Problem 801
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )+2 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-5 x_{1} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= \frac {{\mathrm e}^{\frac {t}{2}} \left (\sin \left (\frac {\sqrt {39}\, t}{2}\right ) \sqrt {39}\, c_{2} -\cos \left (\frac {\sqrt {39}\, t}{2}\right ) \sqrt {39}\, c_{1} -\sin \left (\frac {\sqrt {39}\, t}{2}\right ) c_{1} -\cos \left (\frac {\sqrt {39}\, t}{2}\right ) c_{2} \right )}{10} \\ x_{2} \left (t \right ) &= {\mathrm e}^{\frac {t}{2}} \left (\sin \left (\frac {\sqrt {39}\, t}{2}\right ) c_{1} +\cos \left (\frac {\sqrt {39}\, t}{2}\right ) c_{2} \right ) \\ \end{align*} |
Problem 802
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=-x_{1} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-x_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= c_{2} {\mathrm e}^{-t} \\ x_{2} \left (t \right ) &= {\mathrm e}^{-t} c_{1} \\ \end{align*} |
Problem 803
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )-\frac {5 x_{2} \left (t \right )}{2}\\ x_{2}^{\prime }\left (t \right )&=\frac {9 x_{1} \left (t \right )}{5}-x_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{\frac {t}{2}} \left (\sin \left (\frac {3 t}{2}\right ) c_{1} +\cos \left (\frac {3 t}{2}\right ) c_{2} \right ) \\ x_{2} \left (t \right ) &= \frac {3 \,{\mathrm e}^{\frac {t}{2}} \left (\sin \left (\frac {3 t}{2}\right ) c_{1} +\sin \left (\frac {3 t}{2}\right ) c_{2} -\cos \left (\frac {3 t}{2}\right ) c_{1} +\cos \left (\frac {3 t}{2}\right ) c_{2} \right )}{5} \\ \end{align*} |
Problem 804
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )-2\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )-x_{2} \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{\sqrt {2}\, t} c_{2} +{\mathrm e}^{-\sqrt {2}\, t} c_{1} +1 \\ x_{2} \left (t \right ) &= \sqrt {2}\, {\mathrm e}^{\sqrt {2}\, t} c_{2} -\sqrt {2}\, {\mathrm e}^{-\sqrt {2}\, t} c_{1} -{\mathrm e}^{\sqrt {2}\, t} c_{2} -{\mathrm e}^{-\sqrt {2}\, t} c_{1} +1 \\ \end{align*} |
Problem 805
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=-2 x_{1} \left (t \right )+x_{2} \left (t \right )-2\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )-2 x_{2} \left (t \right )+1 \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= c_{2} {\mathrm e}^{-t}+c_{1} {\mathrm e}^{-3 t}-1 \\ x_{2} \left (t \right ) &= c_{2} {\mathrm e}^{-t}-c_{1} {\mathrm e}^{-3 t} \\ \end{align*} |
Problem 806
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ODE |
\begin {align*} x_{1}^{\prime }\left (t \right )&=-x_{1} \left (t \right )-x_{2} \left (t \right )-1\\ x_{2}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )-x_{2} \left (t \right )+5 \end {align*}
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program solution |
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Maple solution |
\begin{align*} x_{1} \left (t \right ) &= -2+{\mathrm e}^{-t} \left (\cos \left (\sqrt {2}\, t \right ) c_{1} +c_{2} \sin \left (\sqrt {2}\, t \right )\right ) \\ x_{2} \left (t \right ) &= 1-{\mathrm e}^{-t} \sqrt {2}\, \left (c_{2} \cos \left (\sqrt {2}\, t \right )-c_{1} \sin \left (\sqrt {2}\, t \right )\right ) \\ \end{align*} |
Problem 807
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right )\\ y^{\prime }\left (t \right )&=-2 y \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 4, y \left (0\right ) = 2] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= 4 \,{\mathrm e}^{-t} \\ y \left (t \right ) &= 2 \,{\mathrm e}^{-2 t} \\ \end{align*} |
Problem 808
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right )\\ y^{\prime }\left (t \right )&=2 y \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 4, y \left (0\right ) = 2] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= 4 \,{\mathrm e}^{-t} \\ y \left (t \right ) &= 2 \,{\mathrm e}^{2 t} \\ \end{align*} |
Problem 809
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right )\\ y^{\prime }\left (t \right )&=2 y \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 4, y \left (0\right ) = 0] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= 4 \,{\mathrm e}^{-t} \\ y \left (t \right ) &= 0 \\ \end{align*} |
Problem 810
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=-y \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 4, y \left (0\right ) = 0] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= 4 \cos \left (t \right ) \\ y \left (t \right ) &= 4 \sin \left (t \right ) \\ \end{align*} |
Problem 811
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=-y \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 0, y \left (0\right ) = 4] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= -4 \sin \left (t \right ) \\ y \left (t \right ) &= 4 \cos \left (t \right ) \\ \end{align*} |
Problem 812
| ODE |
\[ \boxed {y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime }+3 y=t} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{-t}+{\mathrm e}^{\left (-\left (4+2 \sqrt {2}\right )^{\frac {1}{3}}-\frac {2}{\left (4+2 \sqrt {2}\right )^{\frac {1}{3}}}-1\right ) t} c_{2} +{\mathrm e}^{\left (\frac {\left (4+2 \sqrt {2}\right )^{\frac {1}{3}}}{2}+\frac {1}{\left (4+2 \sqrt {2}\right )^{\frac {1}{3}}}-1+\frac {i \sqrt {3}\, \left (-\left (4+2 \sqrt {2}\right )^{\frac {1}{3}}+\frac {2}{\left (4+2 \sqrt {2}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} c_{3} +{\mathrm e}^{\left (\frac {\left (4+2 \sqrt {2}\right )^{\frac {1}{3}}}{2}+\frac {1}{\left (4+2 \sqrt {2}\right )^{\frac {1}{3}}}-1-\frac {i \sqrt {3}\, \left (-\left (4+2 \sqrt {2}\right )^{\frac {1}{3}}+\frac {2}{\left (4+2 \sqrt {2}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} c_{4} +\frac {t}{3} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {t}{3}+{\mathrm e}^{-t} c_{1} +c_{2} {\mathrm e}^{\frac {t \left (\left (\sqrt {2}-2\right ) \left (4+2 \sqrt {2}\right )^{\frac {2}{3}}-2 \left (4+2 \sqrt {2}\right )^{\frac {1}{3}}-2\right )}{2}}+c_{3} {\mathrm e}^{-\frac {t \left (\left (\sqrt {2}-2\right ) \left (4+2 \sqrt {2}\right )^{\frac {2}{3}}-2 \left (4+2 \sqrt {2}\right )^{\frac {1}{3}}+4\right )}{4}} \cos \left (\frac {t \left (4+2 \sqrt {2}\right )^{\frac {1}{3}} \left (2+\left (\sqrt {2}-2\right ) \left (4+2 \sqrt {2}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{4}\right )+c_{4} {\mathrm e}^{-\frac {t \left (\left (\sqrt {2}-2\right ) \left (4+2 \sqrt {2}\right )^{\frac {2}{3}}-2 \left (4+2 \sqrt {2}\right )^{\frac {1}{3}}+4\right )}{4}} \sin \left (\frac {t \left (4+2 \sqrt {2}\right )^{\frac {1}{3}} \left (2+\left (\sqrt {2}-2\right ) \left (4+2 \sqrt {2}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{4}\right ) \] |
Problem 813
| ODE |
\[ \boxed {t \left (-1+t \right ) y^{\prime \prime \prime \prime }+{\mathrm e}^{t} y^{\prime \prime }+4 y t^{2}=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 814
| ODE |
\[ \boxed {y^{\prime \prime \prime \prime }+y^{\prime \prime }=0} \] |
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program solution |
\[ y = c_{2} t +c_{1} +{\mathrm e}^{-i t} c_{3} +{\mathrm e}^{i t} c_{4} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = c_{1} +c_{2} t +c_{3} \sin \left (t \right )+c_{4} \cos \left (t \right ) \] |
Problem 815
| ODE |
\[ \boxed {y^{\prime \prime \prime }+2 y^{\prime \prime }-y^{\prime }-2 y=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{-t}+c_{2} {\mathrm e}^{-2 t}+c_{3} {\mathrm e}^{t} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \left (c_{1} {\mathrm e}^{3 t}+c_{3} {\mathrm e}^{t}+c_{2} \right ) {\mathrm e}^{-2 t} \] |
Problem 816
| ODE |
\[ \boxed {x y^{\prime \prime \prime }-y^{\prime \prime }=0} \] |
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program solution |
\[ y = \frac {c_{3}^{2} {\mathrm e}^{2 c_{2}} x^{3}}{3}-\frac {c_{1} x}{2}+c_{4} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{3} x^{3}+c_{2} x +c_{1} \] |
Problem 817
| ODE |
\[ \boxed {x^{3} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }-2 y^{\prime } x +2 y=0} \] |
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program solution |
\[ y = \frac {c_{1}}{x}+c_{2} x +c_{3} x^{2} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{2} x^{3}+c_{1} x^{2}+c_{3}}{x} \] |
Problem 818
| ODE |
\[ \boxed {y^{\prime \prime \prime }+2 y^{\prime \prime }-y^{\prime }-3 y=0} \] |
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program solution |
\[ y = {\mathrm e}^{\left (-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{1} +{\mathrm e}^{\left (-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{2} +{\mathrm e}^{\left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}+\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}\right ) x} c_{3} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {2 x \left (-\frac {\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}}{4}+\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}-7\right )}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}}-c_{2} {\mathrm e}^{-\frac {\left (28+\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}+8 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}\right )+c_{3} {\mathrm e}^{-\frac {\left (28+\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}+8 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}\right ) \] |
Problem 819
| ODE |
\[ \boxed {t y^{\prime \prime \prime }+2 y^{\prime \prime }-y^{\prime }+y t=0} \] |
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program solution |
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Maple solution | \[ y \left (t \right ) = {\mathrm e}^{-\frac {t \left (i \sqrt {3}-1\right )}{2}} \left (\operatorname {KummerM}\left (\frac {1}{2}-\frac {i \sqrt {3}}{6}, 1, i \sqrt {3}\, t \right ) \left (\int \operatorname {KummerU}\left (\frac {1}{2}-\frac {i \sqrt {3}}{6}, 1, i \sqrt {3}\, t \right ) {\mathrm e}^{-\frac {t \left (i \sqrt {3}+3\right )}{2}}d t \right ) c_{3} -\operatorname {KummerU}\left (\frac {1}{2}-\frac {i \sqrt {3}}{6}, 1, i \sqrt {3}\, t \right ) \left (\int \operatorname {KummerM}\left (\frac {1}{2}-\frac {i \sqrt {3}}{6}, 1, i \sqrt {3}\, t \right ) {\mathrm e}^{-\frac {t \left (i \sqrt {3}+3\right )}{2}}d t \right ) c_{3} +c_{1} \operatorname {KummerM}\left (\frac {1}{2}-\frac {i \sqrt {3}}{6}, 1, i \sqrt {3}\, t \right )+c_{2} \operatorname {KummerU}\left (\frac {1}{2}-\frac {i \sqrt {3}}{6}, 1, i \sqrt {3}\, t \right )\right ) \] |
Problem 820
| ODE |
\[ \boxed {\left (-t +2\right ) y^{\prime \prime \prime }+\left (-3+2 t \right ) y^{\prime \prime }-t y^{\prime }+y=0} \] |
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program solution |
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Maple solution | \[ y \left (t \right ) = {\mathrm e}^{t} \left (c_{3} t +c_{2} \right )+c_{1} t \] |
Problem 821
| ODE |
\[ \boxed {t^{2} \left (t +3\right ) y^{\prime \prime \prime }-3 t \left (2+t \right ) y^{\prime \prime }+6 \left (t +1\right ) y^{\prime }-6 y=0} \] |
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program solution |
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Maple solution | \[ y \left (t \right ) = c_{2} t^{3}+c_{1} t^{2}+c_{3} t +c_{3} \] |
Problem 822
| ODE |
\[ \boxed {y^{\prime \prime \prime }-y^{\prime \prime }-y^{\prime }+y=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{-x}+c_{2} {\mathrm e}^{x}+x \,{\mathrm e}^{x} c_{3} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-x} c_{1} +\left (c_{3} x +c_{2} \right ) {\mathrm e}^{x} \] |
Problem 823
| ODE |
\[ \boxed {y^{\prime \prime \prime }-3 y^{\prime \prime }+3 y^{\prime }+y=0} \] |
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program solution |
\[ y = {\mathrm e}^{\left (\frac {2^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{2}+1\right ) x} c_{1} +{\mathrm e}^{\left (\frac {2^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{2}+1\right ) x} c_{2} +{\mathrm e}^{\left (-2^{\frac {1}{3}}+1\right ) x} c_{3} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{-\left (2^{\frac {1}{3}}-1\right ) x}+c_{2} {\mathrm e}^{\frac {\left (2^{\frac {1}{3}}+2\right ) x}{2}} \sin \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )+c_{3} {\mathrm e}^{\frac {\left (2^{\frac {1}{3}}+2\right ) x}{2}} \cos \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right ) \] |
Problem 824
| ODE |
\[ \boxed {y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime }+4 y^{\prime \prime }=0} \] |
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program solution |
\[ y = c_{2} x +c_{1} +c_{3} {\mathrm e}^{2 x}+x \,{\mathrm e}^{2 x} c_{4} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (c_{4} x +c_{3} \right ) {\mathrm e}^{2 x}+c_{2} x +c_{1} \] |
Problem 825
| ODE |
\[ \boxed {y^{\left (6\right )}+y=0} \] |
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program solution |
\[ y = {\mathrm e}^{-i x} c_{1} +{\mathrm e}^{i x} c_{2} +{\mathrm e}^{-\frac {\sqrt {-2 i \sqrt {3}+2}\, x}{2}} c_{3} +{\mathrm e}^{\frac {\sqrt {2+2 i \sqrt {3}}\, x}{2}} c_{4} +{\mathrm e}^{\frac {\sqrt {-2 i \sqrt {3}+2}\, x}{2}} c_{5} +{\mathrm e}^{-\frac {\sqrt {2+2 i \sqrt {3}}\, x}{2}} c_{6} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (-\sin \left (\frac {x}{2}\right ) c_{4} +c_{6} \cos \left (\frac {x}{2}\right )\right ) {\mathrm e}^{-\frac {\sqrt {3}\, x}{2}}+\left (\sin \left (\frac {x}{2}\right ) c_{3} +\cos \left (\frac {x}{2}\right ) c_{5} \right ) {\mathrm e}^{\frac {\sqrt {3}\, x}{2}}+c_{1} \sin \left (x \right )+c_{2} \cos \left (x \right ) \] |
Problem 826
| ODE |
\[ \boxed {y^{\left (6\right )}-3 y^{\prime \prime \prime \prime }+3 y^{\prime \prime }-y=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{-x}+x \,{\mathrm e}^{-x} c_{2} +x^{2} {\mathrm e}^{-x} c_{3} +{\mathrm e}^{x} c_{4} +x \,{\mathrm e}^{x} c_{5} +x^{2} {\mathrm e}^{x} c_{6} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (c_{6} x^{2}+c_{5} x +c_{4} \right ) {\mathrm e}^{-x}+{\mathrm e}^{x} \left (c_{3} x^{2}+c_{2} x +c_{1} \right ) \] |
Problem 827
| ODE |
\[ \boxed {y^{\left (6\right )}-y^{\prime \prime }=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{-x}+c_{2} +c_{3} x +{\mathrm e}^{x} c_{4} +{\mathrm e}^{-i x} c_{5} +{\mathrm e}^{i x} c_{6} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} +c_{2} x +c_{3} {\mathrm e}^{x}+c_{4} {\mathrm e}^{-x}+c_{5} \sin \left (x \right )+c_{6} \cos \left (x \right ) \] |
Problem 828
| ODE |
\[ \boxed {y^{\left (5\right )}-3 y^{\prime \prime \prime \prime }+3 y^{\prime \prime \prime }-3 y^{\prime \prime }+2 y^{\prime }=0} \] |
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program solution |
\[ y = c_{1} +c_{2} {\mathrm e}^{x}+c_{3} {\mathrm e}^{2 x}+{\mathrm e}^{-i x} c_{4} +{\mathrm e}^{i x} c_{5} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} +{\mathrm e}^{x} c_{2} +c_{3} {\mathrm e}^{2 x}+c_{4} \sin \left (x \right )+c_{5} \cos \left (x \right ) \] |
Problem 829
| ODE |
\[ \boxed {y^{\left (8\right )}+8 y^{\prime \prime \prime \prime }+16 y=0} \] |
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program solution |
\[ y = {\mathrm e}^{\left (-1-i\right ) x} c_{1} +x \,{\mathrm e}^{\left (-1-i\right ) x} c_{2} +{\mathrm e}^{\left (1+i\right ) x} c_{3} +x \,{\mathrm e}^{\left (1+i\right ) x} c_{4} +{\mathrm e}^{\left (1-i\right ) x} c_{5} +x \,{\mathrm e}^{\left (1-i\right ) x} c_{6} +{\mathrm e}^{\left (-1+i\right ) x} c_{7} +x \,{\mathrm e}^{\left (-1+i\right ) x} c_{8} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (\left (c_{4} x +c_{2} \right ) \cos \left (x \right )+\sin \left (x \right ) \left (c_{3} x +c_{1} \right )\right ) {\mathrm e}^{-x}+\left (\left (c_{8} x +c_{6} \right ) \cos \left (x \right )+\sin \left (x \right ) \left (c_{7} x +c_{5} \right )\right ) {\mathrm e}^{x} \] |
Problem 830
| ODE |
\[ \boxed {y^{\prime \prime \prime \prime }+2 y^{\prime \prime }+y=0} \] |
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program solution |
\[ y = {\mathrm e}^{-i x} c_{1} +x \,{\mathrm e}^{-i x} c_{2} +{\mathrm e}^{i x} c_{3} +x \,{\mathrm e}^{i x} c_{4} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (c_{4} x +c_{2} \right ) \cos \left (x \right )+\sin \left (x \right ) \left (c_{3} x +c_{1} \right ) \] |
Problem 831
| ODE |
\[ \boxed {y^{\prime \prime \prime }+5 y^{\prime \prime }+6 y^{\prime }+2 y=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{-x}+{\mathrm e}^{\left (-2-\sqrt {2}\right ) x} c_{2} +{\mathrm e}^{\left (-2+\sqrt {2}\right ) x} c_{3} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-x} c_{1} +c_{2} {\mathrm e}^{\left (\sqrt {2}-2\right ) x}+c_{3} {\mathrm e}^{-\left (2+\sqrt {2}\right ) x} \] |
Problem 832
| ODE |
\[ \boxed {y^{\prime \prime \prime \prime }-7 y^{\prime \prime \prime }+6 y^{\prime \prime }+30 y^{\prime }-36 y=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{-2 x}+c_{2} {\mathrm e}^{3 x}+{\mathrm e}^{\left (3+\sqrt {3}\right ) x} c_{3} +{\mathrm e}^{\left (3-\sqrt {3}\right ) x} c_{4} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (c_{1} {\mathrm e}^{5 x}+c_{3} {\mathrm e}^{x \left (5+\sqrt {3}\right )}+c_{4} {\mathrm e}^{-x \left (-5+\sqrt {3}\right )}+c_{2} \right ) {\mathrm e}^{-2 x} \] |
Problem 833
| ODE |
\[ \boxed {y^{\prime \prime }-y^{\prime }-6 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = -1] \end {align*}
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program solution |
\[ y = \frac {\left ({\mathrm e}^{5 t}+4\right ) {\mathrm e}^{-2 t}}{5} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {\left ({\mathrm e}^{5 t}+4\right ) {\mathrm e}^{-2 t}}{5} \] |
Problem 834
| ODE |
\[ \boxed {y^{\prime \prime }+3 y^{\prime }+2 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = -{\mathrm e}^{-2 t}+2 \,{\mathrm e}^{-t} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = 2 \,{\mathrm e}^{-t}-{\mathrm e}^{-2 t} \] |
Problem 835
| ODE |
\[ \boxed {y^{\prime \prime }-2 y^{\prime }+2 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
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program solution |
\[ y = \sin \left (t \right ) {\mathrm e}^{t} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = {\mathrm e}^{t} \sin \left (t \right ) \] |
Problem 836
| ODE |
\[ \boxed {y^{\prime \prime }-2 y^{\prime }+4 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = -\frac {2 \,{\mathrm e}^{t} \left (\sin \left (\sqrt {3}\, t \right ) \sqrt {3}-3 \cos \left (\sqrt {3}\, t \right )\right )}{3} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = -\frac {2 \left (\sqrt {3}\, \sin \left (\sqrt {3}\, t \right )-3 \cos \left (\sqrt {3}\, t \right )\right ) {\mathrm e}^{t}}{3} \] |
Problem 837
| ODE |
\[ \boxed {y^{\prime \prime }+2 y^{\prime }+5 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = -1] \end {align*}
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program solution |
\[ y = \frac {{\mathrm e}^{-t} \left (4 \cos \left (2 t \right )+\sin \left (2 t \right )\right )}{2} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {{\mathrm e}^{-t} \left (4 \cos \left (2 t \right )+\sin \left (2 t \right )\right )}{2} \] |
Problem 838
| ODE |
\[ \boxed {y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime }+6 y^{\prime \prime }-4 y^{\prime }+y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1, y^{\prime \prime }\left (0\right ) = 0, y^{\prime \prime \prime }\left (0\right ) = 1] \end {align*}
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program solution |
\[ y = \frac {{\mathrm e}^{t} \left (2 t^{3}-3 t^{2}+3 t \right )}{3} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {{\mathrm e}^{t} t \left (2 t^{2}-3 t +3\right )}{3} \] |
Problem 839
| ODE |
\[ \boxed {y^{\prime \prime \prime \prime }-4 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = 1, y^{\prime \prime \prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = \frac {\cos \left (\sqrt {2}\, t \right )}{4}+\frac {3 \cosh \left (\sqrt {2}\, t \right )}{4} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {\cos \left (t \sqrt {2}\right )}{4}+\frac {3 \cosh \left (t \sqrt {2}\right )}{4} \] |
Problem 840
| ODE |
\[ \boxed {y^{\prime \prime }+\omega ^{2} y=\cos \left (2 t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = \frac {\cos \left (2 t \right )+\cos \left (\omega t \right ) \left (\omega ^{2}-5\right )}{\omega ^{2}-4} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {\cos \left (2 t \right )+\cos \left (\omega t \right ) \left (\omega ^{2}-5\right )}{\omega ^{2}-4} \] |
Problem 841
| ODE |
\[ \boxed {y^{\prime \prime }-2 y^{\prime }+2 y={\mathrm e}^{-t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
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program solution |
\[ y = \frac {{\mathrm e}^{-t}}{5}+\frac {\left (-\cos \left (t \right )+7 \sin \left (t \right )\right ) {\mathrm e}^{t}}{5} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {{\mathrm e}^{-t}}{5}+\frac {\left (-\cos \left (t \right )+7 \sin \left (t \right )\right ) {\mathrm e}^{t}}{5} \] |
Problem 842
| ODE |
\[ \boxed {y^{\prime \prime }+4 y=\left \{\begin {array}{cc} 1 & 0\le t <\pi \\ 0 & \pi \le t <\infty \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = \left \{\begin {array}{cc} \frac {3 \cos \left (2 t \right )}{4}+\frac {1}{4} & t <\pi \\ \cos \left (2 t \right ) & \pi \le t \end {array}\right . \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \left \{\begin {array}{cc} \frac {3 \cos \left (2 t \right )}{4}+\frac {1}{4} & t <\pi \\ \cos \left (2 t \right ) & \pi \le t \end {array}\right . \] |
Problem 843
| ODE |
\[ \boxed {y^{\prime \prime }+4 y=\left \{\begin {array}{cc} 1 & 0\le t <1 \\ 0 & 1\le t <\infty \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = -\frac {\cos \left (2 t \right )}{4}+\frac {\left (\left \{\begin {array}{cc} 1 & t <1 \\ \cos \left (2 t -2\right ) & 1\le t \end {array}\right .\right )}{4} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {\left (\left \{\begin {array}{cc} 1 & t <1 \\ \cos \left (2 t -2\right ) & 1\le t \end {array}\right .\right )}{4}-\frac {\cos \left (2 t \right )}{4} \] |
Problem 844
| ODE |
\[ \boxed {y^{\prime \prime }+y=\left \{\begin {array}{cc} t & 0\le t <1 \\ -t +2 & 1\le t <2 \\ 0 & 2\le t <\infty \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = -\sin \left (t \right )+\cos \left (t \right )+\left (\left \{\begin {array}{cc} t & t <1 \\ 2-t +2 \sin \left (-1+t \right ) & t <2 \\ 2 \sin \left (-1+t \right )-\sin \left (t -2\right ) & 2\le t \end {array}\right .\right ) \] Verified OK.
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Maple solution | \[ y \left (t \right ) = -\sin \left (t \right )+\cos \left (t \right )+\left (\left \{\begin {array}{cc} t & t <1 \\ 2-t +2 \sin \left (t -1\right ) & t <2 \\ -\sin \left (t -2\right )+2 \sin \left (t -1\right ) & 2\le t \end {array}\right .\right ) \] |
Problem 845
| ODE |
\[ \boxed {y^{\prime \prime }+y=\left \{\begin {array}{cc} 1 & 0\le t <3 \pi \\ 0 & 3 \pi \le t <\infty \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
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program solution |
\[ y = \sin \left (t \right )-\left (\left \{\begin {array}{cc} -1+\cos \left (t \right ) & t <3 \pi \\ 2 \cos \left (t \right ) & 3 \pi \le t \end {array}\right .\right ) \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \sin \left (t \right )-\left (\left \{\begin {array}{cc} \cos \left (t \right )-1 & t <3 \pi \\ 2 \cos \left (t \right ) & 3 \pi \le t \end {array}\right .\right ) \] |
Problem 846
| ODE |
\[ \boxed {y^{\prime \prime }+2 y^{\prime }+2 y=\left \{\begin {array}{cc} 1 & \pi \le t <2 \pi \\ 0 & \operatorname {otherwise} \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
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program solution |
\[ y = {\mathrm e}^{-t} \sin \left (t \right )+\frac {\left (\left \{\begin {array}{cc} 0 & t <\pi \\ 1+{\mathrm e}^{\pi -t} \left (\cos \left (t \right )+\sin \left (t \right )\right ) & t <2 \pi \\ \left (\cos \left (t \right )+\sin \left (t \right )\right ) \left ({\mathrm e}^{\pi -t}+{\mathrm e}^{2 \pi -t}\right ) & 2 \pi \le t \end {array}\right .\right )}{2} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \sin \left (t \right ) {\mathrm e}^{-t}+\frac {\left (\left \{\begin {array}{cc} 0 & t <\pi \\ 1+{\mathrm e}^{\pi -t} \left (\cos \left (t \right )+\sin \left (t \right )\right ) & t <2 \pi \\ \left (\cos \left (t \right )+\sin \left (t \right )\right ) \left ({\mathrm e}^{\pi -t}+{\mathrm e}^{2 \pi -t}\right ) & 2 \pi \le t \end {array}\right .\right )}{2} \] |
Problem 847
| ODE |
\[ \boxed {y^{\prime \prime }+4 y=\sin \left (t \right )-\operatorname {Heaviside}\left (t -2 \pi \right ) \sin \left (t \right )} \] 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} \frac {\sin \left (t \right )}{3}-\frac {\sin \left (2 t \right )}{6} & t \le 2 \pi \\ 0 & 2 \pi |
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Maple solution | \[ y \left (t \right ) = \frac {\sin \left (t \right ) \left (\cos \left (t \right )-1\right ) \left (-1+\operatorname {Heaviside}\left (t -2 \pi \right )\right )}{3} \] |
Problem 848
| ODE |
\[ \boxed {y^{\prime \prime }+3 y^{\prime }+2 y=\left \{\begin {array}{cc} 1 & 0\le t <10 \\ 0 & \operatorname {otherwise} \end {array}\right .} \] 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 = \frac {\left (\left \{\begin {array}{cc} {\mathrm e}^{-2 t}-2 \,{\mathrm e}^{-t}+1 & t <10 \\ {\mathrm e}^{-20}-2 \,{\mathrm e}^{-10}+2 & t =10 \\ {\mathrm e}^{-2 t}-2 \,{\mathrm e}^{-t}-{\mathrm e}^{-2 t +20}+2 \,{\mathrm e}^{10-t} & 10 |
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Maple solution | \[
y \left (t \right ) = \frac {\left (\left \{\begin {array}{cc} 1-2 \,{\mathrm e}^{-t}+{\mathrm e}^{-2 t} & t <10 \\ -2 \,{\mathrm e}^{-10}+{\mathrm e}^{-20}+2 & t =10 \\ 2 \,{\mathrm e}^{10-t}-{\mathrm e}^{20-2 t}-2 \,{\mathrm e}^{-t}+{\mathrm e}^{-2 t} & 10 |
Problem 849
| ODE |
\[ \boxed {y^{\prime \prime }+y^{\prime }+\frac {5 y}{4}=t -\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (t -\frac {\pi }{2}\right )} \] 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 {2 \left (\left \{\begin {array}{cc} 8-10 t +\left (-8 \cos \left (t \right )+6 \sin \left (t \right )\right ) {\mathrm e}^{-\frac {t}{2}} & t <\frac {\pi }{2} \\ 6 \,{\mathrm e}^{-\frac {\pi }{4}}-10 \pi +16 & t =\frac {\pi }{2} \\ -5 \pi +\left (6 \cos \left (t \right )+8 \sin \left (t \right )\right ) {\mathrm e}^{-\frac {t}{2}+\frac {\pi }{4}}+\left (-8 \cos \left (t \right )+6 \sin \left (t \right )\right ) {\mathrm e}^{-\frac {t}{2}} & \frac {\pi }{2} |
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Maple solution | \[ y \left (t \right ) = -\frac {16}{25}-\frac {12 \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (\cos \left (t \right )+\frac {4 \sin \left (t \right )}{3}\right ) {\mathrm e}^{-\frac {t}{2}+\frac {\pi }{4}}}{25}+\frac {2 \left (8-10 t +5 \pi \right ) \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )}{25}+\frac {4 \left (4 \cos \left (t \right )-3 \sin \left (t \right )\right ) {\mathrm e}^{-\frac {t}{2}}}{25}+\frac {4 t}{5} \] |
Problem 850
| ODE |
\[ \boxed {y^{\prime \prime }+y^{\prime }+\frac {5 y}{4}=\left \{\begin {array}{cc} \sin \left (t \right ) & 0\le t <\pi \\ 0 & \operatorname {otherwise} \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = \frac {4 \left (\left \{\begin {array}{cc} -8 \,{\mathrm e}^{-\frac {t}{4}} \left (\cos \left (t \right ) \sinh \left (\frac {t}{4}\right )-\frac {\sin \left (t \right ) \cosh \left (\frac {t}{4}\right )}{4}\right ) & t <\pi \\ \left ({\mathrm e}^{-\frac {t}{2}}-{\mathrm e}^{\frac {\pi }{2}-\frac {t}{2}}\right ) \left (4 \cos \left (t \right )+\sin \left (t \right )\right ) & \pi \le t \end {array}\right .\right )}{17} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {4 \left (\left \{\begin {array}{cc} -8 \,{\mathrm e}^{-\frac {t}{4}} \left (\cos \left (t \right ) \sinh \left (\frac {t}{4}\right )-\frac {\sin \left (t \right ) \cosh \left (\frac {t}{4}\right )}{4}\right ) & t <\pi \\ \left (-{\mathrm e}^{-\frac {t}{2}+\frac {\pi }{2}}+{\mathrm e}^{-\frac {t}{2}}\right ) \left (4 \cos \left (t \right )+\sin \left (t \right )\right ) & \pi \le t \end {array}\right .\right )}{17} \] |
Problem 851
| ODE |
\[ \boxed {y^{\prime \prime }+4 y=\operatorname {Heaviside}\left (t -\pi \right )-\operatorname {Heaviside}\left (t -3 \pi \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = \left \{\begin {array}{cc} 0 & t <\pi \\ \frac {\sin \left (t \right )^{2}}{2} & t <3 \pi \\ 0 & 3 \pi \le t \end {array}\right . \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {\left (\operatorname {Heaviside}\left (t -\pi \right )-\operatorname {Heaviside}\left (t -3 \pi \right )\right ) \sin \left (t \right )^{2}}{2} \] |
Problem 852
| ODE |
\[ \boxed {y^{\prime \prime \prime \prime }+5 y^{\prime \prime }+4 y=1-\operatorname {Heaviside}\left (t -\pi \right )} \] 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} -\frac {2 \cos \left (t \right )}{3}+\frac {\left (1+\cos \left (t \right )\right )^{2}}{6} & t \le \pi \\ -\frac {2 \cos \left (t \right )}{3} & \pi |
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Maple solution | \[ y \left (t \right ) = -\frac {\left (\cos \left (t \right )+1\right )^{2} \operatorname {Heaviside}\left (t -\pi \right )}{6}+\frac {\left (\cos \left (t \right )-1\right )^{2}}{6} \] |
Problem 853
| ODE |
\[ \boxed {u^{\prime \prime }+\frac {u^{\prime }}{4}+u=k \left (\operatorname {Heaviside}\left (t -\frac {3}{2}\right )-\operatorname {Heaviside}\left (t -\frac {5}{2}\right )\right )} \] With initial conditions \begin {align*} [u \left (0\right ) = 0, u^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ u = \frac {\left (i \sqrt {7}+21\right ) k \left (\left \{\begin {array}{cc} 0 & t <\frac {3}{2} \\ 3 i \sqrt {7}\, {\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (-3+2 t \right )}{16}}-3 i \sqrt {7}-32 \,{\mathrm e}^{\frac {3}{16}+\frac {3 i \left (-2 t +3\right ) \sqrt {7}}{16}-\frac {t}{8}}-31 \,{\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (-3+2 t \right )}{16}}+63 & t <\frac {5}{2} \\ \left (31-3 i \sqrt {7}\right ) {\mathrm e}^{\frac {3 i \left (2 t -5\right ) \sqrt {7}}{16}-\frac {t}{8}+\frac {5}{16}}+3 i \sqrt {7}\, {\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (-3+2 t \right )}{16}}-31 \,{\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (-3+2 t \right )}{16}}-32 \,{\mathrm e}^{\frac {3}{16}+\frac {3 i \left (-2 t +3\right ) \sqrt {7}}{16}-\frac {t}{8}}+32 \,{\mathrm e}^{-\frac {3 i \left (2 t -5\right ) \sqrt {7}}{16}-\frac {t}{8}+\frac {5}{16}} & \frac {5}{2}\le t \end {array}\right .\right )}{1344} \] Verified OK.
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Maple solution | \[ u \left (t \right ) = -\frac {k \left (\operatorname {Heaviside}\left (t -\frac {5}{2}\right ) \left (-21+i \sqrt {7}\right ) {\mathrm e}^{\frac {3 i \sqrt {7}\, \left (2 t -5\right )}{16}-\frac {t}{8}+\frac {5}{16}}+\left (-i \sqrt {7}-21\right ) \operatorname {Heaviside}\left (t -\frac {5}{2}\right ) {\mathrm e}^{-\frac {3 i \sqrt {7}\, \left (2 t -5\right )}{16}-\frac {t}{8}+\frac {5}{16}}+\left (i \sqrt {7}+21\right ) \operatorname {Heaviside}\left (t -\frac {3}{2}\right ) {\mathrm e}^{\frac {3}{16}+\frac {3 i \left (-2 t +3\right ) \sqrt {7}}{16}-\frac {t}{8}}+\left (-i \sqrt {7}+21\right ) \operatorname {Heaviside}\left (t -\frac {3}{2}\right ) {\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (2 t -3\right )}{16}}+42 \operatorname {Heaviside}\left (t -\frac {5}{2}\right )-42 \operatorname {Heaviside}\left (t -\frac {3}{2}\right )\right )}{42} \] |
Problem 854
| ODE |
\[ \boxed {u^{\prime \prime }+\frac {u^{\prime }}{4}+u=\frac {\operatorname {Heaviside}\left (t -\frac {3}{2}\right )}{2}-\frac {\operatorname {Heaviside}\left (t -\frac {5}{2}\right )}{2}} \] With initial conditions \begin {align*} [u \left (0\right ) = 0, u^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ u = \frac {\left (i \sqrt {7}+21\right ) \left (\left \{\begin {array}{cc} 0 & t <\frac {3}{2} \\ 3 i \sqrt {7}\, {\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (-3+2 t \right )}{16}}-3 i \sqrt {7}-32 \,{\mathrm e}^{\frac {3}{16}+\frac {3 i \left (-2 t +3\right ) \sqrt {7}}{16}-\frac {t}{8}}-31 \,{\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (-3+2 t \right )}{16}}+63 & t <\frac {5}{2} \\ \left (31-3 i \sqrt {7}\right ) {\mathrm e}^{\frac {3 i \left (2 t -5\right ) \sqrt {7}}{16}-\frac {t}{8}+\frac {5}{16}}+3 i \sqrt {7}\, {\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (-3+2 t \right )}{16}}-31 \,{\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (-3+2 t \right )}{16}}-32 \,{\mathrm e}^{\frac {3}{16}+\frac {3 i \left (-2 t +3\right ) \sqrt {7}}{16}-\frac {t}{8}}+32 \,{\mathrm e}^{-\frac {3 i \left (2 t -5\right ) \sqrt {7}}{16}-\frac {t}{8}+\frac {5}{16}} & \frac {5}{2}\le t \end {array}\right .\right )}{2688} \] Verified OK.
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Maple solution | \[ u \left (t \right ) = \frac {\left (-i \sqrt {7}+21\right ) \operatorname {Heaviside}\left (t -\frac {5}{2}\right ) {\mathrm e}^{\frac {3 i \sqrt {7}\, \left (2 t -5\right )}{16}-\frac {t}{8}+\frac {5}{16}}}{84}+\frac {\operatorname {Heaviside}\left (t -\frac {5}{2}\right ) {\mathrm e}^{-\frac {3 i \sqrt {7}\, \left (2 t -5\right )}{16}-\frac {t}{8}+\frac {5}{16}} \left (i \sqrt {7}+21\right )}{84}+\frac {\left (-i \sqrt {7}-21\right ) \operatorname {Heaviside}\left (t -\frac {3}{2}\right ) {\mathrm e}^{\frac {3}{16}+\frac {3 i \left (-2 t +3\right ) \sqrt {7}}{16}-\frac {t}{8}}}{84}+\frac {\left (-21+i \sqrt {7}\right ) \operatorname {Heaviside}\left (t -\frac {3}{2}\right ) {\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (2 t -3\right )}{16}}}{84}-\frac {\operatorname {Heaviside}\left (t -\frac {5}{2}\right )}{2}+\frac {\operatorname {Heaviside}\left (t -\frac {3}{2}\right )}{2} \] |
Problem 855
| ODE |
\[ \boxed {u^{\prime \prime }+\frac {u^{\prime }}{4}+u=\frac {\operatorname {Heaviside}\left (t -5\right ) \left (t -5\right )-\operatorname {Heaviside}\left (t -5-k \right ) \left (t -5-k \right )}{k}} \] With initial conditions \begin {align*} [u \left (0\right ) = 0, u^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ u = \frac {-21 \left (\operatorname {Heaviside}\left (5+k \right )+\operatorname {Heaviside}\left (t -5-k \right )-1\right ) \left (\frac {31 \sqrt {7}\, \sin \left (\frac {3 \sqrt {7}\, \left (-t +5+k \right )}{8}\right )}{21}+\cos \left (\frac {3 \sqrt {7}\, \left (-t +5+k \right )}{8}\right )\right ) {\mathrm e}^{-\frac {t}{8}+\frac {5}{8}+\frac {k}{8}}+21 \operatorname {Heaviside}\left (t -5\right ) \cos \left (\frac {3 \sqrt {7}\, \left (t -5\right )}{8}\right ) {\mathrm e}^{-\frac {t}{8}+\frac {5}{8}}-31 \sqrt {7}\, \operatorname {Heaviside}\left (t -5\right ) \sin \left (\frac {3 \sqrt {7}\, \left (t -5\right )}{8}\right ) {\mathrm e}^{-\frac {t}{8}+\frac {5}{8}}+\left (84 k -84 t +441\right ) \operatorname {Heaviside}\left (t -5-k \right )+84 \,{\mathrm e}^{-\frac {t}{8}} \left (-1+\operatorname {Heaviside}\left (5+k \right )\right ) \left (k +\frac {21}{4}\right ) \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )+4 \sqrt {7}\, {\mathrm e}^{-\frac {t}{8}} \left (-1+\operatorname {Heaviside}\left (5+k \right )\right ) \left (k -\frac {11}{4}\right ) \sin \left (\frac {3 \sqrt {7}\, t}{8}\right )+\left (84 t -441\right ) \operatorname {Heaviside}\left (t -5\right )}{84 k} \] Verified OK.
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Maple solution | \[ u \left (t \right ) = \frac {-21 \left (\frac {31 \sin \left (\frac {3 \sqrt {7}\, \left (-t +5+k \right )}{8}\right ) \sqrt {7}}{21}+\cos \left (\frac {3 \sqrt {7}\, \left (-t +5+k \right )}{8}\right )\right ) \left (\operatorname {Heaviside}\left (5+k \right )+\operatorname {Heaviside}\left (t -5-k \right )-1\right ) {\mathrm e}^{-\frac {t}{8}+\frac {5}{8}+\frac {k}{8}}+21 \,{\mathrm e}^{\frac {5}{8}-\frac {t}{8}} \operatorname {Heaviside}\left (t -5\right ) \cos \left (\frac {3 \sqrt {7}\, \left (t -5\right )}{8}\right )-31 \sqrt {7}\, {\mathrm e}^{\frac {5}{8}-\frac {t}{8}} \operatorname {Heaviside}\left (t -5\right ) \sin \left (\frac {3 \sqrt {7}\, \left (t -5\right )}{8}\right )+\left (84 k -84 t +441\right ) \operatorname {Heaviside}\left (t -5-k \right )+84 \,{\mathrm e}^{-\frac {t}{8}} \left (-1+\operatorname {Heaviside}\left (5+k \right )\right ) \left (k +\frac {21}{4}\right ) \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )+4 \,{\mathrm e}^{-\frac {t}{8}} \left (-1+\operatorname {Heaviside}\left (5+k \right )\right ) \left (k -\frac {11}{4}\right ) \sqrt {7}\, \sin \left (\frac {3 \sqrt {7}\, t}{8}\right )+\left (84 t -441\right ) \operatorname {Heaviside}\left (t -5\right )}{84 k} \] |
Problem 856
| ODE |
\[ \boxed {y^{\prime \prime }+2 y^{\prime }+2 y=\delta \left (t -\pi \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = {\mathrm e}^{-t} \left (\cos \left (t \right )+\sin \left (t \right )\right )-\left (\left \{\begin {array}{cc} 0 & t <\pi \\ {\mathrm e}^{\pi -t} \sin \left (t \right ) & \pi \le t \end {array}\right .\right ) \] Verified OK.
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Maple solution | \[ y \left (t \right ) = {\mathrm e}^{-t} \left (\cos \left (t \right )+\sin \left (t \right )\right )-\sin \left (t \right ) \operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{\pi -t} \] |
Problem 857
| ODE |
\[ \boxed {y^{\prime \prime }+4 y=\delta \left (t -\pi \right )-\delta \left (t -2 \pi \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = \left \{\begin {array}{cc} 0 & t <\pi \\ \frac {\sin \left (2 t \right )}{2} & t <2 \pi \\ 0 & 2 \pi \le t \end {array}\right . \] Verified OK.
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Maple solution | \[ y \left (t \right ) = -\frac {\left (\operatorname {Heaviside}\left (t -2 \pi \right )-\operatorname {Heaviside}\left (t -\pi \right )\right ) \sin \left (2 t \right )}{2} \] |
Problem 858
| ODE |
\[ \boxed {y^{\prime \prime }+3 y^{\prime }+2 y=\delta \left (t -5\right )+\operatorname {Heaviside}\left (t -10\right )} \] With initial conditions \begin {align*} \left [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = {\frac {1}{2}}\right ] \end {align*}
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program solution |
\[ y = -\frac {{\mathrm e}^{-2 t}}{2}+\frac {{\mathrm e}^{-t}}{2}+\left (\left \{\begin {array}{cc} 0 & t <5 \\ -{\mathrm e}^{-2 t +10}+{\mathrm e}^{-t +5} & t <10 \\ \frac {1}{2}+\frac {{\mathrm e}^{-2 t +20}}{2}-{\mathrm e}^{10-t}-{\mathrm e}^{-2 t +10}+{\mathrm e}^{-t +5} & 10\le t \end {array}\right .\right ) \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {{\mathrm e}^{-t}}{2}-\frac {{\mathrm e}^{-2 t}}{2}-\operatorname {Heaviside}\left (t -10\right ) {\mathrm e}^{10-t}+\frac {\operatorname {Heaviside}\left (t -10\right ) {\mathrm e}^{20-2 t}}{2}+\frac {\operatorname {Heaviside}\left (t -10\right )}{2}+\operatorname {Heaviside}\left (t -5\right ) {\mathrm e}^{-t +5}-\operatorname {Heaviside}\left (t -5\right ) {\mathrm e}^{10-2 t} \] |
Problem 859
| ODE |
\[ \boxed {y^{\prime \prime }+2 y^{\prime }+3 y=\sin \left (t \right )+\delta \left (t -3 \pi \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = -\frac {\cos \left (t \right )}{4}+\frac {\sin \left (t \right )}{4}+\frac {{\mathrm e}^{-t} \cos \left (\sqrt {2}\, t \right )}{4}-\left (\left \{\begin {array}{cc} 0 & t <3 \pi \\ -\frac {\sqrt {2}\, {\mathrm e}^{3 \pi -t} \sin \left (\sqrt {2}\, \left (t -3 \pi \right )\right )}{2} & 3 \pi \le t \end {array}\right .\right ) \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {\sqrt {2}\, {\mathrm e}^{3 \pi -t} \operatorname {Heaviside}\left (t -3 \pi \right ) \sin \left (\sqrt {2}\, \left (t -3 \pi \right )\right )}{2}-\frac {\cos \left (t \right )}{4}+\frac {\sin \left (t \right )}{4}+\frac {{\mathrm e}^{-t} \cos \left (t \sqrt {2}\right )}{4} \] |
Problem 860
| ODE |
\[ \boxed {y^{\prime \prime }+y=\delta \left (t -2 \pi \right ) \cos \left (t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
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program solution |
\[ y = \sin \left (t \right ) \left (\left \{\begin {array}{cc} 1 & t <2 \pi \\ 2 & 2 \pi \le t \end {array}\right .\right ) \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \sin \left (t \right ) \left (\operatorname {Heaviside}\left (t -2 \pi \right )+1\right ) \] |
Problem 861
| ODE |
\[ \boxed {y^{\prime \prime }+4 y=2 \delta \left (t -\frac {\pi }{4}\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = -\left (\left \{\begin {array}{cc} 0 & t <\frac {\pi }{4} \\ \cos \left (2 t \right ) & \frac {\pi }{4}\le t \end {array}\right .\right ) \] Verified OK.
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Maple solution | \[ y \left (t \right ) = -\operatorname {Heaviside}\left (t -\frac {\pi }{4}\right ) \cos \left (2 t \right ) \] |
Problem 862
| ODE |
\[ \boxed {y^{\prime \prime }+2 y^{\prime }+2 y=\cos \left (t \right )+\delta \left (t -\frac {\pi }{2}\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = -\left (\left \{\begin {array}{cc} 0 & t <\frac {\pi }{2} \\ {\mathrm e}^{-t +\frac {\pi }{2}} \cos \left (t \right ) & \frac {\pi }{2}\le t \end {array}\right .\right )+\frac {\left (-\cos \left (t \right )-3 \sin \left (t \right )\right ) {\mathrm e}^{-t}}{5}+\frac {\cos \left (t \right )}{5}+\frac {2 \sin \left (t \right )}{5} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = -\cos \left (t \right ) \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) {\mathrm e}^{-t +\frac {\pi }{2}}+\frac {\left (-\cos \left (t \right )-3 \sin \left (t \right )\right ) {\mathrm e}^{-t}}{5}+\frac {\cos \left (t \right )}{5}+\frac {2 \sin \left (t \right )}{5} \] |
Problem 863
| ODE |
\[ \boxed {y^{\prime \prime \prime \prime }-y=\delta \left (-1+t \right )} \] 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 |
\[ y = \left \{\begin {array}{cc} 0 & t <1 \\ -\frac {\sin \left (-1+t \right )}{2}+\frac {\sinh \left (-1+t \right )}{2} & 1\le t \end {array}\right . \] Verified OK.
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Maple solution | \[ y \left (t \right ) = -\frac {\operatorname {Heaviside}\left (t -1\right ) \left (\sin \left (t -1\right )-\sinh \left (t -1\right )\right )}{2} \] |
Problem 864
| ODE |
\[ \boxed {y^{\prime \prime }+\frac {y^{\prime }}{2}+y=\delta \left (-1+t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = \left \{\begin {array}{cc} 0 & t <1 \\ \frac {4 \sqrt {15}\, {\mathrm e}^{\frac {1}{4}-\frac {t}{4}} \sin \left (\frac {\sqrt {15}\, \left (-1+t \right )}{4}\right )}{15} & 1\le t \end {array}\right . \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {4 \,{\mathrm e}^{\frac {1}{4}-\frac {t}{4}} \operatorname {Heaviside}\left (t -1\right ) \sqrt {15}\, \sin \left (\frac {\sqrt {15}\, \left (t -1\right )}{4}\right )}{15} \] |
Problem 865
| ODE |
\[ \boxed {y^{\prime \prime }+\frac {y^{\prime }}{4}+y=\delta \left (-1+t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = \left \{\begin {array}{cc} 0 & t <1 \\ \frac {8 \sqrt {7}\, {\mathrm e}^{\frac {1}{8}-\frac {t}{8}} \sin \left (\frac {3 \sqrt {7}\, \left (-1+t \right )}{8}\right )}{21} & 1\le t \end {array}\right . \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {8 \,{\mathrm e}^{\frac {1}{8}-\frac {t}{8}} \operatorname {Heaviside}\left (t -1\right ) \sqrt {7}\, \sin \left (\frac {3 \sqrt {7}\, \left (t -1\right )}{8}\right )}{21} \] |
Problem 866
| ODE |
\[ \boxed {y^{\prime \prime }+y=\frac {\operatorname {Heaviside}\left (t -4+k \right )-\operatorname {Heaviside}\left (t -4-k \right )}{2 k}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = \frac {\left (\operatorname {Heaviside}\left (4+k \right )+\operatorname {Heaviside}\left (t -4-k \right )-1\right ) \cos \left (-t +4+k \right )-\operatorname {Heaviside}\left (t -4-k \right )+\left (-\cos \left (t -4+k \right )+1\right ) \operatorname {Heaviside}\left (t -4+k \right )-\operatorname {Heaviside}\left (-4+k \right ) \cos \left (t \right )-\cos \left (t \right ) \operatorname {Heaviside}\left (4+k \right )+\operatorname {Heaviside}\left (-4+k \right ) \cos \left (t -4+k \right )+\cos \left (t \right )}{2 k} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {\left (\operatorname {Heaviside}\left (4+k \right )+\operatorname {Heaviside}\left (t -4-k \right )-1\right ) \cos \left (-t +4+k \right )-\operatorname {Heaviside}\left (t -4-k \right )+\left (-\cos \left (t -4+k \right )+1\right ) \operatorname {Heaviside}\left (t -4+k \right )-\operatorname {Heaviside}\left (-4+k \right ) \cos \left (t \right )-\cos \left (t \right ) \operatorname {Heaviside}\left (4+k \right )+\operatorname {Heaviside}\left (-4+k \right ) \cos \left (t -4+k \right )+\cos \left (t \right )}{2 k} \] |
Problem 867
| ODE |
\[ \boxed {y^{\prime \prime }+2 y^{\prime }+2 y=f \left (t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = {\mathrm e}^{-t} \left (-\left (\int _{0}^{t}f \left (\alpha \right ) \sin \left (\alpha \right ) {\mathrm e}^{\alpha }d \alpha \right ) \cos \left (t \right )+\left (\int _{0}^{t}f \left (\alpha \right ) \cos \left (\alpha \right ) {\mathrm e}^{\alpha }d \alpha \right ) \sin \left (t \right )\right ) \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \left (-\cos \left (t \right ) \left (\int _{0}^{t}f \left (\textit {\_z1} \right ) \sin \left (\textit {\_z1} \right ) {\mathrm e}^{\textit {\_z1}}d \textit {\_z1} \right )+\sin \left (t \right ) \left (\int _{0}^{t}f \left (\textit {\_z1} \right ) \cos \left (\textit {\_z1} \right ) {\mathrm e}^{\textit {\_z1}}d \textit {\_z1} \right )\right ) {\mathrm e}^{-t} \] |
Problem 868
| ODE |
\[ \boxed {y^{\prime \prime }+2 y^{\prime }+2 y=\delta \left (t -\pi \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = -\left (\left \{\begin {array}{cc} 0 & t <\pi \\ {\mathrm e}^{\pi -t} \sin \left (t \right ) & \pi \le t \end {array}\right .\right ) \] Verified OK.
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Maple solution | \[ y \left (t \right ) = -\sin \left (t \right ) \operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{\pi -t} \] |
Problem 869
| ODE |
\[ \boxed {y^{\prime }-2 y=0} \] |
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program solution |
\[ y = {\mathrm e}^{2 x} c_{1}^{2} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{2 x} c_{1} \] |
Problem 870
| ODE |
\[ \boxed {y+y^{\prime } x=x^{2}} \] |
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program solution |
\[ y = \frac {x^{3}+3 c_{1}}{3 x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {x^{3}+3 c_{1}}{3 x} \] |
Problem 871
| ODE |
\[ \boxed {2 y x +y^{\prime }=x} \] |
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program solution |
\[ y = \frac {{\mathrm e}^{-x^{2}-2 c_{1}}}{2}+\frac {1}{2} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {1}{2}+{\mathrm e}^{-x^{2}} c_{1} \] |
Problem 872
| ODE |
\[ \boxed {2 y^{\prime }+x \left (-1+y^{2}\right )=0} \] |
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program solution |
\[ y = \frac {c_{3} \cosh \left (\frac {x^{2}}{4}\right )+\sinh \left (\frac {x^{2}}{4}\right )}{c_{3} \sinh \left (\frac {x^{2}}{4}\right )+\cosh \left (\frac {x^{2}}{4}\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \tanh \left (\frac {x^{2}}{4}+\frac {c_{1}}{2}\right ) \] |
Problem 873
| ODE |
\[ \boxed {y^{\prime }-\left (1+y^{2}\right ) x^{2}=0} \] |
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program solution |
\[ y = \frac {-c_{3} \cos \left (\frac {x^{3}}{3}\right )+\sin \left (\frac {x^{3}}{3}\right )}{c_{3} \sin \left (\frac {x^{3}}{3}\right )+\cos \left (\frac {x^{3}}{3}\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \tan \left (\frac {x^{3}}{3}+c_{1} \right ) \] |
Problem 874
| ODE |
\[ \boxed {y^{\prime }=-x} \] |
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program solution |
\[ y = -\frac {x^{2}}{2}+c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {x^{2}}{2}+c_{1} \] |
Problem 875
| ODE |
\[ \boxed {y^{\prime }=-\sin \left (x \right ) x} \] |
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program solution |
\[ y = x \cos \left (x \right )-\sin \left (x \right )+c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\sin \left (x \right )+\cos \left (x \right ) x +c_{1} \] |
Problem 876
| ODE |
\[ \boxed {y^{\prime }=x \ln \left (x \right )} \] |
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program solution |
\[ y = \frac {\ln \left (x \right ) x^{2}}{2}-\frac {x^{2}}{4}+c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\ln \left (x \right ) x^{2}}{2}-\frac {x^{2}}{4}+c_{1} \] |
Problem 877
| ODE |
\[ \boxed {y^{\prime }=-x \,{\mathrm e}^{x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
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program solution |
\[ y = -x \,{\mathrm e}^{x}+{\mathrm e}^{x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\left (x -1\right ) {\mathrm e}^{x} \] |
Problem 878
| ODE |
\[ \boxed {y^{\prime }=x \sin \left (x^{2}\right )} \] With initial conditions \begin {align*} \left [y \left (\frac {\sqrt {2}\, \sqrt {\pi }}{2}\right ) = 1\right ] \end {align*}
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program solution |
\[ y = -\frac {\cos \left (x^{2}\right )}{2}+1 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {\cos \left (x^{2}\right )}{2}+1 \] |
Problem 879
| ODE |
\[ \boxed {y^{\prime }=\tan \left (x \right )} \] With initial conditions \begin {align*} \left [y \left (\frac {\pi }{4}\right ) = 3\right ] \end {align*}
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program solution |
\[ y = -\ln \left (\cos \left (x \right )\right )-\frac {\ln \left (2\right )}{2}+3 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\ln \left (\cos \left (x \right )\right )+3-\frac {\ln \left (2\right )}{2} \] |
Problem 880
| ODE |
\[ \boxed {y^{\prime }+\tan \left (x \right ) y=\cos \left (x \right )} \] With initial conditions \begin {align*} \left [y \left (\frac {\pi }{4}\right ) = \frac {\sqrt {2}\, \pi }{8}\right ] \end {align*}
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program solution |
\[ y = x \cos \left (x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \cos \left (x \right ) x \] |
Problem 881
| ODE |
\[ \boxed {y^{\prime }-\frac {x^{2}-2 x^{2} y+2}{x^{3}}=0} \] With initial conditions \begin {align*} \left [y \left (1\right ) = {\frac {3}{2}}\right ] \end {align*}
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program solution |
\[ y = \frac {x^{2}+4 \ln \left (x \right )+2}{2 x^{2}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\frac {x^{2}}{2}+2 \ln \left (x \right )+1}{x^{2}} \] |
Problem 882
| ODE |
\[ \boxed {y^{\prime }-x \left (1+y^{2}\right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
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program solution |
\[ y = \frac {\sin \left (\frac {x^{2}}{2}\right )}{\cos \left (\frac {x^{2}}{2}\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \tan \left (\frac {x^{2}}{2}\right ) \] |
Problem 883
| ODE |
\[ \boxed {y^{\prime }+\frac {y \left (1+y\right )}{x}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = -2] \end {align*}
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program solution |
\[ y = \frac {2}{-2+x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {2}{-2+x} \] |
Problem 884
| ODE |
\[ \boxed {y^{\prime }-a y^{\frac {a -1}{a}}=0} \] |
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program solution |
\[ y \,y^{-\frac {a -1}{a}} = x +c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (c_{1} +x \right )^{a} \] |
Problem 885
| ODE |
\[ \boxed {y^{\prime }-{| y|}=1} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
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program solution |
\[
\left \{\begin {array}{cc} -\ln \left (1-y\right ) & y\le 0 \\ \ln \left (1+y\right ) & 0 |
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Maple solution | \begin{align*} y \left (x \right ) &= {\mathrm e}^{x}-1 \\ y \left (x \right ) &= 1-{\mathrm e}^{-x} \\ \end{align*} |
Problem 886
| ODE |
\[ \boxed {y^{\prime }-\frac {\sqrt {x^{2}+4 x +4 y}}{2}=-\frac {x}{2}-1} \] |
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program solution |
\[ y = c_{1}^{2}+c_{1} x -x \] Verified OK.
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Maple solution | \[ x -\sqrt {x^{2}+4 x +4 y \left (x \right )}-c_{1} = 0 \] |
Problem 887
| ODE |
\[ \boxed {y^{\prime }+a y=0} \] |
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program solution |
\[ y = \frac {{\mathrm e}^{-x a}}{c_{1}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{-a x} \] |
Problem 888
| ODE |
\[ \boxed {y^{\prime }+3 x^{2} y=0} \] |
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program solution |
\[ y = {\mathrm e}^{-x^{3}-3 c_{1}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{-x^{3}} \] |
Problem 889
| ODE |
\[ \boxed {y^{\prime } x +\ln \left (x \right ) y=0} \] |
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program solution |
\[ y = {\mathrm e}^{-\frac {\ln \left (x \right )^{2}}{2}-c_{1}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {\ln \left (x \right )^{2}}{2}} \] |
Problem 890
| ODE |
\[ \boxed {3 y+y^{\prime } x=0} \] |
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program solution |
\[ y = \frac {{\mathrm e}^{-3 c_{1}}}{x^{3}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{1}}{x^{3}} \] |
Problem 891
| ODE |
\[ \boxed {y^{\prime } x^{2}+y=0} \] |
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program solution |
\[ y = {\mathrm e}^{-\frac {c_{1} x -1}{x}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{\frac {1}{x}} \] |
Problem 892
| ODE |
\[ \boxed {y^{\prime }+\frac {\left (x +1\right ) y}{x}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1] \end {align*}
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program solution |
\[ y = \frac {{\mathrm e}^{1-x}}{x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {{\mathrm e}^{1-x}}{x} \] |
Problem 893
| ODE |
\[ \boxed {y^{\prime } x +\left (1+\frac {1}{\ln \left (x \right )}\right ) y=0} \] With initial conditions \begin {align*} [y \left ({\mathrm e}\right ) = 1] \end {align*}
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program solution |
\[ y = \frac {{\mathrm e}}{x \ln \left (x \right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {{\mathrm e}}{x \ln \left (x \right )} \] |
Problem 894
| ODE |
\[ \boxed {y^{\prime } x +\left (1+x \cot \left (x \right )\right ) y=0} \] With initial conditions \begin {align*} \left [y \left (\frac {\pi }{2}\right ) = 2\right ] \end {align*}
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program solution |
\[ y = \frac {\pi }{\sin \left (x \right ) x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\csc \left (x \right ) \pi }{x} \] |
Problem 895
| ODE |
\[ \boxed {y^{\prime }-\frac {2 x y}{x^{2}+1}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2] \end {align*}
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program solution |
\[ y = 2 x^{2}+2 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = 2 x^{2}+2 \] |
Problem 896
| ODE |
\[ \boxed {y^{\prime }+\frac {k y}{x}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 3] \end {align*}
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program solution |
\[ y = 3 x^{-k} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = 3 x^{-k} \] |
Problem 897
| ODE |
\[ \boxed {y^{\prime }+\tan \left (k x \right ) y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2] \end {align*}
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program solution |
\[ y = 2 \left (\frac {2}{1+\cos \left (2 k x \right )}\right )^{-\frac {1}{2 k}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = 2 \left (\sec \left (k x \right )^{2}\right )^{-\frac {1}{2 k}} \] |
Problem 898
| ODE |
\[ \boxed {3 y+y^{\prime }=1} \] |
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program solution |
\[ y = \frac {{\mathrm e}^{-3 x}}{c_{1}^{3}}+\frac {1}{3} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {1}{3}+c_{1} {\mathrm e}^{-3 x} \] |
Problem 899
| ODE |
\[ \boxed {y^{\prime }+\left (\frac {1}{x}-1\right ) y=-\frac {2}{x}} \] |
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program solution |
\[ y = \frac {\left (2 \,{\mathrm e}^{-x}+c_{1} \right ) {\mathrm e}^{x}}{x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {{\mathrm e}^{x} c_{1} +2}{x} \] |
Problem 900
| ODE |
\[ \boxed {2 y x +y^{\prime }=x \,{\mathrm e}^{-x^{2}}} \] |
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program solution |
\[ y = \frac {{\mathrm e}^{-x^{2}} \left (x^{2}+2 c_{1} \right )}{2} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\left (x^{2}+2 c_{1} \right ) {\mathrm e}^{-x^{2}}}{2} \] |