \right )&=-\frac {x \left (t \right )}{t^{3}}+\frac {y \left (t \right )}{t}+\frac {1}{t^{2}}\\ z^{\prime }\left (t \right )&=-\frac {x \left (t \right )}{t^{4}}-\frac {y \left (t \right )}{t^{2}}+\frac {z \left (t \right )}{t}+\frac {1}{t^{3}} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{3} t^{2}+t \\ y \left (t \right ) &= c_{2} t +c_{3} \\ z \left (t \right ) &= \frac {c_{1} t^{2}+c_{2} t +c_{3}}{t} \\ \end{align*} |
Problem 10233
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=-\frac {b c z \left (t \right )}{a t}+\frac {b c y \left (t \right )}{a t}\\ y^{\prime }\left (t \right )&=-\frac {c a x \left (t \right )}{b t}+\frac {c a z \left (t \right )}{b t}\\ z^{\prime }\left (t \right )&=\frac {a b x \left (t \right )}{c t}-\frac {a b y \left (t \right )}{c t} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} +c_{2} \sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right )+c_{3} \cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right ) \\ y \left (t \right ) &= \frac {\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right ) \sqrt {a^{2}+b^{2}+c^{2}}\, c_{2} a c -\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right ) c_{3} a^{2} b -\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right ) \sqrt {a^{2}+b^{2}+c^{2}}\, c_{3} a c -\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right ) c_{2} a^{2} b +c_{1} b^{3}+c_{1} b \,c^{2}}{b \left (b^{2}+c^{2}\right )} \\ z \left (t \right ) &= -\frac {\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right ) \sqrt {a^{2}+b^{2}+c^{2}}\, c_{2} a b +\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right ) c_{3} a^{2} c -\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right ) \sqrt {a^{2}+b^{2}+c^{2}}\, c_{3} a b +\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right ) c_{2} a^{2} c -c_{1} b^{2} c -c_{1} c^{3}}{\left (b^{2}+c^{2}\right ) c} \\ \end{align*} |
Problem 10234
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=a x_{2} \left (t \right )+b x_{3} \left (t \right ) \cos \left (c t \right )+b x_{4} \left (t \right ) \sin \left (c t \right )\\ x_{2}^{\prime }\left (t \right )&=-a x_{1} \left (t \right )+b x_{3} \left (t \right ) \sin \left (c t \right )-b x_{4} \left (t \right ) \cos \left (c t \right )\\ x_{3}^{\prime }\left (t \right )&=-b x_{1} \left (t \right ) \cos \left (c t \right )-b x_{2} \left (t \right ) \sin \left (c t \right )+a x_{4} \left (t \right )\\ x_{4}^{\prime }\left (t \right )&=-b x_{1} \left (t \right ) \sin \left (c t \right )+b x_{2} \left (t \right ) \cos \left (c t \right )-a x_{3} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} \left \{x_{1} \left (t \right ) &= c_{2} +c_{3} \sin \left (c t \right )+c_{4} \cos \left (c t \right ), x_{2} \left (t \right ) &= -\cos \left (c t \right ) c_{3} +\sin \left (c t \right ) c_{4} +c_{1}, x_{3} \left (t \right ) &= \frac {b \left (\cos \left (c t \right ) c_{1} a -\sin \left (c t \right ) c_{2} a -c_{3} a -c_{3} c \right )}{\left (a +c \right ) a}, x_{4} \left (t \right ) &= \frac {b \left (\cos \left (c t \right ) c_{2} a +\sin \left (c t \right ) c_{1} a +c_{4} a +c_{4} c \right )}{\left (a +c \right ) a}\right \} \\ \text {Expression too large to display} \\ \end{align*} |
Problem 10235
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right ) y \left (t \right )-x \left (t \right )^{2}\\ y^{\prime }\left (t \right )&=x \left (t \right ) y \left (t \right )+y \left (t \right )^{2} \end {align*}
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program solution |
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Maple solution |
\begin{align*} \left [\{x \left (t \right ) = 0\}, \left \{y \left (t \right ) &= \frac {1}{-t +c_{1}}\right \}\right ] \\ \left [\left \{x \left (t \right ) &= \frac {\tanh \left (\frac {c_{2} +t}{c_{1}}\right )}{c_{1}}\right \}, \left \{y \left (t \right ) &= -\frac {x \left (t \right )^{2}+\frac {d}{d t}x \left (t \right )}{x \left (t \right )}\right \}\right ] \\ \end{align*} |
Problem 10236
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=y \left (t \right ) x \left (t \right ) a +b x \left (t \right )\\ y^{\prime }\left (t \right )&=y \left (t \right ) x \left (t \right ) c +y \left (t \right ) d \end {align*}
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program solution |
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Maple solution |
\begin{align*} \\ \left [\left \{x \left (t \right ) &= \operatorname {RootOf}\left (-\left (\int _{}^{\textit {\_Z}}\frac {1}{b \textit {\_a} \left (\operatorname {LambertW}\left (\frac {{\mathrm e}^{-1} \textit {\_a}^{\frac {d}{b}} {\mathrm e}^{\frac {\textit {\_a} c}{b}} {\mathrm e}^{\frac {c_{1}}{b}}}{b}\right )+1\right )}d \textit {\_a} \right )+t +c_{2} \right )\right \}, \left \{y \left (t \right ) &= \frac {-b x \left (t \right )+\frac {d}{d t}x \left (t \right )}{a x \left (t \right )}\right \}\right ] \\ \end{align*} |
Problem 10237
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )^{2} a p +x \left (t \right ) y \left (t \right ) a q +x \left (t \right ) \alpha \\ y^{\prime }\left (t \right )&=x \left (t \right ) y \left (t \right ) b p +y \left (t \right )^{2} b q +y \left (t \right ) \beta \end {align*}
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program solution |
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Maple solution |
\[ \text {No solution found} \] |
Problem 10238
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )^{2} h +x \left (t \right ) y \left (t \right ) h -x \left (t \right ) a h -x \left (t \right ) c h -y \left (t \right ) a h +a c h\\ y^{\prime }\left (t \right )&=x \left (t \right ) y \left (t \right ) k -x \left (t \right ) b k +y \left (t \right )^{2} k -y \left (t \right ) b k -y \left (t \right ) c k +b c k \end {align*}
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program solution |
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Maple solution |
\begin{align*} \left [\{x \left (t \right ) = a\}, \left \{y \left (t \right ) &= -\frac {a \,{\mathrm e}^{a c_{1} k +a k t +b c_{1} k +b k t -c c_{1} k -c k t}-c \,{\mathrm e}^{a c_{1} k +a k t +b c_{1} k +b k t -c c_{1} k -c k t}+b}{-1+{\mathrm e}^{a c_{1} k +a k t +b c_{1} k +b k t -c c_{1} k -c k t}}\right \}\right ] \\ \left [\left \{x \left (t \right ) &= \operatorname {RootOf}\left (-\left (\int _{}^{\textit {\_Z}}\frac {\left (\textit {\_a} -a \right )^{-\frac {k}{h}}}{\left (h \left (\textit {\_a} -a \right )^{-\frac {k}{h}} \textit {\_a} +h \left (\textit {\_a} -a \right )^{-\frac {k}{h}} b -h \left (\textit {\_a} -a \right )^{-\frac {k}{h}} c +c_{1} \right ) \left (\textit {\_a} -a \right )}d \textit {\_a} \right )+t +c_{2} \right )\right \}, \left \{y \left (t \right ) &= \frac {-x \left (t \right )^{2} h +x \left (t \right ) a h +x \left (t \right ) c h -a c h +\frac {d}{d t}x \left (t \right )}{x \left (t \right ) h -h a}\right \}\right ] \\ \end{align*} |
Problem 10239
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=y \left (t \right )^{2}-\cos \left (x \left (t \right )\right )\\ y^{\prime }\left (t \right )&=-y \left (t \right ) \sin \left (x \left (t \right )\right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} \left \{x \left (t \right ) &= \operatorname {RootOf}\left (-2 \left (\int _{}^{\textit {\_Z}}\frac {1}{-3 \tan \left (\operatorname {RootOf}\left (-3 \sqrt {-\cos \left (\textit {\_f} \right )^{2}}\, \ln \left (\frac {9 \cos \left (\textit {\_f} \right )^{2} \tan \left (\textit {\_Z} \right )^{2}}{4}+\frac {9 \cos \left (\textit {\_f} \right )^{2}}{4}\right )+c_{1} \sqrt {-\cos \left (\textit {\_f} \right )^{2}}+2 \textit {\_Z} \cos \left (\textit {\_f} \right )\right )\right ) \sqrt {-\cos \left (\textit {\_f} \right )^{2}}+\cos \left (\textit {\_f} \right )}d \textit {\_f} \right )+t +c_{2} \right )\right \} \\ \left \{y \left (t \right ) &= \sqrt {\frac {d}{d t}x \left (t \right )+\cos \left (x \left (t \right )\right )}, y \left (t \right ) &= -\sqrt {\frac {d}{d t}x \left (t \right )+\cos \left (x \left (t \right )\right )}\right \} \\ \end{align*} |
Problem 10240
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right ) y \left (t \right )^{2}+x \left (t \right )+y \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )^{2} y \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 |
\[ \text {No solution found} \] |
Problem 10241
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right ) y \left (t \right )^{2}-x \left (t \right )^{3}+y \left (t \right )+x \left (t \right )\\ y^{\prime }\left (t \right )&=-x \left (t \right )^{2} y \left (t \right )-y \left (t \right )^{3}-x \left (t \right )+y \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\[ \text {No solution found} \] |
Problem 10242
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right ) y \left (t \right )^{2}+x \left (t \right )^{3}-y \left (t \right )-x \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )^{2} y \left (t \right )+y \left (t \right )^{3}+x \left (t \right )-y \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\[ \text {No solution found} \] |
Problem 10243
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right )^{2} y \left (t \right )-y \left (t \right )^{3}\\ y^{\prime }\left (t \right )&=\left \{\begin {array}{cc} x \left (t \right )^{2}+y \left (t \right )^{2} & 2 x \left (t \right )\le x \left (t \right )^{2}+y \left (t \right )^{2} \\ \frac {x \left (t \right )^{3}}{2}-\frac {y \left (t \right )^{4}}{2 x \left (t \right )} & \operatorname {otherwise} \end {array}\right . \end {align*}
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program solution |
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Maple solution |
\[ \text {No solution found} \] |
Problem 10244
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=-y \left (t \right )+\left (\left \{\begin {array}{cc} x \left (t \right )^{3} \sin \left (\frac {1}{x \left (t \right )^{2}+y \left (t \right )^{2}}\right )+x \left (t \right ) \sin \left (\frac {1}{x \left (t \right )^{2}+y \left (t \right )^{2}}\right ) y \left (t \right )^{2}-x \left (t \right ) \sin \left (\frac {1}{x \left (t \right )^{2}+y \left (t \right )^{2}}\right ) & x \left (t \right )^{2}+y \left (t \right )^{2}\neq 1 \\ 0 & \operatorname {otherwise} \end {array}\right .\right )\\ y^{\prime }\left (t \right )&=x \left (t \right )+\left (\left \{\begin {array}{cc} y \left (t \right ) \sin \left (\frac {1}{x \left (t \right )^{2}+y \left (t \right )^{2}}\right ) x \left (t \right )^{2}+y \left (t \right )^{3} \sin \left (\frac {1}{x \left (t \right )^{2}+y \left (t \right )^{2}}\right )-y \left (t \right ) \sin \left (\frac {1}{x \left (t \right )^{2}+y \left (t \right )^{2}}\right ) & x \left (t \right )^{2}+y \left (t \right )^{2}\neq 1 \\ 0 & \operatorname {otherwise} \end {array}\right .\right ) \end {align*}
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program solution |
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Maple solution |
\[ \text {No solution found} \] |
Problem 10245
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=-\frac {x \left (t \right ) t}{t^{2}+1}+\frac {y \left (t \right )}{t^{2}+1}\\ y^{\prime }\left (t \right )&=-\frac {t y \left (t \right )}{t^{2}+1}-\frac {x \left (t \right )}{t^{2}+1} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= \frac {c_{1} t +c_{2}}{t^{2}+1} \\ y \left (t \right ) &= \frac {-c_{2} t +c_{1}}{t^{2}+1} \\ \end{align*} |
Problem 10246
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=-\frac {2 x \left (t \right ) t}{x \left (t \right )^{2}+y \left (t \right )^{2}-t^{2}}\\ y^{\prime }\left (t \right )&=-\frac {2 t y \left (t \right )}{x \left (t \right )^{2}+y \left (t \right )^{2}-t^{2}} \end {align*}
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program solution |
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Maple solution |
\begin{align*} \left [\{x \left (t \right ) = 0\}, \left \{y \left (t \right ) &= \frac {1+\sqrt {-4 c_{1}^{2} t^{2}+1}}{2 c_{1}}, y \left (t \right ) &= -\frac {-1+\sqrt {-4 c_{1}^{2} t^{2}+1}}{2 c_{1}}\right \}\right ] \\ \left [\left \{x \left (t \right ) &= -\frac {-c_{1} +\sqrt {-2 c_{2} t^{2}+c_{1}^{2}}}{2 c_{2}}, x \left (t \right ) &= \frac {c_{1} +\sqrt {-2 c_{2} t^{2}+c_{1}^{2}}}{2 c_{2}}\right \}, \left \{y \left (t \right ) &= \frac {\sqrt {-\left (\frac {d}{d t}x \left (t \right )\right ) \left (\left (\frac {d}{d t}x \left (t \right )\right ) x \left (t \right )^{2}-t^{2} \left (\frac {d}{d t}x \left (t \right )\right )+2 x \left (t \right ) t \right )}}{\frac {d}{d t}x \left (t \right )}, y \left (t \right ) &= -\frac {\sqrt {-\left (\frac {d}{d t}x \left (t \right )\right ) \left (\left (\frac {d}{d t}x \left (t \right )\right ) x \left (t \right )^{2}-t^{2} \left (\frac {d}{d t}x \left (t \right )\right )+2 x \left (t \right ) t \right )}}{\frac {d}{d t}x \left (t \right )}\right \}\right ] \\ \end{align*} |
Problem 10247
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=\operatorname {RootOf}\left (\textit {\_Z}^{3}+2 t \,\textit {\_Z}^{2}+\left (t^{2}-x \left (t \right )\right ) \textit {\_Z} +a y \left (t \right )-x \left (t \right ) t \right )\\ y^{\prime }\left (t \right )&=-\frac {t \operatorname {RootOf}\left (\textit {\_Z}^{3}+2 t \,\textit {\_Z}^{2}+\left (t^{2}-x \left (t \right )\right ) \textit {\_Z} +a y \left (t \right )-x \left (t \right ) t \right )}{a}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+2 t \,\textit {\_Z}^{2}+\left (t^{2}-x \left (t \right )\right ) \textit {\_Z} +a y \left (t \right )-x \left (t \right ) t \right )^{2}}{a}+\frac {x \left (t \right )}{a} \end {align*}
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program solution |
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Maple solution |
\begin{align*} \left [\left \{x \left (t \right ) &= -\frac {t^{2}}{3}\right \}, \left \{y \left (t \right ) &= -\frac {t^{3}}{27 a}\right \}\right ] \\ \left [\{x \left (t \right ) &= c_{1} t +c_{2}\}, \left \{y \left (t \right ) &= \frac {-\left (\frac {d}{d t}x \left (t \right )\right )^{3}-2 \left (\frac {d}{d t}x \left (t \right )\right )^{2} t -t^{2} \left (\frac {d}{d t}x \left (t \right )\right )+x \left (t \right ) \left (\frac {d}{d t}x \left (t \right )\right )+x \left (t \right ) t}{a}\right \}\right ] \\ \left [\left \{x \left (t \right ) &= -\frac {5 t^{2}}{12}-\frac {t \left (-t -\sqrt {3}\, c_{1} \right )}{6}+\frac {c_{1}^{2}}{4}, x \left (t \right ) &= -\frac {5 t^{2}}{12}-\frac {t \left (-t +\sqrt {3}\, c_{1} \right )}{6}+\frac {c_{1}^{2}}{4}, x \left (t \right ) &= -\frac {5 t^{2}}{12}+\frac {t \left (t -\sqrt {3}\, c_{1} \right )}{6}+\frac {c_{1}^{2}}{4}, x \left (t \right ) &= -\frac {5 t^{2}}{12}+\frac {t \left (t +\sqrt {3}\, c_{1} \right )}{6}+\frac {c_{1}^{2}}{4}\right \}, \left \{y \left (t \right ) &= -\frac {-2 t^{2} \left (\frac {d}{d t}x \left (t \right )\right )-2 t^{3}-6 x \left (t \right ) \left (\frac {d}{d t}x \left (t \right )\right )-7 x \left (t \right ) t}{9 a}\right \}\right ] \\ \end{align*} |
Problem 10248
|
ODE |
\begin {align*} x \left (t \right )&=t x^{\prime }\left (t \right )+f \left (x^{\prime }\left (t \right ), y^{\prime }\left (t \right )\right )\\ y \left (t \right )&=y^{\prime }\left (t \right ) t +g \left (x^{\prime }\left (t \right ), y^{\prime }\left (t \right )\right ) \end {align*}
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program solution |
N/A |
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Maple solution |
\begin{align*} \{\int \operatorname {RootOf}\left (f \left (\frac {d}{d t}x \left (t \right ), \textit {\_Z}\right )+t \left (\frac {d}{d t}x \left (t \right )\right )-x \left (t \right )\right )d t +c_{1} &= \operatorname {RootOf}\left (f \left (\frac {d}{d t}x \left (t \right ), \textit {\_Z}\right )+t \left (\frac {d}{d t}x \left (t \right )\right )-x \left (t \right )\right ) t +g \left (\frac {d}{d t}x \left (t \right ), \operatorname {RootOf}\left (f \left (\frac {d}{d t}x \left (t \right ), \textit {\_Z}\right )+t \left (\frac {d}{d t}x \left (t \right )\right )-x \left (t \right )\right )\right )\} \\ \{y \left (t \right ) &= \int \operatorname {RootOf}\left (f \left (\frac {d}{d t}x \left (t \right ), \textit {\_Z}\right )+t \left (\frac {d}{d t}x \left (t \right )\right )-x \left (t \right )\right )d t +c_{1}\} \\ \end{align*} |
Problem 10249
|
ODE |
\begin {align*} x^{\prime \prime }\left (t \right )&=a \,{\mathrm e}^{2 x \left (t \right )}-{\mathrm e}^{-x \left (t \right )}+{\mathrm e}^{-2 x \left (t \right )} \left (\cos ^{2}\left (y \left (t \right )\right )\right )\\ y^{\prime \prime }\left (t \right )&={\mathrm e}^{-2 x \left (t \right )} \sin \left (y \left (t \right )\right ) \cos \left (y \left (t \right )\right )-\frac {\sin \left (y \left (t \right )\right )}{\cos \left (y \left (t \right )\right )^{3}} \end {align*}
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program solution |
N/A |
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Maple solution |
\[ \text {No solution found} \] |
Problem 10250
|
ODE |
\begin {align*} x^{\prime \prime }\left (t \right )&=\frac {k x \left (t \right )}{\left (x \left (t \right )^{2}+y \left (t \right )^{2}\right )^{\frac {3}{2}}}\\ y^{\prime \prime }\left (t \right )&=\frac {k y \left (t \right )}{\left (x \left (t \right )^{2}+y \left (t \right )^{2}\right )^{\frac {3}{2}}} \end {align*}
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program solution |
N/A |
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Maple solution |
\[ \text {No solution found} \] |
Problem 10251
|
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 )^{2}+y \left (t \right )\\ z^{\prime }\left (t \right )&=x \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} +c_{3} {\mathrm e}^{t}\} \\ \{y \left (t \right ) &= \left (\int x \left (t \right )^{2} {\mathrm e}^{-t}d t +c_{1} \right ) {\mathrm e}^{t}\} \\ \{z \left (t \right ) &= -\frac {d}{d t}x \left (t \right )+y \left (t \right )\} \\ \end{align*} |
Problem 10252
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=-\frac {y \left (t \right ) z \left (t \right ) c}{a}+\frac {y \left (t \right ) z \left (t \right ) b}{a}\\ y^{\prime }\left (t \right )&=-\frac {x \left (t \right ) z \left (t \right ) a}{b}+\frac {x \left (t \right ) z \left (t \right ) c}{b}\\ z^{\prime }\left (t \right )&=\frac {x \left (t \right ) y \left (t \right ) a}{c}-\frac {x \left (t \right ) y \left (t \right ) b}{c} \end {align*}
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program solution |
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Maple solution |
\begin{align*} \\ \\ \\ \text {Expression too large to display} \\ \end{align*} |
Problem 10253
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right ) y \left (t \right )-x \left (t \right ) z \left (t \right )\\ y^{\prime }\left (t \right )&=-x \left (t \right ) y \left (t \right )+y \left (t \right ) z \left (t \right )\\ z^{\prime }\left (t \right )&=x \left (t \right ) z \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 |
\[ \text {No solution found} \] |
Problem 10254
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=-\frac {y \left (t \right ) z \left (t \right )}{2}+\frac {x \left (t \right ) z \left (t \right )}{2}+\frac {x \left (t \right ) y \left (t \right )}{2}\\ y^{\prime }\left (t \right )&=\frac {y \left (t \right ) z \left (t \right )}{2}-\frac {x \left (t \right ) z \left (t \right )}{2}+\frac {x \left (t \right ) y \left (t \right )}{2}\\ z^{\prime }\left (t \right )&=\frac {x \left (t \right ) z \left (t \right )}{2}+\frac {y \left (t \right ) z \left (t \right )}{2}-\frac {x \left (t \right ) y \left (t \right )}{2} \end {align*}
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program solution |
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Maple solution |
\begin{align*} \left [\left \{x \left (t \right ) &= \frac {2}{2 c_{2} -t}\right \}, \left \{y \left (t \right ) &= \left (\int -\frac {x \left (t \right )^{2} {\mathrm e}^{-\left (\int x \left (t \right )d t \right )}}{2}d t +c_{1} \right ) {\mathrm e}^{\int x \left (t \right )d t}\right \}, \{z \left (t \right ) = x \left (t \right )\}\right ] \\ \left [\left \{x \left (t \right ) &= \frac {2}{2 c_{2} -t}\right \}, \{y \left (t \right ) = x \left (t \right )\}, \left \{z \left (t \right ) &= \left (\int -\frac {x \left (t \right )^{2} {\mathrm e}^{-\left (\int x \left (t \right )d t \right )}}{2}d t +c_{1} \right ) {\mathrm e}^{\int x \left (t \right )d t}\right \}\right ] \\ \text {Expression too large to display} \\ \end{align*} |
Problem 10255
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=\frac {x \left (t \right )^{2}}{2}-\frac {y \left (t \right )}{24}\\ y^{\prime }\left (t \right )&=2 x \left (t \right ) y \left (t \right )-3 z \left (t \right )\\ z^{\prime }\left (t \right )&=3 x \left (t \right ) z \left (t \right )-\frac {y \left (t \right )^{2}}{6} \end {align*}
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program solution |
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Maple solution |
\[ \text {No solution found} \] |
Problem 10256
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right ) y \left (t \right )^{2}-z \left (t \right )^{2} x \left (t \right )\\ y^{\prime }\left (t \right )&=-x \left (t \right )^{2} y \left (t \right )+y \left (t \right ) z \left (t \right )^{2}\\ z^{\prime }\left (t \right )&=x \left (t \right )^{2} z \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 |
\[ \text {No solution found} \] |
Problem 10257
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right ) y \left (t \right )^{2}-z \left (t \right )^{2} x \left (t \right )\\ y^{\prime }\left (t \right )&=-x \left (t \right )^{2} y \left (t \right )-y \left (t \right ) z \left (t \right )^{2}\\ z^{\prime }\left (t \right )&=x \left (t \right )^{2} z \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*} \\ \left [\{x \left (t \right ) = 0\}, \left \{y \left (t \right ) &= \frac {\sqrt {-\left ({\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}-1\right ) c_{1} {\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}}}{{\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}-1}, y \left (t \right ) &= -\frac {\sqrt {-\left ({\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}-1\right ) c_{1} {\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}}}{{\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}-1}\right \}, \left \{z \left (t \right ) &= \frac {\sqrt {-y \left (t \right ) \left (\frac {d}{d t}y \left (t \right )\right )}}{y \left (t \right )}, z \left (t \right ) &= -\frac {\sqrt {-y \left (t \right ) \left (\frac {d}{d t}y \left (t \right )\right )}}{y \left (t \right )}\right \}\right ] \\ \\ \left [\left \{x \left (t \right ) &= \frac {\sqrt {-\left ({\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}-1\right ) c_{1} {\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}}}{{\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}-1}, x \left (t \right ) &= -\frac {\sqrt {-\left ({\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}-1\right ) c_{1} {\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}}}{{\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}-1}\right \}, \{y \left (t \right ) = 0\}, \left \{z \left (t \right ) &= \frac {\sqrt {-x \left (t \right ) \left (\frac {d}{d t}x \left (t \right )\right )}}{x \left (t \right )}, z \left (t \right ) &= -\frac {\sqrt {-x \left (t \right ) \left (\frac {d}{d t}x \left (t \right )\right )}}{x \left (t \right )}\right \}\right ] \\ \left [\left \{x \left (t \right ) &= \operatorname {RootOf}\left (-\left (\int _{}^{\textit {\_Z}}-\frac {2}{\sqrt {4 \textit {\_a}^{4}-16 \textit {\_a}^{2} c_{2} +16 c_{2}^{2}+c_{1}}\, \textit {\_a}}d \textit {\_a} \right )+t +c_{3} \right ), x \left (t \right ) &= \operatorname {RootOf}\left (-\left (\int _{}^{\textit {\_Z}}\frac {2}{\sqrt {4 \textit {\_a}^{4}-16 \textit {\_a}^{2} c_{2} +16 c_{2}^{2}+c_{1}}\, \textit {\_a}}d \textit {\_a} \right )+t +c_{3} \right )\right \}, \left \{y \left (t \right ) &= -\frac {\sqrt {-2 x \left (t \right ) \left (x \left (t \right )^{3}-\frac {d}{d t}x \left (t \right )-\sqrt {x \left (t \right )^{6}-\left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right ) x \left (t \right )+2 \left (\frac {d}{d t}x \left (t \right )\right )^{2}}\right )}}{2 x \left (t \right )}, y \left (t \right ) &= \frac {\sqrt {-2 x \left (t \right ) \left (x \left (t \right )^{3}-\frac {d}{d t}x \left (t \right )-\sqrt {x \left (t \right )^{6}-\left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right ) x \left (t \right )+2 \left (\frac {d}{d t}x \left (t \right )\right )^{2}}\right )}}{2 x \left (t \right )}, y \left (t \right ) &= -\frac {\sqrt {-2 x \left (t \right ) \left (x \left (t \right )^{3}-\frac {d}{d t}x \left (t \right )+\sqrt {x \left (t \right )^{6}-\left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right ) x \left (t \right )+2 \left (\frac {d}{d t}x \left (t \right )\right )^{2}}\right )}}{2 x \left (t \right )}, y \left (t \right ) &= \frac {\sqrt {-2 x \left (t \right ) \left (x \left (t \right )^{3}-\frac {d}{d t}x \left (t \right )+\sqrt {x \left (t \right )^{6}-\left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right ) x \left (t \right )+2 \left (\frac {d}{d t}x \left (t \right )\right )^{2}}\right )}}{2 x \left (t \right )}\right \}, \left \{z \left (t \right ) &= \frac {\sqrt {x \left (t \right ) \left (x \left (t \right ) y \left (t \right )^{2}-\frac {d}{d t}x \left (t \right )\right )}}{x \left (t \right )}, z \left (t \right ) &= -\frac {\sqrt {x \left (t \right ) \left (x \left (t \right ) y \left (t \right )^{2}-\frac {d}{d t}x \left (t \right )\right )}}{x \left (t \right )}\right \}\right ] \\ \end{align*} |
Problem 10258
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right ) y \left (t \right )^{2}+x \left (t \right )+y \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )^{2} y \left (t \right )-x \left (t \right )-y \left (t \right )\\ z^{\prime }\left (t \right )&=y \left (t \right )^{2}-x \left (t \right )^{2} \end {align*}
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program solution |
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Maple solution |
\[ \text {No solution found} \] |
Problem 10259
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=\frac {f \left (t \right )}{x \left (t \right )^{2}-x \left (t \right ) y \left (t \right )-x \left (t \right ) z \left (t \right )+y \left (t \right ) z \left (t \right )}\\ y^{\prime }\left (t \right )&=-\frac {f \left (t \right )}{x \left (t \right ) y \left (t \right )-x \left (t \right ) z \left (t \right )-y \left (t \right )^{2}+y \left (t \right ) z \left (t \right )}\\ z^{\prime }\left (t \right )&=\frac {f \left (t \right )}{x \left (t \right ) y \left (t \right )-x \left (t \right ) z \left (t \right )-y \left (t \right ) z \left (t \right )+z \left (t \right )^{2}} \end {align*}
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program solution |
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Maple solution |
\begin{align*} \text {Expression too large to display} \\ \left \{y \left (t \right ) &= \frac {4 \left (\frac {d}{d t}x \left (t \right )\right )^{3} x \left (t \right )+f \left (t \right ) \left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right )-\left (\frac {d}{d t}x \left (t \right )\right ) \left (\frac {d}{d t}f \left (t \right )\right )-\sqrt {-16 \left (\frac {d}{d t}x \left (t \right )\right )^{5} f \left (t \right )+\left (\frac {d}{d t}x \left (t \right )\right )^{2} \left (\frac {d}{d t}f \left (t \right )\right )^{2}-2 \left (\frac {d}{d t}x \left (t \right )\right ) \left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right ) f \left (t \right ) \left (\frac {d}{d t}f \left (t \right )\right )+\left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right )^{2} f \left (t \right )^{2}}}{4 \left (\frac {d}{d t}x \left (t \right )\right )^{3}}, y \left (t \right ) &= \frac {4 \left (\frac {d}{d t}x \left (t \right )\right )^{3} x \left (t \right )+f \left (t \right ) \left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right )-\left (\frac {d}{d t}x \left (t \right )\right ) \left (\frac {d}{d t}f \left (t \right )\right )+\sqrt {-16 \left (\frac {d}{d t}x \left (t \right )\right )^{5} f \left (t \right )+\left (\frac {d}{d t}x \left (t \right )\right )^{2} \left (\frac {d}{d t}f \left (t \right )\right )^{2}-2 \left (\frac {d}{d t}x \left (t \right )\right ) \left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right ) f \left (t \right ) \left (\frac {d}{d t}f \left (t \right )\right )+\left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right )^{2} f \left (t \right )^{2}}}{4 \left (\frac {d}{d t}x \left (t \right )\right )^{3}}\right \} \\ \left \{z \left (t \right ) &= \frac {-y \left (t \right ) \left (\frac {d}{d t}x \left (t \right )\right ) x \left (t \right )+\left (\frac {d}{d t}x \left (t \right )\right ) x \left (t \right )^{2}-f \left (t \right )}{x \left (t \right ) \left (\frac {d}{d t}x \left (t \right )\right )-y \left (t \right ) \left (\frac {d}{d t}x \left (t \right )\right )}\right \} \\ \end{align*} |
Problem 10260
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=\frac {x_{4} \left (t \right ) \sin \left (x_{3} \left (t \right )\right )}{\sin \left (x_{2} \left (t \right )\right )}+\frac {x_{5} \left (t \right ) \cos \left (x_{3} \left (t \right )\right )}{\sin \left (x_{2} \left (t \right )\right )}\\ x_{2}^{\prime }\left (t \right )&=x_{4} \left (t \right ) \cos \left (x_{3} \left (t \right )\right )-x_{5} \left (t \right ) \sin \left (x_{3} \left (t \right )\right )\\ x_{3}^{\prime }\left (t \right )&=-\frac {x_{4} \left (t \right ) \sin \left (x_{3} \left (t \right )\right ) \cos \left (x_{2} \left (t \right )\right )}{\sin \left (x_{2} \left (t \right )\right )}-\frac {\cos \left (x_{3} \left (t \right )\right ) x_{5} \left (t \right ) \cos \left (x_{2} \left (t \right )\right )}{\sin \left (x_{2} \left (t \right )\right )}+a\\ x_{4}^{\prime }\left (t \right )&=-m \sin \left (x_{2} \left (t \right )\right ) \cos \left (x_{3} \left (t \right )\right )-a x_{5} \left (t \right ) \lambda +a x_{5} \left (t \right )\\ x_{5}^{\prime }\left (t \right )&=m \sin \left (x_{2} \left (t \right )\right ) \sin \left (x_{3} \left (t \right )\right )+a x_{4} \left (t \right ) \lambda -a x_{4} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\[ \text {No solution found} \] |