Problem 701
| ODE |
\[ \boxed {t^{2} y^{\prime \prime }-2 t y^{\prime }+2 y=4 t^{2}} \] |
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program solution |
\[ y = t \left (c_{2} t +c_{1} \right )-4 t^{2}+4 t^{2} \ln \left (t \right ) \] Verified OK.
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Maple solution | \[ y \left (t \right ) = t \left (4 t \ln \left (t \right )+\left (c_{1} -4\right ) t +c_{2} \right ) \] |
Problem 702
| ODE |
\[ \boxed {t^{2} y^{\prime \prime }+7 t y^{\prime }+5 y=t} \] |
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program solution |
\[ y = \frac {t^{6}+3 c_{1} t^{4}+12 c_{2}}{12 t^{5}} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {t^{6}+3 c_{1} t^{4}-4 c_{1}^{3}+12 c_{2}}{12 t^{5}} \] |
Problem 703
| ODE |
\[ \boxed {t y^{\prime \prime }-\left (t +1\right ) y^{\prime }+y={\mathrm e}^{2 t} t^{2}} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{t}-c_{2} \left (t +1\right )+\frac {{\mathrm e}^{2 t} \left (-1+t \right )}{2} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \left (t +1\right ) c_{2} +{\mathrm e}^{t} c_{1} +\frac {\left (t -1\right ) {\mathrm e}^{2 t}}{2} \] |
Problem 704
| ODE |
\[ \boxed {\left (1-t \right ) y^{\prime \prime }+t y^{\prime }-y=2 \left (-1+t \right ) {\mathrm e}^{-t}} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{t}-c_{2} t +2 \,\operatorname {expIntegral}_{1}\left (2 t -2\right ) {\mathrm e}^{t -2}-2 \,{\mathrm e}^{-1} \operatorname {expIntegral}_{1}\left (-1+t \right ) t +{\mathrm e}^{-t} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = -2 \,{\mathrm e}^{-1} \operatorname {expIntegral}_{1}\left (t -1\right ) t +2 \,\operatorname {expIntegral}_{1}\left (2 t -2\right ) {\mathrm e}^{t -2}+{\mathrm e}^{t} c_{1} +c_{2} t +{\mathrm e}^{-t} \] |
Problem 705
| ODE |
\[ \boxed {u^{\prime \prime }+2 u=0} \] |
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program solution |
\[ u = c_{1} \cos \left (\sqrt {2}\, t \right )+\frac {c_{2} \sqrt {2}\, \sin \left (\sqrt {2}\, t \right )}{2} \] Verified OK.
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Maple solution | \[ u \left (t \right ) = c_{1} \sin \left (t \sqrt {2}\right )+c_{2} \cos \left (t \sqrt {2}\right ) \] |
Problem 706
| ODE |
\[ \boxed {u^{\prime \prime }+\frac {u^{\prime }}{4}+2 u=0} \] With initial conditions \begin {align*} [u \left (0\right ) = 0, u^{\prime }\left (0\right ) = 2] \end {align*}
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program solution |
\[ u = \frac {16 \sqrt {127}\, {\mathrm e}^{-\frac {t}{8}} \sin \left (\frac {\sqrt {127}\, t}{8}\right )}{127} \] Verified OK.
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Maple solution | \[ u \left (t \right ) = \frac {16 \sqrt {127}\, {\mathrm e}^{-\frac {t}{8}} \sin \left (\frac {\sqrt {127}\, t}{8}\right )}{127} \] |
Problem 707
| ODE |
\[ \boxed {u^{\prime \prime }+\frac {u^{\prime }}{8}+4 u=3 \cos \left (\frac {t}{4}\right )} \] With initial conditions \begin {align*} [u \left (0\right ) = 2, u^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ u = \frac {19658 \,{\mathrm e}^{-\frac {t}{16}} \cos \left (\frac {\sqrt {1023}\, t}{16}\right )}{15877}+\frac {19274 \,{\mathrm e}^{-\frac {t}{16}} \sin \left (\frac {\sqrt {1023}\, t}{16}\right ) \sqrt {1023}}{16242171}+\frac {12096 \cos \left (\frac {t}{4}\right )}{15877}+\frac {96 \sin \left (\frac {t}{4}\right )}{15877} \] Verified OK.
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Maple solution | \[ u \left (t \right ) = \frac {19274 \,{\mathrm e}^{-\frac {t}{16}} \sqrt {1023}\, \sin \left (\frac {\sqrt {1023}\, t}{16}\right )}{16242171}+\frac {19658 \,{\mathrm e}^{-\frac {t}{16}} \cos \left (\frac {\sqrt {1023}\, t}{16}\right )}{15877}+\frac {96 \sin \left (\frac {t}{4}\right )}{15877}+\frac {12096 \cos \left (\frac {t}{4}\right )}{15877} \] |
Problem 708
| ODE |
\[ \boxed {u^{\prime \prime }+\frac {u^{\prime }}{8}+4 u=3 \cos \left (2 t \right )} \] With initial conditions \begin {align*} [u \left (0\right ) = 2, u^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ u = -\frac {382 \,{\mathrm e}^{-\frac {t}{16}} \sin \left (\frac {\sqrt {1023}\, t}{16}\right ) \sqrt {1023}}{1023}+2 \,{\mathrm e}^{-\frac {t}{16}} \cos \left (\frac {\sqrt {1023}\, t}{16}\right )+12 \sin \left (2 t \right ) \] Verified OK.
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Maple solution | \[ u \left (t \right ) = -\frac {382 \,{\mathrm e}^{-\frac {t}{16}} \sqrt {1023}\, \sin \left (\frac {\sqrt {1023}\, t}{16}\right )}{1023}+2 \,{\mathrm e}^{-\frac {t}{16}} \cos \left (\frac {\sqrt {1023}\, t}{16}\right )+12 \sin \left (2 t \right ) \] |
Problem 709
| ODE |
\[ \boxed {u^{\prime \prime }+\frac {u^{\prime }}{8}+4 u=3 \cos \left (6 t \right )} \] With initial conditions \begin {align*} [u \left (0\right ) = 2, u^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ u = \frac {34322 \,{\mathrm e}^{-\frac {t}{16}} \cos \left (\frac {\sqrt {1023}\, t}{16}\right )}{16393}+\frac {2806 \,{\mathrm e}^{-\frac {t}{16}} \sin \left (\frac {\sqrt {1023}\, t}{16}\right ) \sqrt {1023}}{1524549}-\frac {1536 \cos \left (6 t \right )}{16393}+\frac {36 \sin \left (6 t \right )}{16393} \] Verified OK.
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Maple solution | \[ u \left (t \right ) = \frac {2806 \,{\mathrm e}^{-\frac {t}{16}} \sqrt {1023}\, \sin \left (\frac {\sqrt {1023}\, t}{16}\right )}{1524549}+\frac {34322 \,{\mathrm e}^{-\frac {t}{16}} \cos \left (\frac {\sqrt {1023}\, t}{16}\right )}{16393}+\frac {36 \sin \left (6 t \right )}{16393}-\frac {1536 \cos \left (6 t \right )}{16393} \] |
Problem 710
| ODE |
\[ \boxed {u^{\prime \prime }+u^{\prime }+\frac {u^{3}}{5}=\cos \left (t \right )} \] With initial conditions \begin {align*} [u \left (0\right ) = 2, u^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 711
| ODE |
\[ \boxed {y^{\prime \prime }-y=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\frac {1}{720} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{120} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{24} x^{4}\right ) c_{1} +\left (x +\frac {1}{6} x^{3}+\frac {1}{120} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = \left (1+\frac {1}{2} x^{2}+\frac {1}{24} x^{4}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{120} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \] |
Problem 712
| ODE |
\[ \boxed {y^{\prime \prime }-y^{\prime } x -y=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{8} x^{4}+\frac {1}{48} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{3} x^{3}+\frac {1}{15} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{8} x^{4}\right ) c_{1} +\left (x +\frac {1}{3} x^{3}+\frac {1}{15} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = \left (1+\frac {1}{2} x^{2}+\frac {1}{8} x^{4}\right ) y \left (0\right )+\left (x +\frac {1}{3} x^{3}+\frac {1}{15} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \] |
Problem 713
| ODE |
\[ \boxed {y^{\prime \prime }+k^{2} x^{2} y=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = \left (1-\frac {x^{4} k^{2}}{12}\right ) y \left (0\right )+\left (x -\frac {1}{20} k^{2} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {x^{4} k^{2}}{12}\right ) c_{1} +\left (x -\frac {1}{20} k^{2} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = \left (1-\frac {k^{2} x^{4}}{12}\right ) y \left (0\right )+\left (x -\frac {1}{20} k^{2} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \] |
Problem 714
| ODE |
\[ \boxed {\left (1-x \right ) y^{\prime \prime }+y=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{6} x^{3}-\frac {1}{24} x^{4}-\frac {1}{60} x^{5}-\frac {7}{720} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}-\frac {1}{12} x^{4}-\frac {1}{24} x^{5}-\frac {1}{40} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{6} x^{3}-\frac {1}{24} x^{4}-\frac {1}{60} x^{5}\right ) c_{1} +\left (x -\frac {1}{6} x^{3}-\frac {1}{12} x^{4}-\frac {1}{24} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}-\frac {1}{6} x^{3}-\frac {1}{24} x^{4}-\frac {1}{60} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}-\frac {1}{12} x^{4}-\frac {1}{24} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \] |
Problem 715
| ODE |
\[ \boxed {\left (x^{2}+2\right ) y^{\prime \prime }-y^{\prime } x +4 y=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = \left (1-x^{2}+\frac {1}{6} x^{4}-\frac {1}{30} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{4} x^{3}+\frac {7}{160} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-x^{2}+\frac {1}{6} x^{4}\right ) c_{1} +\left (x -\frac {1}{4} x^{3}+\frac {7}{160} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = \left (1-x^{2}+\frac {1}{6} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{4} x^{3}+\frac {7}{160} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \] |
Problem 716
| ODE |
\[ \boxed {y^{\prime \prime }+y^{\prime } x +2 y=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = \left (1-x^{2}+\frac {1}{3} x^{4}-\frac {1}{15} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{3}+\frac {1}{8} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-x^{2}+\frac {1}{3} x^{4}\right ) c_{1} +\left (x -\frac {1}{2} x^{3}+\frac {1}{8} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = \left (1-x^{2}+\frac {1}{3} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{3}+\frac {1}{8} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \] |
Problem 717
| ODE |
\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }-4 y^{\prime } x +6 y=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = \left (-3 x^{2}+1\right ) y \left (0\right )+\left (-\frac {1}{3} x^{3}+x \right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (-3 x^{2}+1\right ) c_{1} +\left (-\frac {1}{3} x^{3}+x \right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = y \left (0\right )+D\left (y \right )\left (0\right ) x -3 y \left (0\right ) x^{2}-\frac {D\left (y \right )\left (0\right ) x^{3}}{3} \] |
Problem 718
| ODE |
\[ \boxed {\left (-x^{2}+4\right ) y^{\prime \prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = \left (1-\frac {x^{2}}{4}\right ) y \left (0\right )+\left (x -\frac {1}{12} x^{3}-\frac {1}{240} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {x^{2}}{4}\right ) c_{1} +\left (x -\frac {1}{12} x^{3}-\frac {1}{240} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = \left (1-\frac {x^{2}}{4}\right ) y \left (0\right )+\left (x -\frac {1}{12} x^{3}-\frac {1}{240} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \] |
Problem 719
| ODE |
\[ \boxed {\left (-x^{2}+3\right ) y^{\prime \prime }-3 y^{\prime } x -y=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = \left (1+\frac {1}{6} x^{2}+\frac {1}{24} x^{4}+\frac {5}{432} x^{6}\right ) y \left (0\right )+\left (x +\frac {2}{9} x^{3}+\frac {8}{135} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{6} x^{2}+\frac {1}{24} x^{4}\right ) c_{1} +\left (x +\frac {2}{9} x^{3}+\frac {8}{135} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = \left (1+\frac {1}{6} x^{2}+\frac {1}{24} x^{4}\right ) y \left (0\right )+\left (x +\frac {2}{9} x^{3}+\frac {8}{135} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \] |
Problem 720
| ODE |
\[ \boxed {\left (1-x \right ) y^{\prime \prime }+y^{\prime } x -y=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}\right ) y \left (0\right )+y^{\prime }\left (0\right ) x +O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}\right ) c_{1} +c_{2} x +O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = \left (1+\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}\right ) y \left (0\right )+D\left (y \right )\left (0\right ) x +O\left (x^{6}\right ) \] |
Problem 721
| ODE |
\[ \boxed {2 y^{\prime \prime }+y^{\prime } x +3 y=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = \left (1-\frac {3}{4} x^{2}+\frac {5}{32} x^{4}-\frac {7}{384} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{20} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {3}{4} x^{2}+\frac {5}{32} x^{4}\right ) c_{1} +\left (x -\frac {1}{3} x^{3}+\frac {1}{20} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = \left (1-\frac {3}{4} x^{2}+\frac {5}{32} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{20} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \] |
Problem 722
| ODE |
\[ \boxed {y^{\prime \prime }-y^{\prime } x -y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 1] \end {align*} With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = x^{2}+x +2+\frac {x^{3}}{3}+\frac {x^{4}}{4}+\frac {x^{5}}{15}+\frac {x^{6}}{24}+O\left (x^{6}\right ) \] Verified OK. \[ y = 2+x^{2}+\frac {x^{4}}{4}+x +\frac {x^{3}}{3}+\frac {x^{5}}{15}+O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = 2+x +x^{2}+\frac {1}{3} x^{3}+\frac {1}{4} x^{4}+\frac {1}{15} x^{5}+\operatorname {O}\left (x^{6}\right ) \] |
Problem 723
| ODE |
\[ \boxed {\left (x^{2}+2\right ) y^{\prime \prime }-y^{\prime } x +4 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = 3] \end {align*} With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = x^{2}+3 x -1-\frac {3 x^{3}}{4}-\frac {x^{4}}{6}+\frac {21 x^{5}}{160}+\frac {x^{6}}{30}+O\left (x^{6}\right ) \] Verified OK. \[ y = -1+x^{2}-\frac {x^{4}}{6}+3 x -\frac {3 x^{3}}{4}+\frac {21 x^{5}}{160}+O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = -1+3 x +x^{2}-\frac {3}{4} x^{3}-\frac {1}{6} x^{4}+\frac {21}{160} x^{5}+\operatorname {O}\left (x^{6}\right ) \] |
Problem 724
| ODE |
\[ \boxed {y^{\prime \prime }+y^{\prime } x +2 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 4, y^{\prime }\left (0\right ) = -1] \end {align*} With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = -4 x^{2}-x +4+\frac {x^{3}}{2}+\frac {4 x^{4}}{3}-\frac {x^{5}}{8}-\frac {4 x^{6}}{15}+O\left (x^{6}\right ) \] Verified OK. \[ y = 4-4 x^{2}+\frac {4 x^{4}}{3}-x +\frac {x^{3}}{2}-\frac {x^{5}}{8}+O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = 4-x -4 x^{2}+\frac {1}{2} x^{3}+\frac {4}{3} x^{4}-\frac {1}{8} x^{5}+\operatorname {O}\left (x^{6}\right ) \] |
Problem 725
| ODE |
\[ \boxed {\left (1-x \right ) y^{\prime \prime }+y^{\prime } x -y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -3, y^{\prime }\left (0\right ) = 2] \end {align*} With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = -3+2 x -\frac {3 x^{2}}{2}-\frac {x^{3}}{2}-\frac {x^{4}}{8}-\frac {x^{5}}{40}-\frac {x^{6}}{240}+O\left (x^{6}\right ) \] Verified OK. \[ y = -3-\frac {3 x^{2}}{2}-\frac {x^{3}}{2}-\frac {x^{4}}{8}-\frac {x^{5}}{40}+2 x +O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = -3+2 x -\frac {3}{2} x^{2}-\frac {1}{2} x^{3}-\frac {1}{8} x^{4}-\frac {1}{40} x^{5}+\operatorname {O}\left (x^{6}\right ) \] |
Problem 726
| ODE |
\[ \boxed {y^{\prime \prime }-2 y^{\prime } x +\lambda y=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = \left (1-\frac {1}{2} \lambda \,x^{2}+\frac {1}{24} x^{4} \lambda ^{2}-\frac {1}{6} x^{4} \lambda -\frac {1}{720} x^{6} \lambda ^{3}+\frac {1}{60} x^{6} \lambda ^{2}-\frac {2}{45} x^{6} \lambda \right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3} \lambda +\frac {1}{3} x^{3}+\frac {1}{120} x^{5} \lambda ^{2}-\frac {1}{15} x^{5} \lambda +\frac {1}{10} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {\lambda \,x^{2}}{2}+\left (\frac {1}{24} \lambda ^{2}-\frac {1}{6} \lambda \right ) x^{4}\right ) c_{1} +\left (x +\left (-\frac {\lambda }{6}+\frac {1}{3}\right ) x^{3}+\left (\frac {1}{120} \lambda ^{2}-\frac {1}{15} \lambda +\frac {1}{10}\right ) x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = \left (1-\frac {\lambda \,x^{2}}{2}+\frac {\lambda \left (\lambda -4\right ) x^{4}}{24}\right ) y \left (0\right )+\left (x -\frac {\left (\lambda -2\right ) x^{3}}{6}+\frac {\left (\lambda -2\right ) \left (-6+\lambda \right ) x^{5}}{120}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \] |
Problem 727
| ODE |
\[ \boxed {y^{\prime \prime }-y^{\prime } x -y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*} With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = 1+\frac {x^{2}}{2}+\frac {x^{4}}{8}+\frac {x^{6}}{48}+O\left (x^{6}\right ) \] Verified OK. \[ y = 1+\frac {x^{2}}{2}+\frac {x^{4}}{8}+O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = 1+\frac {1}{2} x^{2}+\frac {1}{8} x^{4}+\operatorname {O}\left (x^{6}\right ) \] |
Problem 728
| ODE |
\[ \boxed {\left (x^{2}+2\right ) y^{\prime \prime }-y^{\prime } x +4 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*} With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = 1-x^{2}+\frac {x^{4}}{6}-\frac {x^{6}}{30}+O\left (x^{6}\right ) \] Verified OK. \[ y = 1-x^{2}+\frac {x^{4}}{6}+O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = 1-x^{2}+\frac {1}{6} x^{4}+\operatorname {O}\left (x^{6}\right ) \] |
Problem 729
| ODE |
\[ \boxed {y^{\prime \prime }+y^{\prime } x +2 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*} With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = x -\frac {x^{3}}{2}+\frac {x^{5}}{8}+O\left (x^{6}\right ) \] Verified OK. \[ y = x -\frac {x^{3}}{2}+\frac {x^{5}}{8}+O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = x -\frac {1}{2} x^{3}+\frac {1}{8} x^{5}+\operatorname {O}\left (x^{6}\right ) \] |
Problem 730
| ODE |
\[ \boxed {\left (-x^{2}+4\right ) y^{\prime \prime }+y^{\prime } x +2 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*} With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = x -\frac {x^{3}}{8}-\frac {x^{5}}{640}+O\left (x^{6}\right ) \] Verified OK. \[ y = x -\frac {x^{3}}{8}-\frac {x^{5}}{640}+O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = x -\frac {1}{8} x^{3}-\frac {1}{640} x^{5}+\operatorname {O}\left (x^{6}\right ) \] |
Problem 731
| ODE |
\[ \boxed {y^{\prime \prime }+x^{2} y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*} With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = 1-\frac {x^{4}}{12}+O\left (x^{6}\right ) \] Verified OK. \[ y = 1-\frac {x^{4}}{12}+O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = 1-\frac {1}{12} x^{4}+\operatorname {O}\left (x^{6}\right ) \] |
Problem 732
| ODE |
\[ \boxed {\left (1-x \right ) y^{\prime \prime }+y^{\prime } x -2 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*} With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = x +\frac {x^{3}}{6}+\frac {x^{4}}{12}+\frac {x^{5}}{24}+\frac {x^{6}}{45}+O\left (x^{6}\right ) \] Verified OK. \[ y = x +\frac {x^{3}}{6}+\frac {x^{4}}{12}+\frac {x^{5}}{24}+O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = x +\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{24} x^{5}+\operatorname {O}\left (x^{6}\right ) \] |
Problem 733
| ODE |
\[ \boxed {y^{\prime \prime }+y^{\prime } x +y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*} With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = -\frac {x^{2}}{2}+1+\frac {x^{4}}{8}-\frac {x^{6}}{48}+O\left (x^{6}\right ) \] Verified OK. \[ y = -\frac {x^{2}}{2}+1+\frac {x^{4}}{8}+O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = 1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}+\operatorname {O}\left (x^{6}\right ) \] |
Problem 734
| ODE |
\[ \boxed {y^{\prime \prime }+\sin \left (x \right ) y^{\prime }+\cos \left (x \right ) y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*} With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = -\frac {x^{3}}{3}+x +\frac {x^{5}}{10}+O\left (x^{6}\right ) \] Verified OK. \[ y = -\frac {x^{3}}{3}+x +\frac {x^{5}}{10}+O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = x -\frac {1}{3} x^{3}+\frac {1}{10} x^{5}+\operatorname {O}\left (x^{6}\right ) \] |
Problem 735
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+\left (x +1\right ) y^{\prime }+3 \ln \left (x \right ) y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 2, y^{\prime }\left (1\right ) = 0] \end {align*} With the expansion point for the power series method at \(x = 1\). |
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program solution |
\[ y = -\left (x -1\right )^{3}+2+\frac {7 \left (x -1\right )^{4}}{4}-\frac {49 \left (x -1\right )^{5}}{20}+\frac {33 \left (x -1\right )^{6}}{10}+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = 2-\left (x -1\right )^{3}+\frac {7}{4} \left (x -1\right )^{4}-\frac {49}{20} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \] |
Problem 736
| ODE |
\[ \boxed {y^{\prime \prime }+y^{\prime } x^{2}+\sin \left (x \right ) y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = a_{0}, y^{\prime }\left (0\right ) = a_{1}] \end {align*} With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = x a_{1} +a_{0} -\frac {a_{0} x^{3}}{6}-\frac {a_{1} x^{4}}{6}+\frac {a_{0} x^{5}}{120}+\frac {x^{6} a_{0}}{45}+\frac {x^{6} a_{1}}{180}+O\left (x^{6}\right ) \] Verified OK. \[ y = a_{0} -\frac {a_{0} x^{3}}{6}+\frac {a_{0} x^{5}}{120}+x a_{1} -\frac {a_{1} x^{4}}{6}+O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = a_{0} +a_{1} x -\frac {1}{6} a_{0} x^{3}-\frac {1}{6} a_{1} x^{4}+\frac {1}{120} a_{0} x^{5}+\operatorname {O}\left (x^{6}\right ) \] |
Problem 737
| ODE |
\[ \boxed {y^{\prime \prime }+4 y^{\prime }+6 y x=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = \left (1-x^{3}+x^{4}-\frac {4}{5} x^{5}+\frac {11}{15} x^{6}\right ) y \left (0\right )+\left (x -2 x^{2}+\frac {8}{3} x^{3}-\frac {19}{6} x^{4}+\frac {47}{15} x^{5}-\frac {118}{45} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-x^{3}+x^{4}-\frac {4}{5} x^{5}\right ) c_{1} +\left (x -2 x^{2}+\frac {8}{3} x^{3}-\frac {19}{6} x^{4}+\frac {47}{15} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = \left (1-x^{3}+x^{4}-\frac {4}{5} x^{5}\right ) y \left (0\right )+\left (x -2 x^{2}+\frac {8}{3} x^{3}-\frac {19}{6} x^{4}+\frac {47}{15} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \] |
Problem 738
| ODE |
\[ \boxed {y^{\prime \prime }+4 y^{\prime }+6 y x=0} \] With the expansion point for the power series method at \(x = 4\). |
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program solution |
\[ y = \left (1-12 \left (x -4\right )^{2}+15 \left (x -4\right )^{3}+9 \left (x -4\right )^{4}-\frac {108 \left (x -4\right )^{5}}{5}+\frac {21 \left (x -4\right )^{6}}{5}\right ) y \left (4\right )+\left (x -4-2 \left (x -4\right )^{2}-\frac {4 \left (x -4\right )^{3}}{3}+\frac {29 \left (x -4\right )^{4}}{6}-\frac {5 \left (x -4\right )^{5}}{3}-\frac {112 \left (x -4\right )^{6}}{45}\right ) y^{\prime }\left (4\right )+O\left (\left (x -4\right )^{6}\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (1-12 \left (x -4\right )^{2}+15 \left (x -4\right )^{3}+9 \left (x -4\right )^{4}-\frac {108 \left (x -4\right )^{5}}{5}\right ) y \left (4\right )+\left (x -4-2 \left (x -4\right )^{2}-\frac {4 \left (x -4\right )^{3}}{3}+\frac {29 \left (x -4\right )^{4}}{6}-\frac {5 \left (x -4\right )^{5}}{3}\right ) D\left (y \right )\left (4\right )+O\left (x^{6}\right ) \] |
Problem 739
| ODE |
\[ \boxed {\left (x^{2}-2 x -3\right ) y^{\prime \prime }+y^{\prime } x +4 y=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = \left (1+\frac {2}{3} x^{2}-\frac {4}{27} x^{3}+\frac {16}{81} x^{4}-\frac {1}{9} x^{5}+\frac {68}{729} x^{6}\right ) y \left (0\right )+\left (x +\frac {5}{18} x^{3}-\frac {5}{54} x^{4}+\frac {7}{72} x^{5}-\frac {31}{486} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {2}{3} x^{2}-\frac {4}{27} x^{3}+\frac {16}{81} x^{4}-\frac {1}{9} x^{5}\right ) c_{1} +\left (x +\frac {5}{18} x^{3}-\frac {5}{54} x^{4}+\frac {7}{72} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = \left (1+\frac {2}{3} x^{2}-\frac {4}{27} x^{3}+\frac {16}{81} x^{4}-\frac {1}{9} x^{5}\right ) y \left (0\right )+\left (x +\frac {5}{18} x^{3}-\frac {5}{54} x^{4}+\frac {7}{72} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \] |
Problem 740
| ODE |
\[ \boxed {\left (x^{2}-2 x -3\right ) y^{\prime \prime }+y^{\prime } x +4 y=0} \] With the expansion point for the power series method at \(x = 4\). |
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program solution |
\[ y = \left (1-\frac {2 \left (x -4\right )^{2}}{5}+\frac {4 \left (x -4\right )^{3}}{15}-\frac {4 \left (x -4\right )^{4}}{25}+\frac {199 \left (x -4\right )^{5}}{1875}-\frac {2186 \left (x -4\right )^{6}}{28125}\right ) y \left (4\right )+\left (x -4-\frac {2 \left (x -4\right )^{2}}{5}+\frac {\left (x -4\right )^{3}}{10}-\frac {2 \left (x -4\right )^{4}}{75}+\frac {157 \left (x -4\right )^{5}}{15000}-\frac {233 \left (x -4\right )^{6}}{37500}\right ) y^{\prime }\left (4\right )+O\left (\left (x -4\right )^{6}\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (1-\frac {2 \left (x -4\right )^{2}}{5}+\frac {4 \left (x -4\right )^{3}}{15}-\frac {4 \left (x -4\right )^{4}}{25}+\frac {199 \left (x -4\right )^{5}}{1875}\right ) y \left (4\right )+\left (x -4-\frac {2 \left (x -4\right )^{2}}{5}+\frac {\left (x -4\right )^{3}}{10}-\frac {2 \left (x -4\right )^{4}}{75}+\frac {157 \left (x -4\right )^{5}}{15000}\right ) D\left (y \right )\left (4\right )+O\left (x^{6}\right ) \] |
Problem 741
| ODE |
\[ \boxed {\left (x^{2}-2 x -3\right ) y^{\prime \prime }+y^{\prime } x +4 y=0} \] With the expansion point for the power series method at \(x = -4\). |
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program solution |
\[ y = \left (1-\frac {2 \left (x +4\right )^{2}}{21}-\frac {4 \left (x +4\right )^{3}}{189}-\frac {4 \left (x +4\right )^{4}}{1323}-\frac {\left (x +4\right )^{5}}{3087}-\frac {10 \left (x +4\right )^{6}}{583443}\right ) y \left (-4\right )+\left (x +4+\frac {2 \left (x +4\right )^{2}}{21}-\frac {\left (x +4\right )^{3}}{54}-\frac {11 \left (x +4\right )^{4}}{1323}-\frac {157 \left (x +4\right )^{5}}{74088}-\frac {1111 \left (x +4\right )^{6}}{2333772}\right ) y^{\prime }\left (-4\right )+O\left (\left (x +4\right )^{6}\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (1-\frac {2 \left (x +4\right )^{2}}{21}-\frac {4 \left (x +4\right )^{3}}{189}-\frac {4 \left (x +4\right )^{4}}{1323}-\frac {\left (x +4\right )^{5}}{3087}\right ) y \left (-4\right )+\left (x +4+\frac {2 \left (x +4\right )^{2}}{21}-\frac {\left (x +4\right )^{3}}{54}-\frac {11 \left (x +4\right )^{4}}{1323}-\frac {157 \left (x +4\right )^{5}}{74088}\right ) D\left (y \right )\left (-4\right )+O\left (x^{6}\right ) \] |
Problem 742
| ODE |
\[ \boxed {\left (x^{3}+1\right ) y^{\prime \prime }+4 y^{\prime } x +y=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = \left (1-\frac {1}{2} x^{2}+\frac {3}{8} x^{4}+\frac {1}{20} x^{5}-\frac {17}{80} x^{6}\right ) y \left (0\right )+\left (x -\frac {5}{6} x^{3}+\frac {13}{24} x^{5}+\frac {1}{6} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}+\frac {3}{8} x^{4}+\frac {1}{20} x^{5}\right ) c_{1} +\left (x -\frac {5}{6} x^{3}+\frac {13}{24} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}+\frac {3}{8} x^{4}+\frac {1}{20} x^{5}\right ) y \left (0\right )+\left (x -\frac {5}{6} x^{3}+\frac {13}{24} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \] |
Problem 743
| ODE |
\[ \boxed {\left (x^{3}+1\right ) y^{\prime \prime }+4 y^{\prime } x +y=0} \] With the expansion point for the power series method at \(x = 2\). |
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program solution |
\[ y = \left (1-\frac {\left (-2+x \right )^{2}}{18}+\frac {10 \left (-2+x \right )^{3}}{243}-\frac {451 \left (-2+x \right )^{4}}{17496}+\frac {1151 \left (-2+x \right )^{5}}{78732}-\frac {322189 \left (-2+x \right )^{6}}{42515280}\right ) y \left (2\right )+\left (-2+x -\frac {4 \left (-2+x \right )^{2}}{9}+\frac {115 \left (-2+x \right )^{3}}{486}-\frac {271 \left (-2+x \right )^{4}}{2187}+\frac {9713 \left (-2+x \right )^{5}}{157464}-\frac {150829 \left (-2+x \right )^{6}}{5314410}\right ) y^{\prime }\left (2\right )+O\left (\left (-2+x \right )^{6}\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (1-\frac {\left (-2+x \right )^{2}}{18}+\frac {10 \left (-2+x \right )^{3}}{243}-\frac {451 \left (-2+x \right )^{4}}{17496}+\frac {1151 \left (-2+x \right )^{5}}{78732}\right ) y \left (2\right )+\left (-2+x -\frac {4 \left (-2+x \right )^{2}}{9}+\frac {115 \left (-2+x \right )^{3}}{486}-\frac {271 \left (-2+x \right )^{4}}{2187}+\frac {9713 \left (-2+x \right )^{5}}{157464}\right ) D\left (y \right )\left (2\right )+O\left (x^{6}\right ) \] |
Problem 744
| ODE |
\[ \boxed {x y^{\prime \prime }+y=0} \] With the expansion point for the power series method at \(x = 1\). |
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program solution |
\[ y = \left (1-\frac {\left (x -1\right )^{2}}{2}+\frac {\left (x -1\right )^{3}}{6}-\frac {\left (x -1\right )^{4}}{24}+\frac {\left (x -1\right )^{5}}{60}-\frac {7 \left (x -1\right )^{6}}{720}\right ) y \left (1\right )+\left (x -1-\frac {\left (x -1\right )^{3}}{6}+\frac {\left (x -1\right )^{4}}{12}-\frac {\left (x -1\right )^{5}}{24}+\frac {\left (x -1\right )^{6}}{40}\right ) y^{\prime }\left (1\right )+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (1-\frac {\left (x -1\right )^{2}}{2}+\frac {\left (x -1\right )^{3}}{6}-\frac {\left (x -1\right )^{4}}{24}+\frac {\left (x -1\right )^{5}}{60}\right ) y \left (1\right )+\left (x -1-\frac {\left (x -1\right )^{3}}{6}+\frac {\left (x -1\right )^{4}}{12}-\frac {\left (x -1\right )^{5}}{24}\right ) D\left (y \right )\left (1\right )+O\left (x^{6}\right ) \] |
Problem 745
| ODE |
\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x +\alpha ^{2} y=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = \left (1-\frac {1}{2} \alpha ^{2} x^{2}+\frac {1}{24} \alpha ^{4} x^{4}-\frac {1}{6} \alpha ^{2} x^{4}-\frac {1}{720} x^{6} \alpha ^{6}+\frac {1}{36} x^{6} \alpha ^{4}-\frac {4}{45} x^{6} \alpha ^{2}\right ) y \left (0\right )+\left (x -\frac {1}{6} \alpha ^{2} x^{3}+\frac {1}{6} x^{3}+\frac {1}{120} \alpha ^{4} x^{5}-\frac {1}{12} \alpha ^{2} x^{5}+\frac {3}{40} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {\alpha ^{2} x^{2}}{2}+\left (\frac {1}{24} \alpha ^{4}-\frac {1}{6} \alpha ^{2}\right ) x^{4}\right ) c_{1} +\left (x +\left (-\frac {\alpha ^{2}}{6}+\frac {1}{6}\right ) x^{3}+\left (\frac {1}{120} \alpha ^{4}-\frac {1}{12} \alpha ^{2}+\frac {3}{40}\right ) x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = \left (1-\frac {\alpha ^{2} x^{2}}{2}+\frac {\alpha ^{2} \left (\alpha ^{2}-4\right ) x^{4}}{24}\right ) y \left (0\right )+\left (x -\frac {\left (\alpha ^{2}-1\right ) x^{3}}{6}+\frac {\left (\alpha ^{4}-10 \alpha ^{2}+9\right ) x^{5}}{120}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \] |
Problem 746
| ODE |
\[ \boxed {y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}\right ) y \left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}\right ) c_{1} +O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}\right ) y \left (0\right )+O\left (x^{6}\right ) \] |
Problem 747
| ODE |
\[ \boxed {-y x +y^{\prime }=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{8} x^{4}\right ) y \left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{8} x^{4}\right ) c_{1} +O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = \left (1+\frac {1}{2} x^{2}+\frac {1}{8} x^{4}\right ) y \left (0\right )+O\left (x^{6}\right ) \] |
Problem 748
| ODE |
\[ \boxed {\left (1-x \right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = \left (x^{5}+x^{4}+x^{3}+x^{2}+x +1\right ) y \left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (x^{5}+x^{4}+x^{3}+x^{2}+x +1\right ) c_{1} +O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = \left (x^{5}+x^{4}+x^{3}+x^{2}+x +1\right ) y \left (0\right )+O\left (x^{6}\right ) \] |
Problem 749
| ODE |
\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +\alpha \left (\alpha +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = \left (1-\frac {1}{2} \alpha ^{2} x^{2}-\frac {1}{2} \alpha \,x^{2}+\frac {1}{24} \alpha ^{4} x^{4}+\frac {1}{12} \alpha ^{3} x^{4}-\frac {5}{24} \alpha ^{2} x^{4}-\frac {1}{4} \alpha \,x^{4}-\frac {1}{720} x^{6} \alpha ^{6}-\frac {1}{240} x^{6} \alpha ^{5}+\frac {23}{720} x^{6} \alpha ^{4}+\frac {17}{240} x^{6} \alpha ^{3}-\frac {47}{360} x^{6} \alpha ^{2}-\frac {1}{6} x^{6} \alpha \right ) y \left (0\right )+\left (x -\frac {1}{6} \alpha ^{2} x^{3}-\frac {1}{6} \alpha \,x^{3}+\frac {1}{3} x^{3}+\frac {1}{120} \alpha ^{4} x^{5}+\frac {1}{60} x^{5} \alpha ^{3}-\frac {13}{120} \alpha ^{2} x^{5}-\frac {7}{60} x^{5} \alpha +\frac {1}{5} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\left (-\frac {1}{2} \alpha ^{2}-\frac {1}{2} \alpha \right ) x^{2}+\left (-\frac {5}{24} \alpha ^{2}-\frac {1}{4} \alpha +\frac {1}{24} \alpha ^{4}+\frac {1}{12} \alpha ^{3}\right ) x^{4}\right ) c_{1} +\left (x +\left (-\frac {1}{6} \alpha ^{2}-\frac {1}{6} \alpha +\frac {1}{3}\right ) x^{3}+\left (-\frac {13}{120} \alpha ^{2}-\frac {7}{60} \alpha +\frac {1}{5}+\frac {1}{120} \alpha ^{4}+\frac {1}{60} \alpha ^{3}\right ) x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = \left (1-\frac {\alpha \left (1+\alpha \right ) x^{2}}{2}+\frac {\alpha \left (\alpha ^{3}+2 \alpha ^{2}-5 \alpha -6\right ) x^{4}}{24}\right ) y \left (0\right )+\left (x -\frac {\left (\alpha ^{2}+\alpha -2\right ) x^{3}}{6}+\frac {\left (\alpha ^{4}+2 \alpha ^{3}-13 \alpha ^{2}-14 \alpha +24\right ) x^{5}}{120}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \] |
Problem 750
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=-\frac {x_{1} \left (t \right )}{10}+\frac {3 x_{2} \left (t \right )}{40}\\ x_{2}^{\prime }\left (t \right )&=\frac {x_{1} \left (t \right )}{10}-\frac {x_{2} \left (t \right )}{5} \end {align*} With initial conditions \[ [x_{1} \left (0\right ) = -17, x_{2} \left (0\right ) = -21] \] |
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= -\frac {165 \,{\mathrm e}^{-\frac {t}{20}}}{8}+\frac {29 \,{\mathrm e}^{-\frac {t}{4}}}{8} \\ x_{2} \left (t \right ) &= -\frac {55 \,{\mathrm e}^{-\frac {t}{20}}}{4}-\frac {29 \,{\mathrm e}^{-\frac {t}{4}}}{4} \\ \end{align*} |
Problem 751
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )-2 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=4 x_{1} \left (t \right )-x_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{t} \left (c_{1} \sin \left (2 t \right )+c_{2} \cos \left (2 t \right )\right ) \\ x_{2} \left (t \right ) &= -{\mathrm e}^{t} \left (c_{1} \cos \left (2 t \right )-c_{2} \cos \left (2 t \right )-c_{1} \sin \left (2 t \right )-c_{2} \sin \left (2 t \right )\right ) \\ \end{align*} |
Problem 752
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=-x_{1} \left (t \right )-4 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )-x_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-t} \left (c_{1} \sin \left (2 t \right )+c_{2} \cos \left (2 t \right )\right ) \\ x_{2} \left (t \right ) &= -\frac {{\mathrm e}^{-t} \left (c_{1} \cos \left (2 t \right )-c_{2} \sin \left (2 t \right )\right )}{2} \\ \end{align*} |
Problem 753
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )-5 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )-2 x_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right ) \\ x_{2} \left (t \right ) &= -\frac {c_{1} \cos \left (t \right )}{5}+\frac {c_{2} \sin \left (t \right )}{5}+\frac {2 c_{1} \sin \left (t \right )}{5}+\frac {2 c_{2} \cos \left (t \right )}{5} \\ \end{align*} |
Problem 754
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )-\frac {5 x_{2} \left (t \right )}{2}\\ x_{2}^{\prime }\left (t \right )&=\frac {9 x_{1} \left (t \right )}{5}-x_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{\frac {t}{2}} \left (\sin \left (\frac {3 t}{2}\right ) c_{1} +\cos \left (\frac {3 t}{2}\right ) c_{2} \right ) \\ x_{2} \left (t \right ) &= \frac {3 \,{\mathrm e}^{\frac {t}{2}} \left (\sin \left (\frac {3 t}{2}\right ) c_{1} +\sin \left (\frac {3 t}{2}\right ) c_{2} -\cos \left (\frac {3 t}{2}\right ) c_{1} +\cos \left (\frac {3 t}{2}\right ) c_{2} \right )}{5} \\ \end{align*} |
Problem 755
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )-x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=5 x_{1} \left (t \right )-3 x_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-t} \left (c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right )\right ) \\ x_{2} \left (t \right ) &= -{\mathrm e}^{-t} \left (c_{1} \cos \left (t \right )-2 c_{2} \cos \left (t \right )-2 c_{1} \sin \left (t \right )-c_{2} \sin \left (t \right )\right ) \\ \end{align*} |
Problem 756
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )+2 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-5 x_{1} \left (t \right )-x_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= c_{1} \sin \left (3 t \right )+c_{2} \cos \left (3 t \right ) \\ x_{2} \left (t \right ) &= \frac {3 c_{1} \cos \left (3 t \right )}{2}-\frac {3 c_{2} \sin \left (3 t \right )}{2}-\frac {c_{1} \sin \left (3 t \right )}{2}-\frac {c_{2} \cos \left (3 t \right )}{2} \\ \end{align*} |
Problem 757
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )+x_{2} \left (t \right )-2 x_{3} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )+2 x_{2} \left (t \right )+x_{3} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= c_{3} {\mathrm e}^{t} \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{t} \left (2 c_{1} \cos \left (2 t \right )-3 c_{3} \cos \left (2 t \right )+2 c_{2} \sin \left (2 t \right )-3 c_{3} \right )}{2} \\ x_{3} \left (t \right ) &= -\frac {{\mathrm e}^{t} \left (2 c_{2} \cos \left (2 t \right )-2 c_{1} \sin \left (2 t \right )+3 c_{3} \sin \left (2 t \right )-2 c_{3} \right )}{2} \\ \end{align*} |
Problem 758
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=-3 x_{1} \left (t \right )+2 x_{3} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )-x_{2} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=-2 x_{1} \left (t \right )-x_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= c_{1} {\mathrm e}^{-2 t}+c_{2} {\mathrm e}^{-t} \sin \left (\sqrt {2}\, t \right )+c_{3} {\mathrm e}^{-t} \cos \left (\sqrt {2}\, t \right ) \\ x_{2} \left (t \right ) &= -c_{1} {\mathrm e}^{-2 t}-\frac {c_{2} {\mathrm e}^{-t} \sqrt {2}\, \cos \left (\sqrt {2}\, t \right )}{2}+\frac {c_{3} {\mathrm e}^{-t} \sqrt {2}\, \sin \left (\sqrt {2}\, t \right )}{2} \\ x_{3} \left (t \right ) &= \frac {c_{1} {\mathrm e}^{-2 t}}{2}+c_{2} {\mathrm e}^{-t} \sin \left (\sqrt {2}\, t \right )+\frac {c_{2} {\mathrm e}^{-t} \sqrt {2}\, \cos \left (\sqrt {2}\, t \right )}{2}+c_{3} {\mathrm e}^{-t} \cos \left (\sqrt {2}\, t \right )-\frac {c_{3} {\mathrm e}^{-t} \sqrt {2}\, \sin \left (\sqrt {2}\, t \right )}{2} \\ \end{align*} |
Problem 759
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )-5 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )-3 x_{2} \left (t \right ) \end {align*} With initial conditions \[ [x_{1} \left (0\right ) = 1, x_{2} \left (0\right ) = 1] \] |
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-t} \left (-3 \sin \left (t \right )+\cos \left (t \right )\right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (5 \cos \left (t \right )-5 \sin \left (t \right )\right )}{5} \\ \end{align*} |
Problem 760
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=-3 x_{1} \left (t \right )+2 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-x_{1} \left (t \right )-x_{2} \left (t \right ) \end {align*} With initial conditions \[ [x_{1} \left (0\right ) = 1, x_{2} \left (0\right ) = -2] \] |
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-2 t} \left (-5 \sin \left (t \right )+\cos \left (t \right )\right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{-2 t} \left (-6 \sin \left (t \right )-4 \cos \left (t \right )\right )}{2} \\ \end{align*} |
Problem 761
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=\frac {3 x_{1} \left (t \right )}{4}-2 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )-\frac {5 x_{2} \left (t \right )}{4} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-\frac {t}{4}} \left (c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right )\right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{-\frac {t}{4}} \left (c_{1} \sin \left (t \right )+c_{2} \sin \left (t \right )-c_{1} \cos \left (t \right )+c_{2} \cos \left (t \right )\right )}{2} \\ \end{align*} |
Problem 762
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=-\frac {4 x_{1} \left (t \right )}{5}+2 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-x_{1} \left (t \right )+\frac {6 x_{2} \left (t \right )}{5} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{\frac {t}{5}} \left (c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right )\right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{\frac {t}{5}} \left (c_{1} \sin \left (t \right )-c_{2} \sin \left (t \right )+c_{1} \cos \left (t \right )+c_{2} \cos \left (t \right )\right )}{2} \\ \end{align*} |
Problem 763
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=-\frac {x_{1} \left (t \right )}{4}+x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-x_{1} \left (t \right )-\frac {x_{2} \left (t \right )}{4}\\ x_{3}^{\prime }\left (t \right )&=-\frac {x_{3} \left (t \right )}{4} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-\frac {t}{4}} \left (c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right )\right ) \\ x_{2} \left (t \right ) &= -{\mathrm e}^{-\frac {t}{4}} \left (c_{2} \sin \left (t \right )-c_{1} \cos \left (t \right )\right ) \\ x_{3} \left (t \right ) &= c_{3} {\mathrm e}^{-\frac {t}{4}} \\ \end{align*} |
Problem 764
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=-\frac {x_{1} \left (t \right )}{4}+x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-x_{1} \left (t \right )-\frac {x_{2} \left (t \right )}{4}\\ x_{3}^{\prime }\left (t \right )&=\frac {x_{3} \left (t \right )}{10} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-\frac {t}{4}} \left (c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right )\right ) \\ x_{2} \left (t \right ) &= -{\mathrm e}^{-\frac {t}{4}} \left (c_{2} \sin \left (t \right )-c_{1} \cos \left (t \right )\right ) \\ x_{3} \left (t \right ) &= c_{3} {\mathrm e}^{\frac {t}{10}} \\ \end{align*} |
Problem 765
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=-\frac {x_{1} \left (t \right )}{2}-\frac {x_{2} \left (t \right )}{8}\\ x_{2}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )-\frac {x_{2} \left (t \right )}{2} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-\frac {t}{2}} \left (c_{2} \cos \left (\frac {t}{2}\right )+c_{1} \sin \left (\frac {t}{2}\right )\right ) \\ x_{2} \left (t \right ) &= -4 \,{\mathrm e}^{-\frac {t}{2}} \left (\cos \left (\frac {t}{2}\right ) c_{1} -\sin \left (\frac {t}{2}\right ) c_{2} \right ) \\ \end{align*} |
Problem 766
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )-4 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )-x_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{t} \left (c_{2} t +c_{1} \right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{t} \left (2 c_{2} t +2 c_{1} -c_{2} \right )}{4} \\ \end{align*} |
Problem 767
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=4 x_{1} \left (t \right )-2 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=8 x_{1} \left (t \right )-4 x_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= c_{1} t +c_{2} \\ x_{2} \left (t \right ) &= -\frac {1}{2} c_{1} +2 c_{1} t +2 c_{2} \\ \end{align*} |
Problem 768
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=-\frac {3 x_{1} \left (t \right )}{2}+x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-\frac {x_{1} \left (t \right )}{4}-\frac {x_{2} \left (t \right )}{2} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-t} \left (c_{2} t +c_{1} \right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (c_{2} t +c_{1} +2 c_{2} \right )}{2} \\ \end{align*} |
Problem 769
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=-3 x_{1} \left (t \right )+\frac {5 x_{2} \left (t \right )}{2}\\ x_{2}^{\prime }\left (t \right )&=-\frac {5 x_{1} \left (t \right )}{2}+2 x_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-\frac {t}{2}} \left (c_{2} t +c_{1} \right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{-\frac {t}{2}} \left (5 c_{2} t +5 c_{1} +2 c_{2} \right )}{5} \\ \end{align*} |
Problem 770
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )+x_{3} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )+x_{2} \left (t \right )-x_{3} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=-x_{2} \left (t \right )+x_{3} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= -\frac {3 \,{\mathrm e}^{-t} c_{1}}{2}-c_{3} {\mathrm e}^{2 t} \\ x_{2} \left (t \right ) &= 2 \,{\mathrm e}^{-t} c_{1} -c_{2} {\mathrm e}^{2 t}-{\mathrm e}^{2 t} c_{3} t -c_{3} {\mathrm e}^{2 t} \\ x_{3} \left (t \right ) &= {\mathrm e}^{-t} c_{1} +c_{2} {\mathrm e}^{2 t}+{\mathrm e}^{2 t} c_{3} t \\ \end{align*} |
Problem 771
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=x_{2} \left (t \right )+x_{3} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )+x_{3} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= c_{2} {\mathrm e}^{-t}+c_{3} {\mathrm e}^{2 t} \\ x_{2} \left (t \right ) &= c_{2} {\mathrm e}^{-t}+c_{3} {\mathrm e}^{2 t}+{\mathrm e}^{-t} c_{1} \\ x_{3} \left (t \right ) &= -2 c_{2} {\mathrm e}^{-t}+c_{3} {\mathrm e}^{2 t}-{\mathrm e}^{-t} c_{1} \\ \end{align*} |
Problem 772
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )-4 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=4 x_{1} \left (t \right )-7 x_{2} \left (t \right ) \end {align*} With initial conditions \[ [x_{1} \left (0\right ) = 3, x_{2} \left (0\right ) = 2] \] |
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-3 t} \left (4 t +3\right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{-3 t} \left (16 t +8\right )}{4} \\ \end{align*} |
Problem 773
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=-\frac {5 x_{1} \left (t \right )}{2}+\frac {3 x_{2} \left (t \right )}{2}\\ x_{2}^{\prime }\left (t \right )&=-\frac {3 x_{1} \left (t \right )}{2}+\frac {x_{2} \left (t \right )}{2} \end {align*} With initial conditions \[ [x_{1} \left (0\right ) = 3, x_{2} \left (0\right ) = -1] \] |
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-t} \left (-6 t +3\right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (-18 t -3\right )}{3} \\ \end{align*} |
Problem 774
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )+\frac {3 x_{2} \left (t \right )}{2}\\ x_{2}^{\prime }\left (t \right )&=-\frac {3 x_{1} \left (t \right )}{2}-x_{2} \left (t \right ) \end {align*} With initial conditions \[ [x_{1} \left (0\right ) = 3, x_{2} \left (0\right ) = -2] \] |
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{\frac {t}{2}} \left (\frac {3 t}{2}+3\right ) \\ x_{2} \left (t \right ) &= -\frac {{\mathrm e}^{\frac {t}{2}} \left (\frac {9 t}{2}+6\right )}{3} \\ \end{align*} |
Problem 775
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )+9 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-x_{1} \left (t \right )-3 x_{2} \left (t \right ) \end {align*} With initial conditions \[ [x_{1} \left (0\right ) = 2, x_{2} \left (0\right ) = 4] \] |
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= 42 t +2 \\ x_{2} \left (t \right ) &= 4-14 t \\ \end{align*} |
Problem 776
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-4 x_{1} \left (t \right )+x_{2} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )+6 x_{2} \left (t \right )+2 x_{3} \left (t \right ) \end {align*} With initial conditions \[ [x_{1} \left (0\right ) = -1, x_{2} \left (0\right ) = 2, x_{3} \left (0\right ) = -30] \] |
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= -{\mathrm e}^{t} \\ x_{2} \left (t \right ) &= \left (4 t +2\right ) {\mathrm e}^{t} \\ x_{3} \left (t \right ) &= -24 \,{\mathrm e}^{t} t -33 \,{\mathrm e}^{t}+3 \,{\mathrm e}^{2 t} \\ \end{align*} |
Problem 777
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=-\frac {5 x_{1} \left (t \right )}{2}+x_{2} \left (t \right )+x_{3} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )-\frac {5 x_{2} \left (t \right )}{2}+x_{3} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )-\frac {5 x_{3} \left (t \right )}{2} \end {align*} With initial conditions \[ [x_{1} \left (0\right ) = 2, x_{2} \left (0\right ) = 3, x_{3} \left (0\right ) = -1] \] |
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= \frac {2 \,{\mathrm e}^{-\frac {7 t}{2}}}{3}+\frac {4 \,{\mathrm e}^{-\frac {t}{2}}}{3} \\ x_{2} \left (t \right ) &= \frac {5 \,{\mathrm e}^{-\frac {7 t}{2}}}{3}+\frac {4 \,{\mathrm e}^{-\frac {t}{2}}}{3} \\ x_{3} \left (t \right ) &= -\frac {7 \,{\mathrm e}^{-\frac {7 t}{2}}}{3}+\frac {4 \,{\mathrm e}^{-\frac {t}{2}}}{3} \\ \end{align*} |
Problem 778
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )-x_{2} \left (t \right )+{\mathrm e}^{t}\\ x_{2}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )-2 x_{2} \left (t \right )+t \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= \frac {{\mathrm e}^{-t} c_{1}}{3}+c_{2} {\mathrm e}^{t}+\frac {3 \,{\mathrm e}^{t} t}{2}-\frac {{\mathrm e}^{t}}{4}+t \\ x_{2} \left (t \right ) &= c_{2} {\mathrm e}^{t}+{\mathrm e}^{-t} c_{1} +\frac {3 \,{\mathrm e}^{t} t}{2}-\frac {3 \,{\mathrm e}^{t}}{4}+2 t -1 \\ \end{align*} |
Problem 779
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )+\sqrt {3}\, x_{2} \left (t \right )+{\mathrm e}^{t}\\ x_{2}^{\prime }\left (t \right )&=\sqrt {3}\, x_{1} \left (t \right )-x_{2} \left (t \right )+\sqrt {3}\, {\mathrm e}^{-t} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= \sinh \left (2 t \right ) c_{2} +\cosh \left (2 t \right ) c_{1} -\frac {5 \cosh \left (t \right )}{3}+\frac {\sinh \left (t \right )}{3} \\ x_{2} \left (t \right ) &= -\frac {\sqrt {3}\, \left (\cosh \left (2 t \right ) c_{1} -2 \cosh \left (2 t \right ) c_{2} -2 \sinh \left (2 t \right ) c_{1} +\sinh \left (2 t \right ) c_{2} +{\mathrm e}^{t}+2 \sinh \left (t \right )-2 \cosh \left (t \right )\right )}{3} \\ \end{align*} |
Problem 780
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )-5 x_{2} \left (t \right )-\cos \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )-2 x_{2} \left (t \right )+\sin \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= c_{2} \sin \left (t \right )-\sin \left (t \right ) t +c_{1} \cos \left (t \right )+2 \cos \left (t \right ) t -\cos \left (t \right ) \\ x_{2} \left (t \right ) &= \frac {c_{1} \sin \left (t \right )}{5}+\frac {2 c_{2} \sin \left (t \right )}{5}+\frac {2 c_{1} \cos \left (t \right )}{5}-\frac {c_{2} \cos \left (t \right )}{5}+\cos \left (t \right ) t -\cos \left (t \right ) \\ \end{align*} |
Problem 781
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )+{\mathrm e}^{-2 t}\\ x_{2}^{\prime }\left (t \right )&=4 x_{1} \left (t \right )-2 x_{2} \left (t \right )-2 \,{\mathrm e}^{t} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= c_{2} {\mathrm e}^{2 t}-\frac {c_{1} {\mathrm e}^{-3 t}}{4}+\frac {{\mathrm e}^{t}}{2} \\ x_{2} \left (t \right ) &= c_{2} {\mathrm e}^{2 t}+c_{1} {\mathrm e}^{-3 t}-{\mathrm e}^{-2 t} \\ \end{align*} |
Problem 782
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=4 x_{1} \left (t \right )-2 x_{2} \left (t \right )+\frac {1}{t^{3}}\\ x_{2}^{\prime }\left (t \right )&=8 x_{1} \left (t \right )-4 x_{2} \left (t \right )-\frac {1}{t^{2}} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= -\frac {1}{2 t^{2}}+\frac {2}{t}-2 \ln \left (t \right )+c_{1} t +c_{2} \\ x_{2} \left (t \right ) &= 2 c_{1} t -4 \ln \left (t \right )-\frac {c_{1}}{2}+2 c_{2} +\frac {5}{t} \\ \end{align*} |
Problem 783
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=-4 x_{1} \left (t \right )+2 x_{2} \left (t \right )+\frac {1}{t}\\ x_{2}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )-x_{2} \left (t \right )+\frac {2}{t}+4 \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= \ln \left (-5 t \right )-\frac {c_{1} {\mathrm e}^{-5 t}}{5}+\frac {8 t}{5}+c_{2} \\ x_{2} \left (t \right ) &= \frac {c_{1} {\mathrm e}^{-5 t}}{10}+2 \ln \left (-5 t \right )+2 c_{2} +\frac {16 t}{5}+\frac {4}{5} \\ \end{align*} |
Problem 784
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )+2 \,{\mathrm e}^{t}\\ x_{2}^{\prime }\left (t \right )&=4 x_{1} \left (t \right )+x_{2} \left (t \right )-{\mathrm e}^{t} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= c_{2} {\mathrm e}^{3 t}+{\mathrm e}^{-t} c_{1} +\frac {{\mathrm e}^{t}}{4} \\ x_{2} \left (t \right ) &= 2 c_{2} {\mathrm e}^{3 t}-2 \,{\mathrm e}^{-t} c_{1} -2 \,{\mathrm e}^{t} \\ \end{align*} |
Problem 785
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )-x_{2} \left (t \right )+{\mathrm e}^{t}\\ x_{2}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )-2 x_{2} \left (t \right )-{\mathrm e}^{t} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= c_{2} {\mathrm e}^{t}+{\mathrm e}^{-t} c_{1} +2 \,{\mathrm e}^{t} t \\ x_{2} \left (t \right ) &= c_{2} {\mathrm e}^{t}+3 \,{\mathrm e}^{-t} c_{1} +2 \,{\mathrm e}^{t} t -{\mathrm e}^{t} \\ \end{align*} |
Problem 786
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=-\frac {5 x_{1} \left (t \right )}{4}+\frac {3 x_{2} \left (t \right )}{4}+2 t\\ x_{2}^{\prime }\left (t \right )&=\frac {3 x_{1} \left (t \right )}{4}-\frac {5 x_{2} \left (t \right )}{4}+{\mathrm e}^{t} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= c_{2} {\mathrm e}^{-2 t}+c_{1} {\mathrm e}^{-\frac {t}{2}}-\frac {17}{4}+\frac {{\mathrm e}^{t}}{6}+\frac {5 t}{2} \\ x_{2} \left (t \right ) &= -c_{2} {\mathrm e}^{-2 t}+c_{1} {\mathrm e}^{-\frac {t}{2}}+\frac {{\mathrm e}^{t}}{2}-\frac {15}{4}+\frac {3 t}{2} \\ \end{align*} |
Problem 787
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=-3 x_{1} \left (t \right )+\sqrt {2}\, x_{2} \left (t \right )+{\mathrm e}^{-t}\\ x_{2}^{\prime }\left (t \right )&=\sqrt {2}\, x_{1} \left (t \right )-2 x_{2} \left (t \right )-{\mathrm e}^{-t} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= c_{2} {\mathrm e}^{-t}+{\mathrm e}^{-4 t} c_{1} -\frac {t \,{\mathrm e}^{-t} \sqrt {2}}{3}+\frac {t \,{\mathrm e}^{-t}}{3} \\ x_{2} \left (t \right ) &= -\frac {2 t \,{\mathrm e}^{-t}}{3}-\frac {{\mathrm e}^{-t}}{3}+\sqrt {2}\, {\mathrm e}^{-t} c_{2} +\frac {t \,{\mathrm e}^{-t} \sqrt {2}}{3}-\frac {\sqrt {2}\, {\mathrm e}^{-4 t} c_{1}}{2}-\frac {\sqrt {2}\, {\mathrm e}^{-t}}{3} \\ \end{align*} |
Problem 788
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )-5 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )-2 x_{2} \left (t \right )+\cos \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= c_{2} \sin \left (t \right )+c_{1} \cos \left (t \right )-\frac {5 \sin \left (t \right ) t}{2} \\ x_{2} \left (t \right ) &= -\frac {c_{2} \cos \left (t \right )}{5}+\frac {c_{1} \sin \left (t \right )}{5}+\frac {\cos \left (t \right ) t}{2}+\frac {\sin \left (t \right )}{2}+\frac {2 c_{2} \sin \left (t \right )}{5}+\frac {2 c_{1} \cos \left (t \right )}{5}-\sin \left (t \right ) t \\ \end{align*} |
Problem 789
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )-5 x_{2} \left (t \right )+\csc \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )-2 x_{2} \left (t \right )+\sec \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= \ln \left (\sin \left (t \right )\right ) \cos \left (t \right )-5 \cos \left (t \right ) \ln \left (\cos \left (t \right )\right )+c_{1} \cos \left (t \right )-2 \cos \left (t \right ) t +2 \ln \left (\sin \left (t \right )\right ) \sin \left (t \right )+c_{2} \sin \left (t \right )-4 \sin \left (t \right ) t +\cos \left (t \right ) \\ x_{2} \left (t \right ) &= -2 \cos \left (t \right ) \ln \left (\cos \left (t \right )\right )+\frac {2 c_{1} \cos \left (t \right )}{5}-\frac {c_{2} \cos \left (t \right )}{5}+\ln \left (\sin \left (t \right )\right ) \sin \left (t \right )-\sin \left (t \right ) \ln \left (\cos \left (t \right )\right )+\frac {c_{1} \sin \left (t \right )}{5}+\frac {2 c_{2} \sin \left (t \right )}{5}-2 \sin \left (t \right ) t -\frac {\cos \left (t \right )^{2}}{5 \sin \left (t \right )}+\frac {2 \cos \left (t \right )}{5}+\frac {\csc \left (t \right )}{5} \\ \end{align*} |
Problem 790
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=-\frac {x_{1} \left (t \right )}{2}-\frac {x_{2} \left (t \right )}{8}+\frac {{\mathrm e}^{-\frac {t}{2}}}{2}\\ x_{2}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )-\frac {x_{2} \left (t \right )}{2} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= \frac {{\mathrm e}^{-\frac {t}{2}} \left (c_{2} \cos \left (\frac {t}{2}\right )-c_{1} \sin \left (\frac {t}{2}\right )\right )}{4} \\ x_{2} \left (t \right ) &= {\mathrm e}^{-\frac {t}{2}} \left (4+\cos \left (\frac {t}{2}\right ) c_{1} +\sin \left (\frac {t}{2}\right ) c_{2} \right ) \\ \end{align*} |
Problem 791
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=-2 x_{1} \left (t \right )+x_{2} \left (t \right )+2 \,{\mathrm e}^{-t}\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )-2 x_{2} \left (t \right )+3 t \end {align*} With initial conditions \[ [x_{1} \left (0\right ) = \alpha _{1}, x_{2} \left (0\right ) = \alpha _{2}] \] |
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= \left (\frac {3}{2}+\frac {\alpha _{2}}{2}+\frac {\alpha _{1}}{2}\right ) {\mathrm e}^{-t}-\left (\frac {2}{3}+\frac {\alpha _{2}}{2}-\frac {\alpha _{1}}{2}\right ) {\mathrm e}^{-3 t}+\frac {{\mathrm e}^{-t}}{2}+t \,{\mathrm e}^{-t}-\frac {4}{3}+t \\ x_{2} \left (t \right ) &= \left (\frac {3}{2}+\frac {\alpha _{2}}{2}+\frac {\alpha _{1}}{2}\right ) {\mathrm e}^{-t}+\left (\frac {2}{3}+\frac {\alpha _{2}}{2}-\frac {\alpha _{1}}{2}\right ) {\mathrm e}^{-3 t}+t \,{\mathrm e}^{-t}+2 t -\frac {5}{3}-\frac {{\mathrm e}^{-t}}{2} \\ \end{align*} |
Problem 792
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )-2 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )-2 x_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-t} c_{1} +c_{2} {\mathrm e}^{2 t} \\ x_{2} \left (t \right ) &= 2 \,{\mathrm e}^{-t} c_{1} +\frac {c_{2} {\mathrm e}^{2 t}}{2} \\ \end{align*} |
Problem 793
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=5 x_{1} \left (t \right )-x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )+x_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= c_{1} {\mathrm e}^{4 t}+c_{2} {\mathrm e}^{2 t} \\ x_{2} \left (t \right ) &= c_{1} {\mathrm e}^{4 t}+3 c_{2} {\mathrm e}^{2 t} \\ \end{align*} |
Problem 794
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )-x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )-2 x_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-t} \\ x_{2} \left (t \right ) &= c_{1} {\mathrm e}^{t}+3 c_{2} {\mathrm e}^{-t} \\ \end{align*} |
Problem 795
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )-4 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=4 x_{1} \left (t \right )-7 x_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-3 t} \left (c_{2} t +c_{1} \right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{-3 t} \left (4 c_{2} t +4 c_{1} -c_{2} \right )}{4} \\ \end{align*} |
Problem 796
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )-5 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )-3 x_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-t} \left (c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right )\right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (-c_{1} \cos \left (t \right )+c_{2} \sin \left (t \right )+2 c_{1} \sin \left (t \right )+2 c_{2} \cos \left (t \right )\right )}{5} \\ \end{align*} |
Problem 797
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )-5 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )-2 x_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right ) \\ x_{2} \left (t \right ) &= -\frac {c_{1} \cos \left (t \right )}{5}+\frac {c_{2} \sin \left (t \right )}{5}+\frac {2 c_{1} \sin \left (t \right )}{5}+\frac {2 c_{2} \cos \left (t \right )}{5} \\ \end{align*} |
Problem 798
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )-2 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=4 x_{1} \left (t \right )-x_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{t} \left (c_{1} \sin \left (2 t \right )+c_{2} \cos \left (2 t \right )\right ) \\ x_{2} \left (t \right ) &= -{\mathrm e}^{t} \left (c_{1} \cos \left (2 t \right )-c_{2} \cos \left (2 t \right )-c_{1} \sin \left (2 t \right )-c_{2} \sin \left (2 t \right )\right ) \\ \end{align*} |
Problem 799
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=-x_{1} \left (t \right )-x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-\frac {5 x_{2} \left (t \right )}{2} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= \frac {2 c_{2} {\mathrm e}^{-\frac {5 t}{2}}}{3}+{\mathrm e}^{-t} c_{1} \\ x_{2} \left (t \right ) &= c_{2} {\mathrm e}^{-\frac {5 t}{2}} \\ \end{align*} |
Problem 800
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )-4 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )-x_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{t} \left (c_{2} t +c_{1} \right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{t} \left (2 c_{2} t +2 c_{1} -c_{2} \right )}{4} \\ \end{align*} |