2.17.154 Problems 15301 to 15400

Problem 15301

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

Problem 15302

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

Problem 15303

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

Problem 15304

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

Problem 15305

ODE	\[ \boxed {4 y^{\prime \prime }+8 y^{\prime }=x \sin \left (x \right )} \]
program solution	\[ y = {\mathrm e}^{-2 x} c_{2} +\frac {\left (-1-5 x \right ) \cos \left (x \right )}{50}+\frac {\left (14-5 x \right ) \sin \left (x \right )}{100}+\frac {c_{1}}{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {{\mathrm e}^{-2 x} c_{1}}{2}+\frac {\left (-1-5 x \right ) \cos \left (x \right )}{50}+\frac {\left (-5 x +14\right ) \sin \left (x \right )}{100}+c_{2} \]

Problem 15306

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

Problem 15307

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }-2 y=x^{2} {\mathrm e}^{4 x}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+\frac {{\mathrm e}^{x} c_{2}}{3}+\frac {x^{2} {\mathrm e}^{4 x}}{18}-\frac {{\mathrm e}^{4 x} x}{18}+\frac {7 \,{\mathrm e}^{4 x}}{324} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (\left (\frac {7}{18}+x^{2}-x \right ) {\mathrm e}^{6 x}+18 c_{1} {\mathrm e}^{3 x}+18 c_{2} \right ) {\mathrm e}^{-2 x}}{18} \]

Problem 15308

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

Problem 15309

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

Problem 15310

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

Problem 15311

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }+y=x^{3}} \]
program solution	\[ y = {\mathrm e}^{x} \left (c_{2} x +c_{1} \right )+x^{3}+6 x^{2}+18 x +24 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} x +c_{2} \right ) {\mathrm e}^{x}+x^{3}+6 x^{2}+18 x +24 \]

Problem 15312

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

Problem 15313

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

Problem 15314

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

Problem 15315

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

Problem 15316

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

Problem 15317

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

Problem 15318

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

Problem 15319

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

Problem 15320

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

Problem 15321

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

Problem 15322

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

Problem 15323

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

Problem 15324

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

Problem 15325

ODE	\[ \boxed {y^{\prime \prime }+4 y=\sin \left (x \right ) \sin \left (2 x \right )} \]
program solution	\[ y = c_{1} \cos \left (2 x \right )+\frac {c_{2} \sin \left (2 x \right )}{2}+\frac {\cos \left (x \right )}{6}+\frac {\cos \left (3 x \right )}{10} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sin \left (2 x \right ) c_{2} +\cos \left (2 x \right ) c_{1} -\frac {2 \cos \left (x \right )}{15}+\frac {2 \cos \left (x \right )^{3}}{5} \]

Problem 15326

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }=2 \cos \left (4 x \right )^{2}} \]
program solution	\[ y = -\frac {1}{16}+c_{2} {\mathrm e}^{4 x}-\frac {c_{1}}{4}-\frac {x}{4}-\frac {\cos \left (8 x \right )}{80}-\frac {\sin \left (8 x \right )}{160} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} {\mathrm e}^{4 x}}{4}-\frac {\sin \left (8 x \right )}{160}-\frac {\cos \left (8 x \right )}{80}-\frac {x}{4}+c_{2} \]

Problem 15327

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

Problem 15328

ODE	\[ \boxed {y^{\prime \prime }-3 y^{\prime }=18 x -10 \cos \left (x \right )} \]
program solution	\[ y = -\frac {2}{3}+{\mathrm e}^{3 x} c_{2} -3 x^{2}-\frac {c_{1}}{3}-2 x +\cos \left (x \right )+3 \sin \left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} {\mathrm e}^{3 x}}{3}-3 x^{2}+3 \sin \left (x \right )+\cos \left (x \right )-2 x +c_{2} \]

Problem 15329

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

Problem 15330

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

Problem 15331

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+5 y=4 \,{\mathrm e}^{-x}+17 \sin \left (2 x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x} \cos \left (2 x \right )+\frac {c_{2} {\mathrm e}^{-x} \sin \left (2 x \right )}{2}+{\mathrm e}^{-x}-4 \cos \left (2 x \right )+\sin \left (2 x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{1} +1\right ) \cos \left (2 x \right )+\sin \left (2 x \right ) c_{2} +1\right ) {\mathrm e}^{-x}-4 \cos \left (2 x \right )+\sin \left (2 x \right ) \]

Problem 15332

ODE	\[ \boxed {2 y^{\prime \prime }-3 y^{\prime }-2 y=5 \,{\mathrm e}^{x} \cosh \left (x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {x}{2}}+\frac {2 c_{2} {\mathrm e}^{2 x}}{5}-\frac {5}{4}+\frac {\left (-\sinh \left (\frac {5 x}{2}\right )-\cosh \left (\frac {5 x}{2}\right )\right ) {\mathrm e}^{-\frac {x}{2}}}{5}+\frac {x \,{\mathrm e}^{2 x}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {5}{4}+{\mathrm e}^{-\frac {x}{2}} c_{2} +\frac {\left (-2+5 x +10 c_{1} \right ) {\mathrm e}^{2 x}}{10} \]

Problem 15333

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

Problem 15334

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }+2 y^{\prime \prime }+2 y^{\prime }+y={\mathrm e}^{x} x +\frac {\cos \left (x \right )}{2}} \]
program solution	\[ y = {\mathrm e}^{-i x} c_{4} +{\mathrm e}^{i x} c_{3} +{\mathrm e}^{-x} \left (c_{2} x +c_{1} \right )+\frac {\left (4+i-4 x \right ) \cos \left (x \right )}{32}+\frac {\left (x -2\right ) {\mathrm e}^{x}}{8}+\frac {5 \sin \left (x \right )}{32} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{4} x +c_{3} \right ) {\mathrm e}^{-x}+\frac {\left (-x +8 c_{1} +1\right ) \cos \left (x \right )}{8}+\frac {\left (x -2\right ) {\mathrm e}^{x}}{8}+\frac {\sin \left (x \right ) \left (4 c_{2} +1\right )}{4} \]

Problem 15335

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }=\cos \left (x \right )^{2}+{\mathrm e}^{x}+x^{2}} \]
program solution	\[ y = \frac {x^{3}}{3}-x^{2}+\frac {{\mathrm e}^{x}}{2}+c_{1} +\frac {5 x}{2}+\frac {\sin \left (2 x \right )}{20}-\frac {\cos \left (2 x \right )}{10}-\frac {5}{2}+c_{2} {\mathrm e}^{-x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -x^{2}+\frac {x^{3}}{3}-c_{1} {\mathrm e}^{-x}+\frac {{\mathrm e}^{x}}{2}-\frac {\cos \left (2 x \right )}{10}+\frac {\sin \left (2 x \right )}{20}+\frac {5 x}{2}+c_{2} \]

Problem 15336

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime }={\mathrm e}^{x}+3 \sin \left (2 x \right )+1} \]
program solution	\[ y = x^{2} c_{3} +x c_{2} +c_{1} +{\mathrm e}^{-4 x} c_{4} +\frac {x^{3}}{24}+\frac {{\mathrm e}^{x}}{5}+\frac {3 \cos \left (2 x \right )}{40}+\frac {3 \sin \left (2 x \right )}{80} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (\left (-\frac {18 \sin \left (x \right )^{2}}{5}+\frac {9 \sin \left (x \right ) \cos \left (x \right )}{5}+x^{3}+\left (12 c_{2} -\frac {18}{5}\right ) x^{2}+\left (24 c_{3} -\frac {9}{5}\right ) x +24 c_{4} \right ) {\mathrm e}^{4 x}+\frac {24 \,{\mathrm e}^{5 x}}{5}-\frac {3 c_{1}}{8}\right ) {\mathrm e}^{-4 x}}{24} \]

Problem 15337

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }+5 y=10 \sin \left (x \right )+17 \sin \left (2 x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{x} \cos \left (2 x \right )+\frac {c_{2} {\mathrm e}^{x} \sin \left (2 x \right )}{2}+\cos \left (x \right )+2 \sin \left (x \right )+4 \cos \left (2 x \right )+\sin \left (2 x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} {\mathrm e}^{x}+4\right ) \cos \left (2 x \right )+{\mathrm e}^{x} \sin \left (2 x \right ) c_{2} +\cos \left (x \right )+2 \sin \left (x \right )+\sin \left (2 x \right ) \]

Problem 15338

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

Problem 15339

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

Problem 15340

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

Problem 15341

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

Problem 15342

ODE	\[ \boxed {y^{\prime \prime }+y=\cos \left (2 x \right )^{2}+\sin \left (\frac {x}{2}\right )^{2}} \]
program solution	\[ y = c_{1} \cos \left (x \right )+c_{2} \sin \left (x \right )+1-\frac {x \sin \left (x \right )}{4}-\frac {\cos \left (4 x \right )}{30} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1-\frac {\cos \left (4 x \right )}{30}+\frac {\left (-1+8 c_{1} \right ) \cos \left (x \right )}{8}+\frac {\left (-x +4 c_{2} \right ) \sin \left (x \right )}{4} \]

Problem 15343

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

Problem 15344

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

Problem 15345

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

Problem 15346

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

Problem 15347

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

Problem 15348

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

Problem 15349

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

Problem 15350

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+9 y=18 \,{\mathrm e}^{-3 x}+8 \sin \left (x \right )+6 \cos \left (x \right )} \]
program solution	\[ y = {\mathrm e}^{-3 x} \left (c_{2} x +c_{1} \right )+9 x^{2} {\mathrm e}^{-3 x}+\sin \left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} x +9 x^{2}+c_{2} \right ) {\mathrm e}^{-3 x}+\sin \left (x \right ) \]

Problem 15351

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

Problem 15352

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

Problem 15353

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

Problem 15354

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

Problem 15355

ODE	\[ \boxed {y^{\prime \prime \prime }-4 y^{\prime }=x \,{\mathrm e}^{2 x}+\sin \left (x \right )+x^{2}} \]
program solution	\[ y = c_{1} +{\mathrm e}^{-2 x} c_{2} +{\mathrm e}^{2 x} c_{3} +\frac {\left (8 x^{2}-12 x +7\right ) {\mathrm e}^{2 x}}{128}-\frac {x^{3}}{12}-\frac {x}{8}+\frac {\cos \left (x \right )}{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (8 x^{2}+64 c_{1} -12 x +7\right ) {\mathrm e}^{2 x}}{128}-\frac {x^{3}}{12}-\frac {c_{2} {\mathrm e}^{-2 x}}{2}-\frac {x}{8}+c_{3} +\frac {\cos \left (x \right )}{5} \]

Problem 15356

ODE	\[ \boxed {y^{\left (5\right )}-y^{\prime \prime \prime \prime }={\mathrm e}^{x} x -1} \]
program solution	\[ y = x^{3} c_{4} +x^{2} c_{3} +x c_{2} +c_{1} +{\mathrm e}^{x} c_{5} +\frac {x^{4}}{24}-4 \,{\mathrm e}^{x} x +\frac {{\mathrm e}^{x} x^{2}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (x^{2}+2 c_{1} -8 x +20\right ) {\mathrm e}^{x}}{2}+\frac {x^{4}}{24}+\frac {c_{2} x^{3}}{6}+\frac {c_{3} x^{2}}{2}+c_{4} x +c_{5} \]

Problem 15357

ODE	\[ \boxed {y^{\left (5\right )}-y^{\prime \prime \prime }=x +2 \,{\mathrm e}^{-x}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+c_{2} +c_{3} x +x^{2} c_{4} +{\mathrm e}^{x} c_{5} +{\mathrm e}^{-x} x -\frac {x^{4}}{24} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (7+2 x -2 c_{1} \right ) {\mathrm e}^{-x}}{2}-\frac {x^{4}}{24}+\frac {c_{3} x^{2}}{2}+c_{4} x +c_{2} {\mathrm e}^{x}+c_{5} \]

Problem 15358

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

Problem 15359

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+9 y=9 x^{2}-12 x +2} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 3] \end {align*}
program solution	\[ y = x^{2}+{\mathrm e}^{3 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{3 x}+x^{2} \]

Problem 15360

ODE	\[ \boxed {y^{\prime \prime }+9 y=36 \,{\mathrm e}^{3 x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 6] \end {align*}
program solution	\[ y = 2 \,{\mathrm e}^{3 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2 \,{\mathrm e}^{3 x} \]

Problem 15361

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

Problem 15362

ODE	\[ \boxed {y^{\prime \prime }-5 y^{\prime }+6 y=\left (12 x -7\right ) {\mathrm e}^{-x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = {\mathrm e}^{-x} x -{\mathrm e}^{3 x}+{\mathrm e}^{2 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{2 x}-{\mathrm e}^{3 x}+{\mathrm e}^{-x} x \]

Problem 15363

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

Problem 15364

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+9 y=10 \sin \left (x \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = x \,{\mathrm e}^{-3 x}+\frac {3 \,{\mathrm e}^{-3 x}}{5}-\frac {3 \cos \left (x \right )}{5}+\frac {4 \sin \left (x \right )}{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {3 \,{\mathrm e}^{-3 x}}{5}+x \,{\mathrm e}^{-3 x}-\frac {3 \cos \left (x \right )}{5}+\frac {4 \sin \left (x \right )}{5} \]

Problem 15365

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

Problem 15366

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

Problem 15367

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

Problem 15368

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

Problem 15369

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+9 y=16 \,{\mathrm e}^{-x}+9 x -6} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = {\mathrm e}^{3 x} x +{\mathrm e}^{-x}+x \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{3 x} x +x +{\mathrm e}^{-x} \]

Problem 15370

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

Problem 15371

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

Problem 15372

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

Problem 15373

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-y=8 \,{\mathrm e}^{x}} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = 1, y^{\prime \prime \prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = {\mathrm e}^{-x}+\left (2 x -3\right ) {\mathrm e}^{x}+\cos \left (x \right )+2 \sin \left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-x}+\left (2 x -3\right ) {\mathrm e}^{x}+\cos \left (x \right )+2 \sin \left (x \right ) \]

Problem 15374

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

Problem 15375

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-y=8 \,{\mathrm e}^{x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 2, y^{\prime \prime }\left (0\right ) = 4, y^{\prime \prime \prime }\left (0\right ) = 6] \end {align*}
program solution	\[ y = 2 \,{\mathrm e}^{x} x \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2 \,{\mathrm e}^{x} x \]

Problem 15376

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

Problem 15377

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

Problem 15378

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

Problem 15379

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

Problem 15380

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }+y=4 \,{\mathrm e}^{-x}} \] With initial conditions \begin {align*} [y \left (\infty \right ) = 0] \end {align*}
program solution	N/A
Maple solution	\[ y \left (x \right ) = -\operatorname {signum}\left (c_{1} {\mathrm e}^{x}\right ) \infty \]

Problem 15381

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+3 y=8 \,{\mathrm e}^{x}+9} \] With initial conditions \begin {align*} [y \left (-\infty \right ) = 3] \end {align*}
program solution	N/A
Maple solution	\[ y \left (x \right ) = -\operatorname {signum}\left (c_{1} {\mathrm e}^{-x}\right ) \infty \]

Problem 15382

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }-5 y=1} \] With initial conditions \begin {align*} \left [y \left (\infty \right ) = -{\frac {1}{5}}\right ] \end {align*}
program solution	N/A
Maple solution	\[ y \left (x \right ) = -\operatorname {signum}\left (c_{2} {\mathrm e}^{-\frac {\left (-1+\sqrt {21}\right ) x}{2}}\right ) \infty \]

Problem 15383

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+4 y=2 \,{\mathrm e}^{x} \left (\sin \left (x \right )+7 \cos \left (x \right )\right )} \] With initial conditions \begin {align*} [y \left (-\infty \right ) = 0] \end {align*}
program solution	N/A
Maple solution	\[ y \left (x \right ) = \operatorname {signum}\left ({\mathrm e}^{-2 x} c_{1} \right ) \infty \]

Problem 15384

ODE	\[ \boxed {y^{\prime \prime }-5 y^{\prime }+6 y=2 \,{\mathrm e}^{-2 x} \left (9 \sin \left (2 x \right )+4 \cos \left (2 x \right )\right )} \] With initial conditions \begin {align*} [y \left (\infty \right ) = 0] \end {align*}
program solution	N/A
Maple solution	\[ y \left (x \right ) = c_{2} {\mathrm e}^{2 x}+\frac {\left (113 \cos \left (2 x \right )+36 \sin \left (2 x \right )\right ) {\mathrm e}^{-2 x}}{145} \]

Problem 15385

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+4 y={\mathrm e}^{-x} \left (9 x^{2}+5 x -12\right )} \] With initial conditions \begin {align*} [y \left (\infty \right ) = 0] \end {align*}
program solution	N/A
Maple solution	\[ y \left (x \right ) = -\operatorname {signum}\left (c_{1} {\mathrm e}^{2 x}\right ) \infty \]

Problem 15386

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

Problem 15387

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

Problem 15388

ODE	\[ \boxed {x^{2} y^{\prime \prime }+2 x y^{\prime }+6 y=0} \]
program solution	\[ y = c_{1} x^{-\frac {1}{2}-\frac {i \sqrt {23}}{2}}-\frac {i c_{2} \sqrt {23}\, x^{-\frac {1}{2}+\frac {i \sqrt {23}}{2}}}{23} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \sin \left (\frac {\sqrt {23}\, \ln \left (x \right )}{2}\right )+c_{2} \cos \left (\frac {\sqrt {23}\, \ln \left (x \right )}{2}\right )}{\sqrt {x}} \]

Problem 15389

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

Problem 15390

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

Problem 15391

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

Problem 15392

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

Problem 15393

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

Problem 15394

ODE	\[ \boxed {\left (x +1\right )^{2} y^{\prime \prime \prime }-12 y^{\prime }=0} \]
program solution	\[ y = \frac {c_{2} x^{5}}{35}+\frac {c_{2} x^{4}}{7}+\frac {2 c_{2} x^{3}}{7}+\frac {2 c_{2} x^{2}}{7}+\frac {x c_{2}}{7}-\frac {c_{1}}{2 \left (x +1\right )^{2}}+c_{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} +\frac {c_{2}}{\left (1+x \right )^{2}}+c_{3} \left (1+x \right )^{5} \]

Problem 15395

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

Problem 15396

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x y^{\prime }+y=x \left (6-\ln \left (x \right )\right )} \]
program solution	\[ y = x^{-i} c_{1} -\frac {i c_{2} x^{i}}{2}-\frac {x \left (-7+\ln \left (x \right )\right )}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sin \left (\ln \left (x \right )\right ) c_{2} +\cos \left (\ln \left (x \right )\right ) c_{1} -\frac {x \left (\ln \left (x \right )-7\right )}{2} \]

Problem 15397

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

Problem 15398

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

Problem 15399

ODE	\[ \boxed {x^{2} y^{\prime \prime }-2 x y^{\prime }-2 y=x^{2}-2 x +2} \]
program solution	\[ y = c_{1} x^{\frac {3}{2}-\frac {\sqrt {17}}{2}}+\frac {c_{2} \sqrt {17}\, x^{\frac {3}{2}+\frac {\sqrt {17}}{2}}}{17}-\frac {x^{2}}{4}+\frac {x}{2}-1 \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{\frac {3}{2}+\frac {\sqrt {17}}{2}} c_{2} +x^{\frac {3}{2}-\frac {\sqrt {17}}{2}} c_{1} -\frac {x^{2}}{4}+\frac {x}{2}-1 \]

Problem 15400

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