Problem 11501
| ODE |
\[ \boxed {x^{\prime \prime \prime }+x^{\prime }=1} \] |
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program solution |
\[ x = c_{1} +{\mathrm e}^{i t} c_{2} +{\mathrm e}^{-i t} c_{3} +t \] Verified OK.
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Maple solution | \[ x \left (t \right ) = c_{1} \sin \left (t \right )-c_{2} \cos \left (t \right )+t +c_{3} \] |
Problem 11502
| ODE |
\[ \boxed {x^{\prime \prime \prime }+x^{\prime \prime }=0} \] |
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program solution |
\[ x = c_{1} {\mathrm e}^{-t}+c_{2} +c_{3} t \] Verified OK.
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Maple solution | \[ x \left (t \right ) = c_{1} +c_{2} t +c_{3} {\mathrm e}^{-t} \] |
Problem 11503
| ODE |
\[ \boxed {x^{\prime \prime \prime }-x^{\prime }-8 x=0} \] |
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program solution |
\[ x = {\mathrm e}^{\left (\frac {\left (108+3 \sqrt {1293}\right )^{\frac {1}{3}}}{3}+\frac {1}{\left (108+3 \sqrt {1293}\right )^{\frac {1}{3}}}\right ) t} c_{1} +{\mathrm e}^{\left (-\frac {\left (108+3 \sqrt {1293}\right )^{\frac {1}{3}}}{6}-\frac {1}{2 \left (108+3 \sqrt {1293}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (108+3 \sqrt {1293}\right )^{\frac {1}{3}}}{3}-\frac {1}{\left (108+3 \sqrt {1293}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} c_{2} +{\mathrm e}^{\left (-\frac {\left (108+3 \sqrt {1293}\right )^{\frac {1}{3}}}{6}-\frac {1}{2 \left (108+3 \sqrt {1293}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (108+3 \sqrt {1293}\right )^{\frac {1}{3}}}{3}-\frac {1}{\left (108+3 \sqrt {1293}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} c_{3} \] Verified OK.
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Maple solution | \[ x \left (t \right ) = c_{1} {\mathrm e}^{\frac {\left (\left (108+3 \sqrt {1293}\right )^{\frac {2}{3}}+3\right ) t}{3 \left (108+3 \sqrt {1293}\right )^{\frac {1}{3}}}}-c_{2} {\mathrm e}^{-\frac {\left (\left (108+3 \sqrt {1293}\right )^{\frac {2}{3}}+3\right ) t}{6 \left (108+3 \sqrt {1293}\right )^{\frac {1}{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (108+3 \sqrt {3}\, \sqrt {431}\right )^{\frac {2}{3}}-3\right ) t}{6 \left (108+3 \sqrt {3}\, \sqrt {431}\right )^{\frac {1}{3}}}\right )+c_{3} {\mathrm e}^{-\frac {\left (\left (108+3 \sqrt {1293}\right )^{\frac {2}{3}}+3\right ) t}{6 \left (108+3 \sqrt {1293}\right )^{\frac {1}{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (108+3 \sqrt {3}\, \sqrt {431}\right )^{\frac {2}{3}}-3\right ) t}{6 \left (108+3 \sqrt {3}\, \sqrt {431}\right )^{\frac {1}{3}}}\right ) \] |
Problem 11504
| ODE |
\[ \boxed {x^{\prime \prime \prime }+x^{\prime \prime }=2 \,{\mathrm e}^{t}+3 t^{2}} \] |
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program solution |
\[ x = c_{1} {\mathrm e}^{-t}+c_{2} +c_{3} t +{\mathrm e}^{t}+3 t^{2}-t^{3}+\frac {t^{4}}{4} \] Verified OK.
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Maple solution | \[ x \left (t \right ) = \frac {t^{4}}{4}+3 t^{2}-t^{3}+{\mathrm e}^{-t} c_{1} +{\mathrm e}^{t}+c_{2} t +c_{3} \] |
Problem 11505
| ODE |
\[ \boxed {x^{\prime \prime \prime }-8 x=0} \] |
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program solution |
\[ x = {\mathrm e}^{2 t} c_{1} +{\mathrm e}^{\left (-i \sqrt {3}-1\right ) t} c_{2} +{\mathrm e}^{\left (i \sqrt {3}-1\right ) t} c_{3} \] Verified OK.
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Maple solution | \[ x \left (t \right ) = c_{1} {\mathrm e}^{2 t}+c_{2} {\mathrm e}^{-t} \sin \left (\sqrt {3}\, t \right )+c_{3} {\mathrm e}^{-t} \cos \left (\sqrt {3}\, t \right ) \] |
Problem 11506
| ODE |
\[ \boxed {x^{\prime \prime \prime }+x^{\prime \prime }-x^{\prime }-4 x=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 1, x^{\prime }\left (0\right ) = 0, x^{\prime \prime }\left (0\right ) = -1] \end {align*}
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program solution |
\[ x = \frac {\left (\left (2466225 i \sqrt {3}+232025 i \sqrt {339}-9438777-887841 \sqrt {113}\right ) \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}+\left (-317982 i \sqrt {3}-29902 i \sqrt {339}+61863 \sqrt {113}+657999\right ) \left (388+36 \sqrt {113}\right )^{\frac {2}{3}}+20082360 i \sqrt {3}+1889144 i \sqrt {339}+6112920 \sqrt {113}+64982232\right ) {\mathrm e}^{-\frac {\left (i \sqrt {3}\, \left (388+36 \sqrt {113}\right )^{\frac {2}{3}}-16 i \sqrt {3}+\left (388+36 \sqrt {113}\right )^{\frac {2}{3}}+4 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}+16\right ) t}{12 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}}+\left (\left (-2466225 i \sqrt {3}-232025 i \sqrt {339}-887841 \sqrt {113}-9438777\right ) \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}+\left (317982 i \sqrt {3}+29902 i \sqrt {339}+61863 \sqrt {113}+657999\right ) \left (388+36 \sqrt {113}\right )^{\frac {2}{3}}-20082360 i \sqrt {3}-1889144 i \sqrt {339}+6112920 \sqrt {113}+64982232\right ) {\mathrm e}^{\frac {t \left (\left (i \sqrt {3}-1\right ) \left (388+36 \sqrt {113}\right )^{\frac {2}{3}}-16 i \sqrt {3}-4 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}-16\right )}{12 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}}-591894 \left (\left (\sqrt {113}+\frac {9281}{873}\right ) \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}+\left (-\frac {57407 \sqrt {113}}{197298}-\frac {5399}{1746}\right ) \left (388+36 \sqrt {113}\right )^{\frac {2}{3}}+\frac {1248448 \sqrt {113}}{98649}+\frac {117440}{873}\right ) {\mathrm e}^{-\frac {\left (-\frac {\left (388+36 \sqrt {113}\right )^{\frac {2}{3}}}{2}+\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}-8\right ) t}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}}}{339 \left (\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}-4\right )^{2} \left (873 \sqrt {113}+9281\right )} \] Verified OK.
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Maple solution | \[ x \left (t \right ) = \frac {\left (\left (\left (32 \sqrt {113}+352\right ) \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}+\left (-\sqrt {113}-25\right ) \left (388+36 \sqrt {113}\right )^{\frac {2}{3}}+776 \sqrt {113}+8136\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (388+36 \sqrt {113}\right )^{\frac {2}{3}}-16\right ) t}{12 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}\right )+32 \sin \left (\frac {\sqrt {3}\, \left (\left (388+36 \sqrt {113}\right )^{\frac {2}{3}}-16\right ) t}{12 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}\right ) \left (\left (\sqrt {113}+11\right ) \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}+\frac {\left (388+36 \sqrt {113}\right )^{\frac {2}{3}} \left (\sqrt {113}+25\right )}{32}\right ) \sqrt {3}\right ) {\mathrm e}^{-\frac {t \left (4+\frac {\left (388+36 \sqrt {113}\right )^{\frac {2}{3}}}{4}+\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}\right )}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}}-32 \left (\left (\sqrt {113}+11\right ) \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}+\left (-\frac {\sqrt {113}}{32}-\frac {25}{32}\right ) \left (388+36 \sqrt {113}\right )^{\frac {2}{3}}-\frac {97 \sqrt {113}}{8}-\frac {1017}{8}\right ) {\mathrm e}^{-\frac {\left (-\frac {\left (388+36 \sqrt {113}\right )^{\frac {2}{3}}}{2}+\left (388+36 \sqrt {113}\right )^{\frac {1}{3}}-8\right ) t}{3 \left (388+36 \sqrt {113}\right )^{\frac {1}{3}}}}}{1164 \sqrt {113}+12204} \] |
Problem 11507
| ODE |
\[ \boxed {x^{\prime }+5 x=\operatorname {Heaviside}\left (-2+t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 1] \end {align*}
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program solution |
\[ x = \left \{\begin {array}{cc} {\mathrm e}^{-5 t} & t <2 \\ {\mathrm e}^{-5 t}+\frac {1}{5}-\frac {{\mathrm e}^{-5 t +10}}{5} & 2\le t \end {array}\right . \] Verified OK.
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Maple solution | \[ x \left (t \right ) = -\frac {\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-5 t +10}}{5}+\frac {\operatorname {Heaviside}\left (t -2\right )}{5}+{\mathrm e}^{-5 t} \] |
Problem 11508
| ODE |
\[ \boxed {x^{\prime }+x=\sin \left (2 t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0] \end {align*}
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program solution |
\[ x = \frac {2 \,{\mathrm e}^{-t}}{5}-\frac {2 \cos \left (2 t \right )}{5}+\frac {\sin \left (2 t \right )}{5} \] Verified OK.
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Maple solution | \[ x \left (t \right ) = \frac {2 \,{\mathrm e}^{-t}}{5}-\frac {2 \cos \left (2 t \right )}{5}+\frac {\sin \left (2 t \right )}{5} \] |
Problem 11509
| ODE |
\[ \boxed {x^{\prime \prime }-x^{\prime }-6 x=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 2, x^{\prime }\left (0\right ) = -1] \end {align*}
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program solution |
\[ x = \frac {\left (3 \,{\mathrm e}^{5 t}+7\right ) {\mathrm e}^{-2 t}}{5} \] Verified OK.
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Maple solution | \[ x \left (t \right ) = \frac {\left (3 \,{\mathrm e}^{5 t}+7\right ) {\mathrm e}^{-2 t}}{5} \] |
Problem 11510
| ODE |
\[ \boxed {x^{\prime \prime }-2 x^{\prime }+2 x=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 1] \end {align*}
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program solution |
\[ x = {\mathrm e}^{t} \sin \left (t \right ) \] Verified OK.
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Maple solution | \[ x \left (t \right ) = {\mathrm e}^{t} \sin \left (t \right ) \] |
Problem 11511
| ODE |
\[ \boxed {x^{\prime \prime }-2 x^{\prime }+2 x={\mathrm e}^{-t}} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 1] \end {align*}
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program solution |
\[ x = \frac {{\mathrm e}^{-t}}{5}+\frac {\left (-\cos \left (t \right )+7 \sin \left (t \right )\right ) {\mathrm e}^{t}}{5} \] Verified OK.
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Maple solution | \[ x \left (t \right ) = \frac {{\mathrm e}^{-t}}{5}+\frac {\left (-\cos \left (t \right )+7 \sin \left (t \right )\right ) {\mathrm e}^{t}}{5} \] |
Problem 11512
| ODE |
\[ \boxed {x^{\prime \prime }-x^{\prime }=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 1, x^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ x = 1 \] Verified OK.
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Maple solution | \[ x \left (t \right ) = 1 \] |
Problem 11513
| ODE |
\[ \boxed {x^{\prime \prime }+\frac {2 x^{\prime }}{5}+2 x=1-\operatorname {Heaviside}\left (t -5\right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ x = \frac {\left (\left \{\begin {array}{cc} 1 & t <5 \\ \left (\frac {1}{2}+\frac {i}{14}\right ) {\mathrm e}^{\left (-\frac {1}{5}-\frac {7 i}{5}\right ) \left (t -5\right )}+\left (\frac {1}{2}-\frac {i}{14}\right ) {\mathrm e}^{\left (-\frac {1}{5}+\frac {7 i}{5}\right ) \left (t -5\right )} & 5\le t \end {array}\right .\right )}{2}+\frac {\left (-7 \cos \left (\frac {7 t}{5}\right )-\sin \left (\frac {7 t}{5}\right )\right ) {\mathrm e}^{-\frac {t}{5}}}{14} \] Verified OK.
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Maple solution | \[ x \left (t \right ) = \frac {1}{2}+\left (\frac {1}{4}+\frac {i}{28}\right ) \operatorname {Heaviside}\left (t -5\right ) {\mathrm e}^{\left (-\frac {1}{5}-\frac {7 i}{5}\right ) \left (t -5\right )}+\left (\frac {1}{4}-\frac {i}{28}\right ) \operatorname {Heaviside}\left (t -5\right ) {\mathrm e}^{\left (-\frac {1}{5}+\frac {7 i}{5}\right ) \left (t -5\right )}+\frac {\left (-7 \cos \left (\frac {7 t}{5}\right )-\sin \left (\frac {7 t}{5}\right )\right ) {\mathrm e}^{-\frac {t}{5}}}{14}-\frac {\operatorname {Heaviside}\left (t -5\right )}{2} \] |
Problem 11514
| ODE |
\[ \boxed {x^{\prime \prime }+9 x=\sin \left (3 t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ x = \frac {\sin \left (3 t \right )}{18}-\frac {t \cos \left (3 t \right )}{6} \] Verified OK.
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Maple solution | \[ x \left (t \right ) = \frac {\sin \left (3 t \right )}{18}-\frac {\cos \left (3 t \right ) t}{6} \] |
Problem 11515
| ODE |
\[ \boxed {x^{\prime \prime }-2 x=1} \] With initial conditions \begin {align*} [x \left (0\right ) = 1, x^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ x = -\frac {1}{2}+\frac {3 \cosh \left (\sqrt {2}\, t \right )}{2} \] Verified OK.
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Maple solution | \[ x \left (t \right ) = -\frac {1}{2}+\frac {3 \cosh \left (\sqrt {2}\, t \right )}{2} \] |
Problem 11516
| ODE |
\[ \boxed {-2 x+x^{\prime }=\operatorname {Heaviside}\left (t -1\right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0] \end {align*}
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program solution |
\[ x = \left \{\begin {array}{cc} 0 & t <1 \\ -\frac {1}{2}+\frac {{\mathrm e}^{-2+2 t}}{2} & 1\le t \end {array}\right . \] Verified OK.
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Maple solution | \[ x \left (t \right ) = \frac {\operatorname {Heaviside}\left (t -1\right ) \left (-1+{\mathrm e}^{2 t -2}\right )}{2} \] |
Problem 11517
| ODE |
\[ \boxed {x^{\prime \prime }+4 x=\cos \left (2 t \right ) \operatorname {Heaviside}\left (2 \pi -t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ x = \frac {\sin \left (2 t \right ) \left (\left \{\begin {array}{cc} \frac {t}{2} & t <2 \pi \\ \pi & 2 \pi \le t \end {array}\right .\right )}{2} \] Verified OK.
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Maple solution | \[ \text {No solution found} \] |
Problem 11518
| ODE |
\[ \boxed {-x+x^{\prime }=-2 \operatorname {Heaviside}\left (t -1\right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 1] \end {align*}
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program solution |
\[
x = \left \{\begin {array}{cc} {\mathrm e}^{t} & t <1 \\ {\mathrm e}+2 & t =1 \\ {\mathrm e}^{t}+2-2 \,{\mathrm e}^{t -1} & 1 |
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Maple solution | \[ x \left (t \right ) = \left (-2 \,{\mathrm e}^{t -1}+2\right ) \operatorname {Heaviside}\left (t -1\right )+{\mathrm e}^{t} \] |
Problem 11519
| ODE |
\[ \boxed {x^{\prime }+x=\operatorname {Heaviside}\left (t -1\right )-\operatorname {Heaviside}\left (-2+t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 1] \end {align*}
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program solution |
\[ x = \left \{\begin {array}{cc} {\mathrm e}^{-t} & t <1 \\ {\mathrm e}^{-t}+1-{\mathrm e}^{-t +1} & t <2 \\ {\mathrm e}^{-t}-{\mathrm e}^{-t +1}+{\mathrm e}^{-t +2} & 2\le t \end {array}\right . \] Verified OK.
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Maple solution | \[ x \left (t \right ) = \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{2-t}-\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-t +1}+{\mathrm e}^{-t}-\operatorname {Heaviside}\left (t -2\right )+\operatorname {Heaviside}\left (t -1\right ) \] |
Problem 11520
| ODE |
\[ \boxed {x^{\prime \prime }+\pi ^{2} x=\pi ^{2} \operatorname {Heaviside}\left (-t +1\right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 1, x^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ x = \left \{\begin {array}{cc} 1 & t <1 \\ -\cos \left (\pi t \right ) & 1\le t \end {array}\right . \] Verified OK.
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Maple solution | \[ x \left (t \right ) = 1+\left (-\cos \left (\pi t \right )-1\right ) \operatorname {Heaviside}\left (t -1\right ) \] |
Problem 11521
| ODE |
\[ \boxed {x^{\prime \prime }-4 x=1-\operatorname {Heaviside}\left (t -1\right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ x = -\frac {\left (\left \{\begin {array}{cc} 1 & t <1 \\ \cosh \left (-2+2 t \right ) & 1\le t \end {array}\right .\right )}{4}+\frac {\cosh \left (2 t \right )}{4} \] Verified OK.
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Maple solution | \[ x \left (t \right ) = -\frac {1}{4}+\frac {\cosh \left (2 t \right )}{4}-\frac {\operatorname {Heaviside}\left (t -1\right ) \sinh \left (t -1\right )^{2}}{2} \] |
Problem 11522
| ODE |
\[ \boxed {x^{\prime \prime }+3 x^{\prime }+2 x={\mathrm e}^{-4 t}} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ x = -\frac {{\mathrm e}^{-2 t}}{2}+\frac {{\mathrm e}^{-4 t}}{6}+\frac {{\mathrm e}^{-t}}{3} \] Verified OK.
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Maple solution | \[ x \left (t \right ) = \frac {{\mathrm e}^{-4 t}}{6}+\frac {{\mathrm e}^{-t}}{3}-\frac {{\mathrm e}^{-2 t}}{2} \] |
Problem 11523
| ODE |
\[ \boxed {x^{\prime }+3 x=\delta \left (t -1\right )+\operatorname {Heaviside}\left (t -4\right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 1] \end {align*}
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program solution |
\[ x = \left \{\begin {array}{cc} {\mathrm e}^{-3 t} & t <1 \\ {\mathrm e}^{-3 t}+{\mathrm e}^{-3 t +3} & t <4 \\ {\mathrm e}^{-3 t}+{\mathrm e}^{-3 t +3}+\frac {1}{3}-\frac {{\mathrm e}^{-3 t +12}}{3} & 4\le t \end {array}\right . \] Verified OK.
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Maple solution | \[ x \left (t \right ) = \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-3 t +3}-\frac {\operatorname {Heaviside}\left (t -4\right ) {\mathrm e}^{-3 t +12}}{3}+\frac {\operatorname {Heaviside}\left (t -4\right )}{3}+{\mathrm e}^{-3 t} \] |
Problem 11524
| ODE |
\[ \boxed {x^{\prime \prime }-x=\delta \left (t -5\right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ x = \left \{\begin {array}{cc} 0 & t <5 \\ \sinh \left (t -5\right ) & 5\le t \end {array}\right . \] Verified OK.
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Maple solution | \[ x \left (t \right ) = \operatorname {Heaviside}\left (t -5\right ) \sinh \left (t -5\right ) \] |
Problem 11525
| ODE |
\[ \boxed {x^{\prime \prime }+x=\delta \left (-2+t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ x = \left \{\begin {array}{cc} 0 & t <2 \\ \sin \left (-2+t \right ) & 2\le t \end {array}\right . \] Verified OK.
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Maple solution | \[ x \left (t \right ) = \operatorname {Heaviside}\left (t -2\right ) \sin \left (t -2\right ) \] |
Problem 11526
| ODE |
\[ \boxed {x^{\prime \prime }+4 x=\delta \left (-2+t \right )-\delta \left (t -5\right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ x = \frac {\left (\left \{\begin {array}{cc} 0 & t <2 \\ \sin \left (-4+2 t \right ) & t <5 \\ \sin \left (-4+2 t \right )-\sin \left (2 t -10\right ) & 5\le t \end {array}\right .\right )}{2} \] Verified OK.
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Maple solution | \[ x \left (t \right ) = -\frac {\operatorname {Heaviside}\left (t -5\right ) \sin \left (2 t -10\right )}{2}+\frac {\operatorname {Heaviside}\left (t -2\right ) \sin \left (2 t -4\right )}{2} \] |
Problem 11527
| ODE |
\[ \boxed {x^{\prime \prime }+x=3 \delta \left (-2 \pi +t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 1] \end {align*}
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program solution |
\[ x = \sin \left (t \right ) \left (\left \{\begin {array}{cc} 1 & t <2 \pi \\ 4 & 2 \pi \le t \end {array}\right .\right ) \] Verified OK.
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Maple solution | \[ x \left (t \right ) = \sin \left (t \right ) \left (3 \operatorname {Heaviside}\left (-2 \pi +t \right )+1\right ) \] |
Problem 11528
| ODE |
\[ \boxed {y^{\prime \prime }+y^{\prime }+y=\delta \left (t -1\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = \left \{\begin {array}{cc} 0 & t <1 \\ \frac {2 \sqrt {3}\, {\mathrm e}^{-\frac {t}{2}+\frac {1}{2}} \sin \left (\frac {\sqrt {3}\, \left (t -1\right )}{2}\right )}{3} & 1\le t \end {array}\right . \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {2 \sqrt {3}\, {\mathrm e}^{\frac {1}{2}-\frac {t}{2}} \operatorname {Heaviside}\left (t -1\right ) \sin \left (\frac {\sqrt {3}\, \left (t -1\right )}{2}\right )}{3} \] |
Problem 11529
| ODE |
\[ \boxed {x^{\prime \prime }+4 x=\frac {\left (t -5\right ) \operatorname {Heaviside}\left (t -5\right )}{5}+\left (2-\frac {t}{5}\right ) \operatorname {Heaviside}\left (-10+t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ x = \frac {\left (\left \{\begin {array}{cc} 0 & t <5 \\ 2 t -10-\sin \left (2 t -10\right ) & t <10 \\ 10+\sin \left (-20+2 t \right )-\sin \left (2 t -10\right ) & 10\le t \end {array}\right .\right )}{40} \] Verified OK.
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Maple solution | \[ x \left (t \right ) = \frac {\operatorname {Heaviside}\left (t -10\right ) \sin \left (2 t -20\right )}{40}-\frac {\operatorname {Heaviside}\left (t -5\right ) \sin \left (2 t -10\right )}{40}+\frac {\left (-2 t +20\right ) \operatorname {Heaviside}\left (t -10\right )}{40}+\frac {\left (t -5\right ) \operatorname {Heaviside}\left (t -5\right )}{20} \] |
Problem 11530
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ODE |
\begin {align*} x^{\prime }&=-3 y \left (t \right )\\ y^{\prime }\left (t \right )&=2 x \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} \sin \left (\sqrt {6}\, t \right )+c_{2} \cos \left (\sqrt {6}\, t \right ) \\ y \left (t \right ) &= -\frac {\sqrt {6}\, \left (\cos \left (\sqrt {6}\, t \right ) c_{1} -\sin \left (\sqrt {6}\, t \right ) c_{2} \right )}{3} \\ \end{align*} |
Problem 11531
|
ODE |
\begin {align*} x^{\prime }&=-2 y \left (t \right )\\ y^{\prime }\left (t \right )&=-4 x \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{2 \sqrt {2}\, t}+c_{2} {\mathrm e}^{-2 \sqrt {2}\, t} \\ y \left (t \right ) &= -\sqrt {2}\, \left (c_{1} {\mathrm e}^{2 \sqrt {2}\, t}-c_{2} {\mathrm e}^{-2 \sqrt {2}\, t}\right ) \\ \end{align*} |
Problem 11532
|
ODE |
\begin {align*} x^{\prime }&=-3 x\\ y^{\prime }\left (t \right )&=2 y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{-3 t} \\ y \left (t \right ) &= c_{1} {\mathrm e}^{2 t} \\ \end{align*} |
Problem 11533
|
ODE |
\begin {align*} x^{\prime }&=4 y \left (t \right )\\ y^{\prime }\left (t \right )&=2 y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= 2 c_{2} {\mathrm e}^{2 t}+c_{1} \\ y \left (t \right ) &= c_{2} {\mathrm e}^{2 t} \\ \end{align*} |
Problem 11534
|
ODE |
\begin {align*} x^{\prime }&=x\\ y^{\prime }\left (t \right )&=x+2 y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{t} \\ y \left (t \right ) &= -c_{2} {\mathrm e}^{t}+c_{1} {\mathrm e}^{2 t} \\ \end{align*} |
Problem 11535
|
ODE |
\begin {align*} x^{\prime }&=x-y \left (t \right )\\ y^{\prime }\left (t \right )&=x+y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= {\mathrm e}^{t} \left (c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right )\right ) \\ y \left (t \right ) &= -{\mathrm e}^{t} \left (c_{1} \cos \left (t \right )-c_{2} \sin \left (t \right )\right ) \\ \end{align*} |
Problem 11536
|
ODE |
\begin {align*} x^{\prime }&=x+2 y \left (t \right )\\ y^{\prime }\left (t \right )&=x \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= -{\mathrm e}^{-t} c_{1} +2 c_{2} {\mathrm e}^{2 t} \\ y \left (t \right ) &= {\mathrm e}^{-t} c_{1} +c_{2} {\mathrm e}^{2 t} \\ \end{align*} |
Problem 11537
|
ODE |
\begin {align*} x^{\prime }&=-x-2 y \left (t \right )\\ y^{\prime }\left (t \right )&=2 x-y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} \left (c_{1} \sin \left (2 t \right )+c_{2} \cos \left (2 t \right )\right ) \\ y \left (t \right ) &= -{\mathrm e}^{-t} \left (c_{1} \cos \left (2 t \right )-c_{2} \sin \left (2 t \right )\right ) \\ \end{align*} |
Problem 11538
|
ODE |
\begin {align*} x^{\prime }&=-2 x-3 y \left (t \right )\\ y^{\prime }\left (t \right )&=-x+4 y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{\left (1+2 \sqrt {3}\right ) t}+c_{2} {\mathrm e}^{-\left (-1+2 \sqrt {3}\right ) t} \\ y \left (t \right ) &= -\frac {2 c_{1} {\mathrm e}^{\left (1+2 \sqrt {3}\right ) t} \sqrt {3}}{3}+\frac {2 c_{2} {\mathrm e}^{-\left (-1+2 \sqrt {3}\right ) t} \sqrt {3}}{3}-c_{1} {\mathrm e}^{\left (1+2 \sqrt {3}\right ) t}-c_{2} {\mathrm e}^{-\left (-1+2 \sqrt {3}\right ) t} \\ \end{align*} |
Problem 11539
|
ODE |
\begin {align*} x^{\prime }&=-3 y \left (t \right )\\ y^{\prime }\left (t \right )&=-2 x+y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{3 t}+c_{2} {\mathrm e}^{-2 t} \\ y \left (t \right ) &= -c_{1} {\mathrm e}^{3 t}+\frac {2 c_{2} {\mathrm e}^{-2 t}}{3} \\ \end{align*} |
Problem 11540
|
ODE |
\begin {align*} x^{\prime }&=-2 x\\ y^{\prime }\left (t \right )&=x \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{-2 t} \\ y \left (t \right ) &= -\frac {c_{2} {\mathrm e}^{-2 t}}{2}+c_{1} \\ \end{align*} |
Problem 11541
|
ODE |
\begin {align*} x^{\prime }&=-2 x-y \left (t \right )\\ y^{\prime }\left (t \right )&=-4 y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= \frac {c_{2} {\mathrm e}^{-4 t}}{2}+c_{1} {\mathrm e}^{-2 t} \\ y \left (t \right ) &= c_{2} {\mathrm e}^{-4 t} \\ \end{align*} |
Problem 11542
|
ODE |
\begin {align*} x^{\prime }&=x-2 y \left (t \right )\\ y^{\prime }\left (t \right )&=-2 x+4 y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} +c_{2} {\mathrm e}^{5 t} \\ y \left (t \right ) &= -2 c_{2} {\mathrm e}^{5 t}+\frac {c_{1}}{2} \\ \end{align*} |
Problem 11543
|
ODE |
\begin {align*} x^{\prime }&=-6 y \left (t \right )\\ y^{\prime }\left (t \right )&=6 y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= -c_{2} {\mathrm e}^{6 t}+c_{1} \\ y \left (t \right ) &= c_{2} {\mathrm e}^{6 t} \\ \end{align*} |
Problem 11544
|
ODE |
\begin {align*} x^{\prime }&=2 x+3 y \left (t \right )\\ y^{\prime }\left (t \right )&=-x-14 \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= -14+{\mathrm e}^{t} \left (\sqrt {2}\, \sin \left (\sqrt {2}\, t \right ) c_{1} -\sqrt {2}\, \cos \left (\sqrt {2}\, t \right ) c_{2} -\sin \left (\sqrt {2}\, t \right ) c_{2} -\cos \left (\sqrt {2}\, t \right ) c_{1} \right ) \\ y \left (t \right ) &= \frac {28}{3}+{\mathrm e}^{t} \left (\sin \left (\sqrt {2}\, t \right ) c_{2} +\cos \left (\sqrt {2}\, t \right ) c_{1} \right ) \\ \end{align*} |
Problem 11545
|
ODE |
\begin {align*} x^{\prime }&=-3 x+3 y \left (t \right )\\ y^{\prime }\left (t \right )&=x+2 y \left (t \right )-1 \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= {\mathrm e}^{\frac {\left (-1+\sqrt {37}\right ) t}{2}} c_{2} +{\mathrm e}^{-\frac {\left (1+\sqrt {37}\right ) t}{2}} c_{1} +\frac {1}{3} \\ y \left (t \right ) &= \frac {{\mathrm e}^{\frac {\left (-1+\sqrt {37}\right ) t}{2}} c_{2} \sqrt {37}}{6}-\frac {{\mathrm e}^{-\frac {\left (1+\sqrt {37}\right ) t}{2}} c_{1} \sqrt {37}}{6}+\frac {5 \,{\mathrm e}^{\frac {\left (-1+\sqrt {37}\right ) t}{2}} c_{2}}{6}+\frac {5 \,{\mathrm e}^{-\frac {\left (1+\sqrt {37}\right ) t}{2}} c_{1}}{6}+\frac {1}{3} \\ \end{align*} |
Problem 11546
|
ODE |
\begin {align*} x^{\prime }&=-x+y \left (t \right )\\ y^{\prime }\left (t \right )&=-3 y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= -\frac {c_{2} {\mathrm e}^{-3 t}}{2}+{\mathrm e}^{-t} c_{1} \\ y \left (t \right ) &= c_{2} {\mathrm e}^{-3 t} \\ \end{align*} |
Problem 11547
|
ODE |
\begin {align*} x^{\prime }&=x\\ y^{\prime }\left (t \right )&=3 x-4 y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{t} \\ y \left (t \right ) &= \frac {3 c_{2} {\mathrm e}^{t}}{5}+{\mathrm e}^{-4 t} c_{1} \\ \end{align*} |
Problem 11548
|
ODE |
\begin {align*} x^{\prime }&=-x+y \left (t \right )\\ y^{\prime }\left (t \right )&=x-2 y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{\frac {\left (\sqrt {5}-3\right ) t}{2}}+c_{2} {\mathrm e}^{-\frac {\left (3+\sqrt {5}\right ) t}{2}} \\ y \left (t \right ) &= \frac {c_{1} {\mathrm e}^{\frac {\left (\sqrt {5}-3\right ) t}{2}} \sqrt {5}}{2}-\frac {c_{2} {\mathrm e}^{-\frac {\left (3+\sqrt {5}\right ) t}{2}} \sqrt {5}}{2}-\frac {c_{1} {\mathrm e}^{\frac {\left (\sqrt {5}-3\right ) t}{2}}}{2}-\frac {c_{2} {\mathrm e}^{-\frac {\left (3+\sqrt {5}\right ) t}{2}}}{2} \\ \end{align*} |
Problem 11549
|
ODE |
\begin {align*} x^{\prime }&=x+y \left (t \right )\\ y^{\prime }\left (t \right )&=-3 x+3 y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= {\mathrm e}^{2 t} \left (c_{1} \sin \left (\sqrt {2}\, t \right )+c_{2} \cos \left (\sqrt {2}\, t \right )\right ) \\ y \left (t \right ) &= -{\mathrm e}^{2 t} \left (\sin \left (\sqrt {2}\, t \right ) \sqrt {2}\, c_{2} -\cos \left (\sqrt {2}\, t \right ) \sqrt {2}\, c_{1} -c_{1} \sin \left (\sqrt {2}\, t \right )-c_{2} \cos \left (\sqrt {2}\, t \right )\right ) \\ \end{align*} |
Problem 11550
|
ODE |
\begin {align*} x^{\prime }&=x-2 y \left (t \right )\\ y^{\prime }\left (t \right )&=3 x-4 y \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 3, y \left (0\right ) = 1] \] |
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= -4 \,{\mathrm e}^{-2 t}+7 \,{\mathrm e}^{-t} \\ y \left (t \right ) &= -6 \,{\mathrm e}^{-2 t}+7 \,{\mathrm e}^{-t} \\ \end{align*} |
Problem 11551
|
ODE |
\begin {align*} x^{\prime }&=5 x-y \left (t \right )\\ y^{\prime }\left (t \right )&=3 x+y \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 2, y \left (0\right ) = -1] \] |
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= \frac {7 \,{\mathrm e}^{4 t}}{2}-\frac {3 \,{\mathrm e}^{2 t}}{2} \\ y \left (t \right ) &= \frac {7 \,{\mathrm e}^{4 t}}{2}-\frac {9 \,{\mathrm e}^{2 t}}{2} \\ \end{align*} |
Problem 11552
|
ODE |
\begin {align*} x^{\prime }&=-3 x+y \left (t \right )\\ y^{\prime }\left (t \right )&=-3 y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= \left (c_{2} t +c_{1} \right ) {\mathrm e}^{-3 t} \\ y \left (t \right ) &= c_{2} {\mathrm e}^{-3 t} \\ \end{align*} |
Problem 11553
|
ODE |
\begin {align*} x^{\prime }&=x-y \left (t \right )\\ y^{\prime }\left (t \right )&=x+3 y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= {\mathrm e}^{2 t} \left (c_{2} t +c_{1} \right ) \\ y \left (t \right ) &= -{\mathrm e}^{2 t} \left (c_{2} t +c_{1} +c_{2} \right ) \\ \end{align*} |
Problem 11554
|
ODE |
\begin {align*} x^{\prime }&=x+2 y \left (t \right )\\ y^{\prime }\left (t \right )&=3 x+2 y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{4 t}+c_{2} {\mathrm e}^{-t} \\ y \left (t \right ) &= \frac {3 c_{1} {\mathrm e}^{4 t}}{2}-c_{2} {\mathrm e}^{-t} \\ \end{align*} |
Problem 11555
|
ODE |
\begin {align*} x^{\prime }&=-3 x+4 y \left (t \right )\\ y^{\prime }\left (t \right )&=-3 y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= \left (4 c_{2} t +c_{1} \right ) {\mathrm e}^{-3 t} \\ y \left (t \right ) &= c_{2} {\mathrm e}^{-3 t} \\ \end{align*} |
Problem 11556
|
ODE |
\begin {align*} x^{\prime }&=2 x+2 y \left (t \right )\\ y^{\prime }\left (t \right )&=6 x+3 y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= {\mathrm e}^{6 t} c_{1} +c_{2} {\mathrm e}^{-t} \\ y \left (t \right ) &= 2 \,{\mathrm e}^{6 t} c_{1} -\frac {3 c_{2} {\mathrm e}^{-t}}{2} \\ \end{align*} |
Problem 11557
|
ODE |
\begin {align*} x^{\prime }&=-5 x+3 y \left (t \right )\\ y^{\prime }\left (t \right )&=2 x-10 y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= {\mathrm e}^{-4 t} c_{1} +c_{2} {\mathrm e}^{-11 t} \\ y \left (t \right ) &= \frac {{\mathrm e}^{-4 t} c_{1}}{3}-2 c_{2} {\mathrm e}^{-11 t} \\ \end{align*} |
Problem 11558
|
ODE |
\begin {align*} x^{\prime }&=2 x\\ y^{\prime }\left (t \right )&=2 y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{2 t} \\ y \left (t \right ) &= c_{1} {\mathrm e}^{2 t} \\ \end{align*} |
Problem 11559
|
ODE |
\begin {align*} x^{\prime }&=3 x-2 y \left (t \right )\\ y^{\prime }\left (t \right )&=4 x-y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= {\mathrm e}^{t} \left (c_{1} \sin \left (2 t \right )+c_{2} \cos \left (2 t \right )\right ) \\ y \left (t \right ) &= -{\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 11560
|
ODE |
\begin {align*} x^{\prime }&=5 x-4 y \left (t \right )\\ y^{\prime }\left (t \right )&=x+y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= {\mathrm e}^{3 t} \left (c_{2} t +c_{1} \right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{3 t} \left (2 c_{2} t +2 c_{1} -c_{2} \right )}{4} \\ \end{align*} |
Problem 11561
|
ODE |
\begin {align*} x^{\prime }&=9 y \left (t \right )\\ y^{\prime }\left (t \right )&=-x \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} \sin \left (3 t \right )+c_{2} \cos \left (3 t \right ) \\ y \left (t \right ) &= \frac {c_{1} \cos \left (3 t \right )}{3}-\frac {c_{2} \sin \left (3 t \right )}{3} \\ \end{align*} |
Problem 11562
|
ODE |
\begin {align*} x^{\prime }&=2 x+y \left (t \right )\\ y^{\prime }\left (t \right )&=-x \end {align*} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = -1] \] |
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= {\mathrm e}^{t} \\ y \left (t \right ) &= -{\mathrm e}^{t} \\ \end{align*} |
Problem 11563
|
ODE |
\begin {align*} x^{\prime }&=x-2 y \left (t \right )\\ y^{\prime }\left (t \right )&=-2 x+4 y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} +c_{2} {\mathrm e}^{5 t} \\ y \left (t \right ) &= -2 c_{2} {\mathrm e}^{5 t}+\frac {c_{1}}{2} \\ \end{align*} |
Problem 11564
|
ODE |
\begin {align*} x^{\prime }&=3 x-y \left (t \right )+1\\ y^{\prime }\left (t \right )&=x+y \left (t \right )+2 \end {align*} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 2] \] |
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= -\frac {3}{4}+{\mathrm e}^{2 t} \left (-\frac {3 t}{2}+\frac {7}{4}\right ) \\ y \left (t \right ) &= -\frac {5}{4}+{\mathrm e}^{2 t} \left (-\frac {3 t}{2}+\frac {13}{4}\right ) \\ \end{align*} |
Problem 11565
|
ODE |
\begin {align*} x^{\prime }&=-5 x+3 y \left (t \right )+{\mathrm e}^{-t}\\ y^{\prime }\left (t \right )&=2 x-10 y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= 3 c_{2} {\mathrm e}^{-4 t}-\frac {{\mathrm e}^{-11 t} c_{1}}{2}+\frac {3 \,{\mathrm e}^{-t}}{10} \\ y \left (t \right ) &= c_{2} {\mathrm e}^{-4 t}+{\mathrm e}^{-11 t} c_{1} +\frac {{\mathrm e}^{-t}}{15} \\ \end{align*} |
Problem 11566
|
ODE |
\begin {align*} x^{\prime }&=y \left (t \right )\\ y^{\prime }\left (t \right )&=-x+\cos \left (w t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= c_{2} \sin \left (t \right )+c_{1} \cos \left (t \right )-\frac {\cos \left (w t \right )}{w^{2}-1} \\ y \left (t \right ) &= \frac {\cos \left (t \right ) c_{2} w^{2}-\sin \left (t \right ) c_{1} w^{2}+w \sin \left (w t \right )-c_{2} \cos \left (t \right )+c_{1} \sin \left (t \right )}{\left (-1+w \right ) \left (1+w \right )} \\ \end{align*} |
Problem 11567
|
ODE |
\begin {align*} x^{\prime }&=3 x+2 y \left (t \right )+3\\ y^{\prime }\left (t \right )&=7 x+5 y \left (t \right )+2 t \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= {\mathrm e}^{\left (4+\sqrt {15}\right ) t} c_{2} +{\mathrm e}^{-\left (-4+\sqrt {15}\right ) t} c_{1} +4 t +17 \\ y \left (t \right ) &= \frac {{\mathrm e}^{\left (4+\sqrt {15}\right ) t} c_{2} \sqrt {15}}{2}-\frac {{\mathrm e}^{-\left (-4+\sqrt {15}\right ) t} c_{1} \sqrt {15}}{2}+\frac {{\mathrm e}^{\left (4+\sqrt {15}\right ) t} c_{2}}{2}+\frac {{\mathrm e}^{-\left (-4+\sqrt {15}\right ) t} c_{1}}{2}-6 t -25 \\ \end{align*} |
Problem 11568
|
ODE |
\begin {align*} x^{\prime }&=x-3 y \left (t \right )\\ y^{\prime }\left (t \right )&=3 x+7 y \left (t \right ) \end {align*}
|
|
program solution |
|
|
Maple solution |
\begin{align*} x \left (t \right ) &= {\mathrm e}^{4 t} \left (c_{2} t +c_{1} \right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{4 t} \left (3 c_{2} t +3 c_{1} +c_{2} \right )}{3} \\ \end{align*} |
Problem 11569
| ODE |
\[ \boxed {y^{\prime }+y=x +1} \] |
|
program solution |
\[ y = \left (x \,{\mathrm e}^{x}+c_{1} \right ) {\mathrm e}^{-x} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = x +c_{1} {\mathrm e}^{-x} \] |
Problem 11570
| ODE |
\[ \boxed {y^{\prime \prime }-7 y^{\prime }+12 y=0} \] |
|
program solution |
\[ y = c_{1} {\mathrm e}^{3 x}+c_{2} {\mathrm e}^{4 x} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{4 x}+c_{2} {\mathrm e}^{3 x} \] |
Problem 11571
| ODE |
\[ \boxed {y^{\prime \prime }-3 y^{\prime }+2 y=4 x^{2}} \] |
|
program solution |
\[ y = c_{1} {\mathrm e}^{x}+c_{2} {\mathrm e}^{2 x}+2 x^{2}+6 x +7 \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = {\mathrm e}^{2 x} c_{1} +c_{2} {\mathrm e}^{x}+2 x^{2}+6 x +7 \] |
Problem 11572
| ODE |
\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+4 y^{\prime } x +2 y=0} \] |
|
program solution |
\[ y = \frac {c_{1} x +c_{2}}{x^{2}+1} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {c_{1} x +c_{2}}{x^{2}+1} \] |
Problem 11573
| ODE |
\[ \boxed {2 y y^{\prime } x +y^{2}=-x^{2}} \] |
|
program solution |
\[ \frac {x^{3}}{3}+y^{2} x = c_{1} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (x \right ) &= -\frac {\sqrt {3}\, \sqrt {-x \left (x^{3}-3 c_{1} \right )}}{3 x} \\ y \left (x \right ) &= \frac {\sqrt {3}\, \sqrt {-x \left (x^{3}-3 c_{1} \right )}}{3 x} \\ \end{align*} |
Problem 11574
| ODE |
\[ \boxed {y^{\prime } x +y-x^{3} y^{3}=0} \] |
|
program solution |
\[ \frac {-y^{2} x -\frac {1}{2 x^{2}}}{y^{2}} = c_{1} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (x \right ) &= \frac {1}{\sqrt {-2 x +c_{1}}\, x} \\ y \left (x \right ) &= -\frac {1}{\sqrt {-2 x +c_{1}}\, x} \\ \end{align*} |
Problem 11575
| ODE |
\[ \boxed {y^{\prime }+3 y=3 x^{2} {\mathrm e}^{-3 x}} \] |
|
program solution |
\[ y = {\mathrm e}^{-3 x} \left (x^{3}+c_{1} \right ) \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \left (x^{3}+c_{1} \right ) {\mathrm e}^{-3 x} \] |
Problem 11576
| ODE |
\[ \boxed {y^{\prime }+4 x y=8 x} \] |
|
program solution |
\[ y = {\mathrm e}^{-2 x^{2}-4 c_{1}}+2 \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = 2+{\mathrm e}^{-2 x^{2}} c_{1} \] |
Problem 11577
| ODE |
\[ \boxed {y^{\prime \prime }-2 y^{\prime }-8 y=0} \] |
|
program solution |
\[ y = c_{1} {\mathrm e}^{-2 x}+\frac {c_{2} {\mathrm e}^{4 x}}{6} \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{4 x}+{\mathrm e}^{-2 x} c_{2} \] |
Problem 11578
| ODE |
\[ \boxed {y^{\prime \prime \prime }-2 y^{\prime \prime }-4 y^{\prime }+8 y=0} \] |
|
program solution |
\[ y = c_{1} {\mathrm e}^{-2 x}+c_{2} {\mathrm e}^{2 x}+x \,{\mathrm e}^{2 x} c_{3} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \left (c_{3} x +c_{2} \right ) {\mathrm e}^{2 x}+{\mathrm e}^{-2 x} c_{1} \] |
Problem 11579
| ODE |
\[ \boxed {y^{\prime \prime \prime }-3 y^{\prime \prime }-4 y^{\prime }+12 y=0} \] |
|
program solution |
\[ y = c_{1} {\mathrm e}^{-2 x}+c_{2} {\mathrm e}^{2 x}+{\mathrm e}^{3 x} c_{3} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \left (c_{1} {\mathrm e}^{5 x}+c_{2} {\mathrm e}^{4 x}+c_{3} \right ) {\mathrm e}^{-2 x} \] |
Problem 11580
| ODE |
\[ \boxed {x^{3} y^{\prime \prime \prime }+2 x^{2} y^{\prime \prime }-10 y^{\prime } x -8 y=0} \] |
|
program solution |
\[ y = \frac {c_{1}}{x}+\frac {c_{2}}{x^{2}}+c_{3} x^{4} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {c_{1} x^{6}+c_{2} x +c_{3}}{x^{2}} \] |
Problem 11581
| ODE |
\[ \boxed {y^{\prime }+2 y=6 \,{\mathrm e}^{x}+4 x \,{\mathrm e}^{-2 x}} \] |
|
program solution |
\[ y = \left (2 \,{\mathrm e}^{3 x}+2 x^{2}+c_{1} \right ) {\mathrm e}^{-2 x} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \left (2 x^{2}+2 \,{\mathrm e}^{3 x}+c_{1} \right ) {\mathrm e}^{-2 x} \] |
Problem 11582
| ODE |
\[ \boxed {y^{\prime \prime }-4 y^{\prime }+4 y=-8 \sin \left (2 x \right )} \] |
|
program solution |
\[ y = {\mathrm e}^{2 x} \left (c_{2} x +c_{1} \right )-\cos \left (2 x \right ) \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = \left (c_{1} x +c_{2} \right ) {\mathrm e}^{2 x}-\cos \left (2 x \right ) \] |
Problem 11583
| ODE |
\[ \boxed {{y^{\prime }}^{2}-4 y=0} \] |
|
program solution |
\[ \sqrt {y} = x +c_{1} \] Verified OK. \[ -\sqrt {y} = x +c_{2} \] Verified OK.
|
|
Maple solution |
\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \left (x -c_{1} \right )^{2} \\ \end{align*} |
Problem 11584
| ODE |
\[ \boxed {y^{\prime \prime }+y^{\prime }-6 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 6, y^{\prime }\left (0\right ) = 2] \end {align*}
|
|
program solution |
\[ y = 4 \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{-3 x} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \left (4 \,{\mathrm e}^{5 x}+2\right ) {\mathrm e}^{-3 x} \] |
Problem 11585
| ODE |
\[ \boxed {y^{\prime }+y=2 x \,{\mathrm e}^{-x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 2] \end {align*}
|
|
program solution |
\[ y = x^{2} {\mathrm e}^{-x}+2 \,{\mathrm e}^{-x} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \left (x^{2}+2\right ) {\mathrm e}^{-x} \] |
Problem 11586
| ODE |
\[ \boxed {y^{\prime }+y=2 x \,{\mathrm e}^{-x}} \] With initial conditions \begin {align*} [y \left (-1\right ) = {\mathrm e}+3] \end {align*}
|
|
program solution |
\[ y = x^{2} {\mathrm e}^{-x}+3 \,{\mathrm e}^{-x -1} \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = \left (x^{2}+3 \,{\mathrm e}^{-1}\right ) {\mathrm e}^{-x} \] |
Problem 11587
| ODE |
\[ \boxed {y^{\prime \prime }-y^{\prime }-12 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 5, y^{\prime }\left (0\right ) = 6] \end {align*}
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|
program solution |
\[ y = 3 \,{\mathrm e}^{4 x}+2 \,{\mathrm e}^{-3 x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (3 \,{\mathrm e}^{7 x}+2\right ) {\mathrm e}^{-3 x} \] |
Problem 11588
| ODE |
\[ \boxed {y^{\prime \prime }+y=0} \] With initial conditions \begin {align*} \left [y \left (0\right ) = 0, y^{\prime }\left (\frac {\pi }{2}\right ) = 1\right ] \end {align*}
|
|
program solution |
N/A |
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Maple solution | \[ \text {No solution found} \] |
Problem 11589
| ODE |
\[ \boxed {y^{\prime \prime }+y=0} \] With initial conditions \begin {align*} \left [y \left (0\right ) = 0, y^{\prime }\left (\frac {\pi }{2}\right ) = -1\right ] \end {align*}
|
|
program solution |
N/A |
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Maple solution | \[ \text {No solution found} \] |
Problem 11590
| ODE |
\[ \boxed {y^{\prime \prime }+y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (\pi \right ) = 1] \end {align*}
|
|
program solution |
\[ y = -\sin \left (x \right ) \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = -\sin \left (x \right ) \] |
Problem 11591
| ODE |
\[ \boxed {x^{3} y^{\prime \prime \prime }-3 x^{2} y^{\prime \prime }+6 y^{\prime } x -6 y=0} \] With initial conditions \begin {align*} [y \left (2\right ) = 0, y^{\prime }\left (2\right ) = 2, y^{\prime \prime }\left (2\right ) = 6] \end {align*}
|
|
program solution |
\[ y = x^{3}-3 x^{2}+2 x \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x^{3}-3 x^{2}+2 x \] |
Problem 11592
| ODE |
\[ \boxed {y^{\prime }-x^{2} \sin \left (y\right )=0} \] With initial conditions \begin {align*} [y \left (1\right ) = -2] \end {align*}
|
|
program solution |
\[ -\frac {x^{3}}{3}-\ln \left (\csc \left (y\right )+\cot \left (y\right )\right ) = -\frac {1}{3}-\ln \left (1+\cos \left (2\right )\right )+\ln \left (\sin \left (2\right )\right )-i \pi \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = \arctan \left (\frac {2 \sin \left (2\right ) {\mathrm e}^{\frac {\left (-1+x \right ) \left (x^{2}+x +1\right )}{3}}}{\left (-1+\cos \left (2\right )\right ) {\mathrm e}^{\frac {2 \left (-1+x \right ) \left (x^{2}+x +1\right )}{3}}-1-\cos \left (2\right )}, \frac {\left (1-\cos \left (2\right )\right ) {\mathrm e}^{\frac {2 \left (-1+x \right ) \left (x^{2}+x +1\right )}{3}}-1-\cos \left (2\right )}{\left (-1+\cos \left (2\right )\right ) {\mathrm e}^{\frac {2 \left (-1+x \right ) \left (x^{2}+x +1\right )}{3}}-1-\cos \left (2\right )}\right ) \] |
Problem 11593
| ODE |
\[ \boxed {y^{\prime }-\frac {y^{2}}{x -2}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 0] \end {align*}
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|
program solution |
\[ y = 0 \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = 0 \] |
Problem 11594
| ODE |
\[ \boxed {y^{\prime }-y^{\frac {1}{3}}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
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|
program solution |
\[ y = 0 \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = 0 \] |
Problem 11595
| ODE |
\[ \boxed {2 y+\left (2 x +y\right ) y^{\prime }=-3 x} \] |
|
program solution |
\[ \frac {x \left (3 x +4 y\right )}{2}+\frac {y^{2}}{2} = c_{1} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (x \right ) &= \frac {-2 c_{1} x -\sqrt {c_{1}^{2} x^{2}+1}}{c_{1}} \\ y \left (x \right ) &= \frac {-2 c_{1} x +\sqrt {c_{1}^{2} x^{2}+1}}{c_{1}} \\ \end{align*} |
Problem 11596
| ODE |
\[ \boxed {y^{2}+\left (2 x y-4\right ) y^{\prime }=-3} \] |
|
program solution |
\[ \left (y^{2}+3\right ) x -4 y = c_{1} \] Verified OK.
|
|
Maple solution | \[ \frac {-i c_{1} \left (y \left (x \right )^{2} x +3 x -4 y \left (x \right )\right ) \sqrt {3}+12 c_{1} +i}{\left (-y \left (x \right ) \sqrt {3}\, x +4 \sqrt {3}-3 i x \right ) \left (\sqrt {3}+i y \left (x \right )\right )} = 0 \] |
Problem 11597
| ODE |
\[ \boxed {2 x y+\left (x^{2}+4 y\right ) y^{\prime }=-1} \] |
|
program solution |
\[ x^{2} y+2 y^{2}+x = c_{1} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (x \right ) &= -\frac {x^{2}}{4}-\frac {\sqrt {x^{4}-8 c_{1} -8 x}}{4} \\ y \left (x \right ) &= -\frac {x^{2}}{4}+\frac {\sqrt {x^{4}-8 c_{1} -8 x}}{4} \\ \end{align*} |
Problem 11598
| ODE |
\[ \boxed {3 x^{2} y-\left (x^{3}+y\right ) y^{\prime }=-2} \] |
|
program solution |
|
|
Maple solution | \[ \text {No solution found} \] |
Problem 11599
| ODE |
\[ \boxed {6 x y+2 y^{2}+\left (3 x^{2}+4 x y-6\right ) y^{\prime }=5} \] |
|
program solution |
\[ 3 x^{2} y+2 y^{2} x -5 x -6 y = c_{1} \] Verified OK.
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|
Maple solution | \begin{align*} y \left (x \right ) &= \frac {-3 x^{2}+6+\sqrt {9 x^{4}-8 c_{1} x +4 x^{2}+36}}{4 x} \\ y \left (x \right ) &= \frac {-3 x^{2}+6-\sqrt {9 x^{4}-8 c_{1} x +4 x^{2}+36}}{4 x} \\ \end{align*} |
Problem 11600
| ODE |
\[ \boxed {y \sec \left (x \right )^{2}+\left (\tan \left (x \right )+2 y\right ) y^{\prime }=-\sec \left (x \right ) \tan \left (x \right )} \] |
|
program solution |
\[ y \tan \left (x \right )+\sec \left (x \right )+y^{2} = c_{1} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (x \right ) &= -\frac {\tan \left (x \right )}{2}+\frac {\sec \left (x \right ) \sqrt {-4 \cos \left (x \right )^{2} c_{1} +\sin \left (x \right )^{2}-4 \cos \left (x \right )}}{2} \\ y \left (x \right ) &= -\frac {\tan \left (x \right )}{2}-\frac {\sec \left (x \right ) \sqrt {-4 \cos \left (x \right )^{2} c_{1} +\sin \left (x \right )^{2}-4 \cos \left (x \right )}}{2} \\ \end{align*} |