2.17.133 Problems 13201 to 13300

Problem 13201

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }+2 y=5 \cos \left (t \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 t}+c_{2} {\mathrm e}^{-t}+\frac {\cos \left (t \right )}{2}+\frac {3 \sin \left (t \right )}{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -{\mathrm e}^{-2 t} c_{1} +\frac {\cos \left (t \right )}{2}+\frac {3 \sin \left (t \right )}{2}+c_{2} {\mathrm e}^{-t} \]

Problem 13202

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }+2 y=\sin \left (t \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 t}+c_{2} {\mathrm e}^{-t}-\frac {3 \cos \left (t \right )}{10}+\frac {\sin \left (t \right )}{10} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -{\mathrm e}^{-2 t} c_{1} -\frac {3 \cos \left (t \right )}{10}+\frac {\sin \left (t \right )}{10}+c_{2} {\mathrm e}^{-t} \]

Problem 13203

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }+2 y=2 \sin \left (t \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 t}+c_{2} {\mathrm e}^{-t}-\frac {3 \cos \left (t \right )}{5}+\frac {\sin \left (t \right )}{5} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -{\mathrm e}^{-2 t} c_{1} -\frac {3 \cos \left (t \right )}{5}+\frac {\sin \left (t \right )}{5}+c_{2} {\mathrm e}^{-t} \]

Problem 13204

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+8 y=\cos \left (t \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-4 t}+\frac {c_{2} {\mathrm e}^{-2 t}}{2}+\frac {7 \cos \left (t \right )}{85}+\frac {6 \sin \left (t \right )}{85} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {{\mathrm e}^{-4 t} c_{1}}{2}+\frac {7 \cos \left (t \right )}{85}+\frac {6 \sin \left (t \right )}{85}+c_{2} {\mathrm e}^{-2 t} \]

Problem 13205

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+8 y=-4 \cos \left (3 t \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-4 t}+\frac {c_{2} {\mathrm e}^{-2 t}}{2}+\frac {4 \cos \left (3 t \right )}{325}-\frac {72 \sin \left (3 t \right )}{325} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {{\mathrm e}^{-4 t} c_{1}}{2}+c_{2} {\mathrm e}^{-2 t}+\frac {4 \cos \left (3 t \right )}{325}-\frac {72 \sin \left (3 t \right )}{325} \]

Problem 13206

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+13 y=3 \cos \left (2 t \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 t} \cos \left (3 t \right )+\frac {c_{2} {\mathrm e}^{-2 t} \sin \left (3 t \right )}{3}+\frac {27 \cos \left (2 t \right )}{145}+\frac {24 \sin \left (2 t \right )}{145} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{2} {\mathrm e}^{-2 t} \sin \left (3 t \right )+c_{1} {\mathrm e}^{-2 t} \cos \left (3 t \right )+\frac {24 \sin \left (2 t \right )}{145}+\frac {27 \cos \left (2 t \right )}{145} \]

Problem 13207

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+20 y=-\cos \left (5 t \right )} \]
program solution	\[ y = c_{1} \cos \left (4 t \right ) {\mathrm e}^{-2 t}+\frac {c_{2} \sin \left (4 t \right ) {\mathrm e}^{-2 t}}{4}+\frac {\cos \left (5 t \right )}{85}-\frac {4 \sin \left (5 t \right )}{85} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \sin \left (4 t \right ) {\mathrm e}^{-2 t} c_{2} +\cos \left (4 t \right ) {\mathrm e}^{-2 t} c_{1} +\frac {\cos \left (5 t \right )}{85}-\frac {4 \sin \left (5 t \right )}{85} \]

Problem 13208

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+20 y=-3 \sin \left (2 t \right )} \]
program solution	\[ y = c_{1} \cos \left (4 t \right ) {\mathrm e}^{-2 t}+\frac {c_{2} \sin \left (4 t \right ) {\mathrm e}^{-2 t}}{4}+\frac {3 \cos \left (2 t \right )}{40}-\frac {3 \sin \left (2 t \right )}{20} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \sin \left (4 t \right ) {\mathrm e}^{-2 t} c_{2} +\cos \left (4 t \right ) {\mathrm e}^{-2 t} c_{1} -\frac {3 \sin \left (2 t \right )}{20}+\frac {3 \cos \left (2 t \right )}{40} \]

Problem 13209

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+y=\cos \left (3 t \right )} \]
program solution	\[ y = {\mathrm e}^{-t} \left (c_{2} t +c_{1} \right )-\frac {2 \cos \left (3 t \right )}{25}+\frac {3 \sin \left (3 t \right )}{50} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (c_{1} t +c_{2} \right ) {\mathrm e}^{-t}-\frac {2 \cos \left (3 t \right )}{25}+\frac {3 \sin \left (3 t \right )}{50} \]

Problem 13210

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+8 y=\cos \left (t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = -\frac {{\mathrm e}^{-2 t}}{5}+\frac {2 \,{\mathrm e}^{-4 t}}{17}+\frac {7 \cos \left (t \right )}{85}+\frac {6 \sin \left (t \right )}{85} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {2 \,{\mathrm e}^{-4 t}}{17}+\frac {7 \cos \left (t \right )}{85}+\frac {6 \sin \left (t \right )}{85}-\frac {{\mathrm e}^{-2 t}}{5} \]

Problem 13211

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+8 y=2 \cos \left (3 t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = -\frac {2 \,{\mathrm e}^{-2 t}}{13}+\frac {4 \,{\mathrm e}^{-4 t}}{25}-\frac {2 \cos \left (3 t \right )}{325}+\frac {36 \sin \left (3 t \right )}{325} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {4 \,{\mathrm e}^{-4 t}}{25}-\frac {2 \,{\mathrm e}^{-2 t}}{13}-\frac {2 \cos \left (3 t \right )}{325}+\frac {36 \sin \left (3 t \right )}{325} \]

Problem 13212

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+20 y=-3 \sin \left (2 t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = -\frac {9 \,{\mathrm e}^{-3 t} \cos \left (\sqrt {11}\, t \right )}{100}-\frac {3 \,{\mathrm e}^{-3 t} \sin \left (\sqrt {11}\, t \right ) \sqrt {11}}{1100}+\frac {9 \cos \left (2 t \right )}{100}-\frac {3 \sin \left (2 t \right )}{25} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {3 \,{\mathrm e}^{-3 t} \sqrt {11}\, \sin \left (\sqrt {11}\, t \right )}{1100}-\frac {9 \,{\mathrm e}^{-3 t} \cos \left (\sqrt {11}\, t \right )}{100}+\frac {9 \cos \left (2 t \right )}{100}-\frac {3 \sin \left (2 t \right )}{25} \]

Problem 13213

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+y=2 \cos \left (2 t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {2 \left (3-5 t \right ) {\mathrm e}^{-t}}{25}-\frac {6 \cos \left (2 t \right )}{25}+\frac {8 \sin \left (2 t \right )}{25} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {2 \left (3-5 t \right ) {\mathrm e}^{-t}}{25}-\frac {6 \cos \left (2 t \right )}{25}+\frac {8 \sin \left (2 t \right )}{25} \]

Problem 13214

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }+y=\cos \left (3 t \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {\left (3+\sqrt {5}\right ) t}{2}}+\frac {c_{2} {\mathrm e}^{\frac {\left (\sqrt {5}-3\right ) t}{2}} \sqrt {5}}{5}-\frac {8 \cos \left (3 t \right )}{145}+\frac {9 \sin \left (3 t \right )}{145} \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{\frac {\left (\sqrt {5}-3\right ) t}{2}} c_{2} +{\mathrm e}^{-\frac {\left (3+\sqrt {5}\right ) t}{2}} c_{1} -\frac {8 \cos \left (3 t \right )}{145}+\frac {9 \sin \left (3 t \right )}{145} \]

Problem 13215

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+20 y=3+2 \cos \left (2 t \right )} \]
program solution	\[ y = c_{1} \cos \left (4 t \right ) {\mathrm e}^{-2 t}+\frac {c_{2} \sin \left (4 t \right ) {\mathrm e}^{-2 t}}{4}+\frac {3}{20}+\frac {\cos \left (2 t \right )}{10}+\frac {\sin \left (2 t \right )}{20} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \sin \left (4 t \right ) {\mathrm e}^{-2 t} c_{2} +\cos \left (4 t \right ) {\mathrm e}^{-2 t} c_{1} +\frac {3}{20}+\frac {\sin \left (2 t \right )}{20}+\frac {\cos \left (2 t \right )}{10} \]

Problem 13216

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+20 y={\mathrm e}^{-t} \cos \left (t \right )} \]
program solution	\[ y = c_{1} \cos \left (4 t \right ) {\mathrm e}^{-2 t}+\frac {c_{2} \sin \left (4 t \right ) {\mathrm e}^{-2 t}}{4}+\frac {4 \,{\mathrm e}^{-t} \cos \left (t \right )}{65}+\frac {{\mathrm e}^{-t} \sin \left (t \right )}{130} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (c_{1} \cos \left (4 t \right )+c_{2} \sin \left (4 t \right )\right ) {\mathrm e}^{-2 t}+\frac {4 \left (\cos \left (t \right )+\frac {\sin \left (t \right )}{8}\right ) {\mathrm e}^{-t}}{65} \]

Problem 13217

ODE	\[ \boxed {y^{\prime \prime }+9 y=\cos \left (t \right )} \]
program solution	\[ y = c_{1} \cos \left (3 t \right )+\frac {c_{2} \sin \left (3 t \right )}{3}+\frac {\cos \left (t \right )}{8} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{2} \sin \left (3 t \right )+c_{1} \cos \left (3 t \right )+\frac {\cos \left (t \right )}{8} \]

Problem 13218

ODE	\[ \boxed {y^{\prime \prime }+9 y=5 \sin \left (2 t \right )} \]
program solution	\[ y = c_{1} \cos \left (3 t \right )+\frac {c_{2} \sin \left (3 t \right )}{3}+\sin \left (2 t \right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{2} \sin \left (3 t \right )+c_{1} \cos \left (3 t \right )+\sin \left (2 t \right ) \]

Problem 13219

ODE	\[ \boxed {y^{\prime \prime }+4 y=-\cos \left (\frac {t}{2}\right )} \]
program solution	\[ y = c_{1} \cos \left (2 t \right )+\frac {c_{2} \sin \left (2 t \right )}{2}-\frac {4 \cos \left (\frac {t}{2}\right )}{15} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \sin \left (2 t \right ) c_{2} +\cos \left (2 t \right ) c_{1} -\frac {4 \cos \left (\frac {t}{2}\right )}{15} \]

Problem 13220

ODE	\[ \boxed {y^{\prime \prime }+4 y=3 \cos \left (2 t \right )} \]
program solution	\[ y = c_{1} \cos \left (2 t \right )+\frac {c_{2} \sin \left (2 t \right )}{2}+\frac {3 \sin \left (2 t \right ) t}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (6 t +8 c_{2} \right ) \sin \left (2 t \right )}{8}+\frac {\left (8 c_{1} +3\right ) \cos \left (2 t \right )}{8} \]

Problem 13221

ODE	\[ \boxed {y^{\prime \prime }+9 y=2 \cos \left (3 t \right )} \]
program solution	\[ y = c_{1} \cos \left (3 t \right )+\frac {c_{2} \sin \left (3 t \right )}{3}+\frac {t \sin \left (3 t \right )}{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (9 c_{1} +1\right ) \cos \left (3 t \right )}{9}+\frac {\left (t +3 c_{2} \right ) \sin \left (3 t \right )}{3} \]

Problem 13222

ODE	\[ \boxed {y^{\prime \prime }+4 y=8} \] With initial conditions \begin {align*} [y \left (0\right ) = 11, y^{\prime }\left (0\right ) = 5] \end {align*}
program solution	\[ y = 2+9 \cos \left (2 t \right )+\frac {5 \sin \left (2 t \right )}{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = 9 \cos \left (2 t \right )+\frac {5 \sin \left (2 t \right )}{2}+2 \]

Problem 13223

ODE	\[ \boxed {y^{\prime \prime }-4 y={\mathrm e}^{2 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = -1] \end {align*}
program solution	\[ y = \frac {13 \,{\mathrm e}^{-2 t}}{16}+\frac {{\mathrm e}^{2 t} \left (3+4 t \right )}{16} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {13 \,{\mathrm e}^{-2 t}}{16}+\frac {{\mathrm e}^{2 t} \left (4 t +3\right )}{16} \]

Problem 13224

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+5 y=2 \,{\mathrm e}^{t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = {\mathrm e}^{t}+\left (2 \cos \left (t \right )-4 \sin \left (t \right )\right ) {\mathrm e}^{2 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{t}+\left (2 \cos \left (t \right )-4 \sin \left (t \right )\right ) {\mathrm e}^{2 t} \]

Problem 13225

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+13 y=13 \operatorname {Heaviside}\left (t -4\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = {\mathrm e}^{-3 t} \left (3 \cos \left (2 t \right )+5 \sin \left (2 t \right )\right )-\left (\left \{\begin {array}{cc} 0 & t <4 \\ -1+\left (\frac {1}{2}+\frac {3 i}{4}\right ) {\mathrm e}^{\left (-3-2 i\right ) \left (t -4\right )}+\left (\frac {1}{2}-\frac {3 i}{4}\right ) {\mathrm e}^{\left (-3+2 i\right ) \left (t -4\right )} & 4\le t \end {array}\right .\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (-\frac {1}{2}-\frac {3 i}{4}\right ) \operatorname {Heaviside}\left (t -4\right ) {\mathrm e}^{\left (-3-2 i\right ) \left (t -4\right )}+\left (-\frac {1}{2}+\frac {3 i}{4}\right ) \operatorname {Heaviside}\left (t -4\right ) {\mathrm e}^{\left (-3+2 i\right ) \left (t -4\right )}+\operatorname {Heaviside}\left (t -4\right )+{\mathrm e}^{-3 t} \left (3 \cos \left (2 t \right )+5 \sin \left (2 t \right )\right ) \]

Problem 13226

ODE	\[ \boxed {y^{\prime \prime }+4 y=\cos \left (2 t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = -2, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {\sin \left (2 t \right ) t}{4}-2 \cos \left (2 t \right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = -2 \cos \left (2 t \right )+\frac {t \sin \left (2 t \right )}{4} \]

Problem 13227

ODE	\[ \boxed {y^{\prime \prime }+3 y=\operatorname {Heaviside}\left (t -4\right ) \cos \left (5 t -20\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = -2] \end {align*}
program solution	\[ y = -\frac {2 \sin \left (\sqrt {3}\, t \right ) \sqrt {3}}{3}-\left (\left \{\begin {array}{cc} 0 & t <4 \\ \frac {\cos \left (5 t -20\right )}{22}-\frac {\cos \left (\sqrt {3}\, \left (t -4\right )\right )}{22} & 4\le t \end {array}\right .\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {2 \sqrt {3}\, \sin \left (\sqrt {3}\, t \right )}{3}-\frac {\operatorname {Heaviside}\left (t -4\right ) \cos \left (5 t -20\right )}{22}+\frac {\operatorname {Heaviside}\left (t -4\right ) \cos \left (\sqrt {3}\, \left (t -4\right )\right )}{22} \]

Problem 13228

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+9 y=20 \operatorname {Heaviside}\left (-2+t \right ) \sin \left (-2+t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 2] \end {align*}
program solution	\[ y = \frac {\left (\left \{\begin {array}{cc} {\mathrm e}^{-2 t} \left (4 \sqrt {5}\, \sin \left (t \sqrt {5}\right )+5 \cos \left (t \sqrt {5}\right )\right ) & t <2 \\ 4 \,{\mathrm e}^{-2 t} \sqrt {5}\, \sin \left (t \sqrt {5}\right )+5 \,{\mathrm e}^{-2 t} \cos \left (t \sqrt {5}\right )+5 \,{\mathrm e}^{4-2 t} \cos \left (\sqrt {5}\, \left (-2+t \right )\right )+10 \sin \left (-2+t \right )-5 \cos \left (-2+t \right ) & 2\le t \end {array}\right .\right )}{5} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \cos \left (\sqrt {5}\, \left (t -2\right )\right ) \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-2 t +4}+{\mathrm e}^{-2 t} \cos \left (t \sqrt {5}\right )+\frac {4 \,{\mathrm e}^{-2 t} \sqrt {5}\, \sin \left (t \sqrt {5}\right )}{5}-\operatorname {Heaviside}\left (t -2\right ) \left (\cos \left (t -2\right )-2 \sin \left (t -2\right )\right ) \]

Problem 13229

ODE	\[ \boxed {y^{\prime \prime }+3 y=\left \{\begin {array}{cc} t & 0\le t <1 \\ 1 & 1\le t \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = 2 \cos \left (\sqrt {3}\, t \right )-\frac {\sin \left (\sqrt {3}\, t \right ) \sqrt {3}}{9}+\frac {\left (\left \{\begin {array}{cc} t & t <1 \\ 1+\frac {\sin \left (\sqrt {3}\, \left (t -1\right )\right ) \sqrt {3}}{3} & 1\le t \end {array}\right .\right )}{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = 2 \cos \left (\sqrt {3}\, t \right )-\frac {\sqrt {3}\, \sin \left (\sqrt {3}\, t \right )}{9}+\frac {\left (\left \{\begin {array}{cc} t & t <1 \\ 1+\frac {\sqrt {3}\, \sin \left (\sqrt {3}\, \left (t -1\right )\right )}{3} & 1\le t \end {array}\right .\right )}{3} \]

Problem 13230

ODE	\[ \boxed {y^{\prime \prime }+3 y=5 \delta \left (-2+t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} 0 & t <2 \\ \frac {5 \sin \left (\sqrt {3}\, \left (-2+t \right )\right ) \sqrt {3}}{3} & 2\le t \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {5 \sqrt {3}\, \operatorname {Heaviside}\left (t -2\right ) \sin \left (\sqrt {3}\, \left (t -2\right )\right )}{3} \]

Problem 13231

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+5 y=\delta \left (t -3\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = {\mathrm e}^{-t} \left (\sin \left (2 t \right )+\cos \left (2 t \right )\right )+\left (\left \{\begin {array}{cc} 0 & t <3 \\ \frac {{\mathrm e}^{-t +3} \sin \left (2 t -6\right )}{2} & 3\le t \end {array}\right .\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{-t} \left (\cos \left (2 t \right )+\sin \left (2 t \right )\right )+\frac {{\mathrm e}^{-t +3} \operatorname {Heaviside}\left (t -3\right ) \sin \left (2 t -6\right )}{2} \]

Problem 13232

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+2 y=-2 \delta \left (-2+t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \left (2 \cos \left (t \right )+2 \sin \left (t \right )\right ) {\mathrm e}^{-t}-\left (\left \{\begin {array}{cc} 0 & t <2 \\ 2 \,{\mathrm e}^{2-t} \sin \left (-2+t \right ) & 2\le t \end {array}\right .\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = -2 \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{2-t} \sin \left (t -2\right )+2 \,{\mathrm e}^{-t} \left (\sin \left (t \right )+\cos \left (t \right )\right ) \]

Problem 13233

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+3 y=\delta \left (t -1\right )-3 \delta \left (t -4\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {\sqrt {2}\, \left (\left \{\begin {array}{cc} 0 & t <1 \\ {\mathrm e}^{1-t} \sin \left (\sqrt {2}\, \left (t -1\right )\right ) & t <4 \\ {\mathrm e}^{1-t} \sin \left (\sqrt {2}\, \left (t -1\right )\right )-3 \,{\mathrm e}^{4-t} \sin \left (\sqrt {2}\, \left (t -4\right )\right ) & 4\le t \end {array}\right .\right )}{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {3 \sqrt {2}\, \left (\operatorname {Heaviside}\left (t -4\right ) {\mathrm e}^{4-t} \sin \left (\sqrt {2}\, \left (t -4\right )\right )-\frac {\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-t +1} \sin \left (\sqrt {2}\, \left (t -1\right )\right )}{3}\right )}{2} \]

Problem 13234

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+2 y=\sin \left (4 t \right ) {\mathrm e}^{-2 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = -2] \end {align*}
program solution	\[ y = \frac {{\mathrm e}^{-2 t} \left (4 \cos \left (4 t \right )-7 \sin \left (4 t \right )\right )}{130}+\frac {128 \left (\cos \left (t \right )+\frac {\sin \left (t \right )}{8}\right ) {\mathrm e}^{-t}}{65} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {{\mathrm e}^{-2 t} \left (-7 \sin \left (4 t \right )+4 \cos \left (4 t \right )\right )}{130}+\frac {128 \left (\cos \left (t \right )+\frac {\sin \left (t \right )}{8}\right ) {\mathrm e}^{-t}}{65} \]

Problem 13235

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }+5 y=\operatorname {Heaviside}\left (-2+t \right ) \sin \left (-8+4 t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = -2, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} -\frac {2 \,{\mathrm e}^{-\frac {t}{2}} \left (\sqrt {19}\, \sin \left (\frac {\sqrt {19}\, t}{2}\right )+19 \cos \left (\frac {\sqrt {19}\, t}{2}\right )\right )}{19} & t <2 \\ -\frac {2 \sin \left (\frac {\sqrt {19}\, t}{2}\right ) \sqrt {19}\, {\mathrm e}^{-\frac {t}{2}}}{19}-2 \cos \left (\frac {\sqrt {19}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}-\frac {11 \sin \left (-8+4 t \right )}{137}-\frac {4 \cos \left (-8+4 t \right )}{137}+\frac {92 \sqrt {19}\, {\mathrm e}^{-\frac {t}{2}+1} \sin \left (\frac {\sqrt {19}\, \left (-2+t \right )}{2}\right )}{2603}+\frac {4 \,{\mathrm e}^{-\frac {t}{2}+1} \cos \left (\frac {\sqrt {19}\, \left (-2+t \right )}{2}\right )}{137} & 2\le t \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {4 \cos \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right ) \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{1-\frac {t}{2}}}{137}+\frac {92 \sin \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right ) \operatorname {Heaviside}\left (t -2\right ) \sqrt {19}\, {\mathrm e}^{1-\frac {t}{2}}}{2603}-2 \,{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {19}\, t}{2}\right )-\frac {2 \,{\mathrm e}^{-\frac {t}{2}} \sqrt {19}\, \sin \left (\frac {\sqrt {19}\, t}{2}\right )}{19}-\frac {4 \left (\cos \left (4 t -8\right )+\frac {11 \sin \left (4 t -8\right )}{4}\right ) \operatorname {Heaviside}\left (t -2\right )}{137} \]

Problem 13236

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }+8 y=\left (1-\operatorname {Heaviside}\left (t -4\right )\right ) \cos \left (t -4\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {\left (\left \{\begin {array}{cc} \left (7 \cos \left (t \right )+\sin \left (t \right )\right ) \cos \left (4\right )-\sin \left (4\right ) \left (\cos \left (t \right )-7 \sin \left (t \right )\right ) & t <4 \\ \frac {{\mathrm e}^{-\frac {t}{2}+2} \left (9 \sin \left (\frac {\sqrt {31}\, t}{2}\right ) \sqrt {31}\, \cos \left (2 \sqrt {31}\right )-9 \cos \left (\frac {\sqrt {31}\, t}{2}\right ) \sqrt {31}\, \sin \left (2 \sqrt {31}\right )+217 \cos \left (\frac {\sqrt {31}\, t}{2}\right ) \cos \left (2 \sqrt {31}\right )+217 \sin \left (\frac {\sqrt {31}\, t}{2}\right ) \sin \left (2 \sqrt {31}\right )\right )}{31} & 4\le t \end {array}\right .\right )}{50}+\frac {\left (-7 \cos \left (4\right )+\sin \left (4\right )\right ) {\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {31}\, t}{2}\right )}{50}+\frac {\left (-9 \cos \left (4\right )-13 \sin \left (4\right )\right ) \sqrt {31}\, {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {31}\, t}{2}\right )}{1550} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {9 \operatorname {Heaviside}\left (t -4\right ) \left (\left (\sin \left (2 \sqrt {31}\right ) \sqrt {31}-\frac {217 \cos \left (2 \sqrt {31}\right )}{9}\right ) \cos \left (\frac {\sqrt {31}\, t}{2}\right )-\frac {217 \sin \left (\frac {\sqrt {31}\, t}{2}\right ) \left (\frac {9 \sqrt {31}\, \cos \left (2 \sqrt {31}\right )}{217}+\sin \left (2 \sqrt {31}\right )\right )}{9}\right ) {\mathrm e}^{-\frac {t}{2}+2}}{1550}-\frac {7 \,{\mathrm e}^{-\frac {t}{2}} \left (\cos \left (4\right )-\frac {\sin \left (4\right )}{7}\right ) \cos \left (\frac {\sqrt {31}\, t}{2}\right )}{50}-\frac {9 \left (\cos \left (4\right )+\frac {13 \sin \left (4\right )}{9}\right ) \sqrt {31}\, {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {31}\, t}{2}\right )}{1550}-\frac {7 \left (\left (\cos \left (t \right )+\frac {\sin \left (t \right )}{7}\right ) \cos \left (4\right )-\frac {\sin \left (4\right ) \left (-7 \sin \left (t \right )+\cos \left (t \right )\right )}{7}\right ) \left (-1+\operatorname {Heaviside}\left (t -4\right )\right )}{50} \]

Problem 13237

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }+3 y=\left (1-\operatorname {Heaviside}\left (-2+t \right )\right ) {\mathrm e}^{\frac {1}{5}-\frac {t}{10}} \sin \left (-2+t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 2] \end {align*}
program solution	\[ y = \frac {50 \left (\left \{\begin {array}{cc} \left (-80+191 i\right ) {\mathrm e}^{\left (-\frac {1}{10}-i\right ) \left (-2+t \right )}+\left (-80-191 i\right ) {\mathrm e}^{\left (-\frac {1}{10}+i\right ) \left (-2+t \right )} & t <2 \\ \frac {4 \,{\mathrm e}^{-\frac {t}{2}+1} \left (159 \sin \left (\frac {\sqrt {11}\, t}{2}\right ) \sqrt {11}\, \cos \left (\sqrt {11}\right )-159 \cos \left (\frac {\sqrt {11}\, t}{2}\right ) \sqrt {11}\, \sin \left (\sqrt {11}\right )-440 \cos \left (\frac {\sqrt {11}\, t}{2}\right ) \cos \left (\sqrt {11}\right )-440 \sin \left (\frac {\sqrt {11}\, t}{2}\right ) \sin \left (\sqrt {11}\right )\right )}{11} & 2\le t \end {array}\right .\right )}{42881}+\frac {\left (\left (88000 \cos \left (2\right )+210100 \sin \left (2\right )\right ) \cos \left (\frac {\sqrt {11}\, t}{2}\right )+\left (-31800 \cos \left (2\right )+31280 \sin \left (2\right )\right ) \sqrt {11}\, \sin \left (\frac {\sqrt {11}\, t}{2}\right )\right ) {\mathrm e}^{\frac {1}{5}-\frac {t}{2}}}{471691}+\frac {5 \sqrt {11}\, \sin \left (\frac {\sqrt {11}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}}{11}+\cos \left (\frac {\sqrt {11}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {8000 \left (\left (\cos \left (t \right )-\frac {191 \sin \left (t \right )}{80}\right ) \cos \left (2\right )+\frac {191 \sin \left (2\right ) \left (\cos \left (t \right )+\frac {80 \sin \left (t \right )}{191}\right )}{80}\right ) \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-\frac {t}{10}+\frac {1}{5}}}{42881}+\frac {100 \left (11 \left (80 \cos \left (2\right )+191 \sin \left (2\right )\right ) \cos \left (\frac {\sqrt {11}\, t}{2}\right )-318 \left (\cos \left (2\right )-\frac {782 \sin \left (2\right )}{795}\right ) \sin \left (\frac {\sqrt {11}\, t}{2}\right ) \sqrt {11}\right ) {\mathrm e}^{\frac {1}{5}-\frac {t}{2}}}{471691}+\left (-\frac {4000}{42881}+\frac {9550 i}{42881}\right ) {\mathrm e}^{\left (-\frac {1}{10}-i\right ) \left (t -2\right )}+\left (-\frac {4000}{42881}-\frac {9550 i}{42881}\right ) {\mathrm e}^{\left (-\frac {1}{10}+i\right ) \left (t -2\right )}+\frac {200 \operatorname {Heaviside}\left (t -2\right ) \left (\left (-159 \sqrt {11}\, \sin \left (\sqrt {11}\right )-440 \cos \left (\sqrt {11}\right )\right ) \cos \left (\frac {\sqrt {11}\, t}{2}\right )+\left (159 \cos \left (\sqrt {11}\right ) \sqrt {11}-440 \sin \left (\sqrt {11}\right )\right ) \sin \left (\frac {\sqrt {11}\, t}{2}\right )\right ) {\mathrm e}^{1-\frac {t}{2}}}{471691}+\frac {5 \,{\mathrm e}^{-\frac {t}{2}} \sqrt {11}\, \sin \left (\frac {\sqrt {11}\, t}{2}\right )}{11}+{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {11}\, t}{2}\right ) \]

Problem 13238

ODE	\[ \boxed {y^{\prime \prime }+16 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \cos \left (4 t \right )+\frac {\sin \left (4 t \right )}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \cos \left (4 t \right )+\frac {\sin \left (4 t \right )}{4} \]

Problem 13239

ODE	\[ \boxed {y^{\prime \prime }+4 y=\sin \left (2 t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {\sin \left (2 t \right )}{8}-\frac {\cos \left (2 t \right ) t}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\sin \left (2 t \right )}{8}-\frac {t \cos \left (2 t \right )}{4} \]

Problem 13240

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 2] \end {align*}
program solution	\[ y = \left (3 t +1\right ) {\mathrm e}^{-t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (3 t +1\right ) {\mathrm e}^{-t} \]

Problem 13241

ODE	\[ \boxed {y^{\prime \prime }+16 y=t} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \frac {t}{16}+\cos \left (4 t \right )+\frac {15 \sin \left (4 t \right )}{64} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \cos \left (4 t \right )+\frac {15 \sin \left (4 t \right )}{64}+\frac {t}{16} \]

Problem 13242

ODE	\[ \boxed {y^{\prime }=3-\sin \left (x \right )} \]
program solution	\[ y = 3 x +\cos \left (x \right )+c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \cos \left (x \right )+3 x +c_{1} \]

Problem 13243

ODE	\[ \boxed {y^{\prime }+\sin \left (y\right )=3} \]
program solution	\[ y = 2 \arctan \left (\frac {\left (\sqrt {2}+4 \tan \left (\left (x +c_{1} \right ) \sqrt {2}\right )\right ) \sqrt {2}}{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2 \arctan \left (\frac {1}{3}+\frac {2 \sqrt {2}\, \tan \left (\left (c_{1} +x \right ) \sqrt {2}\right )}{3}\right ) \]

Problem 13244

ODE	\[ \boxed {y^{\prime }+4 y={\mathrm e}^{2 x}} \]
program solution	\[ y = \frac {\left ({\mathrm e}^{6 x}+6 c_{1} \right ) {\mathrm e}^{-4 x}}{6} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left ({\mathrm e}^{6 x}+6 c_{1} \right ) {\mathrm e}^{-4 x}}{6} \]

Problem 13245

ODE	\[ \boxed {y^{\prime } x=\arcsin \left (x^{2}\right )} \]
program solution	\[ y = -\frac {i \arcsin \left (x^{2}\right )^{2}}{4}+\frac {\arcsin \left (x^{2}\right ) \ln \left (1-i x^{2}-\sqrt {-x^{4}+1}\right )}{2}-\frac {i \operatorname {polylog}\left (2, i x^{2}+\sqrt {-x^{4}+1}\right )}{2}+\frac {\arcsin \left (x^{2}\right ) \ln \left (1+i x^{2}+\sqrt {-x^{4}+1}\right )}{2}-\frac {i \operatorname {polylog}\left (2, -i x^{2}-\sqrt {-x^{4}+1}\right )}{2}+c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {i \arcsin \left (x^{2}\right )^{2}}{4}+\frac {\arcsin \left (x^{2}\right ) \ln \left (i x^{2}+\sqrt {-x^{4}+1}+1\right )}{2}-\frac {i \operatorname {polylog}\left (2, \left (i x^{2}+\sqrt {-x^{4}+1}\right )^{2}\right )}{4}+\frac {\arcsin \left (x^{2}\right ) \ln \left (1-i x^{2}-\sqrt {-x^{4}+1}\right )}{2}+c_{1} \]

Problem 13246

ODE	\[ \boxed {y y^{\prime }=2 x} \]
program solution	\[ -\frac {x^{2}}{2}+\frac {y^{2}}{4} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \sqrt {2 x^{2}+c_{1}} \\ y \left (x \right ) &= -\sqrt {2 x^{2}+c_{1}} \\ \end{align*}

Problem 13247

ODE	\[ \boxed {y^{\prime \prime }=\frac {1+x}{x -1}} \]
program solution	\[ y = \frac {x^{2}}{2}+2 \ln \left (x -1\right ) \left (x -1\right )-2 x +2+c_{1} x +c_{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2+2 \ln \left (-1+x \right ) \left (-1+x \right )+\frac {x^{2}}{2}+\left (c_{1} -2\right ) x +c_{2} \]

Problem 13248

ODE	\[ \boxed {x^{2} y^{\prime \prime }=1} \]
program solution	\[ y = -\ln \left (x \right )+c_{1} x +c_{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\ln \left (x \right )+c_{1} x +c_{2} \]

Problem 13249

ODE	\[ \boxed {y^{2} y^{\prime \prime }=8 x^{2}} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 13250

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }+8 y={\mathrm e}^{-x^{2}}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {3 x}{2}} \cos \left (\frac {\sqrt {23}\, x}{2}\right )+\frac {2 c_{2} {\mathrm e}^{-\frac {3 x}{2}} \sqrt {23}\, \sin \left (\frac {\sqrt {23}\, x}{2}\right )}{23}+\frac {\sqrt {23}\, {\mathrm e}^{-\frac {3 x}{2}} \sqrt {\pi }\, \left (\left (i \cos \left (\frac {\sqrt {23}\, x}{2}\right )+\sin \left (\frac {\sqrt {23}\, x}{2}\right )\right ) {\mathrm e}^{\frac {\left (i \sqrt {23}+3\right )^{2}}{16}} \operatorname {erf}\left (x -\frac {3}{4}-\frac {i \sqrt {23}}{4}\right )+\left (-i \cos \left (\frac {\sqrt {23}\, x}{2}\right )+\sin \left (\frac {\sqrt {23}\, x}{2}\right )\right ) {\mathrm e}^{\frac {\left (-3+i \sqrt {23}\right )^{2}}{16}} \operatorname {erf}\left (x -\frac {3}{4}+\frac {i \sqrt {23}}{4}\right )+i \left (\operatorname {erf}\left (\frac {3}{4}+\frac {i \sqrt {23}}{4}\right ) {\mathrm e}^{\frac {\left (i \sqrt {23}+3\right )^{2}}{16}}-{\mathrm e}^{\frac {\left (-3+i \sqrt {23}\right )^{2}}{16}} \operatorname {erf}\left (\frac {3}{4}-\frac {i \sqrt {23}}{4}\right )\right ) \cos \left (\frac {\sqrt {23}\, x}{2}\right )+\sin \left (\frac {\sqrt {23}\, x}{2}\right ) \left (\operatorname {erf}\left (\frac {3}{4}+\frac {i \sqrt {23}}{4}\right ) {\mathrm e}^{\frac {\left (i \sqrt {23}+3\right )^{2}}{16}}+{\mathrm e}^{\frac {\left (-3+i \sqrt {23}\right )^{2}}{16}} \operatorname {erf}\left (\frac {3}{4}-\frac {i \sqrt {23}}{4}\right )\right )\right )}{46} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\operatorname {erf}\left (x -\frac {3}{4}+\frac {i \sqrt {23}}{4}\right ) \left (i \cos \left (\frac {\sqrt {23}\, x}{2}\right )-\sin \left (\frac {\sqrt {23}\, x}{2}\right )\right ) \sqrt {23}\, \sqrt {\pi }\, {\mathrm e}^{-\frac {3 x}{2}-\frac {7}{8}-\frac {3 i \sqrt {23}}{8}}}{46}+\frac {\sqrt {23}\, \operatorname {erf}\left (x -\frac {3}{4}-\frac {i \sqrt {23}}{4}\right ) \left (i \cos \left (\frac {\sqrt {23}\, x}{2}\right )+\sin \left (\frac {\sqrt {23}\, x}{2}\right )\right ) \sqrt {\pi }\, {\mathrm e}^{-\frac {3 x}{2}-\frac {7}{8}+\frac {3 i \sqrt {23}}{8}}}{46}+{\mathrm e}^{-\frac {3 x}{2}} \left (c_{1} \cos \left (\frac {\sqrt {23}\, x}{2}\right )+c_{2} \sin \left (\frac {\sqrt {23}\, x}{2}\right )\right ) \]

Problem 13251

ODE	\[ \boxed {x^{2} y^{\prime \prime }+3 y^{\prime } x=0} \]
program solution	\[ y = \frac {c_{1}}{x^{2}}+\frac {c_{2}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} +\frac {c_{2}}{x^{2}} \]

Problem 13252

ODE	\[ \boxed {y^{\prime }=4 x^{3}} \]
program solution	\[ y = x^{4}+c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{4}+c_{1} \]

Problem 13253

ODE	\[ \boxed {y^{\prime }=20 \,{\mathrm e}^{-4 x}} \]
program solution	\[ y = -5 \,{\mathrm e}^{-4 x}+c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -5 \,{\mathrm e}^{-4 x}+c_{1} \]

Problem 13254

ODE	\[ \boxed {y^{\prime } x=-\sqrt {x}+2} \]
program solution	\[ y = -2 \sqrt {x}+2 \ln \left (x \right )+c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2 \ln \left (x \right )-2 \sqrt {x}+c_{1} \]

Problem 13255

ODE	\[ \boxed {\sqrt {x +4}\, y^{\prime }=1} \]
program solution	\[ y = 2 \sqrt {x +4}+c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2 \sqrt {x +4}+c_{1} \]

Problem 13256

ODE	\[ \boxed {y^{\prime }=x \cos \left (x^{2}\right )} \]
program solution	\[ y = \frac {\sin \left (x^{2}\right )}{2}+c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\sin \left (x^{2}\right )}{2}+c_{1} \]

Problem 13257

ODE	\[ \boxed {y^{\prime }=\cos \left (x \right ) x} \]
program solution	\[ y = \cos \left (x \right )+\sin \left (x \right ) x +c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x \sin \left (x \right )+\cos \left (x \right )+c_{1} \]

Problem 13258

ODE	\[ \boxed {-\left (x^{2}-9\right ) y^{\prime }=-x} \]
program solution	\[ y = \frac {\ln \left (x^{2}-9\right )}{2}+c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\ln \left (x^{2}-9\right )}{2}+c_{1} \]

Problem 13259

ODE	\[ \boxed {-\left (x^{2}-9\right ) y^{\prime }=-1} \]
program solution	\[ y = -\frac {\operatorname {arctanh}\left (\frac {x}{3}\right )}{3}+c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\ln \left (-3+x \right )}{6}-\frac {\ln \left (x +3\right )}{6}+c_{1} \]

Problem 13260

ODE	\[ \boxed {9 y^{\prime }=x^{2}-1} \]
program solution	\[ y = \frac {x \left (x^{2}-3\right )}{27}+c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {1}{27} x^{3}-\frac {1}{9} x +c_{1} \]

Problem 13261

ODE	\[ \boxed {y^{\prime \prime }=\sin \left (2 x \right )} \]
program solution	\[ y = -\frac {\sin \left (2 x \right )}{4}+c_{1} x +c_{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\sin \left (2 x \right )}{4}+c_{1} x +c_{2} \]

Problem 13262

ODE	\[ \boxed {y^{\prime \prime }=3+x} \]
program solution	\[ y = \frac {x \left (x^{2}+6 c_{1} +9 x \right )}{6}+c_{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {1}{6} x^{3}+\frac {3}{2} x^{2}+c_{1} x +c_{2} \]

Problem 13263

ODE	\[ \boxed {y^{\prime \prime \prime \prime }=1} \]
program solution	\[ y = c_{4} x^{3}+c_{3} x^{2}+c_{2} x +c_{1} +\frac {1}{24} x^{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {1}{24} x^{4}+\frac {1}{6} c_{1} x^{3}+\frac {1}{2} c_{2} x^{2}+x c_{3} +c_{4} \]

Problem 13264

ODE	\[ \boxed {y^{\prime }=40 x \,{\mathrm e}^{2 x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 4] \end {align*}
program solution	\[ y = 20 x \,{\mathrm e}^{2 x}-10 \,{\mathrm e}^{2 x}+14 \] Verified OK.
Maple solution	\[ y \left (x \right ) = 14+\left (20 x -10\right ) {\mathrm e}^{2 x} \]

Problem 13265

ODE	\[ \boxed {\left (x +6\right )^{\frac {1}{3}} y^{\prime }=1} \] With initial conditions \begin {align*} [y \left (2\right ) = 10] \end {align*}
program solution	\[ y = \frac {3 \left (x +6\right )^{\frac {2}{3}}}{2}+4 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {3 \left (x +6\right )^{\frac {2}{3}}}{2}+4 \]

Problem 13266

ODE	\[ \boxed {y^{\prime }=\frac {x -1}{1+x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 8] \end {align*}
program solution	\[ y = x -2 \ln \left (1+x \right )+8 \] Verified OK.
Maple solution	\[ y \left (x \right ) = x -2 \ln \left (1+x \right )+8 \]

Problem 13267

ODE	\[ \boxed {y^{\prime } x=\sqrt {x}-2} \] With initial conditions \begin {align*} [y \left (1\right ) = 6] \end {align*}
program solution	\[ y = 2 \sqrt {x}-2 \ln \left (x \right )+4 \] Verified OK.
Maple solution	\[ y \left (x \right ) = -2 \ln \left (x \right )+2 \sqrt {x}+4 \]

Problem 13268

ODE	\[ \boxed {\cos \left (x \right ) y^{\prime }=\sin \left (x \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 3] \end {align*}
program solution	\[ y = -\ln \left (\cos \left (x \right )\right )+3 \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\ln \left (\cos \left (x \right )\right )+3 \]

Problem 13269

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime }=1} \] With initial conditions \begin {align*} [y \left (0\right ) = 3] \end {align*}
program solution	\[ y = \arctan \left (x \right )+3 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \arctan \left (x \right )+3 \]

Problem 13270

ODE	\[ \boxed {y^{\prime \prime } x=\sqrt {x}-2} \] With initial conditions \begin {align*} [y \left (1\right ) = 8, y^{\prime }\left (1\right ) = 6] \end {align*}
program solution	\[ y = \frac {4 x^{\frac {3}{2}}}{3}-2 x \ln \left (x \right )+6 x +\frac {2}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {4 x^{\frac {3}{2}}}{3}-2 \ln \left (x \right ) x +6 x +\frac {2}{3} \]

Problem 13271

ODE	\[ \boxed {y^{\prime }=\sin \left (\frac {x}{2}\right )} \]
program solution	\[ y = -2 \cos \left (\frac {x}{2}\right )+c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -2 \cos \left (\frac {x}{2}\right )+c_{1} \]

Problem 13272

ODE	\[ \boxed {y^{\prime }=\sin \left (\frac {x}{2}\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ y = -2 \cos \left (\frac {x}{2}\right )+2 \] Verified OK.
Maple solution	\[ y \left (x \right ) = -2 \cos \left (\frac {x}{2}\right )+2 \]

Problem 13273

ODE	\[ \boxed {y^{\prime }=\sin \left (\frac {x}{2}\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 3] \end {align*}
program solution	\[ y = -2 \cos \left (\frac {x}{2}\right )+5 \] Verified OK.
Maple solution	\[ y \left (x \right ) = -2 \cos \left (\frac {x}{2}\right )+5 \]

Problem 13274

ODE	\[ \boxed {y^{\prime }=3 \sqrt {3+x}} \]
program solution	\[ y = 2 \left (3+x \right )^{\frac {3}{2}}+c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (2 x +6\right ) \sqrt {x +3}+c_{1} \]

Problem 13275

ODE	\[ \boxed {y^{\prime }=3 \sqrt {3+x}} \] With initial conditions \begin {align*} [y \left (1\right ) = 16] \end {align*}
program solution	\[ y = 2 \left (3+x \right )^{\frac {3}{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2 \left (x +3\right )^{\frac {3}{2}} \]

Problem 13276

ODE	\[ \boxed {y^{\prime }=3 \sqrt {3+x}} \] With initial conditions \begin {align*} [y \left (1\right ) = 20] \end {align*}
program solution	\[ y = 2 \left (3+x \right )^{\frac {3}{2}}+4 \] Verified OK.
Maple solution	\[ y \left (x \right ) = 4+\left (2 x +6\right ) \sqrt {x +3} \]

Problem 13277

ODE	\[ \boxed {y^{\prime }=3 \sqrt {3+x}} \] With initial conditions \begin {align*} [y \left (1\right ) = 0] \end {align*}
program solution	\[ y = 2 \left (3+x \right )^{\frac {3}{2}}-16 \] Verified OK.
Maple solution	\[ y \left (x \right ) = -16+\left (2 x +6\right ) \sqrt {x +3} \]

Problem 13278

ODE	\[ \boxed {y^{\prime }=x \,{\mathrm e}^{-x^{2}}} \] With initial conditions \begin {align*} [y \left (0\right ) = 3] \end {align*}
program solution	\[ y = -\frac {{\mathrm e}^{-x^{2}}}{2}+\frac {7}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {{\mathrm e}^{-x^{2}}}{2}+\frac {7}{2} \]

Problem 13279

ODE	\[ \boxed {y^{\prime }=\frac {x}{\sqrt {x^{2}+5}}} \] With initial conditions \begin {align*} [y \left (2\right ) = 7] \end {align*}
program solution	\[ y = \sqrt {x^{2}+5}+4 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x^{2}+5}+4 \]

Problem 13280

ODE	\[ \boxed {y^{\prime }=\frac {1}{x^{2}+1}} \] With initial conditions \begin {align*} [y \left (1\right ) = 0] \end {align*}
program solution	\[ y = \arctan \left (x \right )-\frac {\pi }{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \arctan \left (x \right )-\frac {\pi }{4} \]

Problem 13281

ODE	\[ \boxed {y^{\prime }={\mathrm e}^{-9 x^{2}}} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
program solution	\[ y = \frac {\sqrt {\pi }\, \operatorname {erf}\left (3 x \right )}{6}+1 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\sqrt {\pi }\, \operatorname {erf}\left (3 x \right )}{6}+1 \]

Problem 13282

ODE	\[ \boxed {y^{\prime } x=\sin \left (x \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 4] \end {align*}
program solution	\[ y = \operatorname {Si}\left (x \right )+4 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \operatorname {Si}\left (x \right )+4 \]

Problem 13283

ODE	\[ \boxed {y^{\prime } x=\sin \left (x^{2}\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {\operatorname {Si}\left (x^{2}\right )}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\operatorname {Si}\left (x^{2}\right )}{2} \]

Problem 13284

ODE	\[ \boxed {y^{\prime }=\left \{\begin {array}{cc} 0 & x <0 \\ 1 & 0\le x \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} 0 & x \le 0 \\ x & 0
Maple solution	\[ y \left (x \right ) = \left \{\begin {array}{cc} 0 & x <0 \\ x & 0\le x \end {array}\right . \]

Problem 13285

ODE	\[ \boxed {y^{\prime }=\left \{\begin {array}{cc} 0 & x <1 \\ 1 & 1\le x \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 2] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} 2 & x \le 1 \\ 1+x & 1
Maple solution	\[ y \left (x \right ) = \left \{\begin {array}{cc} 2 & x <1 \\ 1+x & 1\le x \end {array}\right . \]

Problem 13286

ODE	\[ \boxed {y^{\prime }=\left \{\begin {array}{cc} 0 & x <1 \\ 1 & 1\le x <2 \\ 0 & 2\le x \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} 0 & x \le 1 \\ x -1 & x \le 2 \\ 1 & 2
Maple solution	\[ y \left (x \right ) = \left \{\begin {array}{cc} 0 & x <1 \\ -1+x & x <2 \\ 1 & 2\le x \end {array}\right . \]

Problem 13287

ODE	\[ \boxed {y^{\prime }+3 y x=6 x} \]
program solution	\[ y = {\mathrm e}^{-\frac {3 x^{2}}{2}-3 c_{1}}+2 \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2+{\mathrm e}^{-\frac {3 x^{2}}{2}} c_{1} \]

Problem 13288

ODE	\[ \boxed {\sin \left (x +y\right )-y y^{\prime }=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 13289

ODE	\[ \boxed {y^{\prime }-y^{3}=8} \]
program solution	\[ \int _{}^{y}\frac {1}{\textit {\_a}^{3}+8}d \textit {\_a} = x +c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {3}\, \tan \left (\operatorname {RootOf}\left (-\sqrt {3}\, \ln \left (\cos \left (\textit {\_Z} \right )^{2}\right )-2 \sqrt {3}\, \ln \left (\sqrt {3}+\tan \left (\textit {\_Z} \right )\right )+24 \sqrt {3}\, c_{1} +24 \sqrt {3}\, x -6 \textit {\_Z} \right )\right )+1 \]

Problem 13290

ODE	\[ \boxed {x^{2} y^{\prime }+x y^{2}=x} \]
program solution	\[ y = \frac {c_{3} x^{2}-1}{c_{3} x^{2}+1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \tanh \left (\ln \left (x \right )+c_{1} \right ) \]

Problem 13291

ODE	\[ \boxed {y^{\prime }-y^{2}=x} \]
program solution	\[ y = \frac {c_{3} \operatorname {AiryAi}\left (1, -x \right )+\operatorname {AiryBi}\left (1, -x \right )}{c_{3} \operatorname {AiryAi}\left (-x \right )+\operatorname {AiryBi}\left (-x \right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \operatorname {AiryAi}\left (1, -x \right )+\operatorname {AiryBi}\left (1, -x \right )}{c_{1} \operatorname {AiryAi}\left (-x \right )+\operatorname {AiryBi}\left (-x \right )} \]

Problem 13292

ODE	\[ \boxed {y^{3}-25 y+y^{\prime }=0} \]
program solution	\[ y = -\frac {5 \sqrt {\left ({\mathrm e}^{50 x} c_{1}^{50}-1\right ) {\mathrm e}^{50 x} c_{1}^{50}}}{{\mathrm e}^{50 x} c_{1}^{50}-1} \] Verified OK. \[ y = \frac {5 \sqrt {\left ({\mathrm e}^{50 x} c_{1}^{50}-1\right ) {\mathrm e}^{50 x} c_{1}^{50}}}{{\mathrm e}^{50 x} c_{1}^{50}-1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {5}{\sqrt {25 \,{\mathrm e}^{-50 x} c_{1} +1}} \\ y \left (x \right ) &= \frac {5}{\sqrt {25 \,{\mathrm e}^{-50 x} c_{1} +1}} \\ \end{align*}

Problem 13293

ODE	\[ \boxed {\left (x -2\right ) y^{\prime }-y=3} \]
program solution	\[ y = {\mathrm e}^{c_{1}} x -2 \,{\mathrm e}^{c_{1}}-3 \] Verified OK.
Maple solution	\[ y \left (x \right ) = -3+c_{1} \left (x -2\right ) \]

Problem 13294

ODE	\[ \boxed {\left (y-2\right ) y^{\prime }=x -3} \]
program solution	\[ \frac {y^{2}}{2}-\frac {x^{2}}{2}-2 y+3 x = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 2-\sqrt {x^{2}+2 c_{1} -6 x +4} \\ y \left (x \right ) &= 2+\sqrt {x^{2}+2 c_{1} -6 x +4} \\ \end{align*}

Problem 13295

ODE	\[ \boxed {y^{\prime }+2 y-y^{2}=-2} \]
program solution	\[ y = -\frac {\left (-\sqrt {3}+3 \tanh \left (\left (x +c_{1} \right ) \sqrt {3}\right )\right ) \sqrt {3}}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1-\sqrt {3}\, \tanh \left (\left (c_{1} +x \right ) \sqrt {3}\right ) \]

Problem 13296

ODE	\[ \boxed {y^{\prime }+\left (8-x \right ) y-y^{2}=-8 x} \]
program solution	\[ y = -\frac {i {\mathrm e}^{\frac {\left (x +8\right )^{2}}{2}} \sqrt {2}-8 \sqrt {\pi }\, \left (c_{3} +\operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +8\right )}{2}\right )\right )}{\sqrt {\pi }\, \left (c_{3} +\operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +8\right )}{2}\right )\right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {8 i \sqrt {\pi }\, {\mathrm e}^{-32} \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +8\right )}{2}\right )+2 \,{\mathrm e}^{\frac {x \left (x +16\right )}{2}}+16 c_{1}}{i \sqrt {\pi }\, {\mathrm e}^{-32} \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +8\right )}{2}\right )+2 c_{1}} \]

Problem 13297

ODE	\[ \boxed {y^{\prime }-2 \sqrt {y}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 0] \end {align*}
program solution	\[ y = 0 \] Verified OK.
Maple solution	\[ y \left (x \right ) = 0 \]

Problem 13298

ODE	\[ \boxed {y^{\prime }-3 y^{2}+\sin \left (x \right ) y^{2}=0} \]
program solution	\[ y = \frac {1}{c_{3} -3 x -\cos \left (x \right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {1}{\cos \left (x \right )-c_{1} +3 x} \]

Problem 13299

ODE	\[ \boxed {y \sin \left (x \right )+y^{\prime }=3 x} \]
program solution	\[ \int _{}^{x}\left (-3 \textit {\_a} +y \sin \left (\textit {\_a} \right )\right ) {\mathrm e}^{-\cos \left (\textit {\_a} \right )}d \textit {\_a} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (3 \left (\int x \,{\mathrm e}^{-\cos \left (x \right )}d x \right )+c_{1} \right ) {\mathrm e}^{\cos \left (x \right )} \]

Problem 13300

ODE	\[ \boxed {y^{\prime } x -\left (x -y\right )^{2}=0} \]
program solution	\[ y = \frac {\left (\sqrt {x}\, \operatorname {BesselI}\left (0, 2 \sqrt {x}\right ) c_{3} +\sqrt {x}\, \operatorname {BesselK}\left (0, 2 \sqrt {x}\right )-\operatorname {BesselI}\left (1, 2 \sqrt {x}\right ) c_{3} +\operatorname {BesselK}\left (1, 2 \sqrt {x}\right )\right ) \sqrt {x}}{\operatorname {BesselI}\left (0, 2 \sqrt {x}\right ) c_{3} +\operatorname {BesselK}\left (0, 2 \sqrt {x}\right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\sqrt {x}\, \left (\left (\operatorname {BesselK}\left (0, 2 \sqrt {x}\right ) c_{1} +\operatorname {BesselI}\left (0, 2 \sqrt {x}\right )\right ) \sqrt {x}+\operatorname {BesselK}\left (1, 2 \sqrt {x}\right ) c_{1} -\operatorname {BesselI}\left (1, 2 \sqrt {x}\right )\right )}{\operatorname {BesselK}\left (0, 2 \sqrt {x}\right ) c_{1} +\operatorname {BesselI}\left (0, 2 \sqrt {x}\right )} \]