Problem 12301
| ODE |
\[ \boxed {y^{\prime \prime \prime }+8 y^{\prime \prime }+16 y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 1, y^{\prime \prime }\left (0\right ) = -8] \end {align*}
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program solution |
\[ y = 1+t \,{\mathrm e}^{-4 t} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = t \,{\mathrm e}^{-4 t}+1 \] |
Problem 12302
| ODE |
\[ \boxed {y^{\prime \prime \prime }+6 y^{\prime \prime }+13 y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 1, y^{\prime \prime }\left (0\right ) = -6] \end {align*}
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program solution |
\[ y = \frac {{\mathrm e}^{-3 t} \sin \left (2 t \right )}{2}+1 \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {{\mathrm e}^{-3 t} \sin \left (2 t \right )}{2}+1 \] |
Problem 12303
| ODE |
\[ \boxed {y^{\prime \prime \prime }-6 y^{\prime \prime }+13 y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 1, y^{\prime \prime }\left (0\right ) = 6] \end {align*}
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program solution |
\[ y = 1+\frac {{\mathrm e}^{3 t} \sin \left (2 t \right )}{2} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {{\mathrm e}^{3 t} \sin \left (2 t \right )}{2}+1 \] |
Problem 12304
| ODE |
\[ \boxed {y^{\prime \prime \prime }+4 y^{\prime \prime }+29 y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 5, y^{\prime \prime }\left (0\right ) = -20] \end {align*}
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program solution |
\[ y = {\mathrm e}^{-2 t} \sin \left (5 t \right )+1 \] Verified OK.
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Maple solution | \[ y \left (t \right ) = 1+{\mathrm e}^{-2 t} \sin \left (5 t \right ) \] |
Problem 12305
| ODE |
\[ \boxed {y^{\prime \prime \prime }+6 y^{\prime \prime }+25 y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 4, y^{\prime \prime }\left (0\right ) = -24] \end {align*}
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program solution |
\[ y = 1+{\mathrm e}^{-3 t} \sin \left (4 t \right ) \] Verified OK.
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Maple solution | \[ y \left (t \right ) = {\mathrm e}^{-3 t} \sin \left (4 t \right )+1 \] |
Problem 12306
| ODE |
\[ \boxed {y^{\prime \prime \prime }-6 y^{\prime \prime }+10 y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 3, y^{\prime \prime }\left (0\right ) = 8] \end {align*}
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program solution |
\[ y = {\mathrm e}^{3 t} \cos \left (t \right ) \] Verified OK.
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Maple solution | \[ y \left (t \right ) = {\mathrm e}^{3 t} \cos \left (t \right ) \] |
Problem 12307
| ODE |
\[ \boxed {y^{\prime \prime \prime \prime }+13 y^{\prime \prime }+36 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = -1, y^{\prime \prime }\left (0\right ) = 5, y^{\prime \prime \prime }\left (0\right ) = 19] \end {align*}
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program solution |
\[ y = -\cos \left (3 t \right )-\sin \left (3 t \right )+\cos \left (2 t \right )+\sin \left (2 t \right ) \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \cos \left (2 t \right )+\sin \left (2 t \right )-\cos \left (3 t \right )-\sin \left (3 t \right ) \] |
Problem 12308
| ODE |
\[ \boxed {y^{\prime \prime }+2 y^{\prime }+3 y=9 t} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
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program solution |
\[ y = 2 \,{\mathrm e}^{-t} \cos \left (\sqrt {2}\, t \right )-2+3 t \] Verified OK.
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Maple solution | \[ y \left (t \right ) = 2 \,{\mathrm e}^{-t} \cos \left (\sqrt {2}\, t \right )+3 t -2 \] |
Problem 12309
| ODE |
\[ \boxed {4 y^{\prime \prime }+16 y^{\prime }+17 y=17 t -1} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = 2] \end {align*}
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program solution |
\[ y = -1+t +2 \,{\mathrm e}^{-2 t} \sin \left (\frac {t}{2}\right ) \] Verified OK.
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Maple solution | \[ y \left (t \right ) = 2 \,{\mathrm e}^{-2 t} \sin \left (\frac {t}{2}\right )+t -1 \] |
Problem 12310
| ODE |
\[ \boxed {4 y^{\prime \prime }+5 y^{\prime }+4 y=3 \,{\mathrm e}^{-t}} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = 1] \end {align*}
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program solution |
\[ y = \frac {2 \sin \left (\frac {\sqrt {39}\, t}{8}\right ) {\mathrm e}^{-\frac {5 t}{8}} \sqrt {39}}{13}-2 \cos \left (\frac {\sqrt {39}\, t}{8}\right ) {\mathrm e}^{-\frac {5 t}{8}}+{\mathrm e}^{-t} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {2 \sqrt {39}\, {\mathrm e}^{-\frac {5 t}{8}} \sin \left (\frac {\sqrt {39}\, t}{8}\right )}{13}-2 \,{\mathrm e}^{-\frac {5 t}{8}} \cos \left (\frac {\sqrt {39}\, t}{8}\right )+{\mathrm e}^{-t} \] |
Problem 12311
| ODE |
\[ \boxed {y^{\prime \prime }-4 y^{\prime }+4 y=t^{2} {\mathrm e}^{2 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 2] \end {align*}
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program solution |
\[ y = \frac {{\mathrm e}^{2 t} \left (t^{4}+12\right )}{12} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {{\mathrm e}^{2 t} \left (t^{4}+12\right )}{12} \] |
Problem 12312
| ODE |
\[ \boxed {y^{\prime \prime }+9 y={\mathrm e}^{-2 t}} \] With initial conditions \begin {align*} \left [y \left (0\right ) = -{\frac {2}{13}}, y^{\prime }\left (0\right ) = {\frac {1}{13}}\right ] \end {align*}
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program solution |
\[ y = -\frac {3 \cos \left (3 t \right )}{13}+\frac {\sin \left (3 t \right )}{13}+\frac {{\mathrm e}^{-2 t}}{13} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = -\frac {3 \cos \left (3 t \right )}{13}+\frac {\sin \left (3 t \right )}{13}+\frac {{\mathrm e}^{-2 t}}{13} \] |
Problem 12313
| ODE |
\[ \boxed {2 y^{\prime \prime }-3 y^{\prime }+17 y=17 t -1} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = 2] \end {align*}
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program solution |
\[ y = \frac {125 \,{\mathrm e}^{\frac {3 t}{4}} \sin \left (\frac {\sqrt {127}\, t}{4}\right ) \sqrt {127}}{2159}-\frac {19 \,{\mathrm e}^{\frac {3 t}{4}} \cos \left (\frac {\sqrt {127}\, t}{4}\right )}{17}+t +\frac {2}{17} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {125 \sqrt {127}\, {\mathrm e}^{\frac {3 t}{4}} \sin \left (\frac {\sqrt {127}\, t}{4}\right )}{2159}-\frac {19 \,{\mathrm e}^{\frac {3 t}{4}} \cos \left (\frac {\sqrt {127}\, t}{4}\right )}{17}+t +\frac {2}{17} \] |
Problem 12314
| ODE |
\[ \boxed {y^{\prime \prime }+2 y^{\prime }+y={\mathrm e}^{-t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = -1] \end {align*}
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program solution |
\[ y = \frac {{\mathrm e}^{-t} \left (t^{2}+2\right )}{2} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {{\mathrm e}^{-t} \left (t^{2}+2\right )}{2} \] |
Problem 12315
| ODE |
\[ \boxed {y^{\prime \prime }-2 y^{\prime }+5 y=2+t} \] With initial conditions \begin {align*} [y \left (0\right ) = 4, y^{\prime }\left (0\right ) = 1] \end {align*}
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program solution |
\[ y = \frac {88 \,{\mathrm e}^{t} \cos \left (2 t \right )}{25}-\frac {34 \,{\mathrm e}^{t} \sin \left (2 t \right )}{25}+\frac {t}{5}+\frac {12}{25} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {88 \cos \left (2 t \right ) {\mathrm e}^{t}}{25}-\frac {34 \sin \left (2 t \right ) {\mathrm e}^{t}}{25}+\frac {t}{5}+\frac {12}{25} \] |
Problem 12316
| ODE |
\[ \boxed {2 y^{\prime }+y={\mathrm e}^{-\frac {t}{2}}} \] With initial conditions \begin {align*} [y \left (0\right ) = -1] \end {align*}
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program solution |
\[ y = \frac {\left (-2+t \right ) {\mathrm e}^{-\frac {t}{2}}}{2} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {{\mathrm e}^{-\frac {t}{2}} \left (t -2\right )}{2} \] |
Problem 12317
| ODE |
\[ \boxed {y^{\prime \prime }+8 y^{\prime }+20 y=\sin \left (2 t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = -4] \end {align*}
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program solution |
\[ y = \frac {\left (-1+33 \,{\mathrm e}^{-4 t}\right ) \cos \left (2 t \right )}{32}+\frac {\sin \left (2 t \right ) \left (1+{\mathrm e}^{-4 t}\right )}{32} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {\left (-1+33 \,{\mathrm e}^{-4 t}\right ) \cos \left (2 t \right )}{32}+\frac {\sin \left (2 t \right ) \left (1+{\mathrm e}^{-4 t}\right )}{32} \] |
Problem 12318
| ODE |
\[ \boxed {4 y^{\prime \prime }-4 y^{\prime }+y=t^{2}} \] With initial conditions \begin {align*} [y \left (0\right ) = -12, y^{\prime }\left (0\right ) = 7] \end {align*}
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program solution |
\[ y = t^{2}+8 t +24+{\mathrm e}^{\frac {t}{2}} \left (17 t -36\right ) \] Verified OK.
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Maple solution | \[ y \left (t \right ) = t^{2}+8 t +24+{\mathrm e}^{\frac {t}{2}} \left (-36+17 t \right ) \] |
Problem 12319
| ODE |
\[ \boxed {2 y^{\prime \prime }+y^{\prime }-y=4 \sin \left (t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = -4] \end {align*}
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program solution |
\[ y = -\frac {2 \left (4 \,{\mathrm e}^{\frac {3 t}{2}}-5+\left (\cos \left (t \right )+3 \sin \left (t \right )\right ) {\mathrm e}^{t}\right ) {\mathrm e}^{-t}}{5} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = -\frac {2 \,{\mathrm e}^{-t} \left (4 \,{\mathrm e}^{\frac {3 t}{2}}-5+\left (\cos \left (t \right )+3 \sin \left (t \right )\right ) {\mathrm e}^{t}\right )}{5} \] |
Problem 12320
| ODE |
\[ \boxed {y^{\prime }-y={\mathrm e}^{2 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
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program solution |
\[ y = {\mathrm e}^{2 t} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = {\mathrm e}^{2 t} \] |
Problem 12321
| ODE |
\[ \boxed {3 y^{\prime \prime }+5 y^{\prime }-2 y=7 \,{\mathrm e}^{-2 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = -\left (-3 \,{\mathrm e}^{\frac {7 t}{3}}+t \right ) {\mathrm e}^{-2 t} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = -\left (-3 \,{\mathrm e}^{\frac {7 t}{3}}+t \right ) {\mathrm e}^{-2 t} \] |
Problem 12322
| ODE |
\[ \boxed {y^{\prime }+y=\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (-2+t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
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program solution |
\[
y = \left \{\begin {array}{cc} 1 & t <2 \\ 2 & t =2 \\ {\mathrm e}^{2-t} & 2 |
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Maple solution | \[ y \left (t \right ) = 1-\operatorname {Heaviside}\left (t -2\right )+\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{2-t} \] |
Problem 12323
| ODE |
\[ \boxed {y^{\prime }-2 y=4 t \left (\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (-2+t \right )\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
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program solution |
\[
y = \left \{\begin {array}{cc} 2 \,{\mathrm e}^{2 t}-1-2 t & t \le 2 \\ 2 \,{\mathrm e}^{2 t}-5 \,{\mathrm e}^{2 t -4} & 2 |
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Maple solution | \[ y \left (t \right ) = -5 \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{2 t -4}+2 t \operatorname {Heaviside}\left (t -2\right )-2 t +2 \,{\mathrm e}^{2 t}-1+\operatorname {Heaviside}\left (t -2\right ) \] |
Problem 12324
| ODE |
\[ \boxed {y^{\prime \prime }+9 y=24 \sin \left (t \right ) \left (\operatorname {Heaviside}\left (t \right )+\operatorname {Heaviside}\left (t -\pi \right )\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = \left (\left \{\begin {array}{cc} 4 & t <\pi \\ 8 & \pi \le t \end {array}\right .\right ) \sin \left (t \right )^{3} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = 4 \left (1+\operatorname {Heaviside}\left (t -\pi \right )\right ) \sin \left (t \right )^{3} \] |
Problem 12325
| ODE |
\[ \boxed {y^{\prime \prime }+2 y^{\prime }+y=\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -1\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = -1] \end {align*}
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program solution |
\[
y = \left \{\begin {array}{cc} 1-t \,{\mathrm e}^{-t} & t <1 \\ 2-{\mathrm e}^{-1} & t =1 \\ t \left (-{\mathrm e}^{-t}+{\mathrm e}^{1-t}\right ) & 1 |
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Maple solution | \[ y \left (t \right ) = t \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-t +1}-t \,{\mathrm e}^{-t}+1-\operatorname {Heaviside}\left (t -1\right ) \] |
Problem 12326
| ODE |
\[ \boxed {y^{\prime \prime }+2 y^{\prime }+2 y=5 \cos \left (t \right ) \left (\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = -1] \end {align*}
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program solution |
\[ y = -3 \,{\mathrm e}^{-t} \sin \left (t \right )-\left (\left \{\begin {array}{cc} -2 \sin \left (t \right )-\cos \left (t \right ) & t <\frac {\pi }{2} \\ {\mathrm e}^{-t +\frac {\pi }{2}} \left (\cos \left (t \right )-2 \sin \left (t \right )\right ) & \frac {\pi }{2}\le t \end {array}\right .\right ) \] Verified OK.
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Maple solution | \[ y \left (t \right ) = -\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (\cos \left (t \right )-2 \sin \left (t \right )\right ) {\mathrm e}^{-t +\frac {\pi }{2}}+\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (-\cos \left (t \right )-2 \sin \left (t \right )\right )-3 \sin \left (t \right ) {\mathrm e}^{-t}+\cos \left (t \right )+2 \sin \left (t \right ) \] |
Problem 12327
| ODE |
\[ \boxed {y^{\prime \prime }+5 y^{\prime }+6 y=36 t \left (\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -1\right )\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = -2] \end {align*}
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program solution |
\[
y = \left \{\begin {array}{cc} 4 \,{\mathrm e}^{-2 t}-5+6 t & t <1 \\ 4 \,{\mathrm e}^{-2}+2 & t =1 \\ 4 \,{\mathrm e}^{-2 t}-8 \,{\mathrm e}^{-3 t +3}+9 \,{\mathrm e}^{-2 t +2} & 1 |
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Maple solution | \[ y \left (t \right ) = -8 \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-3 t +3}+9 \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-2 t +2}+\left (-6 t +5\right ) \operatorname {Heaviside}\left (t -1\right )+6 t +4 \,{\mathrm e}^{-2 t}-5 \] |
Problem 12328
| ODE |
\[ \boxed {y^{\prime \prime }+4 y^{\prime }+13 y=39 \operatorname {Heaviside}\left (t \right )-507 \left (-2+t \right ) \operatorname {Heaviside}\left (-2+t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = 1] \end {align*}
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program solution |
\[ y = \frac {{\mathrm e}^{-2 t} \sin \left (3 t \right )}{3}+\left (\left \{\begin {array}{cc} 3 & t <2 \\ 93-39 t +{\mathrm e}^{4-2 t} \left (5 \sin \left (-6+3 t \right )-12 \cos \left (-6+3 t \right )\right ) & 2\le t \end {array}\right .\right ) \] Verified OK.
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Maple solution | \[ y \left (t \right ) = 3-12 \left (\left (-\frac {5 \cos \left (6\right )}{12}+\sin \left (6\right )\right ) \sin \left (3 t \right )+\cos \left (3 t \right ) \left (\cos \left (6\right )+\frac {5 \sin \left (6\right )}{12}\right )\right ) \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-2 t +4}+3 \left (30-13 t \right ) \operatorname {Heaviside}\left (t -2\right )+\frac {{\mathrm e}^{-2 t} \sin \left (3 t \right )}{3} \] |
Problem 12329
| ODE |
\[ \boxed {y^{\prime \prime }+4 y=3 \operatorname {Heaviside}\left (t \right )-3 \operatorname {Heaviside}\left (t -4\right )+\left (2 t -5\right ) \operatorname {Heaviside}\left (t -4\right )} \] With initial conditions \begin {align*} \left [y \left (0\right ) = {\frac {3}{4}}, y^{\prime }\left (0\right ) = 2\right ] \end {align*}
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program solution |
\[ y = \sin \left (2 t \right )-\frac {\left (\left \{\begin {array}{cc} -3 & t <4 \\ 5-2 t +\sin \left (2 t -8\right ) & 4\le t \end {array}\right .\right )}{4} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = -\frac {\operatorname {Heaviside}\left (t -4\right ) \sin \left (2 t -8\right )}{4}+\frac {\operatorname {Heaviside}\left (t -4\right ) t}{2}+\sin \left (2 t \right )-2 \operatorname {Heaviside}\left (t -4\right )+\frac {3}{4} \] |
Problem 12330
| ODE |
\[ \boxed {4 y^{\prime \prime }+4 y^{\prime }+5 y=25 t \left (\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 2] \end {align*}
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program solution |
\[
y = \left \{\begin {array}{cc} 6 \cos \left (t \right ) {\mathrm e}^{-\frac {t}{2}}+5 t -4 & t <\frac {\pi }{2} \\ -8+5 \pi & t =\frac {\pi }{2} \\ \left (\frac {5}{4}-\frac {5 i}{8}\right ) \pi \,{\mathrm e}^{\left (\frac {1}{4}-\frac {i}{2}\right ) \left (-2 t +\pi \right )}+\left (\frac {5}{4}+\frac {5 i}{8}\right ) \pi \,{\mathrm e}^{\left (\frac {1}{4}+\frac {i}{2}\right ) \left (-2 t +\pi \right )}+\left (-3 \cos \left (t \right )-4 \sin \left (t \right )\right ) {\mathrm e}^{-\frac {t}{2}+\frac {\pi }{4}}+6 \cos \left (t \right ) {\mathrm e}^{-\frac {t}{2}} & \frac {\pi }{2} |
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Maple solution | \[ y \left (t \right ) = -4+\left (\frac {5}{4}-\frac {5 i}{8}\right ) \pi \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) {\mathrm e}^{\left (\frac {1}{4}-\frac {i}{2}\right ) \left (-2 t +\pi \right )}+\left (\frac {5}{4}+\frac {5 i}{8}\right ) \pi \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) {\mathrm e}^{\left (\frac {1}{4}+\frac {i}{2}\right ) \left (-2 t +\pi \right )}-3 \left (\cos \left (t \right )+\frac {4 \sin \left (t \right )}{3}\right ) \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) {\mathrm e}^{-\frac {t}{2}+\frac {\pi }{4}}+\left (4-5 t \right ) \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )+6 \cos \left (t \right ) {\mathrm e}^{-\frac {t}{2}}+5 t \] |
Problem 12331
| ODE |
\[ \boxed {y^{\prime \prime }+4 y^{\prime }+3 y=\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -1\right )+\operatorname {Heaviside}\left (-2+t \right )-\operatorname {Heaviside}\left (t -3\right )} \] With initial conditions \begin {align*} \left [y \left (0\right ) = -{\frac {2}{3}}, y^{\prime }\left (0\right ) = 1\right ] \end {align*}
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program solution |
\[
y = -\frac {\left (\left \{\begin {array}{cc} 3 \,{\mathrm e}^{-t}-1 & t <1 \\ 3 \,{\mathrm e}^{-t}-\frac {3 \,{\mathrm e}^{1-t}}{2}+\frac {{\mathrm e}^{-3 t +3}}{2} & t <2 \\ 3 \,{\mathrm e}^{-t}-1+\frac {3 \,{\mathrm e}^{2-t}}{2}-\frac {{\mathrm e}^{6-3 t}}{2}-\frac {3 \,{\mathrm e}^{1-t}}{2}+\frac {{\mathrm e}^{-3 t +3}}{2} & t <3 \\ \frac {5 \,{\mathrm e}^{-3}}{2}-2+\frac {3 \,{\mathrm e}^{-1}}{2}-\frac {3 \,{\mathrm e}^{-2}}{2}+\frac {{\mathrm e}^{-6}}{2} & t =3 \\ 3 \,{\mathrm e}^{-t}+\frac {3 \,{\mathrm e}^{2-t}}{2}-\frac {{\mathrm e}^{6-3 t}}{2}-\frac {3 \,{\mathrm e}^{-t +3}}{2}+\frac {{\mathrm e}^{-3 t +9}}{2}-\frac {3 \,{\mathrm e}^{1-t}}{2}+\frac {{\mathrm e}^{-3 t +3}}{2} & 3 |
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Maple solution | \[ y \left (t \right ) = \frac {1}{3}-\frac {\operatorname {Heaviside}\left (t -3\right )}{3}-{\mathrm e}^{-t}-\frac {\operatorname {Heaviside}\left (t -3\right ) {\mathrm e}^{-3 t +9}}{6}+\frac {\operatorname {Heaviside}\left (t -3\right ) {\mathrm e}^{-t +3}}{2}+\frac {\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{6-3 t}}{6}-\frac {\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{2-t}}{2}+\frac {\operatorname {Heaviside}\left (t -2\right )}{3}-\frac {\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-3 t +3}}{6}+\frac {\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-t +1}}{2}-\frac {\operatorname {Heaviside}\left (t -1\right )}{3} \] |
Problem 12332
| ODE |
\[ \boxed {y^{\prime \prime }-2 y^{\prime }=\left \{\begin {array}{cc} 4 & 0\le t <1 \\ 6 & 1\le t \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = -6, y^{\prime }\left (0\right ) = 1] \end {align*}
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program solution |
\[
y = -\frac {\left (\left \{\begin {array}{cc} 15+4 t -3 \,{\mathrm e}^{2 t} & t <1 \\ 20-3 \,{\mathrm e}^{2} & t =1 \\ 14+6 t -3 \,{\mathrm e}^{2 t}-{\mathrm e}^{2 t -2} & 1 |
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Maple solution | \[
y \left (t \right ) = -\frac {\left (\left \{\begin {array}{cc} 15+4 t -3 \,{\mathrm e}^{2 t} & t <1 \\ 20-3 \,{\mathrm e}^{2} & t =1 \\ 14+6 t -3 \,{\mathrm e}^{2 t}-{\mathrm e}^{2 t -2} & 1 |
Problem 12333
| ODE |
\[ \boxed {y^{\prime \prime }-3 y^{\prime }+2 y=\left \{\begin {array}{cc} 0 & 0\le t <1 \\ 1 & 1\le t <2 \\ -1 & 2\le t \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = -1] \end {align*}
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program solution |
\[
y = \left \{\begin {array}{cc} 7 \,{\mathrm e}^{t}-4 \,{\mathrm e}^{2 t} & t <1 \\ 7 \,{\mathrm e}-4 \,{\mathrm e}^{2}+\frac {1}{2} & t =1 \\ 7 \,{\mathrm e}^{t}-4 \,{\mathrm e}^{2 t}-{\mathrm e}^{t -1}+\frac {{\mathrm e}^{2 t -2}}{2}+\frac {1}{2} & t <2 \\ \frac {15 \,{\mathrm e}^{2}}{2}-4 \,{\mathrm e}^{4}-{\mathrm e}-\frac {1}{2} & t =2 \\ 7 \,{\mathrm e}^{t}-4 \,{\mathrm e}^{2 t}+2 \,{\mathrm e}^{-2+t}-{\mathrm e}^{t -1}-{\mathrm e}^{2 t -4}+\frac {{\mathrm e}^{2 t -2}}{2}-\frac {1}{2} & 2 |
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Maple solution | \[
y \left (t \right ) = \left \{\begin {array}{cc} 7 \,{\mathrm e}^{t}-4 \,{\mathrm e}^{2 t} & t <1 \\ 7 \,{\mathrm e}-4 \,{\mathrm e}^{2}+\frac {1}{2} & t =1 \\ 7 \,{\mathrm e}^{t}-4 \,{\mathrm e}^{2 t}-{\mathrm e}^{t -1}+\frac {{\mathrm e}^{2 t -2}}{2}+\frac {1}{2} & t <2 \\ \frac {15 \,{\mathrm e}^{2}}{2}-4 \,{\mathrm e}^{4}-{\mathrm e}-\frac {1}{2} & t =2 \\ 7 \,{\mathrm e}^{t}-4 \,{\mathrm e}^{2 t}+2 \,{\mathrm e}^{t -2}-{\mathrm e}^{t -1}-{\mathrm e}^{2 t -4}+\frac {{\mathrm e}^{2 t -2}}{2}-\frac {1}{2} & 2 |
Problem 12334
| ODE |
\[ \boxed {y^{\prime \prime }+3 y^{\prime }+2 y=\left \{\begin {array}{cc} 1 & 0\le t <2 \\ -1 & 2\le t \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = -{\mathrm e}^{-t}+\frac {{\mathrm e}^{-2 t}}{2}-\frac {\left (\left \{\begin {array}{cc} -1 & t <2 \\ 1-4 \,{\mathrm e}^{2-t}+2 \,{\mathrm e}^{4-2 t} & 2\le t \end {array}\right .\right )}{2} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = -{\mathrm e}^{-t}+\frac {{\mathrm e}^{-2 t}}{2}-\frac {\left (\left \{\begin {array}{cc} -1 & t <2 \\ 1-4 \,{\mathrm e}^{2-t}+2 \,{\mathrm e}^{-2 t +4} & 2\le t \end {array}\right .\right )}{2} \] |
Problem 12335
| ODE |
\[ \boxed {y^{\prime \prime }+y=\left \{\begin {array}{cc} t & 0\le t <\pi \\ -t & \pi \le t \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = \left \{\begin {array}{cc} -\sin \left (t \right )+t & t <\pi \\ -t -2 \pi \cos \left (t \right )-3 \sin \left (t \right ) & \pi \le t \end {array}\right . \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \left \{\begin {array}{cc} t -\sin \left (t \right ) & t <\pi \\ -2 \cos \left (t \right ) \pi -3 \sin \left (t \right )-t & \pi \le t \end {array}\right . \] |
Problem 12336
| ODE |
\[ \boxed {y^{\prime \prime }+4 y=\left \{\begin {array}{cc} 8 t & 0\le t <\frac {\pi }{2} \\ 8 \pi & \frac {\pi }{2}\le t \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = \left \{\begin {array}{cc} -\sin \left (2 t \right )+2 t & t <\frac {\pi }{2} \\ -2 \sin \left (2 t \right )+2 \pi \cos \left (t \right )^{2}+\pi & \frac {\pi }{2}\le t \end {array}\right . \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \left \{\begin {array}{cc} -\sin \left (2 t \right )+2 t & t <\frac {\pi }{2} \\ -2 \sin \left (2 t \right )+2 \cos \left (t \right )^{2} \pi +\pi & \frac {\pi }{2}\le t \end {array}\right . \] |
Problem 12337
| ODE |
\[ \boxed {y^{\prime \prime }+4 \pi ^{2} y=3 \delta \left (t -\frac {1}{3}\right )-\delta \left (t -1\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = -\frac {\left \{\begin {array}{cc} 0 & t <\frac {1}{3} \\ 3 \sin \left (\pi \left (2 t +\frac {1}{3}\right )\right ) & t <1 \\ \frac {3 \cos \left (2 \pi t \right ) \sqrt {3}}{2}+\frac {5 \sin \left (2 \pi t \right )}{2} & 1\le t \end {array}\right .}{2 \pi } \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {\left (-3 \sqrt {3}\, \cos \left (2 \pi t \right )-3 \sin \left (2 \pi t \right )\right ) \operatorname {Heaviside}\left (t -\frac {1}{3}\right )-2 \sin \left (2 \pi t \right ) \operatorname {Heaviside}\left (t -1\right )}{4 \pi } \] |
Problem 12338
| ODE |
\[ \boxed {y^{\prime \prime }+2 y^{\prime }+2 y=3 \delta \left (t -1\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = \left \{\begin {array}{cc} 0 & t <1 \\ 3 \,{\mathrm e}^{1-t} \sin \left (t -1\right ) & 1\le t \end {array}\right . \] Verified OK.
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Maple solution | \[ y \left (t \right ) = 3 \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-t +1} \sin \left (t -1\right ) \] |
Problem 12339
| ODE |
\[ \boxed {y^{\prime \prime }+4 y^{\prime }+29 y=5 \delta \left (t -\pi \right )-5 \delta \left (t -2 \pi \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = -\sin \left (5 t \right ) \left (\left \{\begin {array}{cc} 0 & t <\pi \\ {\mathrm e}^{-2 t +2 \pi } & t <2 \pi \\ {\mathrm e}^{-2 t +2 \pi }+{\mathrm e}^{-2 t +4 \pi } & 2 \pi \le t \end {array}\right .\right ) \] Verified OK.
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Maple solution | \[ y \left (t \right ) = -\sin \left (5 t \right ) \left ({\mathrm e}^{-2 t +2 \pi } \operatorname {Heaviside}\left (t -\pi \right )+\operatorname {Heaviside}\left (-2 \pi +t \right ) {\mathrm e}^{4 \pi -2 t}\right ) \] |
Problem 12340
| ODE |
\[ \boxed {y^{\prime \prime }+3 y^{\prime }+2 y=1-\delta \left (t -1\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = \frac {1}{2}-{\mathrm e}^{-t}+\frac {{\mathrm e}^{-2 t}}{2}-\left (\left \{\begin {array}{cc} 0 & t <1 \\ {\mathrm e}^{1-t}-{\mathrm e}^{-2 t +2} & 1\le t \end {array}\right .\right ) \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-2 t +2}-\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-t +1}-{\mathrm e}^{-t}+\frac {{\mathrm e}^{-2 t}}{2}+\frac {1}{2} \] |
Problem 12341
| ODE |
\[ \boxed {4 y^{\prime \prime }+4 y^{\prime }+y={\mathrm e}^{-\frac {t}{2}} \delta \left (t -1\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = \left \{\begin {array}{cc} 0 & t <1 \\ \frac {\left (t -1\right ) {\mathrm e}^{-\frac {t}{2}}}{4} & 1\le t \end {array}\right . \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {\operatorname {Heaviside}\left (t -1\right ) \left (t -1\right ) {\mathrm e}^{-\frac {t}{2}}}{4} \] |
Problem 12342
| ODE |
\[ \boxed {y^{\prime \prime }-7 y^{\prime }+6 y=\delta \left (t -1\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = \left \{\begin {array}{cc} 0 & t <1 \\ \frac {{\mathrm e}^{6 t -6}}{5}-\frac {{\mathrm e}^{t -1}}{5} & 1\le t \end {array}\right . \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {\operatorname {Heaviside}\left (t -1\right ) \left ({\mathrm e}^{-6+6 t}-{\mathrm e}^{t -1}\right )}{5} \] |
Problem 12343
| ODE |
\[ \boxed {10 Q^{\prime }+100 Q=\operatorname {Heaviside}\left (t -1\right )-\operatorname {Heaviside}\left (-2+t \right )} \] With initial conditions \begin {align*} [Q \left (0\right ) = 0] \end {align*}
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program solution |
\[ Q = \left \{\begin {array}{cc} 0 & t <1 \\ \frac {1}{100}-\frac {{\mathrm e}^{-10 t +10}}{100} & t <2 \\ -\frac {{\mathrm e}^{-10 t +10}}{100}+\frac {{\mathrm e}^{-10 t +20}}{100} & 2\le t \end {array}\right . \] Verified OK.
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Maple solution | \[ Q \left (t \right ) = \frac {\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-10 t +20}}{100}-\frac {\operatorname {Heaviside}\left (t -2\right )}{100}-\frac {\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-10 t +10}}{100}+\frac {\operatorname {Heaviside}\left (t -1\right )}{100} \] |
Problem 12344
| ODE |
\[ \boxed {y^{\prime \prime \prime }+y^{\prime \prime }+4 y^{\prime }+4 y=8} \] With initial conditions \begin {align*} [y \left (0\right ) = 4, y^{\prime }\left (0\right ) = -3, y^{\prime \prime }\left (0\right ) = -3] \end {align*}
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program solution |
\[ y = 2+{\mathrm e}^{-t}+\cos \left (2 t \right )-\sin \left (2 t \right ) \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \cos \left (2 t \right )-\sin \left (2 t \right )+{\mathrm e}^{-t}+2 \] |
Problem 12345
| ODE |
\[ \boxed {y^{\prime \prime \prime }-2 y^{\prime \prime }-y^{\prime }+2 y=4 t} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = -2, y^{\prime \prime }\left (0\right ) = 4] \end {align*}
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program solution |
\[ y = 1-6 \sinh \left (t \right )+{\mathrm e}^{2 t}+2 t \] Verified OK.
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Maple solution | \[ y \left (t \right ) = -6 \sinh \left (t \right )+2 t +{\mathrm e}^{2 t}+1 \] |
Problem 12346
| ODE |
\[ \boxed {y^{\prime \prime \prime }-y^{\prime \prime }+4 y^{\prime }-4 y=8 \,{\mathrm e}^{2 t}-5 \,{\mathrm e}^{t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = 3] \end {align*}
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program solution |
\[ y = {\mathrm e}^{2 t}-\sin \left (2 t \right )-{\mathrm e}^{t} \left (t -1\right ) \] Verified OK.
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Maple solution | \[ y \left (t \right ) = -t \,{\mathrm e}^{t}+{\mathrm e}^{2 t}+{\mathrm e}^{t}-\sin \left (2 t \right ) \] |
Problem 12347
| ODE |
\[ \boxed {y^{\prime \prime \prime }-5 y^{\prime \prime }+y^{\prime }-y=-t^{2}+2 t -10} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = t^{2}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{3}-5 \textit {\_Z}^{2}+\textit {\_Z} -1\right )}{\sum }\left (\underline {\hspace {1.25 ex}}\alpha ^{2}-11 \underline {\hspace {1.25 ex}}\alpha +28\right ) {\mathrm e}^{\underline {\hspace {1.25 ex}}\alpha t}\right )}{26} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{3}-5 \textit {\_Z}^{2}+\textit {\_Z} -1\right )}{\sum }\left (\underline {\hspace {1.25 ex}}\alpha -4\right ) \left (\underline {\hspace {1.25 ex}}\alpha -7\right ) {\mathrm e}^{\underline {\hspace {1.25 ex}}\alpha t}\right )}{26}+t^{2} \] |
Problem 12348
| ODE |
\[ \boxed {y^{\prime \prime \prime \prime }-5 y^{\prime \prime }+4 y=12 \operatorname {Heaviside}\left (t \right )-12 \operatorname {Heaviside}\left (t -1\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = 0, y^{\prime \prime \prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[
y = \left \{\begin {array}{cc} -4 \cosh \left (t \right )+\cosh \left (2 t \right )+3 & t <1 \\ -4 \cosh \left (1\right )+\cosh \left (2\right )+\frac {9}{2} & t =1 \\ -4 \cosh \left (t \right )+\cosh \left (2 t \right )+2 \,{\mathrm e}^{t -1}-\frac {{\mathrm e}^{2 t -2}}{2}+2 \,{\mathrm e}^{1-t}-\frac {{\mathrm e}^{-2 t +2}}{2} & 1 |
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Maple solution | \[ y \left (t \right ) = 2 \left (-1+\cosh \left (t \right )\right )^{2}-\frac {\operatorname {Heaviside}\left (t -1\right ) \left ({\mathrm e}^{-2 t +2}-4 \,{\mathrm e}^{-t +1}+{\mathrm e}^{2 t -2}+6-4 \,{\mathrm e}^{t -1}\right )}{2} \] |
Problem 12349
| ODE |
\[ \boxed {y^{\prime \prime \prime \prime }-16 y=-32 \operatorname {Heaviside}\left (t -\pi \right )+32 \operatorname {Heaviside}\left (t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = 0, y^{\prime \prime \prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[
y = \left \{\begin {array}{cc} \cosh \left (2 t \right )+2 \cos \left (t \right )^{2}-3 & t <\pi \\ \cosh \left (2 \pi \right )-2 & t =\pi \\ \cosh \left (2 t \right )-\cosh \left (2 t -2 \pi \right ) & \pi |
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Maple solution | \[ y \left (t \right ) = -\operatorname {Heaviside}\left (t -\pi \right ) \cosh \left (2 t -2 \pi \right )+\left (-\cos \left (2 t \right )+2\right ) \operatorname {Heaviside}\left (t -\pi \right )+\cos \left (2 t \right )+\cosh \left (2 t \right )-2 \] |
Problem 12350
| ODE |
\[ \boxed {t^{2} y^{\prime \prime }+3 t y^{\prime }+y=t^{7}} \] |
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program solution |
\[ y = \frac {\frac {t^{8}}{64}+c_{1} \ln \left (t \right )+c_{2}}{t} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {c_{2}}{t}+\frac {t^{7}}{64}+\frac {c_{1} \ln \left (t \right )}{t} \] |
Problem 12351
| ODE |
\[ \boxed {t^{2} y^{\prime \prime }-6 t y^{\prime }+\sin \left (2 t \right ) y=\ln \left (t \right )} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 12352
| ODE |
\[ \boxed {y^{\prime \prime }+3 y^{\prime }+\frac {y}{t}=t} \] |
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program solution |
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Maple solution | \[ y \left (t \right ) = \frac {\left (7 \,{\mathrm e}^{-3 t} \operatorname {KummerU}\left (\frac {2}{3}, 2, 3 t \right ) c_{1} +7 \,{\mathrm e}^{-3 t} \operatorname {KummerM}\left (\frac {2}{3}, 2, 3 t \right ) c_{2} +t -\frac {1}{2}\right ) t}{7} \] |
Problem 12353
| ODE |
\[ \boxed {y^{\prime \prime }+t y^{\prime }-y \ln \left (t \right )=\cos \left (2 t \right )} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 12354
| ODE |
\[ \boxed {t^{3} y^{\prime \prime }-2 t y^{\prime }+y=t^{4}} \] |
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program solution |
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Maple solution | \[ y \left (t \right ) = -\left (\left (-\operatorname {BesselI}\left (0, \frac {1}{t}\right )-\operatorname {BesselI}\left (1, \frac {1}{t}\right )\right ) \left (\int t \left (\operatorname {BesselK}\left (0, \frac {1}{t}\right )-\operatorname {BesselK}\left (1, \frac {1}{t}\right )\right ) {\mathrm e}^{\frac {1}{t}}d t \right )+\left (\int t \left (\operatorname {BesselI}\left (0, \frac {1}{t}\right )+\operatorname {BesselI}\left (1, \frac {1}{t}\right )\right ) {\mathrm e}^{\frac {1}{t}}d t \right ) \left (\operatorname {BesselK}\left (0, \frac {1}{t}\right )-\operatorname {BesselK}\left (1, \frac {1}{t}\right )\right )-\operatorname {BesselK}\left (0, \frac {1}{t}\right ) c_{1} +\operatorname {BesselK}\left (1, \frac {1}{t}\right ) c_{1} -\operatorname {BesselI}\left (0, \frac {1}{t}\right ) c_{2} -\operatorname {BesselI}\left (1, \frac {1}{t}\right ) c_{2} \right ) {\mathrm e}^{-\frac {1}{t}} \] |
Problem 12355
| ODE |
\[ \boxed {y^{\prime \prime }+2 y^{\prime }+y=1} \] |
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program solution |
\[ y = {\mathrm e}^{-t} \left (c_{2} t +c_{1} \right )+1 \] Verified OK.
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Maple solution | \[ y \left (t \right ) = 1+\left (c_{1} t +c_{2} \right ) {\mathrm e}^{-t} \] |
Problem 12356
| ODE |
\[ \boxed {y^{\prime \prime }-2 y^{\prime }+5 y={\mathrm e}^{t}} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{t} \cos \left (2 t \right )+\frac {c_{2} {\mathrm e}^{t} \sin \left (2 t \right )}{2}+\frac {{\mathrm e}^{t}}{4} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = {\mathrm e}^{t} \left (\frac {1}{4}+\sin \left (2 t \right ) c_{2} +\cos \left (2 t \right ) c_{1} \right ) \] |
Problem 12357
| ODE |
\[ \boxed {y^{\prime \prime }-3 y^{\prime }-7 y=4} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{-\frac {\left (-3+\sqrt {37}\right ) t}{2}}+\frac {c_{2} \sqrt {37}\, {\mathrm e}^{\frac {\left (3+\sqrt {37}\right ) t}{2}}}{37}-\frac {4}{7} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = {\mathrm e}^{\frac {\left (3+\sqrt {37}\right ) t}{2}} c_{2} +{\mathrm e}^{-\frac {\left (-3+\sqrt {37}\right ) t}{2}} c_{1} -\frac {4}{7} \] |
Problem 12358
| ODE |
\[ \boxed {y^{\prime \prime \prime }+3 y^{\prime \prime }+3 y^{\prime }+y=5} \] |
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program solution |
\[ y = {\mathrm e}^{-t} \left (c_{3} t^{2}+c_{2} t +c_{1} \right )+5 \] Verified OK.
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Maple solution | \[ y \left (t \right ) = 5+\left (c_{3} t^{2}+c_{2} t +c_{1} \right ) {\mathrm e}^{-t} \] |
Problem 12359
| ODE |
\[ \boxed {3 y^{\prime \prime }+5 y^{\prime }-2 y=3 t^{2}} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{-2 t}+\frac {3 c_{2} {\mathrm e}^{\frac {t}{3}}}{7}-\frac {3 t^{2}}{2}-\frac {15 t}{2}-\frac {93}{4} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = -\frac {3 \,{\mathrm e}^{-2 t} \left (-\frac {2 \,{\mathrm e}^{\frac {7 t}{3}} c_{1}}{3}+\left (t^{2}+5 t +\frac {31}{2}\right ) {\mathrm e}^{2 t}-\frac {2 c_{2}}{3}\right )}{2} \] |
Problem 12360
| ODE |
\[ \boxed {y^{\prime \prime \prime }-2 y^{\prime \prime }+4 y^{\prime }=\sin \left (t \right )} \] |
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program solution |
\[ y = c_{1} +{\mathrm e}^{\left (1+i \sqrt {3}\right ) t} c_{2} +{\mathrm e}^{\left (1-i \sqrt {3}\right ) t} c_{3} -\frac {3 \cos \left (t \right )}{13}+\frac {2 \sin \left (t \right )}{13} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {{\mathrm e}^{t} \left (-c_{2} \sqrt {3}+c_{1} \right ) \cos \left (\sqrt {3}\, t \right )}{4}+\frac {{\mathrm e}^{t} \left (\sqrt {3}\, c_{1} +c_{2} \right ) \sin \left (\sqrt {3}\, t \right )}{4}+c_{3} -\frac {3 \cos \left (t \right )}{13}+\frac {2 \sin \left (t \right )}{13} \] |
Problem 12361
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )-2 y\\ y^{\prime }&=3 x \left (t \right )-4 y \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} c_{1} +c_{2} {\mathrm e}^{-2 t} \\ y \left (t \right ) &= {\mathrm e}^{-t} c_{1} +\frac {3 c_{2} {\mathrm e}^{-2 t}}{2} \\ \end{align*} |
Problem 12362
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=\frac {5 x \left (t \right )}{4}+\frac {3 y}{4}\\ y^{\prime }&=\frac {x \left (t \right )}{2}-\frac {3 y}{2} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{\frac {\left (-1+\sqrt {145}\right ) t}{8}}+c_{2} {\mathrm e}^{-\frac {\left (1+\sqrt {145}\right ) t}{8}} \\ y \left (t \right ) &= \frac {c_{1} {\mathrm e}^{\frac {\left (-1+\sqrt {145}\right ) t}{8}} \sqrt {145}}{6}-\frac {c_{2} {\mathrm e}^{-\frac {\left (1+\sqrt {145}\right ) t}{8}} \sqrt {145}}{6}-\frac {11 c_{1} {\mathrm e}^{\frac {\left (-1+\sqrt {145}\right ) t}{8}}}{6}-\frac {11 c_{2} {\mathrm e}^{-\frac {\left (1+\sqrt {145}\right ) t}{8}}}{6} \\ \end{align*} |
Problem 12363
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )-2 y\\ y^{\prime }&=-y+x \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right ) \\ y \left (t \right ) &= -\frac {c_{1} \cos \left (t \right )}{2}+\frac {c_{2} \sin \left (t \right )}{2}+\frac {c_{1} \sin \left (t \right )}{2}+\frac {c_{2} \cos \left (t \right )}{2} \\ \end{align*} |
Problem 12364
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=-5 x \left (t \right )+2 y\\ y^{\prime }&=-2 x \left (t \right )+y \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{\left (-2+\sqrt {5}\right ) t}+c_{2} {\mathrm e}^{-\left (2+\sqrt {5}\right ) t} \\ y \left (t \right ) &= \frac {c_{1} {\mathrm e}^{\left (-2+\sqrt {5}\right ) t} \sqrt {5}}{2}-\frac {c_{2} {\mathrm e}^{-\left (2+\sqrt {5}\right ) t} \sqrt {5}}{2}+\frac {3 c_{1} {\mathrm e}^{\left (-2+\sqrt {5}\right ) t}}{2}+\frac {3 c_{2} {\mathrm e}^{-\left (2+\sqrt {5}\right ) t}}{2} \\ \end{align*} |
Problem 12365
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=3 x \left (t \right )-2 y\\ y^{\prime }&=x \left (t \right )-3 y \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{\sqrt {7}\, t}+c_{2} {\mathrm e}^{-\sqrt {7}\, t} \\ y \left (t \right ) &= -\frac {c_{1} \sqrt {7}\, {\mathrm e}^{\sqrt {7}\, t}}{2}+\frac {c_{2} \sqrt {7}\, {\mathrm e}^{-\sqrt {7}\, t}}{2}+\frac {3 c_{1} {\mathrm e}^{\sqrt {7}\, t}}{2}+\frac {3 c_{2} {\mathrm e}^{-\sqrt {7}\, t}}{2} \\ \end{align*} |
Problem 12366
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right )+z \left (t \right )\\ y^{\prime }&=y-x \left (t \right )\\ z^{\prime }\left (t \right )&=-x \left (t \right )-2 y+3 z \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} +c_{2} t +c_{3} {\mathrm e}^{3 t} \\ y \left (t \right ) &= -\frac {c_{3} {\mathrm e}^{3 t}}{2}+c_{2} +c_{1} +c_{2} t \\ z \left (t \right ) &= c_{2} +4 c_{3} {\mathrm e}^{3 t}+c_{1} +c_{2} t \\ \end{align*} |
Problem 12367
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=-\frac {x \left (t \right )}{2}+2 y-3 z \left (t \right )\\ y^{\prime }&=y-\frac {z \left (t \right )}{2}\\ z^{\prime }\left (t \right )&=-2 x \left (t \right )+z \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= -\frac {c_{2} {\mathrm e}^{\frac {\left (-3+\sqrt {33}\right ) t}{4}} \sqrt {33}}{8}+\frac {c_{3} {\mathrm e}^{-\frac {\left (3+\sqrt {33}\right ) t}{4}} \sqrt {33}}{8}+\frac {7 c_{2} {\mathrm e}^{\frac {\left (-3+\sqrt {33}\right ) t}{4}}}{8}+\frac {7 c_{3} {\mathrm e}^{-\frac {\left (3+\sqrt {33}\right ) t}{4}}}{8}-c_{1} {\mathrm e}^{3 t} \\ y \left (t \right ) &= \frac {c_{2} {\mathrm e}^{\frac {\left (-3+\sqrt {33}\right ) t}{4}} \sqrt {33}}{8}-\frac {c_{3} {\mathrm e}^{-\frac {\left (3+\sqrt {33}\right ) t}{4}} \sqrt {33}}{8}+\frac {7 c_{2} {\mathrm e}^{\frac {\left (-3+\sqrt {33}\right ) t}{4}}}{8}+\frac {7 c_{3} {\mathrm e}^{-\frac {\left (3+\sqrt {33}\right ) t}{4}}}{8}-\frac {c_{1} {\mathrm e}^{3 t}}{4} \\ z \left (t \right ) &= c_{1} {\mathrm e}^{3 t}+c_{2} {\mathrm e}^{\frac {\left (-3+\sqrt {33}\right ) t}{4}}+c_{3} {\mathrm e}^{-\frac {\left (3+\sqrt {33}\right ) t}{4}} \\ \end{align*} |
Problem 12368
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=\frac {y}{2}+\frac {x \left (t \right )}{2}\\ y^{\prime }&=\frac {y}{2}-\frac {x \left (t \right )}{2} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \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 ) \\ y \left (t \right ) &= {\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 12369
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=\frac {t}{3}+\frac {2 x \left (t \right )}{3}+\frac {2 y}{3}\\ y^{\prime }&=\frac {t}{3}-\frac {x \left (t \right )}{3}-\frac {y}{3} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= 3 c_{1} {\mathrm e}^{\frac {t}{3}}-\frac {t^{2}}{2}-4 t +c_{2} \\ y \left (t \right ) &= -\frac {3 c_{1} {\mathrm e}^{\frac {t}{3}}}{2}+2 t -6+\frac {t^{2}}{2}-c_{2} \\ \end{align*} |
Problem 12370
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=\frac {6}{5}+\frac {3 y}{5}-\frac {3 t}{5}+x \left (t \right )\\ y^{\prime }&=\frac {6}{5}-\frac {2 y}{5}+\frac {2 t}{5} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{t}+{\mathrm e}^{-\frac {2 t}{5}} c_{1} -\frac {3}{2} \\ y \left (t \right ) &= -\frac {7 \,{\mathrm e}^{-\frac {2 t}{5}} c_{1}}{3}+\frac {1}{2}+t \\ \end{align*} |
Problem 12371
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=\frac {2 t}{7}+\frac {y}{7}\\ y^{\prime }&=-\frac {3 t}{7}+\frac {2 y}{7} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= \frac {t^{2}}{4}+\frac {7 \,{\mathrm e}^{\frac {2 t}{7}} c_{1}}{2}+\frac {3 t}{4}+c_{2} \\ y \left (t \right ) &= \frac {3 t}{2}+7 \,{\mathrm e}^{\frac {2 t}{7}} c_{1} +\frac {21}{4} \\ \end{align*} |
Problem 12372
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=\frac {3 t}{4}-\frac {x \left (t \right )}{4}-\frac {y}{4}\\ y^{\prime }&=\frac {5 t}{4}-\frac {3 x \left (t \right )}{4}-\frac {3 y}{4} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= \frac {t^{2}}{8}-{\mathrm e}^{-t} c_{1} +\frac {t}{2}+c_{2} \\ y \left (t \right ) &= \frac {3 t}{2}-3 \,{\mathrm e}^{-t} c_{1} -2-\frac {t^{2}}{8}-c_{2} \\ \end{align*} |
Problem 12373
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=\frac {4 y}{5}+\frac {4 t}{5}\\ y^{\prime }&=\frac {y}{5}+\frac {t}{5} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= 5 \,{\mathrm e}^{\frac {t}{5}} c_{1} -4 t +c_{2} \\ y \left (t \right ) &= \frac {5 \,{\mathrm e}^{\frac {t}{5}} c_{1}}{4}-5-t \\ \end{align*} |
Problem 12374
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=\frac {\sin \left (t \right )}{4}+\frac {x \left (t \right )}{4}+\frac {y}{4}+\frac {t}{4}\\ y^{\prime }&=\frac {\sin \left (t \right )}{8}-\frac {3 x \left (t \right )}{8}-\frac {3 y}{8}-\frac {3 t}{8} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= -8 \,{\mathrm e}^{-\frac {t}{8}} c_{1} -\frac {6 \sin \left (t \right )}{65}-\frac {17 \cos \left (t \right )}{65}+2 t +c_{2} \\ y \left (t \right ) &= 12 \,{\mathrm e}^{-\frac {t}{8}} c_{1} -\frac {7 \cos \left (t \right )}{65}+\frac {9 \sin \left (t \right )}{65}+8-3 t -c_{2} \\ \end{align*} |
Problem 12375
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=-4 x \left (t \right )+9 y+12 \,{\mathrm e}^{-t}\\ y^{\prime }&=-5 x \left (t \right )+2 y \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (6 \sin \left (6 t \right ) c_{1} +3 \sin \left (6 t \right ) c_{2} +3 \cos \left (6 t \right ) c_{1} -6 \cos \left (6 t \right ) c_{2} -5\right )}{5} \\ y \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (-5+3 \sin \left (6 t \right ) c_{2} +3 \cos \left (6 t \right ) c_{1} \right )}{3} \\ \end{align*} |
Problem 12376
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=-7 x \left (t \right )+6 y+6 \,{\mathrm e}^{-t}\\ y^{\prime }&=-12 x \left (t \right )+5 y+37 \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= 6+\frac {{\mathrm e}^{-t} \left (-2+\sin \left (6 t \right ) c_{1} +\sin \left (6 t \right ) c_{2} +\cos \left (6 t \right ) c_{1} -\cos \left (6 t \right ) c_{2} -2 \sin \left (6 t \right )-2 \cos \left (6 t \right )\right )}{2} \\ y \left (t \right ) &= 7+{\mathrm e}^{-t} \left (-2+\sin \left (6 t \right ) c_{2} +\cos \left (6 t \right ) c_{1} -2 \cos \left (6 t \right )\right ) \\ \end{align*} |
Problem 12377
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=-7 x \left (t \right )+10 y+18 \,{\mathrm e}^{t}\\ y^{\prime }&=-10 x \left (t \right )+9 y+37 \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= 10+\frac {{\mathrm e}^{t} \left (-20+3 \sin \left (6 t \right ) c_{1} +4 \sin \left (6 t \right ) c_{2} +4 \cos \left (6 t \right ) c_{1} -3 \cos \left (6 t \right ) c_{2} -15 \sin \left (6 t \right )-20 \cos \left (6 t \right )\right )}{5} \\ y \left (t \right ) &= 7+{\mathrm e}^{t} \left (-5+\sin \left (6 t \right ) c_{2} +\cos \left (6 t \right ) c_{1} -5 \cos \left (6 t \right )\right ) \\ \end{align*} |
Problem 12378
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=-14 x \left (t \right )+39 y+78 \sinh \left (t \right )\\ y^{\prime }&=-6 x \left (t \right )+16 y+6 \cosh \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= {\mathrm e}^{t} \sin \left (3 t \right ) c_{2} +{\mathrm e}^{t} \cos \left (3 t \right ) c_{1} -52 \,{\mathrm e}^{t}+60 \,{\mathrm e}^{-t} \\ y \left (t \right ) &= \frac {5 \,{\mathrm e}^{t} \sin \left (3 t \right ) c_{2}}{13}+\frac {{\mathrm e}^{t} \cos \left (3 t \right ) c_{2}}{13}+\frac {5 \,{\mathrm e}^{t} \cos \left (3 t \right ) c_{1}}{13}-\frac {{\mathrm e}^{t} \sin \left (3 t \right ) c_{1}}{13}-20 \,{\mathrm e}^{t}+20 \,{\mathrm e}^{-t}-2 \sinh \left (t \right ) \\ \end{align*} |
Problem 12379
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )+4 y-2 z \left (t \right )-2 \sinh \left (t \right )\\ y^{\prime }&=4 x \left (t \right )+2 y-2 z \left (t \right )+10 \cosh \left (t \right )\\ z^{\prime }\left (t \right )&=-x \left (t \right )+3 y+z \left (t \right )+5 \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= -1+2 c_{3} {\mathrm e}^{5 t}+\frac {9 c_{1} {\mathrm e}^{-2 t}}{8}-\frac {45 \cosh \left (t \right )}{16}-\frac {3 \sinh \left (t \right )}{16}+\frac {c_{2} {\mathrm e}^{2 t}}{2}-\frac {275 \,{\mathrm e}^{-2 t} \sinh \left (t \right )}{224}-\frac {3 \,{\mathrm e}^{-2 t} \sinh \left (3 t \right )}{14}-\frac {275 \,{\mathrm e}^{-2 t} \cosh \left (t \right )}{224}-\frac {3 \,{\mathrm e}^{-2 t} \cosh \left (3 t \right )}{14}+\frac {3 \,{\mathrm e}^{2 t} \sinh \left (t \right )}{2}+\frac {275 \,{\mathrm e}^{2 t} \sinh \left (3 t \right )}{288}-\frac {3 \,{\mathrm e}^{2 t} \cosh \left (t \right )}{2}-\frac {275 \,{\mathrm e}^{2 t} \cosh \left (3 t \right )}{288}-\frac {3 \,{\mathrm e}^{5 t} \sinh \left (4 t \right )}{14}-\frac {275 \,{\mathrm e}^{5 t} \sinh \left (6 t \right )}{1008}+\frac {3 \,{\mathrm e}^{5 t} \cosh \left (4 t \right )}{14}+\frac {275 \,{\mathrm e}^{5 t} \cosh \left (6 t \right )}{1008} \\ y \left (t \right ) &= -1+2 c_{3} {\mathrm e}^{5 t}-\frac {5 c_{1} {\mathrm e}^{-2 t}}{8}-\frac {15 \cosh \left (t \right )}{16}-\frac {\sinh \left (t \right )}{16}+\frac {c_{2} {\mathrm e}^{2 t}}{2}+\frac {25 \,{\mathrm e}^{-2 t} \sinh \left (t \right )}{32}-\frac {{\mathrm e}^{-2 t} \sinh \left (3 t \right )}{14}+\frac {25 \,{\mathrm e}^{-2 t} \cosh \left (t \right )}{32}-\frac {{\mathrm e}^{-2 t} \cosh \left (3 t \right )}{14}+\frac {{\mathrm e}^{2 t} \sinh \left (t \right )}{2}-\frac {175 \,{\mathrm e}^{2 t} \sinh \left (3 t \right )}{288}-\frac {{\mathrm e}^{2 t} \cosh \left (t \right )}{2}+\frac {175 \,{\mathrm e}^{2 t} \cosh \left (3 t \right )}{288}-\frac {{\mathrm e}^{5 t} \sinh \left (4 t \right )}{14}+\frac {25 \,{\mathrm e}^{5 t} \sinh \left (6 t \right )}{144}+\frac {{\mathrm e}^{5 t} \cosh \left (4 t \right )}{14}-\frac {25 \,{\mathrm e}^{5 t} \cosh \left (6 t \right )}{144} \\ z \left (t \right ) &= -\frac {25 \,{\mathrm e}^{-2 t} \sinh \left (t \right )}{14}-3-\frac {4 \,{\mathrm e}^{-2 t} \sinh \left (3 t \right )}{7}-\frac {25 \,{\mathrm e}^{-2 t} \cosh \left (t \right )}{14}-\frac {4 \,{\mathrm e}^{-2 t} \cosh \left (3 t \right )}{7}+4 \,{\mathrm e}^{2 t} \sinh \left (t \right )+\frac {25 \,{\mathrm e}^{2 t} \sinh \left (3 t \right )}{18}-4 \,{\mathrm e}^{2 t} \cosh \left (t \right )-\frac {25 \,{\mathrm e}^{2 t} \cosh \left (3 t \right )}{18}-\frac {4 \,{\mathrm e}^{5 t} \sinh \left (4 t \right )}{7}-\frac {25 \,{\mathrm e}^{5 t} \sinh \left (6 t \right )}{63}+\frac {4 \,{\mathrm e}^{5 t} \cosh \left (4 t \right )}{7}+\frac {25 \,{\mathrm e}^{5 t} \cosh \left (6 t \right )}{63}+c_{1} {\mathrm e}^{-2 t}+c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{5 t} \\ \end{align*} |
Problem 12380
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )+6 y-2 z \left (t \right )+50 \,{\mathrm e}^{t}\\ y^{\prime }&=6 x \left (t \right )+2 y-2 z \left (t \right )+21 \,{\mathrm e}^{-t}\\ z^{\prime }\left (t \right )&=-x \left (t \right )+6 y+z \left (t \right )+9 \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= -6 \,{\mathrm e}^{-t}+c_{3} {\mathrm e}^{6 t}+{\mathrm e}^{-4 t} c_{1} +\frac {2 c_{2} {\mathrm e}^{3 t}}{5}+12 \,{\mathrm e}^{t}-1 \\ y \left (t \right ) &= {\mathrm e}^{-t}+c_{3} {\mathrm e}^{6 t}-\frac {2 \,{\mathrm e}^{-4 t} c_{1}}{3}+\frac {2 c_{2} {\mathrm e}^{3 t}}{5}+2 \,{\mathrm e}^{t}-1 \\ z \left (t \right ) &= -6 \,{\mathrm e}^{-t}+c_{3} {\mathrm e}^{6 t}+{\mathrm e}^{-4 t} c_{1} +c_{2} {\mathrm e}^{3 t}+37 \,{\mathrm e}^{t}-4 \\ \end{align*} |
Problem 12381
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=-2 x \left (t \right )-2 y+4 z \left (t \right )\\ y^{\prime }&=-2 x \left (t \right )+y+2 z \left (t \right )\\ z^{\prime }\left (t \right )&=-4 x \left (t \right )-2 y+6 z \left (t \right )+{\mathrm e}^{2 t} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= \left ({\mathrm e}^{t} \left (4 t +c_{2} -4\right )+c_{3} \right ) {\mathrm e}^{t} \\ y \left (t \right ) &= \left (\frac {c_{2}}{2}+2 t -2+c_{1} \right ) {\mathrm e}^{2 t}+\frac {c_{3} {\mathrm e}^{t}}{2} \\ z \left (t \right ) &= \frac {5 c_{2} {\mathrm e}^{2 t}}{4}+5 \,{\mathrm e}^{2 t} t -4 \,{\mathrm e}^{2 t}+c_{3} {\mathrm e}^{t}+\frac {c_{1} {\mathrm e}^{2 t}}{2} \\ \end{align*} |
Problem 12382
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=3 x \left (t \right )-2 y+3 z \left (t \right )\\ y^{\prime }&=x \left (t \right )-y+2 z \left (t \right )+2 \,{\mathrm e}^{-t}\\ z^{\prime }\left (t \right )&=-2 x \left (t \right )+2 y-2 z \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= 2 \,{\mathrm e}^{-t}+c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-2 t}+c_{3} {\mathrm e}^{t} t \\ y \left (t \right ) &= {\mathrm e}^{-t}-\frac {c_{1} {\mathrm e}^{t}}{2}+c_{2} {\mathrm e}^{-2 t}-\frac {c_{3} {\mathrm e}^{t} t}{2}+\frac {7 c_{3} {\mathrm e}^{t}}{4} \\ z \left (t \right ) &= -2 \,{\mathrm e}^{-t}-c_{1} {\mathrm e}^{t}-c_{2} {\mathrm e}^{-2 t}-c_{3} {\mathrm e}^{t} t +\frac {3 c_{3} {\mathrm e}^{t}}{2} \\ \end{align*} |
Problem 12383
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=7 x \left (t \right )+y-1-6 \,{\mathrm e}^{t}\\ y^{\prime }&=-4 x \left (t \right )+3 y+4 \,{\mathrm e}^{t}-3 \end {align*} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = -1] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= -2 t \,{\mathrm e}^{5 t}+{\mathrm e}^{t} \\ y \left (t \right ) &= 1-{\mathrm e}^{5 t} \left (-4 t +2\right ) \\ \end{align*} |
Problem 12384
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=3 x \left (t \right )-2 y+24 \sin \left (t \right )\\ y^{\prime }&=9 x \left (t \right )-3 y+12 \cos \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = -1] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= -\frac {4 \sin \left (3 t \right )}{3}+\cos \left (3 t \right )+9 \sin \left (t \right ) \\ y \left (t \right ) &= \frac {7 \cos \left (3 t \right )}{2}-\frac {\sin \left (3 t \right )}{2}-\frac {9 \cos \left (t \right )}{2}+\frac {51 \sin \left (t \right )}{2} \\ \end{align*} |
Problem 12385
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=7 x \left (t \right )-4 y+10 \,{\mathrm e}^{t}\\ y^{\prime }&=3 x \left (t \right )+14 y+6 \,{\mathrm e}^{2 t} \end {align*} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = -1] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= -\frac {14 \,{\mathrm e}^{11 t}}{3}+\frac {67 \,{\mathrm e}^{10 t}}{9}-\frac {{\mathrm e}^{2 t}}{3}-\frac {13 \,{\mathrm e}^{t}}{9} \\ y \left (t \right ) &= \frac {14 \,{\mathrm e}^{11 t}}{3}-\frac {67 \,{\mathrm e}^{10 t}}{12}-\frac {5 \,{\mathrm e}^{2 t}}{12}+\frac {{\mathrm e}^{t}}{3} \\ \end{align*} |
Problem 12386
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=-7 x \left (t \right )+4 y+6 \,{\mathrm e}^{3 t}\\ y^{\prime }&=-5 x \left (t \right )+2 y+6 \,{\mathrm e}^{2 t} \end {align*} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = -1] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= -\frac {46 \,{\mathrm e}^{-2 t}}{5}+\frac {44 \,{\mathrm e}^{-3 t}}{5}+\frac {{\mathrm e}^{3 t}}{5}+\frac {6 \,{\mathrm e}^{2 t}}{5} \\ y \left (t \right ) &= -{\mathrm e}^{3 t}-\frac {23 \,{\mathrm e}^{-2 t}}{2}+\frac {44 \,{\mathrm e}^{-3 t}}{5}+\frac {27 \,{\mathrm e}^{2 t}}{10} \\ \end{align*} |
Problem 12387
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=-3 x \left (t \right )-3 y+z \left (t \right )\\ y^{\prime }&=2 y+2 z \left (t \right )+29 \,{\mathrm e}^{-t}\\ z^{\prime }\left (t \right )&=5 x \left (t \right )+y+z \left (t \right )+39 \,{\mathrm e}^{t} \end {align*} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 2, z \left (0\right ) = 3] \] |
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program solution |
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Maple solution |
\begin{align*} \text {Expression too large to display} \\ \text {Expression too large to display} \\ \text {Expression too large to display} \\ \end{align*} |
Problem 12388
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )+y-z \left (t \right )+5 \sin \left (t \right )\\ y^{\prime }&=y+z \left (t \right )-10 \cos \left (t \right )\\ z^{\prime }\left (t \right )&=x \left (t \right )+z \left (t \right )+2 \end {align*} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 2, z \left (0\right ) = 3] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= -2 \cos \left (t \right )-1-3 \,{\mathrm e}^{t} \sin \left (t \right )+4 \,{\mathrm e}^{t} \cos \left (t \right ) \\ y \left (t \right ) &= -4 \sin \left (t \right )+5 \cos \left (t \right )+1+3 \,{\mathrm e}^{t} \sin \left (t \right )-4 \,{\mathrm e}^{t} \cos \left (t \right ) \\ z \left (t \right ) &= -1-\sin \left (t \right )+\cos \left (t \right )+3 \,{\mathrm e}^{t} \cos \left (t \right )+4 \,{\mathrm e}^{t} \sin \left (t \right ) \\ \end{align*} |
Problem 12389
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=-3 x \left (t \right )+3 y+z \left (t \right )+10 \cos \left (t \right ) \sin \left (t \right )\\ y^{\prime }&=x \left (t \right )-5 y-3 z \left (t \right )+10 \cos \left (t \right )^{2}-5\\ z^{\prime }\left (t \right )&=-3 x \left (t \right )+7 y+3 z \left (t \right )+23 \,{\mathrm e}^{t} \end {align*} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 2, z \left (0\right ) = 3] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= -\frac {69 \,{\mathrm e}^{t}}{26}+\sin \left (2 t \right )+\frac {\cos \left (2 t \right )}{2}+\frac {21 \,{\mathrm e}^{-t}}{2}-\frac {191 \,{\mathrm e}^{-2 t} \cos \left (2 t \right )}{26}+\frac {16 \,{\mathrm e}^{-2 t} \sin \left (2 t \right )}{13} \\ y \left (t \right ) &= -\frac {253 \,{\mathrm e}^{t}}{26}-\frac {5 \sin \left (2 t \right )}{2}+\frac {21 \,{\mathrm e}^{-t}}{2}+\frac {191 \,{\mathrm e}^{-2 t} \sin \left (2 t \right )}{26}+\frac {16 \,{\mathrm e}^{-2 t} \cos \left (2 t \right )}{13} \\ z \left (t \right ) &= \frac {483 \,{\mathrm e}^{t}}{26}+\frac {7 \cos \left (2 t \right )}{2}+\frac {9 \sin \left (2 t \right )}{2}-\frac {21 \,{\mathrm e}^{-t}}{2}-\frac {223 \,{\mathrm e}^{-2 t} \cos \left (2 t \right )}{26}-\frac {159 \,{\mathrm e}^{-2 t} \sin \left (2 t \right )}{26} \\ \end{align*} |
Problem 12390
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=-3 x \left (t \right )+y-3 z \left (t \right )+2 \,{\mathrm e}^{t}\\ y^{\prime }&=4 x \left (t \right )-y+2 z \left (t \right )+4 \,{\mathrm e}^{t}\\ z^{\prime }\left (t \right )&=4 x \left (t \right )-2 y+3 z \left (t \right )+4 \,{\mathrm e}^{t} \end {align*} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 2, z \left (0\right ) = 3] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= -\frac {3 \,{\mathrm e}^{t}}{2}-2 \,{\mathrm e}^{-t} \sin \left (2 t \right )+\frac {5 \,{\mathrm e}^{-t} \cos \left (2 t \right )}{2} \\ y \left (t \right ) &= \frac {5 \,{\mathrm e}^{t}}{2}+\frac {9 \,{\mathrm e}^{-t} \sin \left (2 t \right )}{2}-\frac {{\mathrm e}^{-t} \cos \left (2 t \right )}{2} \\ z \left (t \right ) &= \frac {7 \,{\mathrm e}^{t}}{2}+\frac {9 \,{\mathrm e}^{-t} \sin \left (2 t \right )}{2}-\frac {{\mathrm e}^{-t} \cos \left (2 t \right )}{2} \\ \end{align*} |
Problem 12391
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )+5 y+10 \sinh \left (t \right )\\ y^{\prime }&=19 x \left (t \right )-13 y+24 \sinh \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= {\mathrm e}^{-18 t} c_{2} +{\mathrm e}^{6 t} c_{1} +\frac {5 \,{\mathrm e}^{-18 t} \left (\left (-\frac {221 \cosh \left (5 t \right )}{60}+\frac {17 \cosh \left (7 t \right )}{7}+\frac {221 \sinh \left (5 t \right )}{60}-\frac {17 \sinh \left (7 t \right )}{7}\right ) {\mathrm e}^{24 t}+\sinh \left (17 t \right )-\frac {221 \sinh \left (19 t \right )}{228}+\cosh \left (17 t \right )-\frac {221 \cosh \left (19 t \right )}{228}\right )}{17} \\ y \left (t \right ) &= -\frac {2 \cosh \left (7 t \right ) {\mathrm e}^{6 t}}{7}+\frac {2 \sinh \left (7 t \right ) {\mathrm e}^{6 t}}{7}-\frac {2 \,{\mathrm e}^{-18 t} \sinh \left (17 t \right )}{17}-\frac {2 \,{\mathrm e}^{-18 t} \cosh \left (17 t \right )}{17}-\frac {19 \,{\mathrm e}^{-18 t} c_{2}}{5}+{\mathrm e}^{6 t} c_{1} -2 \sinh \left (t \right ) \\ \end{align*} |
Problem 12392
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=9 x \left (t \right )-3 y-6 t\\ y^{\prime }&=-x \left (t \right )+11 y+10 t \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{8 t}+{\mathrm e}^{12 t} c_{1} +\frac {3 t}{8}+\frac {1}{64} \\ y \left (t \right ) &= \frac {c_{2} {\mathrm e}^{8 t}}{3}-{\mathrm e}^{12 t} c_{1} -\frac {5}{64}-\frac {7 t}{8} \\ \end{align*} |
Problem 12393
| ODE |
\[ \boxed {\left (x -1\right ) y^{\prime \prime }-y^{\prime } x +y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{x} \end {align*}
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program solution |
\[ y = c_{1} {\mathrm e}^{x}-c_{2} {\mathrm e}^{x} x \,{\mathrm e}^{-x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{2} {\mathrm e}^{x}+c_{1} x \] |
Problem 12394
| ODE |
\[ \boxed {x y^{\prime \prime }+2 y^{\prime }+y x=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \frac {\sin \left (x \right )}{x} \end {align*}
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program solution |
\[ y = \frac {\sin \left (x \right ) c_{1}}{x}-\frac {c_{2} \sin \left (x \right ) \cot \left (x \right )}{x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{1} \sin \left (x \right )+c_{2} \cos \left (x \right )}{x} \] |
Problem 12395
| ODE |
\[ \boxed {y^{\prime \prime }-2 y^{\prime }+y=x^{\frac {3}{2}} {\mathrm e}^{x}} \] |
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program solution |
\[ y = {\mathrm e}^{x} \left (c_{2} x +c_{1} \right )+\frac {4 x^{\frac {7}{2}} {\mathrm e}^{x}}{35} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{x} \left (c_{2} +c_{1} x +\frac {4 x^{\frac {7}{2}}}{35}\right ) \] |
Problem 12396
| ODE |
\[ \boxed {y^{\prime \prime }+4 y=2 \sec \left (2 x \right )} \] |
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program solution |
\[ y = c_{1} \cos \left (2 x \right )+\frac {c_{2} \sin \left (2 x \right )}{2}-\frac {\ln \left (\sec \left (2 x \right )^{2}\right ) \cos \left (2 x \right )}{4}+x \sin \left (2 x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {\ln \left (\sec \left (2 x \right )\right ) \cos \left (2 x \right )}{2}+\cos \left (2 x \right ) c_{1} +\sin \left (2 x \right ) \left (c_{2} +x \right ) \] |
Problem 12397
| ODE |
\[ \boxed {y^{\prime \prime }+\frac {y^{\prime }}{x}+\left (1-\frac {1}{4 x^{2}}\right ) y=x} \] |
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program solution |
\[ y = \frac {\cos \left (x \right ) c_{1} +c_{2} \sin \left (x \right )}{\sqrt {x}}-\frac {3 \left (\sin \left (x \right ) \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {x}}{\sqrt {\pi }}\right )-\cos \left (x \right ) \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {x}}{\sqrt {\pi }}\right )-\frac {4 x^{\frac {3}{2}}}{3}\right )}{4 \sqrt {x}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\sin \left (x \right ) c_{2} +c_{1} \cos \left (x \right )+\frac {3 \cos \left (x \right ) \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {x}}{\sqrt {\pi }}\right ) \sqrt {2}}{4}-\frac {3 \sin \left (x \right ) \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {x}}{\sqrt {\pi }}\right ) \sqrt {2}}{4}+x^{\frac {3}{2}}}{\sqrt {x}} \] |
Problem 12398
| ODE |
\[ \boxed {y^{\prime \prime }+y=f \left (x \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = -\left (\int _{0}^{x}\sin \left (\alpha \right ) f \left (\alpha \right )d \alpha \right ) \cos \left (x \right )+\left (\int _{0}^{x}\cos \left (\alpha \right ) f \left (\alpha \right )d \alpha \right ) \sin \left (x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (\int _{0}^{x}\cos \left (\textit {\_z1} \right ) f \left (\textit {\_z1} \right )d \textit {\_z1} \right ) \sin \left (x \right )-\left (\int _{0}^{x}\sin \left (\textit {\_z1} \right ) f \left (\textit {\_z1} \right )d \textit {\_z1} \right ) \cos \left (x \right ) \] |
Problem 12399
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+x \left (x -\frac {1}{2}\right ) y^{\prime }+\frac {y}{2}=0} \] |
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program solution |
\[ y = c_{3} {\mathrm e}^{\int \frac {-2 \sqrt {\pi }\, \operatorname {erfi}\left (\sqrt {x}\right ) x^{\frac {3}{2}} c_{1} +\sqrt {\pi }\, \operatorname {erfi}\left (\sqrt {x}\right ) \sqrt {x}\, c_{1} +2 c_{1} x \,{\mathrm e}^{x}-2 x^{\frac {3}{2}} c_{2} +\sqrt {x}\, c_{2}}{2 x^{\frac {3}{2}} \left (c_{1} \sqrt {\pi }\, \operatorname {erfi}\left (\sqrt {x}\right )+c_{2} \right )}d x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {{\mathrm e}^{-x} \left (\operatorname {erf}\left (\sqrt {-x}\right ) \sqrt {\pi }\, c_{1} x +2 c_{2} \sqrt {x}\, \sqrt {-x}\right )}{2 \sqrt {-x}} \] |
Problem 12400
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-y=0} \] |
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program solution |
\[ y = c_{3} \left (c_{1} -\frac {c_{1}}{x}+\frac {{\mathrm e}^{-x} c_{2}}{x}\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{2} {\mathrm e}^{-x}+c_{1} \left (-1+x \right )}{x} \] |