2.17.119 Problems 11801 to 11900

Problem 11801

ODE	\[ \boxed {y^{\prime \prime }+7 y^{\prime }+10 y=4 x \,{\mathrm e}^{-3 x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = -1] \end {align*}
program solution	\[ y = \left (-2 x -1\right ) {\mathrm e}^{-3 x}+{\mathrm e}^{-2 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-2 x}+\left (-2 x -1\right ) {\mathrm e}^{-3 x} \]

Problem 11802

ODE	\[ \boxed {y^{\prime \prime }+8 y^{\prime }+16 y=8 \,{\mathrm e}^{-2 x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = 4 \,{\mathrm e}^{-4 x} x +2 \,{\mathrm e}^{-2 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 4 \,{\mathrm e}^{-4 x} x +2 \,{\mathrm e}^{-2 x} \]

Problem 11803

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+9 y=27 \,{\mathrm e}^{-6 x}} \] With initial conditions \begin {align*} [y \left (0\right ) = -2, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \left (3 x -5\right ) {\mathrm e}^{-3 x}+3 \,{\mathrm e}^{-6 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (3 x -5\right ) {\mathrm e}^{-3 x}+3 \,{\mathrm e}^{-6 x} \]

Problem 11804

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+13 y=18 \,{\mathrm e}^{-2 x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 4] \end {align*}
program solution	\[ y = -\frac {2 \,{\mathrm e}^{-2 x} \left (-3-2 \sin \left (3 x \right )+3 \cos \left (3 x \right )\right )}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {2 \,{\mathrm e}^{-2 x} \left (2 \sin \left (3 x \right )-3 \cos \left (3 x \right )+3\right )}{3} \]

Problem 11805

ODE	\[ \boxed {y^{\prime \prime }-10 y^{\prime }+29 y=8 \,{\mathrm e}^{5 x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 8] \end {align*}
program solution	\[ y = -2 \,{\mathrm e}^{5 x} \left (-1+\cos \left (2 x \right )-2 \sin \left (2 x \right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -2 \,{\mathrm e}^{5 x} \left (-1-2 \sin \left (2 x \right )+\cos \left (2 x \right )\right ) \]

Problem 11806

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+13 y=8 \sin \left (3 x \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 2] \end {align*}
program solution	\[ y = \frac {\left (2 \,{\mathrm e}^{2 x}+3\right ) \cos \left (3 x \right )}{5}+\frac {\sin \left (3 x \right ) \left ({\mathrm e}^{2 x}+1\right )}{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (2 \,{\mathrm e}^{2 x}+3\right ) \cos \left (3 x \right )}{5}+\frac {\sin \left (3 x \right ) \left ({\mathrm e}^{2 x}+1\right )}{5} \]

Problem 11807

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }-6 y=8 \,{\mathrm e}^{2 x}-5 \,{\mathrm e}^{3 x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = 5] \end {align*}
program solution	\[ y = -\left (\left (-4+x \right ) {\mathrm e}^{5 x}+2 \,{\mathrm e}^{4 x}-1\right ) {\mathrm e}^{-2 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\left (\left (-4+x \right ) {\mathrm e}^{5 x}+2 \,{\mathrm e}^{4 x}-1\right ) {\mathrm e}^{-2 x} \]

Problem 11808

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }+y=2 \,{\mathrm e}^{2 x} x +6 \,{\mathrm e}^{x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \left (2 x -4\right ) {\mathrm e}^{2 x}+{\mathrm e}^{x} \left (3 x^{2}+x +5\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (2 x -4\right ) {\mathrm e}^{2 x}+{\mathrm e}^{x} \left (3 x^{2}+x +5\right ) \]

Problem 11809

ODE	\[ \boxed {y^{\prime \prime }-y=3 x^{2} {\mathrm e}^{x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 2] \end {align*}
program solution	\[ y = -\frac {{\mathrm e}^{-x}}{8}+\frac {\left (4 x^{3}-6 x^{2}+6 x +9\right ) {\mathrm e}^{x}}{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {{\mathrm e}^{-x}}{8}+\frac {\left (4 x^{3}-6 x^{2}+6 x +9\right ) {\mathrm e}^{x}}{8} \]

Problem 11810

ODE	\[ \boxed {y^{\prime \prime }+y=3 x^{2}-4 \sin \left (x \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \left (2 x +6\right ) \cos \left (x \right )+3 x^{2}-\sin \left (x \right )-6 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (2 x +6\right ) \cos \left (x \right )+3 x^{2}-\sin \left (x \right )-6 \]

Problem 11811

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

Problem 11812

ODE	\[ \boxed {y^{\prime \prime \prime }-4 y^{\prime \prime }+y^{\prime }+6 y=3 x \,{\mathrm e}^{x}+2 \,{\mathrm e}^{x}-\sin \left (x \right )} \] With initial conditions \begin {align*} \left [y \left (0\right ) = {\frac {33}{40}}, y^{\prime }\left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = 0\right ] \end {align*}
program solution	\[ y = \frac {7 \,{\mathrm e}^{-x}}{20}-\frac {31 \,{\mathrm e}^{2 x}}{40}+\frac {3 x \,{\mathrm e}^{x}}{4}+\frac {5 \,{\mathrm e}^{x}}{4}-\frac {\sin \left (x \right )}{10} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {7 \,{\mathrm e}^{-x}}{20}-\frac {31 \,{\mathrm e}^{2 x}}{40}+\frac {\left (3 x +5\right ) {\mathrm e}^{x}}{4}-\frac {\sin \left (x \right )}{10} \]

Problem 11813

ODE	\[ \boxed {y^{\prime \prime \prime }-6 y^{\prime \prime }+9 y^{\prime }-4 y=8 x^{2}+3-6 \,{\mathrm e}^{2 x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 7, y^{\prime \prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = -15-\frac {5 \,{\mathrm e}^{4 x}}{3}+2 x \,{\mathrm e}^{x}+\frac {44 \,{\mathrm e}^{x}}{3}+3 \,{\mathrm e}^{2 x}-9 x -2 x^{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -2 x^{2}-9 x +3 \,{\mathrm e}^{2 x}-15+\frac {44 \,{\mathrm e}^{x}}{3}-\frac {5 \,{\mathrm e}^{4 x}}{3}+2 \,{\mathrm e}^{x} x \]

Problem 11814

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+8 y=x^{3}+x +{\mathrm e}^{-2 x}} \]
program solution	\[ y = c_{1} {\mathrm e}^{2 x}+\frac {c_{2} {\mathrm e}^{4 x}}{2}+\frac {{\mathrm e}^{-2 x}}{24}+\frac {69}{256}+\frac {29 x}{64}+\frac {9 x^{2}}{32}+\frac {x^{3}}{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} {\mathrm e}^{4 x}}{2}+\frac {{\mathrm e}^{-2 x}}{24}+\frac {69}{256}+\frac {29 x}{64}+\frac {9 x^{2}}{32}+\frac {x^{3}}{8}+c_{2} {\mathrm e}^{2 x} \]

Problem 11815

ODE	\[ \boxed {y^{\prime \prime }+9 y={\mathrm e}^{3 x}+{\mathrm e}^{-3 x}+{\mathrm e}^{3 x} \sin \left (3 x \right )} \]
program solution	\[ y = c_{1} \cos \left (3 x \right )+\frac {c_{2} \sin \left (3 x \right )}{3}+\frac {{\mathrm e}^{-3 x}}{18}+\frac {{\mathrm e}^{3 x}}{18}-\frac {2 \,{\mathrm e}^{3 x} \cos \left (3 x \right )}{45}+\frac {{\mathrm e}^{3 x} \sin \left (3 x \right )}{45} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{2} \sin \left (3 x \right )+c_{1} \cos \left (3 x \right )+\frac {\left (2 \sin \left (3 x \right )-4 \cos \left (3 x \right )+5\right ) {\mathrm e}^{3 x}}{90}+\frac {{\mathrm e}^{-3 x}}{18} \]

Problem 11816

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

Problem 11817

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+9 y={\mathrm e}^{x} x^{4}+x^{3} {\mathrm e}^{2 x}+{\mathrm e}^{3 x} x^{2}} \]
program solution	\[ y = {\mathrm e}^{3 x} \left (c_{2} x +c_{1} \right )+\frac {x^{4} {\mathrm e}^{3 x}}{12}+6 x^{2} {\mathrm e}^{2 x}+x^{3} {\mathrm e}^{2 x}+18 \,{\mathrm e}^{2 x} x +24 \,{\mathrm e}^{2 x}+3 x \,{\mathrm e}^{x}+x^{3} {\mathrm e}^{x}+\frac {9 x^{2} {\mathrm e}^{x}}{4}+\frac {x^{4} {\mathrm e}^{x}}{4}+\frac {15 \,{\mathrm e}^{x}}{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (x^{3}+6 x^{2}+18 x +24\right ) {\mathrm e}^{2 x}+\frac {\left (x^{4}+12 c_{1} x +12 c_{2} \right ) {\mathrm e}^{3 x}}{12}+\frac {\left (x^{4}+4 x^{3}+9 x^{2}+12 x +\frac {15}{2}\right ) {\mathrm e}^{x}}{4} \]

Problem 11818

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+13 y=x \,{\mathrm e}^{-3 x} \sin \left (2 x \right )+x^{2} {\mathrm e}^{-2 x} \sin \left (3 x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-3 x} \cos \left (2 x \right )+\frac {c_{2} {\mathrm e}^{-3 x} \sin \left (2 x \right )}{2}-\frac {6 \left ({\mathrm e}^{x} \left (x^{2}-\frac {2}{13} x -\frac {180}{169}\right ) \cos \left (x \right )^{3}+\left (\frac {2 \sin \left (x \right ) \left (x^{2}-\frac {41}{13} x +\frac {563}{338}\right ) {\mathrm e}^{x}}{3}+\frac {13 x^{2}}{24}-\frac {39}{32}\right ) \cos \left (x \right )^{2}+\left (\left (-\frac {3}{4} x^{2}+\frac {3}{26} x +\frac {135}{169}\right ) {\mathrm e}^{x}-\frac {13 \sin \left (x \right ) x}{48}\right ) \cos \left (x \right )-\frac {\sin \left (x \right ) \left (x^{2}-\frac {41}{13} x +\frac {563}{338}\right ) {\mathrm e}^{x}}{6}-\frac {13 x^{2}}{48}+\frac {39}{64}\right ) {\mathrm e}^{-3 x}}{13} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {3 \left (\left (\frac {13 x^{2}}{12}-\frac {26 c_{1}}{3}-\frac {39}{16}\right ) \cos \left (2 x \right )+{\mathrm e}^{x} \left (x^{2}-\frac {2}{13} x -\frac {180}{169}\right ) \cos \left (3 x \right )+\frac {2 \,{\mathrm e}^{x} \left (x^{2}-\frac {41}{13} x +\frac {563}{338}\right ) \sin \left (3 x \right )}{3}-\frac {13 \sin \left (2 x \right ) \left (x +16 c_{2} \right )}{24}\right ) {\mathrm e}^{-3 x}}{26} \]

Problem 11819

ODE	\[ \boxed {y^{\prime \prime \prime }-3 y^{\prime \prime }+2 y^{\prime }=x^{2} {\mathrm e}^{x}+3 \,{\mathrm e}^{2 x} x +5 x^{2}} \]
program solution	\[ y = c_{1} +c_{2} {\mathrm e}^{x}+{\mathrm e}^{2 x} c_{3} +\frac {3 x^{2} {\mathrm e}^{2 x}}{4}-\frac {9 \,{\mathrm e}^{2 x} x}{4}+\frac {35 x}{4}+\frac {15 x^{2}}{4}+\frac {5 x^{3}}{6}-2 x \,{\mathrm e}^{x}-\frac {x^{3} {\mathrm e}^{x}}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (6 x^{2}+4 c_{1} -18 x +21\right ) {\mathrm e}^{2 x}}{8}+\frac {\left (-x^{3}+3 c_{2} -6 x +6\right ) {\mathrm e}^{x}}{3}+\frac {5 x^{3}}{6}+\frac {15 x^{2}}{4}+\frac {35 x}{4}+c_{3} \]

Problem 11820

ODE	\[ \boxed {y^{\prime \prime \prime }-6 y^{\prime \prime }+12 y^{\prime }-8 y={\mathrm e}^{2 x} x +{\mathrm e}^{3 x} x^{2}} \]
program solution	\[ y = {\mathrm e}^{2 x} \left (c_{3} x^{2}+c_{2} x +c_{1} \right )+\frac {{\mathrm e}^{2 x} x^{4}}{24}-6 x \,{\mathrm e}^{3 x}+{\mathrm e}^{3 x} x^{2}+12 \,{\mathrm e}^{3 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (x^{4}+24 c_{3} x^{2}+24 c_{2} x +24 c_{1} \right ) {\mathrm e}^{2 x}}{24}+{\mathrm e}^{3 x} \left (x^{2}-6 x +12\right ) \]

Problem 11821

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

Problem 11822

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-16 y=x^{2} \sin \left (2 x \right )+{\mathrm e}^{2 x} x^{4}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+c_{2} {\mathrm e}^{2 x}+{\mathrm e}^{-2 i x} c_{3} +{\mathrm e}^{2 i x} c_{4} +\frac {\left (128 x^{5}-480 x^{4}+800 x^{3}-600 x^{2}+60 x +105\right ) {\mathrm e}^{2 x}}{20480}+\frac {\left (8 x^{3}-15 x \right ) \cos \left (2 x \right )}{768}+\frac {\left (-24 x^{2}+5\right ) \sin \left (2 x \right )}{1024} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (128 x^{5}-480 x^{4}+800 x^{3}-600 x^{2}+20480 c_{3} +60 x +105\right ) {\mathrm e}^{2 x}}{20480}+\frac {\left (8 x^{3}+768 c_{1} -15 x \right ) \cos \left (2 x \right )}{768}+\frac {\left (-6 x^{2}+256 c_{4} -11\right ) \sin \left (2 x \right )}{256}+{\mathrm e}^{-2 x} c_{2} \]

Problem 11823

ODE	\[ \boxed {y^{\left (6\right )}+2 y^{\left (5\right )}+5 y^{\prime \prime \prime \prime }=x^{3}+x^{2} {\mathrm e}^{-x}+{\mathrm e}^{-x} \sin \left (2 x \right )} \]
program solution	\[ y = x^{3} c_{4} +c_{3} x^{2}+c_{2} x +c_{1} +{\mathrm e}^{\left (-1+2 i\right ) x} c_{5} +{\mathrm e}^{\left (-1-2 i\right ) x} c_{6} +\frac {\left (\left (183750 x -1176000\right ) \cos \left (2 x \right )+\left (-630000 x -889875\right ) \sin \left (2 x \right )[phone] x^{2}[phone] x [phone]\right ) {\mathrm e}^{-x}}{65625000}+\frac {x^{7}}{4200}-\frac {x^{6}}{1500}-\frac {x^{5}}{2500}+\frac {3 x^{4}}{625}-\frac {19 x^{3}}{3125}-\frac {66 x^{2}}{15625}+\frac {834 x}{78125}-\frac {1008}{390625} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{5} x +c_{6} +\frac {\left (\int \left (\left (\left (-330 x +1320 c_{1} +240 c_{2} +69\right ) \cos \left (2 x \right )+\left (60 x -240 c_{1} +1320 c_{2} +567\right ) \sin \left (2 x \right )-3750 x^{2}-22500 x -43125\right ) {\mathrm e}^{-x}+25 x^{6}-60 x^{5}-30 x^{4}+288 x^{3}+7500 c_{3} x^{2}+15000 c_{4} x \right )d x \right )}{15000} \]

Problem 11824

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+2 y^{\prime \prime }+y=\cos \left (x \right ) x^{2}} \]
program solution	\[ y = \left (c_{4} x +c_{3} \right ) {\mathrm e}^{-i x}+{\mathrm e}^{i x} \left (c_{2} x +c_{1} \right )+\frac {\cos \left (x \right ) \left (-4 x^{4}+36 x^{2}+6 i x -45\right )}{192}-\frac {5 \sin \left (x \right ) \left (-\frac {16}{15} x^{3}+i+\frac {18}{5} x \right )}{64} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-4 x^{4}+192 c_{4} x +36 x^{2}+192 c_{1} -21\right ) \cos \left (x \right )}{192}+\frac {\left (x^{3}+\left (12 c_{3} -3\right ) x +12 c_{2} \right ) \sin \left (x \right )}{12} \]

Problem 11825

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+16 y=x \,{\mathrm e}^{\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+{\mathrm e}^{-\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right )} \]
program solution	\[ y = {\mathrm e}^{\left (-1-i\right ) \sqrt {2}\, x} c_{1} +{\mathrm e}^{\left (1-i\right ) \sqrt {2}\, x} c_{2} +{\mathrm e}^{\left (-1+i\right ) \sqrt {2}\, x} c_{3} +{\mathrm e}^{\left (1+i\right ) \sqrt {2}\, x} c_{4} +\frac {{\mathrm e}^{-\sqrt {2}\, x} \left (\left (4 \sqrt {2}\, x -1+i\right ) \sin \left (\sqrt {2}\, x \right )+\cos \left (\sqrt {2}\, x \right ) \left (4 \sqrt {2}\, x +7+i\right )\right )}{256}-\frac {{\mathrm e}^{\sqrt {2}\, x} \left (\left (\sqrt {2}\, x^{2}-3 x +\frac {7 \sqrt {2}}{8}\right ) \sin \left (\sqrt {2}\, x \right )+\sqrt {2}\, \cos \left (\sqrt {2}\, x \right ) \left (x^{2}-\frac {3}{8}\right )\right )}{128} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (\left (2 x \sqrt {2}+128 c_{3} +3\right ) \cos \left (x \sqrt {2}\right )+2 \sin \left (x \sqrt {2}\right ) \left (x \sqrt {2}+64 c_{4} \right )\right ) {\mathrm e}^{-x \sqrt {2}}}{128}-\frac {\left (\left (x^{2} \sqrt {2}-128 c_{1} -\frac {5 \sqrt {2}}{8}\right ) \cos \left (x \sqrt {2}\right )+\sin \left (x \sqrt {2}\right ) \left (x^{2} \sqrt {2}-3 x -128 c_{2} +\frac {5 \sqrt {2}}{8}\right )\right ) {\mathrm e}^{x \sqrt {2}}}{128} \]

Problem 11826

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+3 y^{\prime \prime }-4 y=\cos \left (x \right )^{2}-\cosh \left (x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+c_{2} {\mathrm e}^{x}+{\mathrm e}^{-2 i x} c_{3} +{\mathrm e}^{2 i x} c_{4} -\frac {1}{8}+\frac {\left (9+10 x \right ) {\mathrm e}^{-x}}{200}-\frac {\sin \left (2 x \right ) \left (8 x +i\right )}{320}-\frac {37 \cos \left (2 x \right )}{1600}+\frac {\left (9-10 x \right ) {\mathrm e}^{x}}{200} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {1}{8}+\frac {\left (10 x +200 c_{3} +9\right ) {\mathrm e}^{-x}}{200}+\frac {\left (200 c_{2} -9\right ) \cos \left (2 x \right )}{200}+\frac {\left (-x +40 c_{4} \right ) \sin \left (2 x \right )}{40}+\frac {\left (-10 x +200 c_{1} +9\right ) {\mathrm e}^{x}}{200} \]

Problem 11827

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+10 y^{\prime \prime }+9 y=\sin \left (x \right ) \sin \left (2 x \right )} \]
program solution	\[ y = {\mathrm e}^{i x} c_{1} +{\mathrm e}^{-3 i x} c_{2} +{\mathrm e}^{-i x} c_{3} +{\mathrm e}^{3 i x} c_{4} +\frac {\left (\int \sin \left (x \right ) \sin \left (2 x \right ) \sin \left (3 x \right )d x \right ) \cos \left (3 x \right )}{24}-\frac {\left (\int \sin \left (x \right ) \sin \left (2 x \right ) \cos \left (3 x \right )d x \right ) \sin \left (3 x \right )}{24}-\frac {\left (\int \sin \left (2 x \right ) \sin \left (x \right )^{2}d x \right ) \cos \left (x \right )}{8}+\frac {\left (\int \left (1-\cos \left (4 x \right )\right )d x \right ) \sin \left (x \right )}{32} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (11+1152 c_{3} \right ) \cos \left (3 x \right )}{1152}+\frac {\left (x +96 c_{4} \right ) \sin \left (3 x \right )}{96}+\frac {\left (-1+64 c_{1} \right ) \cos \left (x \right )}{64}+\frac {\sin \left (x \right ) \left (x +32 c_{2} \right )}{32} \]

Problem 11828

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

Problem 11829

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

Problem 11830

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

Problem 11831

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

Problem 11832

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

Problem 11833

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

Problem 11834

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

Problem 11835

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

Problem 11836

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

Problem 11837

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }+y=x \,{\mathrm e}^{x} \ln \left (x \right )} \]
program solution	\[ y = {\mathrm e}^{x} \left (c_{2} x +c_{1} \right )+\frac {x^{3} {\mathrm e}^{x} \left (-5+6 \ln \left (x \right )\right )}{36} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (\ln \left (x \right ) x^{3}-\frac {5 x^{3}}{6}+6 c_{1} x +6 c_{2} \right ) {\mathrm e}^{x}}{6} \]

Problem 11838

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

Problem 11839

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

Problem 11840

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

Problem 11841

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

Problem 11842

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

Problem 11843

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

Problem 11844

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

Problem 11845

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

Problem 11846

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

Problem 11847

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

Problem 11848

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

Problem 11849

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

Problem 11850

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

Problem 11851

ODE	\[ \boxed {\left (2 x +1\right ) \left (x +1\right ) y^{\prime \prime }+2 y^{\prime } x -2 y=\left (2 x +1\right )^{2}} \]
program solution	\[ y = \frac {8 x^{3}+\left (48 c_{2} +8\right ) x^{2}+\left (48 c_{2} +2\right ) x -3 c_{1} +12 c_{2}}{12 x +12} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {4 x^{3}+\left (6 c_{1} +24 c_{2} +4\right ) x^{2}+\left (6 c_{1} +24 c_{2} +1\right ) x +6 c_{2}}{6 x +6} \]

Problem 11852

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

Problem 11853

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

Problem 11854

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

Problem 11855

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

Problem 11856

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

Problem 11857

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

Problem 11858

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

Problem 11859

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

Problem 11860

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

Problem 11861

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 11862

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

Problem 11863

ODE	\[ \boxed {x^{2} y^{\prime \prime }-5 y^{\prime } x +10 y=0} \]
program solution	\[ y = c_{1} x^{3-i}-\frac {i c_{2} x^{3+i}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{3} \left (c_{1} \sin \left (\ln \left (x \right )\right )+\cos \left (\ln \left (x \right )\right ) c_{2} \right ) \]

Problem 11864

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

Problem 11865

ODE	\[ \boxed {x^{3} y^{\prime \prime \prime }+2 x^{2} y^{\prime \prime }-10 y^{\prime } x -8 y=0} \]
program solution	\[ y = \frac {c_{1}}{x}+\frac {c_{2}}{x^{2}}+c_{3} x^{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{6}+c_{2} x +c_{3}}{x^{2}} \]

Problem 11866

ODE	\[ \boxed {x^{3} y^{\prime \prime \prime }-x^{2} y^{\prime \prime }-6 y^{\prime } x +18 y=0} \]
program solution	\[ y = \frac {c_{1}}{x^{2}}+c_{2} x^{3}+c_{3} \ln \left (x \right ) x^{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{3} x^{5} \ln \left (x \right )+c_{2} x^{5}+c_{1}}{x^{2}} \]

Problem 11867

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

Problem 11868

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

Problem 11869

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

Problem 11870

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +4 y=2 x \ln \left (x \right )} \]
program solution	\[ y = x^{-2 i} c_{1} -\frac {i c_{2} x^{2 i}}{4}+\frac {2 x \ln \left (x \right )}{5}-\frac {4 x}{25} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sin \left (2 \ln \left (x \right )\right ) c_{2} +\cos \left (2 \ln \left (x \right )\right ) c_{1} +\frac {2 \ln \left (x \right ) x}{5}-\frac {4 x}{25} \]

Problem 11871

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

Problem 11872

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

Problem 11873

ODE	\[ \boxed {x^{2} y^{\prime \prime }-2 y^{\prime } x -10 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 5, y^{\prime }\left (1\right ) = 4] \end {align*}
program solution	\[ y = \frac {2 x^{7}+3}{x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2 x^{5}+\frac {3}{x^{2}} \]

Problem 11874

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

Problem 11875

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

Problem 11876

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

Problem 11877

ODE	\[ \boxed {x^{2} y^{\prime \prime }-4 y^{\prime } x +4 y=-6 x^{3}+4 x^{2}} \] With initial conditions \begin {align*} [y \left (2\right ) = 4, y^{\prime }\left (2\right ) = -1] \end {align*}
program solution	\[ y = -\frac {23}{24} x^{4}+3 x^{3}-2 x^{2}+\frac {5}{3} x \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {23}{24} x^{4}+3 x^{3}-2 x^{2}+\frac {5}{3} x \]

Problem 11878

ODE	\[ \boxed {x^{2} y^{\prime \prime }+2 y^{\prime } x -6 y=10 x^{2}} \] With initial conditions \begin {align*} [y \left (1\right ) = 1, y^{\prime }\left (1\right ) = -6] \end {align*}
program solution	\[ y = \frac {2 \ln \left (x \right ) x^{5}-x^{5}+2}{x^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {2 x^{5} \ln \left (x \right )-x^{5}+2}{x^{3}} \]

Problem 11879

ODE	\[ \boxed {x^{2} y^{\prime \prime }-5 y^{\prime } x +8 y=2 x^{3}} \] With initial conditions \begin {align*} [y \left (2\right ) = 0, y^{\prime }\left (2\right ) = -8] \end {align*}
program solution	\[ y = -2 x^{3}+4 x^{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -2 x^{3}+4 x^{2} \]

Problem 11880

ODE	\[ \boxed {x^{2} y^{\prime \prime }-6 y=\ln \left (x \right )} \] With initial conditions \begin {align*} \left [y \left (1\right ) = {\frac {1}{6}}, y^{\prime }\left (1\right ) = -{\frac {1}{6}}\right ] \end {align*}
program solution	\[ y = \frac {2 x^{5}-6 \ln \left (x \right ) x^{2}+x^{2}+3}{36 x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {1}{12 x^{2}}+\frac {x^{3}}{18}-\frac {\ln \left (x \right )}{6}+\frac {1}{36} \]

Problem 11881

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

Problem 11882

ODE	\[ \boxed {\left (2 x -3\right )^{2} y^{\prime \prime }-6 \left (2 x -3\right ) y^{\prime }+12 y=0} \]
program solution	\[ y = c_{1} \sqrt {2}\, \left (2 x -3\right )+\frac {c_{2} \left (2 x^{3}-9 x^{2}+9 x \right ) \sqrt {2}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (x -\frac {3}{2}\right ) \left (c_{1} +c_{2} \left (x -\frac {3}{2}\right )^{2}\right ) \]

Problem 11883

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime } x +y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}-\frac {1}{48} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{15} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}\right ) c_{1} +\left (x -\frac {1}{3} x^{3}+\frac {1}{15} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{15} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 11884

ODE	\[ \boxed {y^{\prime \prime }+8 y^{\prime } x -4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+2 x^{2}-2 x^{4}+\frac {28}{15} x^{6}\right ) y \left (0\right )+\left (x -\frac {2}{3} x^{3}+\frac {2}{3} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (-2 x^{4}+2 x^{2}+1\right ) c_{1} +\left (x -\frac {2}{3} x^{3}+\frac {2}{3} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (-2 x^{4}+2 x^{2}+1\right ) y \left (0\right )+\left (x -\frac {2}{3} x^{3}+\frac {2}{3} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 11885

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime } x +\left (2 x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{24} x^{4}+\frac {29}{720} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}-\frac {1}{30} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{24} x^{4}\right ) c_{1} +\left (x -\frac {1}{3} x^{3}-\frac {1}{30} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}-\frac {1}{24} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}-\frac {1}{30} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 11886

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime } x +\left (x^{2}-4\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+2 x^{2}+\frac {1}{4} x^{4}-\frac {1}{15} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{3}-\frac {1}{40} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+2 x^{2}+\frac {1}{4} x^{4}\right ) c_{1} +\left (x +\frac {1}{2} x^{3}-\frac {1}{40} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+2 x^{2}+\frac {1}{4} x^{4}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{3}-\frac {1}{40} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 11887

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime } x +\left (3 x +2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-x^{2}-\frac {1}{2} x^{3}+\frac {1}{3} x^{4}+\frac {11}{40} x^{5}-\frac {1}{60} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{3}-\frac {1}{4} x^{4}+\frac {1}{8} x^{5}+\frac {1}{10} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-x^{2}-\frac {1}{2} x^{3}+\frac {1}{3} x^{4}+\frac {11}{40} x^{5}\right ) c_{1} +\left (x -\frac {1}{2} x^{3}-\frac {1}{4} x^{4}+\frac {1}{8} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-x^{2}-\frac {1}{2} x^{3}+\frac {1}{3} x^{4}+\frac {11}{40} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{3}-\frac {1}{4} x^{4}+\frac {1}{8} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 11888

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime } x +\left (3 x -2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+x^{2}-\frac {1}{2} x^{3}+\frac {1}{3} x^{4}-\frac {11}{40} x^{5}+\frac {7}{60} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{3}-\frac {1}{4} x^{4}+\frac {1}{8} x^{5}-\frac {1}{10} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+x^{2}-\frac {1}{2} x^{3}+\frac {1}{3} x^{4}-\frac {11}{40} x^{5}\right ) c_{1} +\left (x +\frac {1}{2} x^{3}-\frac {1}{4} x^{4}+\frac {1}{8} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+x^{2}-\frac {1}{2} x^{3}+\frac {1}{3} x^{4}-\frac {11}{40} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{3}-\frac {1}{4} x^{4}+\frac {1}{8} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 11889

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+y^{\prime } x +x y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{6} x^{3}+\frac {3}{40} x^{5}+\frac {1}{180} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}-\frac {1}{12} x^{4}+\frac {3}{40} x^{5}+\frac {1}{20} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{6} x^{3}+\frac {3}{40} x^{5}\right ) c_{1} +\left (x -\frac {1}{6} x^{3}-\frac {1}{12} x^{4}+\frac {3}{40} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{6} x^{3}+\frac {3}{40} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}-\frac {1}{12} x^{4}+\frac {3}{40} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 11890

ODE	\[ \boxed {\left (x -1\right ) y^{\prime \prime }-\left (3 x -2\right ) y^{\prime }+2 x y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{3} x^{3}+\frac {1}{3} x^{4}+\frac {11}{60} x^{5}+\frac {13}{180} x^{6}\right ) y \left (0\right )+\left (x +x^{2}+\frac {1}{2} x^{3}+\frac {1}{6} x^{4}+\frac {1}{24} x^{5}+\frac {1}{120} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{3} x^{3}+\frac {1}{3} x^{4}+\frac {11}{60} x^{5}\right ) c_{1} +\left (x +x^{2}+\frac {1}{2} x^{3}+\frac {1}{6} x^{4}+\frac {1}{24} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{3} x^{3}+\frac {1}{3} x^{4}+\frac {11}{60} x^{5}\right ) y \left (0\right )+\left (x +x^{2}+\frac {1}{2} x^{3}+\frac {1}{6} x^{4}+\frac {1}{24} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 11891

ODE	\[ \boxed {\left (x^{3}-1\right ) y^{\prime \prime }+y^{\prime } x^{2}+x y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{6} x^{3}+\frac {1}{18} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{4}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {x^{3}}{6}\right ) c_{1} +\left (x +\frac {1}{6} x^{4}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {x^{3}}{6}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{4}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 11892

ODE	\[ \boxed {\left (x +3\right ) y^{\prime \prime }+\left (x +2\right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{6} x^{2}+\frac {1}{18} x^{3}-\frac {1}{216} x^{4}-\frac {7}{3240} x^{5}+\frac {19}{19440} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{2}+\frac {1}{36} x^{4}-\frac {1}{108} x^{5}+\frac {1}{648} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{6} x^{2}+\frac {1}{18} x^{3}-\frac {1}{216} x^{4}-\frac {7}{3240} x^{5}\right ) c_{1} +\left (x -\frac {1}{3} x^{2}+\frac {1}{36} x^{4}-\frac {1}{108} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{6} x^{2}+\frac {1}{18} x^{3}-\frac {1}{216} x^{4}-\frac {7}{3240} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{2}+\frac {1}{36} x^{4}-\frac {1}{108} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 11893

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime } x -y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 1+\frac {x^{2}}{2}+\frac {x^{4}}{8}+\frac {x^{6}}{48}+O\left (x^{6}\right ) \] Verified OK. \[ y = 1+\frac {x^{2}}{2}+\frac {x^{4}}{8}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1+\frac {1}{2} x^{2}+\frac {1}{8} x^{4}+\operatorname {O}\left (x^{6}\right ) \]

Problem 11894

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime } x -2 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = x +\frac {x^{3}}{6}-\frac {x^{5}}{120}+O\left (x^{6}\right ) \] Verified OK. \[ y = x +\frac {x^{3}}{6}-\frac {x^{5}}{120}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x +\frac {1}{6} x^{3}-\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 11895

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+y^{\prime } x +2 x y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 3] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 3 x +2-\frac {7 x^{3}}{6}-\frac {x^{4}}{2}+\frac {21 x^{5}}{40}+\frac {31 x^{6}}{90}+O\left (x^{6}\right ) \] Verified OK. \[ y = 2-\frac {7 x^{3}}{6}+\frac {21 x^{5}}{40}+3 x -\frac {x^{4}}{2}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2+3 x -\frac {7}{6} x^{3}-\frac {1}{2} x^{4}+\frac {21}{40} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 11896

ODE	\[ \boxed {\left (2 x^{2}-3\right ) y^{\prime \prime }-2 y^{\prime } x +y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = 5] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = -1+5 x -\frac {x^{2}}{6}-\frac {5 x^{3}}{18}-\frac {x^{4}}{216}-\frac {7 x^{5}}{216}-\frac {17 x^{6}}{19440}+O\left (x^{6}\right ) \] Verified OK. \[ y = -1-\frac {x^{2}}{6}-\frac {x^{4}}{216}+5 x -\frac {5 x^{3}}{18}-\frac {7 x^{5}}{216}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -1+5 x -\frac {1}{6} x^{2}-\frac {5}{18} x^{3}-\frac {1}{216} x^{4}-\frac {7}{216} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 11897

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +y=0} \] With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = \left (1-\frac {\left (x -1\right )^{2}}{2}+\frac {\left (x -1\right )^{3}}{2}-\frac {5 \left (x -1\right )^{4}}{12}+\frac {\left (x -1\right )^{5}}{3}-\frac {19 \left (x -1\right )^{6}}{72}\right ) y \left (1\right )+\left (x -1-\frac {\left (x -1\right )^{2}}{2}+\frac {\left (x -1\right )^{3}}{6}-\frac {\left (x -1\right )^{5}}{12}+\frac {\left (x -1\right )^{6}}{8}\right ) y^{\prime }\left (1\right )+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {\left (-1+x \right )^{2}}{2}+\frac {\left (-1+x \right )^{3}}{2}-\frac {5 \left (-1+x \right )^{4}}{12}+\frac {\left (-1+x \right )^{5}}{3}\right ) y \left (1\right )+\left (-1+x -\frac {\left (-1+x \right )^{2}}{2}+\frac {\left (-1+x \right )^{3}}{6}-\frac {\left (-1+x \right )^{5}}{12}\right ) D\left (y \right )\left (1\right )+O\left (x^{6}\right ) \]

Problem 11898

ODE	\[ \boxed {x^{2} y^{\prime \prime }+3 y^{\prime } x -y=0} \] With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = \left (1+\frac {\left (x -1\right )^{2}}{2}-\frac {5 \left (x -1\right )^{3}}{6}+\frac {7 \left (x -1\right )^{4}}{6}-\frac {91 \left (x -1\right )^{5}}{60}+\frac {679 \left (x -1\right )^{6}}{360}\right ) y \left (1\right )+\left (x -1-\frac {3 \left (x -1\right )^{2}}{2}+\frac {13 \left (x -1\right )^{3}}{6}-\frac {35 \left (x -1\right )^{4}}{12}+\frac {56 \left (x -1\right )^{5}}{15}-\frac {553 \left (x -1\right )^{6}}{120}\right ) y^{\prime }\left (1\right )+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {\left (-1+x \right )^{2}}{2}-\frac {5 \left (-1+x \right )^{3}}{6}+\frac {7 \left (-1+x \right )^{4}}{6}-\frac {91 \left (-1+x \right )^{5}}{60}\right ) y \left (1\right )+\left (-1+x -\frac {3 \left (-1+x \right )^{2}}{2}+\frac {13 \left (-1+x \right )^{3}}{6}-\frac {35 \left (-1+x \right )^{4}}{12}+\frac {56 \left (-1+x \right )^{5}}{15}\right ) D\left (y \right )\left (1\right )+O\left (x^{6}\right ) \]

Problem 11899

ODE	\[ \boxed {x y^{\prime \prime }+y^{\prime }+2 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 2, y^{\prime }\left (1\right ) = 4] \end {align*} With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = -4 \left (x -1\right )^{2}+4 x -2+\frac {4 \left (x -1\right )^{3}}{3}-\frac {\left (x -1\right )^{4}}{3}+\frac {2 \left (x -1\right )^{5}}{15}-\frac {4 \left (x -1\right )^{6}}{45}+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2+4 \left (-1+x \right )-4 \left (-1+x \right )^{2}+\frac {4}{3} \left (-1+x \right )^{3}-\frac {1}{3} \left (-1+x \right )^{4}+\frac {2}{15} \left (-1+x \right )^{5}+\operatorname {O}\left (\left (-1+x \right )^{6}\right ) \]

Problem 11900

ODE	\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +n \left (n +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} n^{2} x^{2}-\frac {1}{2} n \,x^{2}+\frac {1}{24} n^{4} x^{4}+\frac {1}{12} x^{4} n^{3}-\frac {5}{24} n^{2} x^{4}-\frac {1}{4} n \,x^{4}-\frac {1}{720} n^{6} x^{6}-\frac {1}{240} n^{5} x^{6}+\frac {23}{720} n^{4} x^{6}+\frac {17}{240} n^{3} x^{6}-\frac {47}{360} n^{2} x^{6}-\frac {1}{6} n \,x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{6} n^{2} x^{3}-\frac {1}{6} n \,x^{3}+\frac {1}{3} x^{3}+\frac {1}{120} x^{5} n^{4}+\frac {1}{60} x^{5} n^{3}-\frac {13}{120} x^{5} n^{2}-\frac {7}{60} x^{5} n +\frac {1}{5} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\left (-\frac {1}{2} n^{2}-\frac {1}{2} n \right ) x^{2}+\left (-\frac {5}{24} n^{2}-\frac {1}{4} n +\frac {1}{24} n^{4}+\frac {1}{12} n^{3}\right ) x^{4}\right ) c_{1} +\left (x +\left (-\frac {1}{6} n^{2}-\frac {1}{6} n +\frac {1}{3}\right ) x^{3}+\left (-\frac {13}{120} n^{2}-\frac {7}{60} n +\frac {1}{5}+\frac {1}{120} n^{4}+\frac {1}{60} n^{3}\right ) x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {n \left (n +1\right ) x^{2}}{2}+\frac {n \left (n^{3}+2 n^{2}-5 n -6\right ) x^{4}}{24}\right ) y \left (0\right )+\left (x -\frac {\left (n^{2}+n -2\right ) x^{3}}{6}+\frac {\left (n^{4}+2 n^{3}-13 n^{2}-14 n +24\right ) x^{5}}{120}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]