2.17.3 Problems 201 to 300

Problem 201

ODE	\[ \boxed {4 y^{\prime \prime }-12 y^{\prime }+9 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{\frac {3 x}{2}}+c_{2} x \,{\mathrm e}^{\frac {3 x}{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{\frac {3 x}{2}} \left (c_{2} x +c_{1} \right ) \]

Problem 202

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+13 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{3 x} \cos \left (2 x \right )+\frac {c_{2} {\mathrm e}^{3 x} \sin \left (2 x \right )}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{3 x} \left (\sin \left (2 x \right ) c_{1} +c_{2} \cos \left (2 x \right )\right ) \]

Problem 203

ODE	\[ \boxed {y^{\prime \prime }+8 y^{\prime }+25 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-4 x} \cos \left (3 x \right )+\frac {c_{2} {\mathrm e}^{-4 x} \sin \left (3 x \right )}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-4 x} \left (c_{1} \sin \left (3 x \right )+c_{2} \cos \left (3 x \right )\right ) \]

Problem 204

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+3 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 7, y^{\prime }\left (0\right ) = 11] \end {align*}
program solution	\[ y = 2 \,{\mathrm e}^{3 x}+5 \,{\mathrm e}^{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 5 \,{\mathrm e}^{x}+2 \,{\mathrm e}^{3 x} \]

Problem 205

ODE	\[ \boxed {9 y^{\prime \prime }+6 y^{\prime }+4 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = 4] \end {align*}
program solution	\[ y = \left (5 \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, x}{3}\right )+3 \cos \left (\frac {\sqrt {3}\, x}{3}\right )\right ) {\mathrm e}^{-\frac {x}{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\frac {x}{3}} \left (5 \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, x}{3}\right )+3 \cos \left (\frac {\sqrt {3}\, x}{3}\right )\right ) \]

Problem 206

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+25 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 4, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \frac {{\mathrm e}^{3 x} \left (16 \cos \left (4 x \right )-11 \sin \left (4 x \right )\right )}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {{\mathrm e}^{3 x} \left (11 \sin \left (4 x \right )-16 \cos \left (4 x \right )\right )}{4} \]

Problem 207

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

Problem 208

ODE	\[ \boxed {y^{\prime \prime }-i y^{\prime }+6 y=0} \]
program solution	\[ y = c_{1} \cos \left (\frac {5 x}{2}\right ) {\mathrm e}^{\frac {i x}{2}}+\frac {2 c_{2} \sin \left (\frac {5 x}{2}\right ) {\mathrm e}^{\frac {i x}{2}}}{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{3 i x}+c_{2} {\mathrm e}^{-2 i x} \]

Problem 209

ODE	\[ \boxed {y^{\prime \prime }-\left (-2+2 i \sqrt {3}\right ) y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{\sqrt {2}\, \sqrt {i \sqrt {3}-1}\, x}+\frac {c_{2} \left (i \sqrt {3}-1\right ) {\mathrm e}^{-\left (1+i \sqrt {3}\right ) x}}{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\left (1+i \sqrt {3}\right ) x}+c_{2} {\mathrm e}^{\left (1+i \sqrt {3}\right ) x} \]

Problem 210

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +9 y=0} \]
program solution	\[ y = x^{-3 i} c_{1} -\frac {i c_{2} x^{3 i}}{6} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sin \left (3 \ln \left (x \right )\right )+c_{2} \cos \left (3 \ln \left (x \right )\right ) \]

Problem 211

ODE	\[ \boxed {x^{2} y^{\prime \prime }+7 y^{\prime } x +25 y=0} \]
program solution	\[ y = c_{1} x^{-3-4 i}-\frac {i c_{2} x^{-3+4 i}}{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \sin \left (4 \ln \left (x \right )\right )+c_{2} \cos \left (4 \ln \left (x \right )\right )}{x^{3}} \]

Problem 212

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

Problem 213

ODE	\[ \boxed {3 x^{\prime \prime }+30 x^{\prime }+63 x=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 2, x^{\prime }\left (0\right ) = 2] \end {align*}
program solution	\[ x = 4 \,{\mathrm e}^{-3 t}-2 \,{\mathrm e}^{-7 t} \] Verified OK.
Maple solution	\[ x \left (t \right ) = 4 \,{\mathrm e}^{-3 t}-2 \,{\mathrm e}^{-7 t} \]

Problem 214

ODE	\[ \boxed {x^{\prime \prime }+8 x^{\prime }+16 x=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 5, x^{\prime }\left (0\right ) = -10] \end {align*}
program solution	\[ x = \left (5+10 t \right ) {\mathrm e}^{-4 t} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \left (5+10 t \right ) {\mathrm e}^{-4 t} \]

Problem 215

ODE	\[ \boxed {2 x^{\prime \prime }+12 x^{\prime }+50 x=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = -8] \end {align*}
program solution	\[ x = -2 \,{\mathrm e}^{-3 t} \sin \left (4 t \right ) \] Verified OK.
Maple solution	\[ x \left (t \right ) = -2 \,{\mathrm e}^{-3 t} \sin \left (4 t \right ) \]

Problem 216

ODE	\[ \boxed {4 x^{\prime \prime }+20 x^{\prime }+169 x=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 4, x^{\prime }\left (0\right ) = 16] \end {align*}
program solution	\[ x = \frac {{\mathrm e}^{-\frac {5 t}{2}} \left (12 \cos \left (6 t \right )+13 \sin \left (6 t \right )\right )}{3} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {{\mathrm e}^{-\frac {5 t}{2}} \left (13 \sin \left (6 t \right )+12 \cos \left (6 t \right )\right )}{3} \]

Problem 217

ODE	\[ \boxed {2 x^{\prime \prime }+16 x^{\prime }+40 x=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 5, x^{\prime }\left (0\right ) = 4] \end {align*}
program solution	\[ x = {\mathrm e}^{-4 t} \left (5 \cos \left (2 t \right )+12 \sin \left (2 t \right )\right ) \] Verified OK.
Maple solution	\[ x \left (t \right ) = {\mathrm e}^{-4 t} \left (12 \sin \left (2 t \right )+5 \cos \left (2 t \right )\right ) \]

Problem 218

ODE	\[ \boxed {x^{\prime \prime }+10 x^{\prime }+125 x=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 6, x^{\prime }\left (0\right ) = 50] \end {align*}
program solution	\[ x = 2 \,{\mathrm e}^{-5 t} \left (3 \cos \left (10 t \right )+4 \sin \left (10 t \right )\right ) \] Verified OK.
Maple solution	\[ x \left (t \right ) = 2 \,{\mathrm e}^{-5 t} \left (4 \sin \left (10 t \right )+3 \cos \left (10 t \right )\right ) \]

Problem 219

ODE	\[ \boxed {y^{\prime \prime }+16 y={\mathrm e}^{3 x}} \]
program solution	\[ y = c_{1} \cos \left (4 x \right )+\frac {c_{2} \sin \left (4 x \right )}{4}+\frac {{\mathrm e}^{3 x}}{25} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sin \left (4 x \right ) c_{2} +\cos \left (4 x \right ) c_{1} +\frac {{\mathrm e}^{3 x}}{25} \]

Problem 220

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }-2 y=3 x +4} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+\frac {c_{2} {\mathrm e}^{2 x}}{3}-\frac {3 x}{2}-\frac {5}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{2} {\mathrm e}^{-x}+{\mathrm e}^{2 x} c_{1} -\frac {3 x}{2}-\frac {5}{4} \]

Problem 221

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }-6 y=2 \sin \left (3 x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+\frac {c_{2} {\mathrm e}^{3 x}}{5}+\frac {\cos \left (3 x \right )}{39}-\frac {5 \sin \left (3 x \right )}{39} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-2 x} \left (\frac {\left (\cos \left (3 x \right )-5 \sin \left (3 x \right )\right ) {\mathrm e}^{2 x}}{39}+c_{2} {\mathrm e}^{5 x}+c_{1} \right ) \]

Problem 222

ODE	\[ \boxed {4 y^{\prime \prime }+4 y^{\prime }+y=3 x \,{\mathrm e}^{x}} \]
program solution	\[ y = {\mathrm e}^{-\frac {x}{2}} \left (c_{2} x +c_{1} \right )+\frac {x \,{\mathrm e}^{x}}{3}-\frac {4 \,{\mathrm e}^{x}}{9} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} x +c_{2} \right ) {\mathrm e}^{-\frac {x}{2}}+\frac {\left (3 x -4\right ) {\mathrm e}^{x}}{9} \]

Problem 223

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }+y=\sin \left (x \right )^{2}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )+\frac {2 c_{2} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) {\mathrm e}^{-\frac {x}{2}} \sqrt {3}}{3}+\frac {1}{2}+\frac {3 \cos \left (2 x \right )}{26}-\frac {\sin \left (2 x \right )}{13} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) c_{2} +{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right ) c_{1} -\frac {\sin \left (2 x \right )}{13}+\frac {3 \cos \left (2 x \right )}{26}+\frac {1}{2} \]

Problem 224

ODE	\[ \boxed {2 y^{\prime \prime }+4 y^{\prime }+7 y=x^{2}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x} \cos \left (\frac {\sqrt {10}\, x}{2}\right )+\frac {c_{2} \sin \left (\frac {\sqrt {10}\, x}{2}\right ) {\mathrm e}^{-x} \sqrt {10}}{5}+\frac {x^{2}}{7}-\frac {8 x}{49}+\frac {4}{343} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-x} \sin \left (\frac {\sqrt {10}\, x}{2}\right ) c_{2} +{\mathrm e}^{-x} \cos \left (\frac {\sqrt {10}\, x}{2}\right ) c_{1} +\frac {x^{2}}{7}-\frac {8 x}{49}+\frac {4}{343} \]

Problem 225

ODE	\[ \boxed {y^{\prime \prime }-4 y=\sinh \left (x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+\frac {c_{2} {\mathrm e}^{2 x}}{4}+\frac {\left (-2 \sinh \left (x \right )^{3}-2 \sinh \left (x \right )^{2} \cosh \left (x \right )+\cosh \left (x \right )\right ) {\mathrm e}^{-2 x}}{12}+\frac {\left (-\sinh \left (x \right )^{3}+\sinh \left (x \right )^{2} \cosh \left (x \right )-\frac {\cosh \left (x \right )}{2}\right ) {\mathrm e}^{2 x}}{6} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-2 \sinh \left (x \right )^{2} \cosh \left (x \right )-2 \sinh \left (x \right )^{3}+12 c_{1} +\cosh \left (x \right )\right ) {\mathrm e}^{-2 x}}{12}+{\mathrm e}^{2 x} \left (\frac {\sinh \left (x \right )^{2} \cosh \left (x \right )}{6}-\frac {\sinh \left (x \right )^{3}}{6}+c_{2} -\frac {\cosh \left (x \right )}{12}\right ) \]

Problem 226

ODE	\[ \boxed {y^{\prime \prime }-4 y=\cosh \left (2 x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+\frac {c_{2} {\mathrm e}^{2 x}}{4}+\frac {\left (-4 x -1-\cosh \left (4 x \right )-\sinh \left (4 x \right )\right ) {\mathrm e}^{-2 x}}{32}+\frac {\left (x +\frac {\sinh \left (4 x \right )}{4}-\frac {\cosh \left (4 x \right )}{4}\right ) {\mathrm e}^{2 x}}{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-4 x +32 c_{1} -2\right ) {\mathrm e}^{-2 x}}{32}+\frac {{\mathrm e}^{2 x} \left (x +8 c_{2} -\frac {1}{4}\right )}{8} \]

Problem 227

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }-3 y=1+x \,{\mathrm e}^{x}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-3 x}+\frac {c_{2} {\mathrm e}^{x}}{4}-\frac {1}{3}+\frac {\left (8 x^{2}-4 x +1\right ) {\mathrm e}^{x}}{64} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{-3 x} \left (\left (x^{2}-\frac {1}{2} x +8 c_{2} +\frac {1}{8}\right ) {\mathrm e}^{4 x}+8 c_{1} -\frac {8 \,{\mathrm e}^{3 x}}{3}\right )}{8} \]

Problem 228

ODE	\[ \boxed {y^{\prime \prime }+9 y=2 \cos \left (3 x \right )+3 \sin \left (3 x \right )} \]
program solution	\[ y = c_{1} \cos \left (3 x \right )+\frac {c_{2} \sin \left (3 x \right )}{3}-\frac {x \cos \left (3 x \right )}{2}+\frac {x \sin \left (3 x \right )}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (18 c_{1} -9 x +2\right ) \cos \left (3 x \right )}{18}+\frac {\sin \left (3 x \right ) \left (x +3 c_{2} \right )}{3} \]

Problem 229

ODE	\[ \boxed {y^{\prime \prime }+9 y=2 x^{2} {\mathrm e}^{3 x}+5} \]
program solution	\[ y = c_{1} \cos \left (3 x \right )+\frac {c_{2} \sin \left (3 x \right )}{3}+\frac {5}{9}-\frac {2 x \,{\mathrm e}^{3 x}}{27}+\frac {x^{2} {\mathrm e}^{3 x}}{9}+\frac {{\mathrm e}^{3 x}}{81} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sin \left (3 x \right ) c_{2} +\cos \left (3 x \right ) c_{1} +\frac {5}{9}+\frac {\left (x -\frac {1}{3}\right )^{2} {\mathrm e}^{3 x}}{9} \]

Problem 230

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

Problem 231

ODE	\[ \boxed {y^{\prime \prime }+4 y=3 x \cos \left (2 x \right )} \]
program solution	\[ y = c_{1} \cos \left (2 x \right )+\frac {c_{2} \sin \left (2 x \right )}{2}+\frac {3 x \cos \left (2 x \right )}{16}+\frac {3 x^{2} \sin \left (2 x \right )}{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (24 x^{2}+64 c_{2} -3\right ) \sin \left (2 x \right )}{64}+\frac {3 \left (x +\frac {16 c_{1}}{3}\right ) \cos \left (2 x \right )}{16} \]

Problem 232

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

Problem 233

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+13 y=x \,{\mathrm e}^{3 x} \sin \left (2 x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{3 x} \cos \left (2 x \right )+\frac {c_{2} {\mathrm e}^{3 x} \sin \left (2 x \right )}{2}+\frac {x \,{\mathrm e}^{3 x} \sin \left (2 x \right )}{16}-\frac {x^{2} \cos \left (2 x \right ) {\mathrm e}^{3 x}}{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\left (\left (x^{2}-8 c_{1} \right ) \cos \left (2 x \right )-\frac {\sin \left (2 x \right ) \left (x +16 c_{2} \right )}{2}\right ) {\mathrm e}^{3 x}}{8} \]

Problem 234

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

Problem 235

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }+2 y={\mathrm e}^{x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 3] \end {align*}
program solution	\[ y = \frac {\left ({\mathrm e}^{3 x}+15 \,{\mathrm e}^{x}-16\right ) {\mathrm e}^{-2 x}}{6} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left ({\mathrm e}^{3 x}+15 \,{\mathrm e}^{x}-16\right ) {\mathrm e}^{-2 x}}{6} \]

Problem 236

ODE	\[ \boxed {y^{\prime \prime }+9 y=\sin \left (2 x \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \cos \left (3 x \right )-\frac {2 \sin \left (3 x \right )}{15}+\frac {\sin \left (2 x \right )}{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {2 \sin \left (3 x \right )}{15}+\cos \left (3 x \right )+\frac {\sin \left (2 x \right )}{5} \]

Problem 237

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

Problem 238

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

Problem 239

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }+y=\sin \left (x \right ) \sin \left (3 x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )+\frac {2 c_{2} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) {\mathrm e}^{-\frac {x}{2}} \sqrt {3}}{3}-\frac {3 \cos \left (2 x \right )}{26}+\frac {\sin \left (2 x \right )}{13}+\frac {15 \cos \left (4 x \right )}{482}-\frac {2 \sin \left (4 x \right )}{241} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) c_{2} +{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right ) c_{1} -\frac {3 \cos \left (2 x \right )}{26}+\frac {\sin \left (2 x \right )}{13}-\frac {2 \sin \left (4 x \right )}{241}+\frac {15 \cos \left (4 x \right )}{482} \]

Problem 240

ODE	\[ \boxed {y^{\prime \prime }+9 y=\sin \left (x \right )^{4}} \]
program solution	\[ y = c_{1} \cos \left (3 x \right )+\frac {c_{2} \sin \left (3 x \right )}{3}+\frac {1}{24}-\frac {\cos \left (2 x \right )}{10}-\frac {\cos \left (4 x \right )}{56} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sin \left (3 x \right ) c_{2} +\cos \left (3 x \right ) c_{1} -\frac {\cos \left (2 x \right )}{10}-\frac {\cos \left (2 x \right )^{2}}{28}+\frac {5}{84} \]

Problem 241

ODE	\[ \boxed {y^{\prime \prime }+y=x \cos \left (x \right )^{3}} \]
program solution	\[ y = c_{1} \cos \left (x \right )+c_{2} \sin \left (x \right )+\frac {3 x \cos \left (x \right )}{16}+\frac {3 \sin \left (x \right ) x^{2}}{16}-\frac {x \cos \left (3 x \right )}{32}+\frac {3 \sin \left (3 x \right )}{128} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {x \cos \left (x \right )^{3}}{8}+\frac {3 \sin \left (x \right ) \cos \left (x \right )^{2}}{32}+\frac {\left (9 x +32 c_{1} \right ) \cos \left (x \right )}{32}+\frac {3 \left (x^{2}+\frac {16 c_{2}}{3}+\frac {3}{4}\right ) \sin \left (x \right )}{16} \]

Problem 242

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

Problem 243

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

Problem 244

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

Problem 245

ODE	\[ \boxed {y^{\prime \prime }-4 y=\sinh \left (2 x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+\frac {c_{2} {\mathrm e}^{2 x}}{4}+\frac {\left (4 x -1-\cosh \left (4 x \right )-\sinh \left (4 x \right )\right ) {\mathrm e}^{-2 x}}{32}+\frac {\left (x -\frac {\sinh \left (4 x \right )}{4}+\frac {\cosh \left (4 x \right )}{4}\right ) {\mathrm e}^{2 x}}{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{2 x} \left (4 x +32 c_{2} -1\right )}{32}+\frac {{\mathrm e}^{-2 x} \left (x +8 c_{1} \right )}{8} \]

Problem 246

ODE	\[ \boxed {y^{\prime \prime }+4 y=\cos \left (3 x \right )} \]
program solution	\[ y = c_{1} \cos \left (2 x \right )+\frac {c_{2} \sin \left (2 x \right )}{2}-\frac {\cos \left (3 x \right )}{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sin \left (2 x \right ) c_{2} +\cos \left (2 x \right ) c_{1} -\frac {\cos \left (3 x \right )}{5} \]

Problem 247

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

Problem 248

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

Problem 249

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

Problem 250

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

Problem 251

ODE	\[ \boxed {y^{\prime \prime }-4 y=x \,{\mathrm e}^{x}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+\frac {c_{2} {\mathrm e}^{2 x}}{4}-\frac {x \,{\mathrm e}^{x}}{3}-\frac {2 \,{\mathrm e}^{x}}{9} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\left (-9 \,{\mathrm e}^{4 x} c_{2} +3 x \,{\mathrm e}^{3 x}+2 \,{\mathrm e}^{3 x}-9 c_{1} \right ) {\mathrm e}^{-2 x}}{9} \]

Problem 252

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x -y=72 x^{5}} \]
program solution	\[ y = x \left (3 x^{4}-\frac {c_{1}}{2 x^{2}}\right )+c_{2} x \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {3 x^{6}+c_{2} x^{2}+c_{1}}{x} \]

Problem 253

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

Problem 254

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

Problem 255

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }-4 y^{\prime } x +3 y=8 x^{\frac {4}{3}}} \]
program solution	\[ y = \sqrt {x}\, \left (c_{2} x +c_{1} \right )-\frac {72 x^{\frac {4}{3}}}{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{\frac {3}{2}} c_{2} +c_{1} \sqrt {x}-\frac {72 x^{\frac {4}{3}}}{5} \]

Problem 256

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

Problem 257

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

Problem 258

ODE	\[ \boxed {x^{\prime \prime }+9 x=10 \cos \left (2 t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ x = 2 \cos \left (2 t \right )-2 \cos \left (3 t \right ) \] Verified OK.
Maple solution	\[ x \left (t \right ) = -8 \cos \left (t \right )^{3}+6 \cos \left (t \right )+4 \cos \left (t \right )^{2}-2 \]

Problem 259

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

Problem 260

ODE	\[ \boxed {x^{\prime \prime }+100 x=225 \cos \left (5 t \right )+300 \sin \left (5 t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 375, x^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ x = 4 \sin \left (5 t \right )+3 \cos \left (5 t \right )+372 \cos \left (10 t \right )-2 \sin \left (10 t \right ) \] Verified OK.
Maple solution	\[ x \left (t \right ) = -2 \sin \left (10 t \right )+372 \cos \left (10 t \right )+3 \cos \left (5 t \right )+4 \sin \left (5 t \right ) \]

Problem 261

ODE	\[ \boxed {x^{\prime \prime }+25 x=90 \cos \left (4 t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 90] \end {align*}
program solution	\[ x = 10 \cos \left (4 t \right )-10 \cos \left (5 t \right )+18 \sin \left (5 t \right ) \] Verified OK.
Maple solution	\[ x \left (t \right ) = 18 \sin \left (5 t \right )-10 \cos \left (5 t \right )+10 \cos \left (4 t \right ) \]

Problem 262

ODE	\[ \boxed {m x^{\prime \prime }+k x=F_{0} \cos \left (\omega t \right )} \]
program solution	\[ x = c_{1} {\mathrm e}^{\sqrt {-\frac {k}{m}}\, t}+\frac {c_{2} m \sqrt {-\frac {k}{m}}\, {\mathrm e}^{-\sqrt {-\frac {k}{m}}\, t}}{2 k}+\frac {F_{0} \cos \left (\omega t \right )}{-m \,\omega ^{2}+k} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {c_{1} \left (-m \,\omega ^{2}+k \right ) \cos \left (\frac {\sqrt {k}\, t}{\sqrt {m}}\right )+c_{2} \left (-m \,\omega ^{2}+k \right ) \sin \left (\frac {\sqrt {k}\, t}{\sqrt {m}}\right )+F_{0} \cos \left (\omega t \right )}{-m \,\omega ^{2}+k} \]

Problem 263

ODE	\[ \boxed {x^{\prime \prime }+4 x^{\prime }+4 x=10 \cos \left (3 t \right )} \]
program solution	\[ x = {\mathrm e}^{-2 t} \left (c_{2} t +c_{1} \right )-\frac {50 \cos \left (3 t \right )}{169}+\frac {120 \sin \left (3 t \right )}{169} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \left (c_{1} t +c_{2} \right ) {\mathrm e}^{-2 t}-\frac {50 \cos \left (3 t \right )}{169}+\frac {120 \sin \left (3 t \right )}{169} \]

Problem 264

ODE	\[ \boxed {x^{\prime \prime }+3 x^{\prime }+5 x=-4 \cos \left (5 t \right )} \]
program solution	\[ x = c_{1} {\mathrm e}^{-\frac {3 t}{2}} \cos \left (\frac {\sqrt {11}\, t}{2}\right )+\frac {2 c_{2} \sin \left (\frac {\sqrt {11}\, t}{2}\right ) {\mathrm e}^{-\frac {3 t}{2}} \sqrt {11}}{11}+\frac {16 \cos \left (5 t \right )}{125}-\frac {12 \sin \left (5 t \right )}{125} \] Verified OK.
Maple solution	\[ x \left (t \right ) = {\mathrm e}^{-\frac {3 t}{2}} \sin \left (\frac {\sqrt {11}\, t}{2}\right ) c_{2} +{\mathrm e}^{-\frac {3 t}{2}} \cos \left (\frac {\sqrt {11}\, t}{2}\right ) c_{1} -\frac {12 \sin \left (5 t \right )}{125}+\frac {16 \cos \left (5 t \right )}{125} \]

Problem 265

ODE	\[ \boxed {2 x^{\prime \prime }+2 x^{\prime }+x=3 \sin \left (10 t \right )} \]
program solution	\[ x = {\mathrm e}^{-\frac {t}{2}} \left (c_{1} \cos \left (\frac {t}{2}\right )+2 c_{2} \sin \left (\frac {t}{2}\right )\right )-\frac {60 \cos \left (10 t \right )}{40001}-\frac {597 \sin \left (10 t \right )}{40001} \] Verified OK.
Maple solution	\[ x \left (t \right ) = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {t}{2}\right ) c_{2} +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {t}{2}\right ) c_{1} -\frac {597 \sin \left (10 t \right )}{40001}-\frac {60 \cos \left (10 t \right )}{40001} \]

Problem 266

ODE	\[ \boxed {x^{\prime \prime }+3 x^{\prime }+3 x=8 \cos \left (10 t \right )+6 \sin \left (10 t \right )} \]
program solution	\[ x = c_{1} {\mathrm e}^{-\frac {3 t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right )+\frac {2 c_{2} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) {\mathrm e}^{-\frac {3 t}{2}} \sqrt {3}}{3}-\frac {956 \cos \left (10 t \right )}{10309}-\frac {342 \sin \left (10 t \right )}{10309} \] Verified OK.
Maple solution	\[ x \left (t \right ) = {\mathrm e}^{-\frac {3 t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} +{\mathrm e}^{-\frac {3 t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} -\frac {342 \sin \left (10 t \right )}{10309}-\frac {956 \cos \left (10 t \right )}{10309} \]

Problem 267

ODE	\[ \boxed {x^{\prime \prime }+4 x^{\prime }+5 x=10 \cos \left (3 t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ x = \frac {\left (\cos \left (t \right )-7 \sin \left (t \right )\right ) {\mathrm e}^{-2 t}}{4}-\frac {\cos \left (3 t \right )}{4}+\frac {3 \sin \left (3 t \right )}{4} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {\left (\cos \left (t \right )-7 \sin \left (t \right )\right ) {\mathrm e}^{-2 t}}{4}-\frac {\cos \left (3 t \right )}{4}+\frac {3 \sin \left (3 t \right )}{4} \]

Problem 268

ODE	\[ \boxed {x^{\prime \prime }+6 x^{\prime }+13 x=10 \sin \left (5 t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ x = \frac {25 \left (2 \cos \left (2 t \right )+5 \sin \left (2 t \right )\right ) {\mathrm e}^{-3 t}}{174}-\frac {25 \cos \left (5 t \right )}{87}-\frac {10 \sin \left (5 t \right )}{87} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {25 \left (2 \cos \left (2 t \right )+5 \sin \left (2 t \right )\right ) {\mathrm e}^{-3 t}}{174}-\frac {25 \cos \left (5 t \right )}{87}-\frac {10 \sin \left (5 t \right )}{87} \]

Problem 269

ODE	\[ \boxed {x^{\prime \prime }+6 x^{\prime }+13 x=10 \sin \left (5 t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ x = \frac {25 \left (2 \cos \left (2 t \right )+5 \sin \left (2 t \right )\right ) {\mathrm e}^{-3 t}}{174}-\frac {25 \cos \left (5 t \right )}{87}-\frac {10 \sin \left (5 t \right )}{87} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {25 \left (2 \cos \left (2 t \right )+5 \sin \left (2 t \right )\right ) {\mathrm e}^{-3 t}}{174}-\frac {25 \cos \left (5 t \right )}{87}-\frac {10 \sin \left (5 t \right )}{87} \]

Problem 270

ODE	\[ \boxed {x^{\prime \prime }+2 x^{\prime }+26 x=600 \cos \left (10 t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 10, x^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ x = \frac {\left (25790 \cos \left (5 t \right )-842 \sin \left (5 t \right )\right ) {\mathrm e}^{-t}}{1469}-\frac {11100 \cos \left (10 t \right )}{1469}+\frac {3000 \sin \left (10 t \right )}{1469} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {\left (25790 \cos \left (5 t \right )-842 \sin \left (5 t \right )\right ) {\mathrm e}^{-t}}{1469}-\frac {11100 \cos \left (10 t \right )}{1469}+\frac {3000 \sin \left (10 t \right )}{1469} \]

Problem 271

ODE	\[ \boxed {x^{\prime \prime }+8 x^{\prime }+25 x=200 \cos \left (t \right )+520 \sin \left (t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = -30, x^{\prime }\left (0\right ) = -10] \end {align*}
program solution	\[ x = \left (-31 \cos \left (3 t \right )-52 \sin \left (3 t \right )\right ) {\mathrm e}^{-4 t}+\cos \left (t \right )+22 \sin \left (t \right ) \] Verified OK.
Maple solution	\[ x \left (t \right ) = \left (-31 \cos \left (3 t \right )-52 \sin \left (3 t \right )\right ) {\mathrm e}^{-4 t}+22 \sin \left (t \right )+\cos \left (t \right ) \]

Problem 272

ODE	\begin {align*} x^{\prime }&=-3 y \left (t \right )\\ y^{\prime }\left (t \right )&=3 x \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} \sin \left (3 t \right )+c_{2} \cos \left (3 t \right ) \\ y \left (t \right ) &= -c_{1} \cos \left (3 t \right )+c_{2} \sin \left (3 t \right ) \\ \end{align*}

Problem 273

ODE	\begin {align*} x^{\prime }&=3 x-2 y \left (t \right )\\ y^{\prime }\left (t \right )&=2 x+y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{2 t} \left (\sin \left (\sqrt {3}\, t \right ) c_{1} +\cos \left (\sqrt {3}\, t \right ) c_{2} \right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{2 t} \left (\sin \left (\sqrt {3}\, t \right ) \sqrt {3}\, c_{2} -\cos \left (\sqrt {3}\, t \right ) \sqrt {3}\, c_{1} +\sin \left (\sqrt {3}\, t \right ) c_{1} +\cos \left (\sqrt {3}\, t \right ) c_{2} \right )}{2} \\ \end{align*}

Problem 274

ODE	\begin {align*} x^{\prime }&=2 x+4 y \left (t \right )+3 \,{\mathrm e}^{t}\\ y^{\prime }\left (t \right )&=5 x-y \left (t \right )-t^{2} \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{\frac {\left (1+\sqrt {89}\right ) t}{2}} c_{2} \sqrt {89}}{10}-\frac {{\mathrm e}^{-\frac {\left (-1+\sqrt {89}\right ) t}{2}} c_{1} \sqrt {89}}{10}+\frac {3 \,{\mathrm e}^{\frac {\left (1+\sqrt {89}\right ) t}{2}} c_{2}}{10}+\frac {3 \,{\mathrm e}^{-\frac {\left (-1+\sqrt {89}\right ) t}{2}} c_{1}}{10}+\frac {2 t^{2}}{11}-\frac {3 \,{\mathrm e}^{t}}{11}-\frac {2 t}{121}+\frac {23}{1331} \\ y \left (t \right ) &= {\mathrm e}^{\frac {\left (1+\sqrt {89}\right ) t}{2}} c_{2} +{\mathrm e}^{-\frac {\left (-1+\sqrt {89}\right ) t}{2}} c_{1} -\frac {t^{2}}{11}-\frac {15 \,{\mathrm e}^{t}}{22}+\frac {12 t}{121}-\frac {17}{1331} \\ \end{align*}

Problem 275

ODE	\begin {align*} x^{\prime }&=y \left (t \right )+z \left (t \right )\\ y^{\prime }\left (t \right )&=z \left (t \right )+x\\ z^{\prime }\left (t \right )&=x+y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{-t}+c_{3} {\mathrm e}^{2 t} \\ y \left (t \right ) &= c_{2} {\mathrm e}^{-t}+c_{3} {\mathrm e}^{2 t}+{\mathrm e}^{-t} c_{1} \\ z \left (t \right ) &= -2 c_{2} {\mathrm e}^{-t}+c_{3} {\mathrm e}^{2 t}-{\mathrm e}^{-t} c_{1} \\ \end{align*}

Problem 276

ODE	\begin {align*} x_{1}^{\prime }\left (t \right )&=x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=2 x_{3} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=3 x_{4} \left (t \right )\\ x_{4}^{\prime }\left (t \right )&=4 x_{1} \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x_{1} \left (t \right ) &= c_{1} {\mathrm e}^{-24^{\frac {1}{4}} t}+c_{2} {\mathrm e}^{24^{\frac {1}{4}} t}-c_{3} \sin \left (24^{\frac {1}{4}} t \right )+c_{4} \cos \left (24^{\frac {1}{4}} t \right ) \\ x_{2} \left (t \right ) &= -24^{\frac {1}{4}} \left (c_{1} {\mathrm e}^{-24^{\frac {1}{4}} t}-c_{2} {\mathrm e}^{24^{\frac {1}{4}} t}+\cos \left (24^{\frac {1}{4}} t \right ) c_{3} +\sin \left (24^{\frac {1}{4}} t \right ) c_{4} \right ) \\ x_{3} \left (t \right ) &= \sqrt {6}\, \left (c_{1} {\mathrm e}^{-24^{\frac {1}{4}} t}+c_{2} {\mathrm e}^{24^{\frac {1}{4}} t}-c_{4} \cos \left (24^{\frac {1}{4}} t \right )+c_{3} \sin \left (24^{\frac {1}{4}} t \right )\right ) \\ x_{4} \left (t \right ) &= -\frac {24^{\frac {3}{4}} \left (c_{1} {\mathrm e}^{-24^{\frac {1}{4}} t}-c_{2} {\mathrm e}^{24^{\frac {1}{4}} t}-\cos \left (24^{\frac {1}{4}} t \right ) c_{3} -\sin \left (24^{\frac {1}{4}} t \right ) c_{4} \right )}{6} \\ \end{align*}

Problem 277

ODE	\begin {align*} x_{1}^{\prime }\left (t \right )&=x_{2} \left (t \right )+x_{3} \left (t \right )+1\\ x_{2}^{\prime }\left (t \right )&=x_{3} \left (t \right )+x_{4} \left (t \right )+t\\ x_{3}^{\prime }\left (t \right )&=x_{1} \left (t \right )+x_{4} \left (t \right )+t^{2}\\ x_{4}^{\prime }\left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )+t^{3} \end {align*}
program solution
Maple solution	\begin{align*} x_{1} \left (t \right ) &= \frac {t^{2}}{16}-\frac {7 t^{3}}{24}-\frac {t^{4}}{16}+\frac {c_{1} {\mathrm e}^{2 t}}{2}-\frac {11 t}{16}+c_{4} +\frac {{\mathrm e}^{-t} \sin \left (t \right ) c_{2}}{2}-\frac {{\mathrm e}^{-t} \sin \left (t \right ) c_{3}}{2}-\frac {{\mathrm e}^{-t} \cos \left (t \right ) c_{2}}{2}-\frac {{\mathrm e}^{-t} \cos \left (t \right ) c_{3}}{2} \\ x_{2} \left (t \right ) &= \frac {t^{4}}{16}+\frac {{\mathrm e}^{-t} \sin \left (t \right ) c_{2}}{2}+\frac {{\mathrm e}^{-t} \sin \left (t \right ) c_{3}}{2}+\frac {{\mathrm e}^{-t} \cos \left (t \right ) c_{2}}{2}-\frac {{\mathrm e}^{-t} \cos \left (t \right ) c_{3}}{2}-\frac {11 t^{3}}{24}+\frac {c_{1} {\mathrm e}^{2 t}}{2}+\frac {t^{2}}{16}-c_{4} -\frac {3 t}{16}-\frac {19}{16} \\ x_{3} \left (t \right ) &= -\frac {t^{4}}{16}-\frac {{\mathrm e}^{-t} \sin \left (t \right ) c_{2}}{2}+\frac {{\mathrm e}^{-t} \sin \left (t \right ) c_{3}}{2}+\frac {{\mathrm e}^{-t} \cos \left (t \right ) c_{2}}{2}+\frac {{\mathrm e}^{-t} \cos \left (t \right ) c_{3}}{2}+\frac {5 t^{3}}{24}+\frac {c_{1} {\mathrm e}^{2 t}}{2}-\frac {15 t^{2}}{16}+c_{4} +\frac {5 t}{16}-\frac {1}{2} \\ x_{4} \left (t \right ) &= \frac {t^{4}}{16}-\frac {{\mathrm e}^{-t} \sin \left (t \right ) c_{2}}{2}-\frac {{\mathrm e}^{-t} \sin \left (t \right ) c_{3}}{2}-\frac {{\mathrm e}^{-t} \cos \left (t \right ) c_{2}}{2}+\frac {{\mathrm e}^{-t} \cos \left (t \right ) c_{3}}{2}+\frac {t^{3}}{24}+\frac {c_{1} {\mathrm e}^{2 t}}{2}-\frac {7 t^{2}}{16}-c_{4} -\frac {19 t}{16}+\frac {5}{16} \\ \end{align*}

Problem 278

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x -9 y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= x^{3} \end {align*}
program solution	\[ y = c_{1} x^{3}-\frac {c_{2}}{6 x^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{6}+c_{1}}{x^{3}} \]

Problem 279

ODE	\[ \boxed {4 y^{\prime \prime }-4 y^{\prime }+y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{\frac {x}{2}} \end {align*}
program solution	\[ y = {\mathrm e}^{\frac {x}{2}} c_{1} +c_{2} {\mathrm e}^{\frac {x}{2}} x \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{\frac {x}{2}} \left (c_{2} x +c_{1} \right ) \]

Problem 280

ODE	\[ \boxed {x^{2} y^{\prime \prime }-x \left (2+x \right ) y^{\prime }+\left (2+x \right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= x \end {align*}
program solution	\[ y = c_{1} x +c_{2} x \,{\mathrm e}^{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x \left (c_{1} +{\mathrm e}^{x} c_{2} \right ) \]

Problem 281

ODE	\[ \boxed {\left (x +1\right ) y^{\prime \prime }-\left (2+x \right ) y^{\prime }+y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{x} \end {align*}
program solution	\[ y = c_{1} {\mathrm e}^{x}-c_{2} {\mathrm e}^{x} {\mathrm e}^{-x} \left (2+x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (2+x \right )+{\mathrm e}^{x} c_{2} \]

Problem 282

ODE	\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }+2 y^{\prime } x -2 y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= x \end {align*}
program solution	\[ y = c_{1} x +c_{2} x \left (x +\frac {1}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{2} x^{2}+c_{1} x +c_{2} \]

Problem 283

ODE	\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +2 y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= x \end {align*}
program solution	\[ y = c_{1} x +c_{2} x \left (-\frac {\ln \left (x +1\right )}{2}+\frac {1}{x}+\frac {\ln \left (x -1\right )}{2}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} \ln \left (x -1\right ) x}{2}-\frac {c_{2} \ln \left (x +1\right ) x}{2}+c_{1} x +c_{2} \]

Problem 284

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (x^{2}-\frac {1}{4}\right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \frac {\cos \left (x \right )}{\sqrt {x}} \end {align*}
program solution	\[ y = \frac {\cos \left (x \right ) c_{1}}{\sqrt {x}}+\frac {c_{2} \cos \left (x \right ) \tan \left (x \right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \sin \left (x \right )+c_{2} \cos \left (x \right )}{\sqrt {x}} \]

Problem 285

ODE	\[ \boxed {5 y^{\prime \prime \prime \prime }+3 y^{\prime \prime \prime }=0} \]
program solution	\[ y = c_{3} x^{2}+c_{2} x +c_{1} +{\mathrm e}^{-\frac {3 x}{5}} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} +c_{2} x +c_{3} x^{2}+c_{4} {\mathrm e}^{-\frac {3 x}{5}} \]

Problem 286

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-8 y^{\prime \prime \prime }+16 y^{\prime \prime }=0} \]
program solution	\[ y = c_{2} x +c_{1} +{\mathrm e}^{4 x} c_{3} +x \,{\mathrm e}^{4 x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{4} x +c_{3} \right ) {\mathrm e}^{4 x}+c_{2} x +c_{1} \]

Problem 287

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-3 y^{\prime \prime \prime }+3 y^{\prime \prime }-y^{\prime }=0} \]
program solution	\[ y = c_{1} +c_{2} {\mathrm e}^{x}+x \,{\mathrm e}^{x} c_{3} +x^{2} {\mathrm e}^{x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{4} x^{2}+c_{3} x +c_{2} \right ) {\mathrm e}^{x}+c_{1} \]

Problem 288

ODE	\[ \boxed {9 y^{\prime \prime \prime }+12 y^{\prime \prime }+4 y^{\prime }=0} \]
program solution	\[ y = c_{1} +{\mathrm e}^{-\frac {2 x}{3}} c_{2} +x \,{\mathrm e}^{-\frac {2 x}{3}} c_{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{3} x +c_{2} \right ) {\mathrm e}^{-\frac {2 x}{3}}+c_{1} \]

Problem 289

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+3 y^{\prime \prime }-4 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+c_{2} {\mathrm e}^{x}+{\mathrm e}^{-2 i x} c_{3} +{\mathrm e}^{2 i x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{x} c_{1} +c_{2} {\mathrm e}^{-x}+c_{3} \sin \left (2 x \right )+c_{4} \cos \left (2 x \right ) \]

Problem 290

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-16 y^{\prime \prime }+16 y=0} \]
program solution	\[ y = {\mathrm e}^{\left (-\sqrt {2}-\sqrt {2}\, \sqrt {3}\right ) x} c_{1} +{\mathrm e}^{\left (-\sqrt {2}+\sqrt {2}\, \sqrt {3}\right ) x} c_{2} +{\mathrm e}^{\left (\sqrt {2}-\sqrt {2}\, \sqrt {3}\right ) x} c_{3} +{\mathrm e}^{\left (\sqrt {2}+\sqrt {2}\, \sqrt {3}\right ) x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\sqrt {2}\, \left (1+\sqrt {3}\right ) x}+c_{2} {\mathrm e}^{\sqrt {2}\, \left (1+\sqrt {3}\right ) x}+c_{3} {\mathrm e}^{-\sqrt {2}\, \left (\sqrt {3}-1\right ) x}+c_{4} {\mathrm e}^{\sqrt {2}\, \left (\sqrt {3}-1\right ) x} \]

Problem 291

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+18 y^{\prime \prime }+81 y=0} \]
program solution	\[ y = {\mathrm e}^{-3 i x} c_{1} +x \,{\mathrm e}^{-3 i x} c_{2} +{\mathrm e}^{3 i x} c_{3} +x \,{\mathrm e}^{3 i x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{4} x +c_{2} \right ) \cos \left (3 x \right )+\sin \left (3 x \right ) \left (c_{3} x +c_{1} \right ) \]

Problem 292

ODE	\[ \boxed {6 y^{\prime \prime \prime \prime }+11 y^{\prime \prime }+4 y=0} \]
program solution	\[ y = {\mathrm e}^{\frac {i \sqrt {2}\, x}{2}} c_{1} +{\mathrm e}^{-\frac {2 i \sqrt {3}\, x}{3}} c_{2} +{\mathrm e}^{-\frac {i \sqrt {2}\, x}{2}} c_{3} +{\mathrm e}^{\frac {2 i \sqrt {3}\, x}{3}} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sin \left (\frac {2 \sqrt {3}\, x}{3}\right )+c_{2} \cos \left (\frac {2 \sqrt {3}\, x}{3}\right )+c_{3} \sin \left (\frac {\sqrt {2}\, x}{2}\right )+c_{4} \cos \left (\frac {\sqrt {2}\, x}{2}\right ) \]

Problem 293

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-16 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+c_{2} {\mathrm e}^{2 x}+{\mathrm e}^{-2 i x} c_{3} +{\mathrm e}^{2 i x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{2 x} c_{1} +c_{2} {\mathrm e}^{-2 x}+c_{3} \sin \left (2 x \right )+c_{4} \cos \left (2 x \right ) \]

Problem 294

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

Problem 295

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }+3 y^{\prime \prime }+2 y^{\prime }+y=0} \]
program solution	\[ y = {\mathrm e}^{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x} c_{1} +x \,{\mathrm e}^{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x} c_{2} +{\mathrm e}^{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x} c_{3} +x \,{\mathrm e}^{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\frac {x}{2}} \left (\left (c_{4} x +c_{2} \right ) \cos \left (\frac {\sqrt {3}\, x}{2}\right )+\sin \left (\frac {\sqrt {3}\, x}{2}\right ) \left (c_{3} x +c_{1} \right )\right ) \]

Problem 296

ODE	\[ \boxed {2 y^{\prime \prime \prime }-3 y^{\prime \prime }-2 y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = -1, y^{\prime \prime }\left (0\right ) = 3] \end {align*}
program solution	\[ y = -\frac {7}{2}+\frac {{\mathrm e}^{2 x}}{2}+4 \,{\mathrm e}^{-\frac {x}{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {7}{2}+4 \,{\mathrm e}^{-\frac {x}{2}}+\frac {{\mathrm e}^{2 x}}{2} \]

Problem 297

ODE	\[ \boxed {3 y^{\prime \prime \prime }+2 y^{\prime \prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = -\frac {13}{4}+\frac {3 x}{2}+\frac {9 \,{\mathrm e}^{-\frac {2 x}{3}}}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {13}{4}+\frac {3 x}{2}+\frac {9 \,{\mathrm e}^{-\frac {2 x}{3}}}{4} \]

Problem 298

ODE	\[ \boxed {y^{\prime \prime \prime }+10 y^{\prime \prime }+25 y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = 4, y^{\prime \prime }\left (0\right ) = 5] \end {align*}
program solution	\[ y = \frac {24}{5}-\frac {9 \,{\mathrm e}^{-5 x}}{5}-5 x \,{\mathrm e}^{-5 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {24}{5}-\frac {9 \,{\mathrm e}^{-5 x}}{5}-5 \,{\mathrm e}^{-5 x} x \]

Problem 299

ODE	\[ \boxed {y^{\prime \prime \prime }+3 y^{\prime \prime }-4 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+x \,{\mathrm e}^{-2 x} c_{2} +{\mathrm e}^{x} c_{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} {\mathrm e}^{3 x}+c_{3} x +c_{2} \right ) {\mathrm e}^{-2 x} \]

Problem 300

ODE	\[ \boxed {2 y^{\prime \prime \prime }-y^{\prime \prime }-5 y^{\prime }-2 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+c_{2} {\mathrm e}^{2 x}+{\mathrm e}^{-\frac {x}{2}} c_{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} {\mathrm e}^{3 x}+c_{1} {\mathrm e}^{\frac {x}{2}}+c_{3} \right ) {\mathrm e}^{-x} \]