2.17.140 Problems 13901 to 14000

Problem 13901

ODE	\[ \boxed {y^{\prime \prime }+16 y=\delta \left (-2+t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} 0 & t <2 \\ \frac {\sin \left (-8+4 t \right )}{4} & 2\le t \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\operatorname {Heaviside}\left (t -2\right ) \sin \left (4 t -8\right )}{4} \]

Problem 13902

ODE	\[ \boxed {y^{\prime \prime }-16 y=\delta \left (t -10\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} 0 & t <10 \\ \frac {\sinh \left (4 t -40\right )}{4} & 10\le t \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\operatorname {Heaviside}\left (t -10\right ) \sinh \left (-40+4 t \right )}{4} \]

Problem 13903

ODE	\[ \boxed {y^{\prime \prime }+y=\delta \left (t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = -1] \end {align*}
program solution	\[ y = 0 \] Warning, solution could not be verified
Maple solution	\[ y \left (t \right ) = 0 \]

Problem 13904

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }-12 y=\delta \left (t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {{\mathrm e}^{-2 t} \sinh \left (4 t \right )}{4} \] Warning, solution could not be verified
Maple solution	\[ y \left (t \right ) = \frac {{\mathrm e}^{-2 t} \sinh \left (4 t \right )}{4} \]

Problem 13905

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }-12 y=\delta \left (t -3\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} 0 & t <3 \\ \frac {{\mathrm e}^{-2 t +6} \sinh \left (4 t -12\right )}{4} & 3\le t \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\operatorname {Heaviside}\left (t -3\right ) {\mathrm e}^{-2 t +6} \sinh \left (4 t -12\right )}{4} \]

Problem 13906

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+9 y=\delta \left (t -4\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} 0 & t <4 \\ \left (t -4\right ) {\mathrm e}^{-3 t +12} & 4\le t \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (t -4\right ) {\mathrm e}^{-3 t +12} \operatorname {Heaviside}\left (t -4\right ) \]

Problem 13907

ODE	\[ \boxed {y^{\prime \prime }-12 y^{\prime }+45 y=\delta \left (t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {{\mathrm e}^{6 t} \sin \left (3 t \right )}{3} \] Warning, solution could not be verified
Maple solution	\[ y \left (t \right ) = \frac {{\mathrm e}^{6 t} \sin \left (3 t \right )}{3} \]

Problem 13908

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

Problem 13909

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-16 y=\delta \left (t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = 0, y^{\prime \prime \prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = -\frac {\sin \left (2 t \right )}{16}+\frac {\sinh \left (2 t \right )}{16} \] Warning, solution could not be verified
Maple solution	\[ y \left (t \right ) = -\frac {\sin \left (2 t \right )}{16}+\frac {\sinh \left (2 t \right )}{16} \]

Problem 13910

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

Problem 13911

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

Problem 13912

ODE	\[ \boxed {y^{\prime }+\frac {2 y}{2 x -1}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (32 x^{5}+16 x^{4}+8 x^{3}+4 x^{2}+2 x +1\right ) y \left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (32 x^{5}+16 x^{4}+8 x^{3}+4 x^{2}+2 x +1\right ) c_{1} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (32 x^{5}+16 x^{4}+8 x^{3}+4 x^{2}+2 x +1\right ) y \left (0\right )+O\left (x^{6}\right ) \]

Problem 13913

ODE	\[ \boxed {\left (x -3\right ) y^{\prime }-2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {2}{3} x +\frac {1}{9} x^{2}\right ) y \left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {2}{3} x +\frac {1}{9} x^{2}\right ) c_{1} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {y \left (0\right ) \left (-3+x \right )^{2}}{9} \]

Problem 13914

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

Problem 13915

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

Problem 13916

ODE	\[ \boxed {y^{\prime }+\frac {y}{x -1}=0} \] With the expansion point for the power series method at \(x = 3\).
program solution	\[ y = \left (\frac {5}{2}-\frac {x}{2}+\frac {\left (x -3\right )^{2}}{4}-\frac {\left (x -3\right )^{3}}{8}+\frac {\left (x -3\right )^{4}}{16}-\frac {\left (x -3\right )^{5}}{32}\right ) y \left (3\right )+O\left (\left (x -3\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\frac {5}{2}-\frac {x}{2}+\frac {\left (-3+x \right )^{2}}{4}-\frac {\left (-3+x \right )^{3}}{8}+\frac {\left (-3+x \right )^{4}}{16}-\frac {\left (-3+x \right )^{5}}{32}\right ) y \left (3\right )+O\left (x^{6}\right ) \]

Problem 13917

ODE	\[ \boxed {\left (1-x \right ) y^{\prime }-2 y=0} \] With the expansion point for the power series method at \(x = 5\).
program solution	\[ y = \left (\frac {7}{2}-\frac {x}{2}+\frac {3 \left (x -5\right )^{2}}{16}-\frac {\left (x -5\right )^{3}}{16}+\frac {5 \left (x -5\right )^{4}}{256}-\frac {3 \left (x -5\right )^{5}}{512}\right ) y \left (5\right )+O\left (\left (x -5\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\frac {7}{2}-\frac {x}{2}+\frac {3 \left (x -5\right )^{2}}{16}-\frac {\left (x -5\right )^{3}}{16}+\frac {5 \left (x -5\right )^{4}}{256}-\frac {3 \left (x -5\right )^{5}}{512}\right ) y \left (5\right )+O\left (x^{6}\right ) \]

Problem 13918

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

Problem 13919

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

Problem 13920

ODE	\[ \boxed {\left (x +1\right ) y^{\prime }-y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{2} x^{2}-\frac {1}{3} x^{3}+\frac {3}{8} x^{4}-\frac {11}{30} x^{5}\right ) y \left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{2} x^{2}-\frac {1}{3} x^{3}+\frac {3}{8} x^{4}-\frac {11}{30} x^{5}\right ) c_{1} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{2} x^{2}-\frac {1}{3} x^{3}+\frac {3}{8} x^{4}-\frac {11}{30} x^{5}\right ) y \left (0\right )+O\left (x^{6}\right ) \]

Problem 13921

ODE	\[ \boxed {\left (x +1\right ) y^{\prime }+\left (1-x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-x +\frac {3}{2} x^{2}-\frac {11}{6} x^{3}+\frac {53}{24} x^{4}-\frac {103}{40} x^{5}\right ) y \left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-x +\frac {3}{2} x^{2}-\frac {11}{6} x^{3}+\frac {53}{24} x^{4}-\frac {103}{40} x^{5}\right ) c_{1} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-x +\frac {3}{2} x^{2}-\frac {11}{6} x^{3}+\frac {53}{24} x^{4}-\frac {103}{40} x^{5}\right ) y \left (0\right )+O\left (x^{6}\right ) \]

Problem 13922

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

Problem 13923

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 13924

ODE	\[ \boxed {\left (x^{2}+4\right ) y^{\prime \prime }+2 y^{\prime } x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = y \left (0\right )+\left (x -\frac {1}{12} x^{3}+\frac {1}{80} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = c_{1} +\left (x -\frac {1}{12} x^{3}+\frac {1}{80} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = y \left (0\right )+\left (x -\frac {1}{12} x^{3}+\frac {1}{80} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 13925

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

Problem 13926

ODE	\[ \boxed {\left (-x^{2}+4\right ) y^{\prime \prime }-5 y^{\prime } x -3 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {3}{8} x^{2}+\frac {15}{128} x^{4}+\frac {35}{1024} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{3} x^{3}+\frac {1}{10} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {3}{8} x^{2}+\frac {15}{128} x^{4}\right ) c_{1} +\left (x +\frac {1}{3} x^{3}+\frac {1}{10} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {3}{8} x^{2}+\frac {15}{128} x^{4}\right ) y \left (0\right )+\left (x +\frac {1}{3} x^{3}+\frac {1}{10} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 13927

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

Problem 13928

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

Problem 13929

ODE	\[ \boxed {\left (x^{2}-6 x \right ) y^{\prime \prime }+4 \left (x -3\right ) y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1+\frac {x}{6}+\frac {x^{2}}{36}+\frac {x^{3}}{216}+\frac {x^{4}}{1296}+\frac {x^{5}}{7776}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+\frac {x}{6}+\frac {x^{2}}{36}+\frac {x^{3}}{216}+\frac {x^{4}}{1296}+\frac {x^{5}}{7776}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (1+\frac {1}{6} x +\frac {1}{36} x^{2}+\frac {1}{216} x^{3}+\frac {1}{1296} x^{4}+\frac {1}{7776} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (1+\frac {1}{3} x +\frac {1}{18} x^{2}+\frac {1}{108} x^{3}+\frac {1}{648} x^{4}+\frac {1}{3888} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 13930

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

Problem 13931

ODE	\[ \boxed {\left (x^{2}-2 x +2\right ) y^{\prime \prime }+\left (1-x \right ) y^{\prime }-3 y=0} \] With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = \left (1+\frac {3 \left (x -1\right )^{2}}{2}+\frac {3 \left (x -1\right )^{4}}{8}-\frac {\left (x -1\right )^{6}}{16}\right ) y \left (1\right )+\left (x -1+\frac {2 \left (x -1\right )^{3}}{3}\right ) y^{\prime }\left (1\right )+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {3 \left (-1+x \right )^{2}}{2}+\frac {3 \left (-1+x \right )^{4}}{8}\right ) y \left (1\right )+\left (-1+x +\frac {2 \left (-1+x \right )^{3}}{3}\right ) D\left (y \right )\left (1\right )+O\left (x^{6}\right ) \]

Problem 13932

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }-y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{30} x^{5}+\frac {1}{60} x^{6}\right ) y \left (0\right )+\left (x +x^{2}+\frac {2}{3} x^{3}+\frac {5}{12} x^{4}+\frac {13}{60} x^{5}+\frac {17}{180} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{30} x^{5}\right ) c_{1} +\left (x +x^{2}+\frac {2}{3} x^{3}+\frac {5}{12} x^{4}+\frac {13}{60} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{30} x^{5}\right ) y \left (0\right )+\left (x +x^{2}+\frac {2}{3} x^{3}+\frac {5}{12} x^{4}+\frac {13}{60} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 13933

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime } x -2 y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{3} x^{3}+\frac {1}{20} x^{5}+\frac {1}{45} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{6} x^{4}+\frac {1}{40} x^{5}+\frac {1}{30} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{3} x^{3}+\frac {1}{20} x^{5}\right ) c_{1} +\left (x +\frac {1}{6} x^{3}+\frac {1}{6} x^{4}+\frac {1}{40} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{3} x^{3}+\frac {1}{20} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{6} x^{4}+\frac {1}{40} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 13934

ODE	\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x +\lambda y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} \lambda \,x^{2}+\frac {1}{24} \lambda ^{2} x^{4}-\frac {1}{6} \lambda \,x^{4}-\frac {1}{720} x^{6} \lambda ^{3}+\frac {1}{36} x^{6} \lambda ^{2}-\frac {4}{45} x^{6} \lambda \right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3} \lambda +\frac {1}{6} x^{3}+\frac {1}{120} x^{5} \lambda ^{2}-\frac {1}{12} x^{5} \lambda +\frac {3}{40} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {\lambda \,x^{2}}{2}+\left (\frac {1}{24} \lambda ^{2}-\frac {1}{6} \lambda \right ) x^{4}\right ) c_{1} +\left (x +\left (-\frac {\lambda }{6}+\frac {1}{6}\right ) x^{3}+\left (\frac {1}{120} \lambda ^{2}-\frac {1}{12} \lambda +\frac {3}{40}\right ) x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {x^{2} \lambda }{2}+\frac {\lambda \left (\lambda -4\right ) x^{4}}{24}\right ) y \left (0\right )+\left (x -\frac {\left (\lambda -1\right ) x^{3}}{6}+\frac {\left (\lambda -1\right ) \left (\lambda -9\right ) x^{5}}{120}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 13935

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

Problem 13936

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

Problem 13937

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

Problem 13938

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

Problem 13939

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

Problem 13940

ODE	\[ \boxed {y^{\prime \prime }+y x=\sin \left (x \right )} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {x^{3}}{6}\right ) y \left (0\right )+\left (x -\frac {1}{12} x^{4}\right ) y^{\prime }\left (0\right )+\frac {x^{3}}{6}-\frac {x^{5}}{120}+O\left (x^{5}\right ) \] Verified OK. \[ y = \left (1-\frac {x^{3}}{6}\right ) c_{1} +\left (x -\frac {1}{12} x^{4}\right ) c_{2} +\frac {x^{3}}{6}+O\left (x^{5}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {x^{3}}{6}\right ) y \left (0\right )+\left (x -\frac {1}{12} x^{4}\right ) D\left (y \right )\left (0\right )+\frac {x^{3}}{6}+O\left (x^{5}\right ) \]

Problem 13941

ODE	\[ \boxed {y^{\prime \prime }-\sin \left (x \right ) y^{\prime }-y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{6} x^{3}+\frac {1}{40} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{60} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{5}\right ) \] Verified OK. \[ y = \left (1+\frac {x^{3}}{6}\right ) c_{1} +\left (x +\frac {1}{6} x^{3}+\frac {1}{12} x^{4}\right ) c_{2} +O\left (x^{5}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {x^{3}}{6}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{12} x^{4}\right ) D\left (y \right )\left (0\right )+O\left (x^{5}\right ) \]

Problem 13942

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

Problem 13943

ODE	\[ \boxed {y^{\prime }+\cos \left (y\right )=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = y \left (0\right )-\cos \left (y \left (0\right )\right ) x -\frac {x^{2} \sin \left (y \left (0\right )\right ) \cos \left (y \left (0\right )\right )}{2}+\frac {\cos \left (y \left (0\right )\right )^{3} x^{3}}{3}-\frac {\cos \left (y \left (0\right )\right ) x^{3}}{6}+\frac {x^{4} \cos \left (y \left (0\right )\right )^{3} \sin \left (y \left (0\right )\right )}{4}-\frac {x^{4} \sin \left (y \left (0\right )\right ) \cos \left (y \left (0\right )\right )}{24}+O\left (x^{5}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = y \left (0\right )-\cos \left (y \left (0\right )\right ) x -\frac {\sin \left (2 y \left (0\right )\right ) x^{2}}{4}+\frac {\cos \left (y \left (0\right )\right ) \cos \left (2 y \left (0\right )\right ) x^{3}}{6}+\left (\frac {\sin \left (4 y \left (0\right )\right )}{32}+\frac {\sin \left (2 y \left (0\right )\right )}{24}\right ) x^{4}+O\left (x^{5}\right ) \]

Problem 13944

ODE	\[ \boxed {y^{\prime }-y \,{\mathrm e}^{x}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+x +x^{2}+\frac {5}{6} x^{3}+\frac {5}{8} x^{4}+\frac {13}{30} x^{5}\right ) y \left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+x +x^{2}+\frac {5}{6} x^{3}+\frac {5}{8} x^{4}+\frac {13}{30} x^{5}\right ) c_{1} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+x +x^{2}+\frac {5}{6} x^{3}+\frac {5}{8} x^{4}+\frac {13}{30} x^{5}\right ) y \left (0\right )+O\left (x^{6}\right ) \]

Problem 13945

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

Problem 13946

ODE	\[ \boxed {\sin \left (x \right ) y^{\prime \prime }+x^{2} y^{\prime }-y \,{\mathrm e}^{x}=0} \] With the expansion point for the power series method at \(x = 2\).
program solution	\[ \text {Expression too large to display} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {\csc \left (2\right ) {\mathrm e}^{2} \left (x -2\right )^{2}}{2}-\frac {{\mathrm e}^{2} \left (4+\cos \left (2\right )-\sin \left (2\right )\right ) \csc \left (2\right )^{2} \left (x -2\right )^{3}}{6}+\frac {\csc \left (2\right )^{3} \left (\left (\frac {3}{2}-\frac {\sin \left (4\right )}{12}-\sin \left (2\right )+\cos \left (2\right )\right ) {\mathrm e}^{2}+\frac {{\mathrm e}^{4} \sin \left (2\right )}{12}\right ) \left (x -2\right )^{4}}{2}+\frac {\left (\left (-210+56 \sin \left (4\right )+\sin \left (6\right )+201 \sin \left (2\right )+\cos \left (6\right )-205 \cos \left (2\right )-6 \cos \left (4\right )\right ) {\mathrm e}^{2}-4 \,{\mathrm e}^{4} \left (-1+\sin \left (4\right )+\cos \left (4\right )+4 \sin \left (2\right )\right )\right ) \csc \left (2\right )^{4} \left (x -2\right )^{5}}{240}\right ) y \left (2\right )+\left (x -2-2 \csc \left (2\right ) \left (x -2\right )^{2}-\frac {\left (-{\mathrm e}^{2} \sin \left (2\right )-4 \cos \left (2\right )+4 \sin \left (2\right )-16\right ) \csc \left (2\right )^{2} \left (x -2\right )^{3}}{6}+\frac {\csc \left (2\right )^{3} \left (\left (-\frac {\cos \left (4\right )}{12}-\frac {2 \sin \left (2\right )}{3}-\frac {\sin \left (4\right )}{12}+\frac {1}{12}\right ) {\mathrm e}^{2}-4 \cos \left (2\right )-\frac {\cos \left (4\right )}{12}+4 \sin \left (2\right )+\frac {\sin \left (4\right )}{3}-\frac {71}{12}\right ) \left (x -2\right )^{4}}{2}+\frac {\left (\left (\left (-12 \cos \left (2\right )-72\right ) \sin \left (2\right )^{2}+108 \sin \left (2\right )+36 \sin \left (4\right )\right ) {\mathrm e}^{2}+2 \sin \left (2\right )^{2} {\mathrm e}^{4}-6 \sin \left (6\right )+817 \cos \left (2\right )+32 \cos \left (4\right )-\cos \left (6\right )-798 \sin \left (2\right )-224 \sin \left (4\right )+832\right ) \csc \left (2\right )^{4} \left (x -2\right )^{5}}{240}\right ) D\left (y \right )\left (2\right )+O\left (x^{6}\right ) \]

Problem 13947

ODE	\[ \boxed {\sinh \left (x \right ) y^{\prime \prime }+x^{2} y^{\prime }-y \,{\mathrm e}^{x}=0} \] With the expansion point for the power series method at \(x = 2\).
program solution	\[ \text {Expression too large to display} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {{\mathrm e}^{4} \left (x -2\right )^{2}}{{\mathrm e}^{4}-1}-\frac {2 \left ({\mathrm e}^{2}+4 \,{\mathrm e}^{4}\right ) {\mathrm e}^{2} \left (x -2\right )^{3}}{3 \left ({\mathrm e}^{4}-1\right )^{2}}+\frac {\left ({\mathrm e}^{2}+12 \,{\mathrm e}^{4}+\frac {33 \,{\mathrm e}^{6}}{2}+\frac {{\mathrm e}^{10}}{2}\right ) {\mathrm e}^{2} \left (x -2\right )^{4}}{3 \left ({\mathrm e}^{4}-1\right )^{3}}-\frac {2 \,{\mathrm e}^{2} \left ({\mathrm e}^{2}+\frac {53 \,{\mathrm e}^{4}}{2}+98 \,{\mathrm e}^{6}+79 \,{\mathrm e}^{8}+3 \,{\mathrm e}^{10}+\frac {5 \,{\mathrm e}^{12}}{2}\right ) \left (x -2\right )^{5}}{15 \left ({\mathrm e}^{4}-1\right )^{4}}\right ) y \left (2\right )+\left (x -2-\frac {4 \,{\mathrm e}^{2} \left (x -2\right )^{2}}{{\mathrm e}^{4}-1}-\frac {2 \left (-\frac {31 \,{\mathrm e}^{2}}{2}-\frac {{\mathrm e}^{6}}{2}-4\right ) {\mathrm e}^{2} \left (x -2\right )^{3}}{3 \left ({\mathrm e}^{4}-1\right )^{2}}+\frac {\left (-47 \,{\mathrm e}^{2}-65 \,{\mathrm e}^{4}-{\mathrm e}^{6}-\frac {7 \,{\mathrm e}^{8}}{2}-\frac {7}{2}\right ) {\mathrm e}^{2} \left (x -2\right )^{4}}{3 \left ({\mathrm e}^{4}-1\right )^{3}}-\frac {2 \,{\mathrm e}^{2} \left (-\frac {205 \,{\mathrm e}^{2}}{2}-\frac {1537 \,{\mathrm e}^{4}}{4}-\frac {1249 \,{\mathrm e}^{6}}{4}-\frac {85 \,{\mathrm e}^{8}}{4}-17 \,{\mathrm e}^{10}+\frac {{\mathrm e}^{12}}{4}-\frac {{\mathrm e}^{14}}{4}-\frac {11}{4}\right ) \left (x -2\right )^{5}}{15 \left ({\mathrm e}^{4}-1\right )^{4}}\right ) D\left (y \right )\left (2\right )+O\left (x^{6}\right ) \]

Problem 13948

ODE	\[ \boxed {\sinh \left (x \right ) y^{\prime \prime }+x^{2} y^{\prime }-y \sin \left (x \right )=0} \] With the expansion point for the power series method at \(x = 2\).
program solution	\[ \text {Expression too large to display} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {\sin \left (2\right ) {\mathrm e}^{2} \left (x -2\right )^{2}}{{\mathrm e}^{4}-1}+\frac {{\mathrm e}^{2} \left (\left (-8 \,{\mathrm e}^{2}-{\mathrm e}^{4}-1\right ) \sin \left (2\right )+\cos \left (2\right ) \left ({\mathrm e}^{4}-1\right )\right ) \left (x -2\right )^{3}}{3 \left ({\mathrm e}^{4}-1\right )^{2}}+\frac {2 \left (\frac {\left (-{\mathrm e}^{2}+{\mathrm e}^{6}\right ) \sin \left (2\right )^{2}}{4}+\left (5 \,{\mathrm e}^{2}+9 \,{\mathrm e}^{4}+{\mathrm e}^{6}\right ) \sin \left (2\right )+\cos \left (2\right ) \left ({\mathrm e}^{2}-{\mathrm e}^{6}-\frac {{\mathrm e}^{8}}{4}+\frac {1}{4}\right )\right ) {\mathrm e}^{2} \left (x -2\right )^{4}}{3 \left ({\mathrm e}^{4}-1\right )^{3}}+\frac {2 \left (\left (\left ({\mathrm e}^{2}-2 \,{\mathrm e}^{6}+{\mathrm e}^{10}\right ) \sin \left (2\right )-8 \,{\mathrm e}^{2}-\frac {41 \,{\mathrm e}^{4}}{4}+6 \,{\mathrm e}^{6}+\frac {41 \,{\mathrm e}^{8}}{4}+2 \,{\mathrm e}^{10}+\frac {{\mathrm e}^{12}}{4}-\frac {1}{4}\right ) \cos \left (2\right )+\left (\left ({\mathrm e}^{2}+4 \,{\mathrm e}^{4}-4 \,{\mathrm e}^{8}-{\mathrm e}^{10}\right ) \sin \left (2\right )-\frac {33 \,{\mathrm e}^{2}}{2}-\frac {365 \,{\mathrm e}^{4}}{4}-93 \,{\mathrm e}^{6}-\frac {45 \,{\mathrm e}^{8}}{4}+\frac {3 \,{\mathrm e}^{10}}{2}+\frac {{\mathrm e}^{12}}{4}+\frac {1}{4}\right ) \sin \left (2\right )\right ) {\mathrm e}^{2} \left (x -2\right )^{5}}{15 \left ({\mathrm e}^{4}-1\right )^{4}}\right ) y \left (2\right )+\left (x -2-\frac {4 \,{\mathrm e}^{2} \left (x -2\right )^{2}}{{\mathrm e}^{4}-1}+\frac {{\mathrm e}^{2} \left (\left ({\mathrm e}^{4}-1\right ) \sin \left (2\right )+32 \,{\mathrm e}^{2}+8\right ) \left (x -2\right )^{3}}{3 \left ({\mathrm e}^{4}-1\right )^{2}}+\frac {2 \left (-\frac {7}{4}+\left (2 \,{\mathrm e}^{2}-2 \,{\mathrm e}^{6}-\frac {{\mathrm e}^{8}}{4}+\frac {1}{4}\right ) \sin \left (2\right )+\frac {\left (\frac {1}{2}-{\mathrm e}^{4}+\frac {{\mathrm e}^{8}}{2}\right ) \cos \left (2\right )}{2}-24 \,{\mathrm e}^{2}-\frac {69 \,{\mathrm e}^{4}}{2}+\frac {{\mathrm e}^{8}}{4}\right ) {\mathrm e}^{2} \left (x -2\right )^{4}}{3 \left ({\mathrm e}^{4}-1\right )^{3}}+\frac {2 \left (\left (-\frac {{\mathrm e}^{2}}{4}-\frac {{\mathrm e}^{10}}{4}+\frac {{\mathrm e}^{6}}{2}\right ) \cos \left (2\right )^{2}+\left (-\frac {3}{4}+\frac {3 \,{\mathrm e}^{4}}{4}+\frac {3 \,{\mathrm e}^{8}}{4}-\frac {3 \,{\mathrm e}^{12}}{4}-5 \,{\mathrm e}^{10}+10 \,{\mathrm e}^{6}-5 \,{\mathrm e}^{2}\right ) \cos \left (2\right )+\left (-13 \,{\mathrm e}^{2}-27 \,{\mathrm e}^{4}+8 \,{\mathrm e}^{6}+27 \,{\mathrm e}^{8}+5 \,{\mathrm e}^{10}\right ) \sin \left (2\right )+\frac {11}{4}+\frac {13 \,{\mathrm e}^{8}}{4}-\frac {{\mathrm e}^{12}}{4}-\frac {31 \,{\mathrm e}^{10}}{4}+\frac {1609 \,{\mathrm e}^{4}}{4}+\frac {417 \,{\mathrm e}^{2}}{4}+\frac {671 \,{\mathrm e}^{6}}{2}\right ) {\mathrm e}^{2} \left (x -2\right )^{5}}{15 \left ({\mathrm e}^{4}-1\right )^{4}}\right ) D\left (y \right )\left (2\right )+O\left (x^{6}\right ) \]

Problem 13949

ODE	\[ \boxed {{\mathrm e}^{3 x} y^{\prime \prime }+\sin \left (x \right ) y^{\prime }+\frac {2 y}{x^{2}+4}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{4} x^{2}+\frac {1}{4} x^{3}-\frac {1}{8} x^{4}-\frac {7}{160} x^{5}+\frac {227}{1440} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{4} x^{3}+\frac {3}{8} x^{4}-\frac {67}{240} x^{5}+\frac {5}{96} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{4} x^{2}+\frac {1}{4} x^{3}-\frac {1}{8} x^{4}-\frac {7}{160} x^{5}\right ) c_{1} +\left (x -\frac {1}{4} x^{3}+\frac {3}{8} x^{4}-\frac {67}{240} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{4} x^{2}+\frac {1}{4} x^{3}-\frac {1}{8} x^{4}-\frac {7}{160} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{4} x^{3}+\frac {3}{8} x^{4}-\frac {67}{240} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 13950

ODE	\[ \boxed {y^{\prime \prime }+\frac {\left ({\mathrm e}^{x}+1\right ) y}{1-{\mathrm e}^{x}}=0} \] With the expansion point for the power series method at \(x = 3\).
program solution	\[ \text {Expression too large to display} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {\left ({\mathrm e}^{3}+1\right ) \left (-3+x \right )^{2}}{2 \,{\mathrm e}^{3}-2}-\frac {{\mathrm e}^{3} \left (-3+x \right )^{3}}{3 \left ({\mathrm e}^{3}-1\right )^{2}}+\frac {\left ({\mathrm e}^{3}+3 \,{\mathrm e}^{6}+{\mathrm e}^{9}-1\right ) \left (-3+x \right )^{4}}{24 \left ({\mathrm e}^{3}-1\right )^{3}}+\frac {\left (6 \,{\mathrm e}^{3}-8 \,{\mathrm e}^{6}-10 \,{\mathrm e}^{9}\right ) \left (-3+x \right )^{5}}{120 \left ({\mathrm e}^{3}-1\right )^{4}}\right ) y \left (3\right )+\left (-3+x +\frac {\left ({\mathrm e}^{6}-1\right ) \left (-3+x \right )^{3}}{6 \left ({\mathrm e}^{3}-1\right )^{2}}+\frac {\left ({\mathrm e}^{3}-{\mathrm e}^{6}\right ) \left (-3+x \right )^{4}}{6 \left ({\mathrm e}^{3}-1\right )^{3}}+\frac {\left (-6 \,{\mathrm e}^{3}-2 \,{\mathrm e}^{6}+6 \,{\mathrm e}^{9}+{\mathrm e}^{12}+1\right ) \left (-3+x \right )^{5}}{120 \left ({\mathrm e}^{3}-1\right )^{4}}\right ) D\left (y \right )\left (3\right )+O\left (x^{6}\right ) \]

Problem 13951

ODE	\[ \boxed {\left (x^{2}-4\right ) y^{\prime \prime }+\left (x^{2}+x -6\right ) y=0} \] With the expansion point for the power series method at \(x = 2\).
program solution	\[ y = \left (1-\frac {5 \left (x -2\right )^{2}}{8}+\frac {\left (x -2\right )^{3}}{96}+\frac {49 \left (x -2\right )^{4}}{768}-\frac {37 \left (x -2\right )^{5}}{15360}-\frac {3 \left (x -2\right )^{6}}{1280}\right ) y \left (2\right )+\left (x -2-\frac {5 \left (x -2\right )^{3}}{24}+\frac {\left (x -2\right )^{4}}{192}+\frac {47 \left (x -2\right )^{5}}{3840}-\frac {\left (x -2\right )^{6}}{1920}\right ) y^{\prime }\left (2\right )+O\left (\left (x -2\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {5 \left (x -2\right )^{2}}{8}+\frac {\left (x -2\right )^{3}}{96}+\frac {49 \left (x -2\right )^{4}}{768}-\frac {37 \left (x -2\right )^{5}}{15360}\right ) y \left (2\right )+\left (x -2-\frac {5 \left (x -2\right )^{3}}{24}+\frac {\left (x -2\right )^{4}}{192}+\frac {47 \left (x -2\right )^{5}}{3840}\right ) D\left (y \right )\left (2\right )+O\left (x^{6}\right ) \]

Problem 13952

ODE	\[ \boxed {x y^{\prime \prime }+\left (1-{\mathrm e}^{x}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{12} x^{3}+\frac {1}{18} x^{4}+\frac {3}{160} x^{5}+\frac {17}{2700} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{60} x^{5}+\frac {1}{180} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{12} x^{3}+\frac {1}{18} x^{4}+\frac {3}{160} x^{5}\right ) c_{1} +\left (x +\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{60} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{2} x^{2}+\frac {1}{12} x^{3}+\frac {1}{18} x^{4}+\frac {3}{160} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{60} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 13953

ODE	\[ \boxed {\sin \left (\pi \,x^{2}\right ) y^{\prime \prime }+y x^{2}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {x^{2}}{2 \pi }+\frac {x^{4}}{24 \pi ^{2}}-\frac {\pi \,x^{6}}{180}-\frac {x^{6}}{720 \pi ^{3}}\right ) y \left (0\right )+\left (x -\frac {x^{3}}{6 \pi }+\frac {x^{5}}{120 \pi ^{2}}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {x^{2}}{2 \pi }+\frac {x^{4}}{24 \pi ^{2}}\right ) c_{1} +\left (x -\frac {x^{3}}{6 \pi }+\frac {x^{5}}{120 \pi ^{2}}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {x^{2}}{2 \pi }+\frac {x^{4}}{24 \pi ^{2}}\right ) y \left (0\right )+\left (x -\frac {x^{3}}{6 \pi }+\frac {x^{5}}{120 \pi ^{2}}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 13954

ODE	\[ \boxed {y^{\prime }-y \,{\mathrm e}^{x}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+x +x^{2}+\frac {5}{6} x^{3}+\frac {5}{8} x^{4}+\frac {13}{30} x^{5}\right ) y \left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+x +x^{2}+\frac {5}{6} x^{3}+\frac {5}{8} x^{4}+\frac {13}{30} x^{5}\right ) c_{1} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+x +x^{2}+\frac {5}{6} x^{3}+\frac {5}{8} x^{4}+\frac {13}{30} x^{5}\right ) y \left (0\right )+O\left (x^{6}\right ) \]

Problem 13955

ODE	\[ \boxed {y^{\prime }+{\mathrm e}^{2 x} y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {3}{8} x^{4}+\frac {23}{120} x^{5}\right ) y \left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {3}{8} x^{4}+\frac {23}{120} x^{5}\right ) c_{1} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {3}{8} x^{4}+\frac {23}{120} x^{5}\right ) y \left (0\right )+O\left (x^{6}\right ) \]

Problem 13956

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

Problem 13957

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

Problem 13958

ODE	\[ \boxed {y^{\prime \prime }-y \,{\mathrm e}^{x}=0} \] With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = \left (1+\frac {{\mathrm e} \left (x -1\right )^{2}}{2}+\frac {\left (x -1\right )^{3} {\mathrm e}}{6}+\frac {\left (x -1\right )^{4} {\mathrm e}^{2}}{24}+\frac {\left (x -1\right )^{4} {\mathrm e}}{24}+\frac {\left (x -1\right )^{5} {\mathrm e}^{2}}{30}+\frac {\left (x -1\right )^{5} {\mathrm e}}{120}+\frac {{\mathrm e}^{3} \left (x -1\right )^{6}}{720}+\frac {11 \left (x -1\right )^{6} {\mathrm e}^{2}}{720}+\frac {\left (x -1\right )^{6} {\mathrm e}}{720}\right ) y \left (1\right )+\left (x -1+\frac {\left (x -1\right )^{3} {\mathrm e}}{6}+\frac {\left (x -1\right )^{4} {\mathrm e}}{12}+\frac {\left (x -1\right )^{5} {\mathrm e}^{2}}{120}+\frac {\left (x -1\right )^{5} {\mathrm e}}{40}+\frac {\left (x -1\right )^{6} {\mathrm e}^{2}}{120}+\frac {\left (x -1\right )^{6} {\mathrm e}}{180}\right ) y^{\prime }\left (1\right )+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {{\mathrm e} \left (-1+x \right )^{2}}{2}+\frac {{\mathrm e} \left (-1+x \right )^{3}}{6}+\left (\frac {{\mathrm e}^{2}}{24}+\frac {{\mathrm e}}{24}\right ) \left (-1+x \right )^{4}+\left (\frac {{\mathrm e}^{2}}{30}+\frac {{\mathrm e}}{120}\right ) \left (-1+x \right )^{5}\right ) y \left (1\right )+\left (-1+x +\frac {{\mathrm e} \left (-1+x \right )^{3}}{6}+\frac {{\mathrm e} \left (-1+x \right )^{4}}{12}+\frac {\left (3 \,{\mathrm e}+{\mathrm e}^{2}\right ) \left (-1+x \right )^{5}}{120}\right ) D\left (y \right )\left (1\right )+O\left (x^{6}\right ) \]

Problem 13959

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime } x -y \,{\mathrm e}^{x}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{6} x^{4}-\frac {1}{30} x^{5}+\frac {11}{144} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{12} x^{4}+\frac {19}{120} x^{5}-\frac {13}{360} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{6} x^{4}-\frac {1}{30} x^{5}\right ) c_{1} +\left (x -\frac {1}{3} x^{3}+\frac {1}{12} x^{4}+\frac {19}{120} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{6} x^{4}-\frac {1}{30} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{12} x^{4}+\frac {19}{120} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 13960

ODE	\[ \boxed {x y^{\prime \prime }-3 y^{\prime } x +y \sin \left (x \right )=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{2} x^{3}-\frac {23}{72} x^{4}\right ) y \left (0\right )+\left (x +\frac {3}{2} x^{2}+\frac {4}{3} x^{3}+\frac {7}{8} x^{4}\right ) y^{\prime }\left (0\right )+O\left (x^{4}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{2} x^{3}\right ) c_{1} +\left (x +\frac {3}{2} x^{2}+\frac {4}{3} x^{3}\right ) c_{2} +O\left (x^{4}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}-\frac {1}{2} x^{3}\right ) y \left (0\right )+\left (x +\frac {3}{2} x^{2}+\frac {4}{3} x^{3}\right ) D\left (y \right )\left (0\right )+O\left (x^{4}\right ) \]

Problem 13961

ODE	\[ \boxed {y^{\prime \prime }+\ln \left (x \right ) y=0} \] With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = \left (1-\frac {\left (x -1\right )^{3}}{6}+\frac {\left (x -1\right )^{4}}{24}-\frac {\left (x -1\right )^{5}}{60}+\frac {\left (x -1\right )^{6}}{72}\right ) y \left (1\right )+\left (x -1-\frac {\left (x -1\right )^{4}}{12}+\frac {\left (x -1\right )^{5}}{40}-\frac {\left (x -1\right )^{6}}{90}\right ) y^{\prime }\left (1\right )+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {\left (-1+x \right )^{3}}{6}+\frac {\left (-1+x \right )^{4}}{24}-\frac {\left (-1+x \right )^{5}}{60}\right ) y \left (1\right )+\left (-1+x -\frac {\left (-1+x \right )^{4}}{12}+\frac {\left (-1+x \right )^{5}}{40}\right ) D\left (y \right )\left (1\right )+O\left (x^{6}\right ) \]

Problem 13962

ODE	\[ \boxed {\sqrt {x}\, y^{\prime \prime }+y=0} \] With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = \left (1-\frac {\left (x -1\right )^{2}}{2}+\frac {\left (x -1\right )^{3}}{12}+\frac {\left (x -1\right )^{4}}{96}-\frac {\left (x -1\right )^{5}}{960}-\frac {7 \left (x -1\right )^{6}}{3840}\right ) y \left (1\right )+\left (x -1-\frac {\left (x -1\right )^{3}}{6}+\frac {\left (x -1\right )^{4}}{24}-\frac {\left (x -1\right )^{5}}{96}+\frac {\left (x -1\right )^{6}}{160}\right ) y^{\prime }\left (1\right )+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {\left (-1+x \right )^{2}}{2}+\frac {\left (-1+x \right )^{3}}{12}+\frac {\left (-1+x \right )^{4}}{96}-\frac {\left (-1+x \right )^{5}}{960}\right ) y \left (1\right )+\left (-1+x -\frac {\left (-1+x \right )^{3}}{6}+\frac {\left (-1+x \right )^{4}}{24}-\frac {\left (-1+x \right )^{5}}{96}\right ) D\left (y \right )\left (1\right )+O\left (x^{6}\right ) \]

Problem 13963

ODE	\[ \boxed {y^{\prime \prime }+\left (6 x^{2}+2 x +1\right ) y^{\prime }+\left (2+12 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-x^{2}-\frac {5}{3} x^{3}+\frac {11}{12} x^{4}+\frac {101}{60} x^{5}+\frac {389}{360} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}-\frac {1}{2} x^{3}-\frac {9}{8} x^{4}+\frac {41}{40} x^{5}+\frac {169}{240} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-x^{2}-\frac {5}{3} x^{3}+\frac {11}{12} x^{4}+\frac {101}{60} x^{5}\right ) c_{1} +\left (x -\frac {1}{2} x^{2}-\frac {1}{2} x^{3}-\frac {9}{8} x^{4}+\frac {41}{40} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-x^{2}-\frac {5}{3} x^{3}+\frac {11}{12} x^{4}+\frac {101}{60} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}-\frac {1}{2} x^{3}-\frac {9}{8} x^{4}+\frac {41}{40} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 13964

ODE	\[ \boxed {y^{\prime }-y \,{\mathrm e}^{x}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+x +x^{2}+\frac {5}{6} x^{3}+\frac {5}{8} x^{4}+\frac {13}{30} x^{5}+\frac {203}{720} x^{6}+\frac {877}{5040} x^{7}+\frac {23}{224} x^{8}+\frac {1007}{17280} x^{9}\right ) y \left (0\right )+O\left (x^{10}\right ) \] Verified OK. \[ y = \left (1+x +x^{2}+\frac {5}{6} x^{3}+\frac {5}{8} x^{4}+\frac {13}{30} x^{5}+\frac {203}{720} x^{6}+\frac {877}{5040} x^{7}+\frac {23}{224} x^{8}+\frac {1007}{17280} x^{9}\right ) c_{1} +O\left (x^{10}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+x +x^{2}+\frac {5}{6} x^{3}+\frac {5}{8} x^{4}+\frac {13}{30} x^{5}+\frac {203}{720} x^{6}+\frac {877}{5040} x^{7}+\frac {23}{224} x^{8}+\frac {1007}{17280} x^{9}\right ) y \left (0\right )+O\left (x^{10}\right ) \]

Problem 13965

ODE	\[ \boxed {y^{\prime }+\sqrt {x^{2}+1}\, y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-x +\frac {1}{2} x^{2}-\frac {1}{3} x^{3}+\frac {5}{24} x^{4}-\frac {1}{15} x^{5}+\frac {13}{720} x^{6}-\frac {11}{630} x^{7}\right ) y \left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-x +\frac {1}{2} x^{2}-\frac {1}{3} x^{3}+\frac {5}{24} x^{4}-\frac {1}{15} x^{5}+\frac {13}{720} x^{6}-\frac {11}{630} x^{7}\right ) c_{1} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-x +\frac {1}{2} x^{2}-\frac {1}{3} x^{3}+\frac {5}{24} x^{4}-\frac {1}{15} x^{5}+\frac {13}{720} x^{6}-\frac {11}{630} x^{7}\right ) y \left (0\right )+O\left (x^{8}\right ) \]

Problem 13966

ODE	\[ \boxed {\cos \left (x \right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-x +\frac {1}{2} x^{2}-\frac {1}{3} x^{3}+\frac {5}{24} x^{4}-\frac {2}{15} x^{5}+\frac {61}{720} x^{6}-\frac {17}{315} x^{7}\right ) y \left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-x +\frac {1}{2} x^{2}-\frac {1}{3} x^{3}+\frac {5}{24} x^{4}-\frac {2}{15} x^{5}+\frac {61}{720} x^{6}-\frac {17}{315} x^{7}\right ) c_{1} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-x +\frac {1}{2} x^{2}-\frac {1}{3} x^{3}+\frac {5}{24} x^{4}-\frac {2}{15} x^{5}+\frac {61}{720} x^{6}-\frac {17}{315} x^{7}\right ) y \left (0\right )+O\left (x^{8}\right ) \]

Problem 13967

ODE	\[ \boxed {y^{\prime }+\sqrt {2 x^{2}+1}\, y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-x +\frac {1}{2} x^{2}-\frac {1}{2} x^{3}+\frac {3}{8} x^{4}-\frac {3}{40} x^{5}+\frac {1}{80} x^{6}-\frac {51}{560} x^{7}\right ) y \left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-x +\frac {1}{2} x^{2}-\frac {1}{2} x^{3}+\frac {3}{8} x^{4}-\frac {3}{40} x^{5}+\frac {1}{80} x^{6}-\frac {51}{560} x^{7}\right ) c_{1} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-x +\frac {1}{2} x^{2}-\frac {1}{2} x^{3}+\frac {3}{8} x^{4}-\frac {3}{40} x^{5}+\frac {1}{80} x^{6}-\frac {51}{560} x^{7}\right ) y \left (0\right )+O\left (x^{8}\right ) \]

Problem 13968

ODE	\[ \boxed {y^{\prime \prime }-y \,{\mathrm e}^{x}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{24} x^{5}+\frac {13}{720} x^{6}+\frac {1}{140} x^{7}+\frac {109}{40320} x^{8}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{30} x^{5}+\frac {1}{72} x^{6}+\frac {29}{5040} x^{7}+\frac {1}{448} x^{8}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{24} x^{5}+\frac {13}{720} x^{6}+\frac {1}{140} x^{7}\right ) c_{1} +\left (x +\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{30} x^{5}+\frac {1}{72} x^{6}+\frac {29}{5040} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{24} x^{5}+\frac {13}{720} x^{6}+\frac {1}{140} x^{7}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{30} x^{5}+\frac {1}{72} x^{6}+\frac {29}{5040} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 13969

ODE	\[ \boxed {y^{\prime \prime }+\cos \left (x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{12} x^{4}-\frac {1}{80} x^{6}+\frac {11}{8064} x^{8}-\frac {17}{129600} x^{10}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {1}{30} x^{5}-\frac {19}{5040} x^{7}+\frac {29}{72576} x^{9}\right ) y^{\prime }\left (0\right )+O\left (x^{10}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{12} x^{4}-\frac {1}{80} x^{6}+\frac {11}{8064} x^{8}\right ) c_{1} +\left (x -\frac {1}{6} x^{3}+\frac {1}{30} x^{5}-\frac {19}{5040} x^{7}+\frac {29}{72576} x^{9}\right ) c_{2} +O\left (x^{10}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{12} x^{4}-\frac {1}{80} x^{6}+\frac {11}{8064} x^{8}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {1}{30} x^{5}-\frac {19}{5040} x^{7}+\frac {29}{72576} x^{9}\right ) D\left (y \right )\left (0\right )+O\left (x^{10}\right ) \]

Problem 13970

ODE	\[ \boxed {y^{\prime \prime }+\sin \left (x \right ) y^{\prime }+\cos \left (x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{4}-\frac {31}{720} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{10} x^{5}-\frac {59}{2520} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{7}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{4}-\frac {31}{720} x^{6}\right ) c_{1} +\left (x -\frac {1}{3} x^{3}+\frac {1}{10} x^{5}\right ) c_{2} +O\left (x^{7}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{4}-\frac {31}{720} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{10} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{7}\right ) \]

Problem 13971

ODE	\[ \boxed {\sqrt {x}\, y^{\prime \prime }+y^{\prime }+y x=0} \] With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = \left (1-\frac {\left (x -1\right )^{2}}{2}+\frac {\left (x -1\right )^{3}}{12}-\frac {\left (x -1\right )^{4}}{96}+\frac {31 \left (x -1\right )^{5}}{960}\right ) y \left (1\right )+\left (x -1-\frac {\left (x -1\right )^{2}}{2}+\frac {\left (x -1\right )^{3}}{12}-\frac {3 \left (x -1\right )^{4}}{32}+\frac {71 \left (x -1\right )^{5}}{960}\right ) y^{\prime }\left (1\right )+O\left (\left (x -1\right )^{5}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {\left (-1+x \right )^{2}}{2}+\frac {\left (-1+x \right )^{3}}{12}-\frac {\left (-1+x \right )^{4}}{96}\right ) y \left (1\right )+\left (-1+x -\frac {\left (-1+x \right )^{2}}{2}+\frac {\left (-1+x \right )^{3}}{12}-\frac {3 \left (-1+x \right )^{4}}{32}\right ) D\left (y \right )\left (1\right )+O\left (x^{5}\right ) \]

Problem 13972

ODE	\[ \boxed {\left (x -3\right )^{2} y^{\prime \prime }-2 \left (x -3\right ) y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {x^{2}}{9}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{2}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {x^{2}}{9}\right ) c_{1} +\left (x -\frac {1}{3} x^{2}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-x^{2}+9\right ) y \left (0\right )}{9}-\frac {x D\left (y \right )\left (0\right ) \left (-3+x \right )}{3} \]

Problem 13973

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

Problem 13974

ODE	\[ \boxed {\left (x -1\right )^{2} y^{\prime \prime }-5 y^{\prime } \left (x -1\right )+9 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {9}{2} x^{2}+\frac {9}{2} x^{3}-\frac {3}{4} x^{4}-\frac {3}{20} x^{5}-\frac {1}{20} x^{6}\right ) y \left (0\right )+\left (x -\frac {5}{2} x^{2}+\frac {11}{6} x^{3}-\frac {1}{4} x^{4}-\frac {1}{20} x^{5}-\frac {1}{60} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {9}{2} x^{2}+\frac {9}{2} x^{3}-\frac {3}{4} x^{4}-\frac {3}{20} x^{5}\right ) c_{1} +\left (x -\frac {5}{2} x^{2}+\frac {11}{6} x^{3}-\frac {1}{4} x^{4}-\frac {1}{20} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {9}{2} x^{2}+\frac {9}{2} x^{3}-\frac {3}{4} x^{4}-\frac {3}{20} x^{5}\right ) y \left (0\right )+\left (x -\frac {5}{2} x^{2}+\frac {11}{6} x^{3}-\frac {1}{4} x^{4}-\frac {1}{20} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 13975

ODE	\[ \boxed {\left (x +2\right )^{2} y^{\prime \prime }+y^{\prime } \left (x +2\right )=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = y \left (0\right )+\left (x -\frac {1}{4} x^{2}+\frac {1}{12} x^{3}-\frac {1}{32} x^{4}+\frac {1}{80} x^{5}-\frac {1}{192} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = c_{1} +\left (x -\frac {1}{4} x^{2}+\frac {1}{12} x^{3}-\frac {1}{32} x^{4}+\frac {1}{80} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = y \left (0\right )+\left (x -\frac {1}{4} x^{2}+\frac {1}{12} x^{3}-\frac {1}{32} x^{4}+\frac {1}{80} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 13976

ODE	\[ \boxed {3 \left (x -2\right )^{2} y^{\prime \prime }-4 \left (x -5\right ) y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{12} x^{2}+\frac {1}{54} x^{3}+\frac {1}{648} x^{4}-\frac {1}{4860} x^{5}-\frac {61}{349920} x^{6}\right ) y \left (0\right )+\left (x -\frac {5}{6} x^{2}+\frac {23}{108} x^{3}+\frac {23}{1296} x^{4}-\frac {23}{9720} x^{5}-\frac {1403}{699840} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{12} x^{2}+\frac {1}{54} x^{3}+\frac {1}{648} x^{4}-\frac {1}{4860} x^{5}\right ) c_{1} +\left (x -\frac {5}{6} x^{2}+\frac {23}{108} x^{3}+\frac {23}{1296} x^{4}-\frac {23}{9720} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{12} x^{2}+\frac {1}{54} x^{3}+\frac {1}{648} x^{4}-\frac {1}{4860} x^{5}\right ) y \left (0\right )+\left (x -\frac {5}{6} x^{2}+\frac {23}{108} x^{3}+\frac {23}{1296} x^{4}-\frac {23}{9720} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 13977

ODE	\[ \boxed {\left (x -5\right )^{2} y^{\prime \prime }+\left (x -5\right ) y^{\prime }+4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {2}{25} x^{2}-\frac {2}{125} x^{3}-\frac {7}{3750} x^{4}-\frac {1}{9375} x^{5}+\frac {1}{56250} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{10} x^{2}-\frac {1}{75} x^{3}-\frac {3}{500} x^{4}-\frac {1}{750} x^{5}-\frac {3}{12500} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {2}{25} x^{2}-\frac {2}{125} x^{3}-\frac {7}{3750} x^{4}-\frac {1}{9375} x^{5}\right ) c_{1} +\left (x +\frac {1}{10} x^{2}-\frac {1}{75} x^{3}-\frac {3}{500} x^{4}-\frac {1}{750} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {2}{25} x^{2}-\frac {2}{125} x^{3}-\frac {7}{3750} x^{4}-\frac {1}{9375} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{10} x^{2}-\frac {1}{75} x^{3}-\frac {3}{500} x^{4}-\frac {1}{750} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 13978

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\frac {x y^{\prime }}{x -2}+\frac {2 y}{x +2}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {3}{4}-\frac {i \sqrt {7}}{4}} \left (1+\frac {\left (\sqrt {7}+11 i\right ) x}{8 \sqrt {7}+16 i}+\frac {\left (7 i \sqrt {7}-45\right ) x^{2}}{-64+384 i \sqrt {7}}+\frac {\left (223 i \sqrt {7}+43\right ) x^{3}}{9216+9472 i \sqrt {7}}+\frac {\left (-7577 \sqrt {7}-979 i\right ) x^{4}}{4096 \left (\sqrt {7}+6 i\right ) \left (6 \sqrt {7}+i\right ) \left (\sqrt {7}+8 i\right )}+\frac {\left (-553875 i \sqrt {7}-1249007\right ) x^{5}}{81920 \left (\sqrt {7}+6 i\right ) \left (6 \sqrt {7}+i\right ) \left (\sqrt {7}+8 i\right ) \left (\sqrt {7}+10 i\right )}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {3}{4}+\frac {i \sqrt {7}}{4}} \left (1+\frac {\left (\sqrt {7}-11 i\right ) x}{8 \sqrt {7}-16 i}+\frac {\left (-7 i \sqrt {7}-45\right ) x^{2}}{-64-384 i \sqrt {7}}+\frac {\left (-223 i \sqrt {7}+43\right ) x^{3}}{9216-9472 i \sqrt {7}}+\frac {\left (-7577 \sqrt {7}+979 i\right ) x^{4}}{4096 \left (\sqrt {7}-6 i\right ) \left (6 \sqrt {7}-i\right ) \left (\sqrt {7}-8 i\right )}+\frac {\left (553875 i \sqrt {7}-1249007\right ) x^{5}}{81920 \left (\sqrt {7}-6 i\right ) \left (6 \sqrt {7}-i\right ) \left (\sqrt {7}-8 i\right ) \left (\sqrt {7}-10 i\right )}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{\frac {3}{4}} \left (c_{2} x^{\frac {i \sqrt {7}}{4}} \left (1+\frac {i \sqrt {7}+11}{8 i \sqrt {7}+16} x +\frac {1}{64} \frac {7 i \sqrt {7}+45}{\left (2+i \sqrt {7}\right ) \left (i \sqrt {7}+4\right )} x^{2}+\frac {1}{256} \frac {223 i \sqrt {7}-43}{\left (2+i \sqrt {7}\right ) \left (i \sqrt {7}+4\right ) \left (i \sqrt {7}+6\right )} x^{3}+\frac {1}{4096} \frac {7577 i \sqrt {7}+979}{\left (2+i \sqrt {7}\right ) \left (i \sqrt {7}+4\right ) \left (i \sqrt {7}+6\right ) \left (i \sqrt {7}+8\right )} x^{4}+\frac {1}{81920} \frac {553875 \sqrt {7}+1249007 i}{\left (-2 i+\sqrt {7}\right ) \left (i \sqrt {7}+4\right ) \left (i \sqrt {7}+6\right ) \left (i \sqrt {7}+8\right ) \left (i \sqrt {7}+10\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} x^{-\frac {i \sqrt {7}}{4}} \left (1+\frac {\sqrt {7}+11 i}{8 \sqrt {7}+16 i} x +\frac {7 i \sqrt {7}-45}{-64+384 i \sqrt {7}} x^{2}+\frac {-223 \sqrt {7}+43 i}{9216 i-9472 \sqrt {7}} x^{3}+\frac {7577 \sqrt {7}+979 i}{-2240512 i+1064960 \sqrt {7}} x^{4}+\frac {1}{81920} \frac {553875 \sqrt {7}-1249007 i}{\left (4 i+\sqrt {7}\right ) \left (\sqrt {7}+2 i\right ) \left (\sqrt {7}+6 i\right ) \left (\sqrt {7}+8 i\right ) \left (\sqrt {7}+10 i\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

Problem 13979

ODE	\[ \boxed {x^{3} y^{\prime \prime }+x^{2} y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 2\).
program solution	\[ y = \left (1-\frac {\left (x -2\right )^{2}}{16}+\frac {\left (x -2\right )^{3}}{24}-\frac {35 \left (x -2\right )^{4}}{1536}+\frac {89 \left (x -2\right )^{5}}{7680}-\frac {2089 \left (x -2\right )^{6}}{368640}\right ) y \left (2\right )+\left (x -2-\frac {\left (x -2\right )^{2}}{4}+\frac {\left (x -2\right )^{3}}{16}-\frac {\left (x -2\right )^{4}}{96}-\frac {19 \left (x -2\right )^{5}}{7680}+\frac {43 \left (x -2\right )^{6}}{10240}\right ) y^{\prime }\left (2\right )+O\left (\left (x -2\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {\left (x -2\right )^{2}}{16}+\frac {\left (x -2\right )^{3}}{24}-\frac {35 \left (x -2\right )^{4}}{1536}+\frac {89 \left (x -2\right )^{5}}{7680}\right ) y \left (2\right )+\left (x -2-\frac {\left (x -2\right )^{2}}{4}+\frac {\left (x -2\right )^{3}}{16}-\frac {\left (x -2\right )^{4}}{96}-\frac {19 \left (x -2\right )^{5}}{7680}\right ) D\left (y \right )\left (2\right )+O\left (x^{6}\right ) \]

Problem 13980

ODE	\[ \boxed {\left (-x^{4}+x^{3}\right ) y^{\prime \prime }+\left (3 x -1\right ) y^{\prime }+827 y=0} \] With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = c_{1} \left (x -1\right )^{3} \left (-\frac {407}{2}+\frac {409 x}{2}+\frac {328391 \left (x -1\right )^{2}}{20}+\frac {128327201 \left (x -1\right )^{3}}{180}+\frac {19341852779 \left (x -1\right )^{4}}{1008}+\frac {6949904889503 \left (x -1\right )^{5}}{20160}+O\left (\left (x -1\right )^{6}\right )\right )+c_{2} \left (\frac {283830535 \left (x -1\right )^{3} \left (-\frac {407}{2}+\frac {409 x}{2}+\frac {328391 \left (x -1\right )^{2}}{20}+\frac {128327201 \left (x -1\right )^{3}}{180}+\frac {19341852779 \left (x -1\right )^{4}}{1008}+\frac {6949904889503 \left (x -1\right )^{5}}{20160}+O\left (\left (x -1\right )^{6}\right )\right ) \ln \left (x -1\right )}{6}+\frac {829}{2}-\frac {827 x}{2}+\frac {343205 \left (x -1\right )^{2}}{2}-\frac {587257732295 \left (x -1\right )^{4}}{48}-\frac {246980325926441 \left (x -1\right )^{5}}{160}+O\left (\left (x -1\right )^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (-1+x \right )^{3} \left (1+\frac {409}{2} \left (-1+x \right )+\frac {328391}{20} \left (-1+x \right )^{2}+\frac {128327201}{180} \left (-1+x \right )^{3}+\frac {19341852779}{1008} \left (-1+x \right )^{4}+\frac {6949904889503}{20160} \left (-1+x \right )^{5}+\operatorname {O}\left (\left (-1+x \right )^{6}\right )\right )+c_{2} \left (\ln \left (-1+x \right ) \left (567661070 \left (-1+x \right )^{3}[phone] \left (-1+x \right )^{4}+\frac {18641478643837}{2} \left (-1+x \right )^{5}+\operatorname {O}\left (\left (-1+x \right )^{6}\right )\right )+\left (12-4962 \left (-1+x \right )+2059230 \left (-1+x \right )^{2}-6162812 \left (-1+x \right )^{3}-\frac {592298912511}{4} \left (-1+x \right )^{4}-\frac {744988601770307}{40} \left (-1+x \right )^{5}+\operatorname {O}\left (\left (-1+x \right )^{6}\right )\right )\right ) \]

Problem 13981

ODE	\[ \boxed {y^{\prime \prime }+\frac {y^{\prime }}{x -3}+\frac {y}{x -4}=0} \] With the expansion point for the power series method at \(x = 3\).
program solution	\[ y = c_{1} \left (1+\frac {\left (x -3\right )^{2}}{4}+\frac {\left (x -3\right )^{3}}{9}+\frac {5 \left (x -3\right )^{4}}{64}+\frac {49 \left (x -3\right )^{5}}{900}+O\left (\left (x -3\right )^{6}\right )\right )+c_{2} \left (\left (1+\frac {\left (x -3\right )^{2}}{4}+\frac {\left (x -3\right )^{3}}{9}+\frac {5 \left (x -3\right )^{4}}{64}+\frac {49 \left (x -3\right )^{5}}{900}+O\left (\left (x -3\right )^{6}\right )\right ) \ln \left (x -3\right )-\frac {\left (x -3\right )^{2}}{4}-\frac {2 \left (x -3\right )^{3}}{27}-\frac {7 \left (x -3\right )^{4}}{128}-\frac {469 \left (x -3\right )^{5}}{13500}+O\left (\left (x -3\right )^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\ln \left (-3+x \right ) c_{2} +c_{1} \right ) \left (1+\frac {1}{4} \left (-3+x \right )^{2}+\frac {1}{9} \left (-3+x \right )^{3}+\frac {5}{64} \left (-3+x \right )^{4}+\frac {49}{900} \left (-3+x \right )^{5}+\operatorname {O}\left (\left (-3+x \right )^{6}\right )\right )+\left (-\frac {1}{4} \left (-3+x \right )^{2}-\frac {2}{27} \left (-3+x \right )^{3}-\frac {7}{128} \left (-3+x \right )^{4}-\frac {469}{13500} \left (-3+x \right )^{5}+\operatorname {O}\left (\left (-3+x \right )^{6}\right )\right ) c_{2} \]

Problem 13982

ODE	\[ \boxed {y^{\prime \prime }+\frac {y^{\prime }}{\left (x -3\right )^{2}}+\frac {y}{\left (x -4\right )^{2}}=0} \] With the expansion point for the power series method at \(x = 4\).
program solution	\[ y = c_{1} \left (x -4\right )^{\frac {1}{2}+\frac {i \sqrt {3}}{2}} \left (3-\frac {x}{2}+\frac {\left (-3 \sqrt {3}-2 i\right ) \left (x -4\right )^{2}}{2 \left (i-\sqrt {3}\right ) \left (i \sqrt {3}+2\right )}+\frac {\left (41 \sqrt {3}+103 i\right ) \left (x -4\right )^{3}}{12 \left (i-\sqrt {3}\right ) \left (i \sqrt {3}+2\right ) \left (i \sqrt {3}+3\right )}+\frac {\left (-112 \sqrt {3}-1201 i\right ) \left (x -4\right )^{4}}{24 \left (i-\sqrt {3}\right ) \left (i \sqrt {3}+2\right ) \left (i \sqrt {3}+3\right ) \left (i \sqrt {3}+4\right )}+\frac {\left (8017 i \sqrt {3}+71929\right ) \left (x -4\right )^{5}}{240 \left (\sqrt {3}-5 i\right ) \left (i-\sqrt {3}\right ) \left (i \sqrt {3}+2\right ) \left (i \sqrt {3}+3\right ) \left (i \sqrt {3}+4\right )}+O\left (\left (x -4\right )^{6}\right )\right )+c_{2} \left (x -4\right )^{\frac {1}{2}-\frac {i \sqrt {3}}{2}} \left (3-\frac {x}{2}+\frac {\left (-3 \sqrt {3}+2 i\right ) \left (x -4\right )^{2}}{2 \left (-i-\sqrt {3}\right ) \left (-i \sqrt {3}+2\right )}+\frac {\left (41 \sqrt {3}-103 i\right ) \left (x -4\right )^{3}}{12 \left (-i-\sqrt {3}\right ) \left (-i \sqrt {3}+2\right ) \left (3-i \sqrt {3}\right )}+\frac {\left (-112 \sqrt {3}+1201 i\right ) \left (x -4\right )^{4}}{24 \left (-i-\sqrt {3}\right ) \left (-i \sqrt {3}+2\right ) \left (3-i \sqrt {3}\right ) \left (-i \sqrt {3}+4\right )}+\frac {\left (-8017 i \sqrt {3}+71929\right ) \left (x -4\right )^{5}}{240 \left (\sqrt {3}+5 i\right ) \left (-i-\sqrt {3}\right ) \left (-i \sqrt {3}+2\right ) \left (3-i \sqrt {3}\right ) \left (-i \sqrt {3}+4\right )}+O\left (\left (x -4\right )^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x -4}\, \left (c_{2} \left (x -4\right )^{\frac {i \sqrt {3}}{2}} \left (1-\frac {1}{2} \left (x -4\right )+\frac {5 i \sqrt {3}+7}{8 i \sqrt {3}+16} \left (x -4\right )^{2}-\frac {1}{12} \frac {5+36 i \sqrt {3}}{\left (i \sqrt {3}+3\right ) \left (i \sqrt {3}+2\right )} \left (x -4\right )^{3}+\frac {1}{96} \frac {1313 i \sqrt {3}-865}{\left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+3\right ) \left (i \sqrt {3}+2\right )} \left (x -4\right )^{4}+\frac {1}{480} \frac {23995 i+15978 \sqrt {3}}{\left (-\frac {\sqrt {3}}{2}+i\right ) \left (i \sqrt {3}+3\right ) \left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+5\right )} \left (x -4\right )^{5}+\operatorname {O}\left (\left (x -4\right )^{6}\right )\right )+c_{1} \left (x -4\right )^{-\frac {i \sqrt {3}}{2}} \left (1-\frac {1}{2} \left (x -4\right )+\frac {5 \sqrt {3}+7 i}{8 \sqrt {3}+16 i} \left (x -4\right )^{2}+\frac {5-36 i \sqrt {3}}{-36+60 i \sqrt {3}} \left (x -4\right )^{3}+\frac {-1313 \sqrt {3}+865 i}{288 i-2208 \sqrt {3}} \left (x -4\right )^{4}+\frac {-23995-15978 i \sqrt {3}}{26880 i \sqrt {3}+20160} \left (x -4\right )^{5}+\operatorname {O}\left (\left (x -4\right )^{6}\right )\right )\right ) \]

Problem 13983

ODE	\[ \boxed {y^{\prime \prime }+\left (\frac {1}{x}-\frac {1}{3}\right ) y^{\prime }+\left (\frac {1}{x}-\frac {1}{4}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-x +\frac {11 x^{2}}{48}-\frac {47 x^{3}}{1296}+\frac {11 x^{4}}{3072}-\frac {653 x^{5}}{2073600}+O\left (x^{6}\right )\right )+c_{2} \left (\left (1-x +\frac {11 x^{2}}{48}-\frac {47 x^{3}}{1296}+\frac {11 x^{4}}{3072}-\frac {653 x^{5}}{2073600}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\frac {7 x}{3}-\frac {101 x^{2}}{144}+\frac {10 x^{3}}{81}-\frac {6721 x^{4}}{497664}+\frac {229213 x^{5}}{186624000}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} +c_{2} \ln \left (x \right )\right ) \left (1-x +\frac {11}{48} x^{2}-\frac {47}{1296} x^{3}+\frac {11}{3072} x^{4}-\frac {653}{2073600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {7}{3} x -\frac {101}{144} x^{2}+\frac {10}{81} x^{3}-\frac {6721}{497664} x^{4}+\frac {229213}{186624000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 13984

ODE	\[ \boxed {\left (4 x^{2}-1\right ) y^{\prime \prime }+\left (4-\frac {2}{x}\right ) y^{\prime }+\frac {\left (-x^{2}+1\right ) y}{x^{2}+1}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1+\frac {x^{2}}{6}+\frac {x^{3}}{9}+\frac {x^{4}}{24}+\frac {31 x^{5}}{270}+O\left (x^{6}\right )\right )+c_{2} \left (\left (-4-\frac {2 x^{2}}{3}-\frac {4 x^{3}}{9}-\frac {x^{4}}{6}-\frac {62 x^{5}}{135}-4 O\left (x^{6}\right )\right ) \ln \left (x \right )+\frac {1-\frac {7 x^{2}}{2}+\frac {8 x^{3}}{9}+\frac {37 x^{4}}{216}+\frac {4 x^{5}}{45}+O\left (x^{6}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\ln \left (x \right ) \left (\left (-4\right ) x -\frac {2}{3} x^{3}-\frac {4}{9} x^{4}-\frac {1}{6} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} +c_{1} \left (1+\frac {1}{6} x^{2}+\frac {1}{9} x^{3}+\frac {1}{24} x^{4}+\frac {31}{270} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) x +\left (1+4 x -\frac {7}{2} x^{2}+\frac {14}{9} x^{3}+\frac {133}{216} x^{4}+\frac {23}{90} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2}}{x} \]

Problem 13985

ODE	\[ \boxed {\left (x^{2}+4\right )^{2} y^{\prime \prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{32} x^{2}+\frac {17}{6144} x^{4}-\frac {253}{589824} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{96} x^{3}+\frac {49}{30720} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{32} x^{2}+\frac {17}{6144} x^{4}\right ) c_{1} +\left (x -\frac {1}{96} x^{3}+\frac {49}{30720} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{32} x^{2}+\frac {17}{6144} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{96} x^{3}+\frac {49}{30720} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 13986

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

Problem 13987

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+\left (1-4 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (x +1+\frac {x^{2}}{4}+\frac {x^{3}}{36}+\frac {x^{4}}{576}+\frac {x^{5}}{14400}+O\left (x^{6}\right )\right )+c_{2} \left (\sqrt {x}\, \left (x +1+\frac {x^{2}}{4}+\frac {x^{3}}{36}+\frac {x^{4}}{576}+\frac {x^{5}}{14400}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\sqrt {x}\, \left (-2 x -\frac {3 x^{2}}{4}-\frac {11 x^{3}}{108}-\frac {25 x^{4}}{3456}-\frac {137 x^{5}}{432000}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{1} +c_{2} \ln \left (x \right )\right ) \left (1+x +\frac {1}{4} x^{2}+\frac {1}{36} x^{3}+\frac {1}{576} x^{4}+\frac {1}{14400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-2\right ) x -\frac {3}{4} x^{2}-\frac {11}{108} x^{3}-\frac {25}{3456} x^{4}-\frac {137}{432000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \sqrt {x} \]

Problem 13988

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (4 x -4\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-\frac {4 x}{5}+\frac {4 x^{2}}{15}-\frac {16 x^{3}}{315}+\frac {2 x^{4}}{315}-\frac {8 x^{5}}{14175}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {16 x^{2} \left (1-\frac {4 x}{5}+\frac {4 x^{2}}{15}-\frac {16 x^{3}}{315}+\frac {2 x^{4}}{315}-\frac {8 x^{5}}{14175}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{9}+\frac {1+\frac {4 x}{3}+\frac {4 x^{2}}{3}+\frac {16 x^{3}}{9}-\frac {128 x^{5}}{75}+O\left (x^{6}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{4} \left (1-\frac {4}{5} x +\frac {4}{15} x^{2}-\frac {16}{315} x^{3}+\frac {2}{315} x^{4}-\frac {8}{14175} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (256 x^{4}-\frac {1024}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-192 x -192 x^{2}-256 x^{3}+\frac {6144}{25} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]

Problem 13989

ODE	\[ \boxed {\left (-9 x^{4}+x^{2}\right ) y^{\prime \prime }-6 y^{\prime } x +10 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{5} \left (1+18 x^{2}+243 x^{4}+O\left (x^{6}\right )\right )+c_{2} x^{2} \left (1-9 x^{2}-243 x^{4}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{5} \left (1+18 x^{2}+243 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{2} \left (12-108 x^{2}-2916 x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 13990

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

Problem 13991

ODE	\[ \boxed {y^{\prime \prime }+\frac {y^{\prime }}{x}+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-\frac {x^{2}}{4}+\frac {x^{4}}{64}+O\left (x^{6}\right )\right )+c_{2} \left (\left (1-\frac {x^{2}}{4}+\frac {x^{4}}{64}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\frac {x^{2}}{4}-\frac {3 x^{4}}{128}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} +c_{2} \ln \left (x \right )\right ) \left (1-\frac {1}{4} x^{2}+\frac {1}{64} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{4} x^{2}-\frac {3}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 13992

ODE	\[ \boxed {y^{\prime \prime }+\frac {y^{\prime }}{x}+\left (1-\frac {1}{x^{2}}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {x^{2}}{8}+\frac {x^{4}}{192}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {x \left (1-\frac {x^{2}}{8}+\frac {x^{4}}{192}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{2}+\frac {1-\frac {3 x^{4}}{64}+O\left (x^{6}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{2} \left (1-\frac {1}{8} x^{2}+\frac {1}{192} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (x^{2}-\frac {1}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+\frac {3}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]

Problem 13993

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }+\left (-2 x^{3}+5 x \right ) y^{\prime }+\left (-x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1+O\left (x^{6}\right )\right )}{\sqrt {x}}+\frac {c_{2} \left (1-\frac {x^{2}}{6}-\frac {x^{4}}{56}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-\frac {1}{6} x^{2}-\frac {1}{56} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x}+\frac {c_{2} \left (1+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

Problem 13994

ODE	\[ \boxed {x^{2} y^{\prime \prime }-\left (2 x^{2}+5 x \right ) y^{\prime }+\left (9+4 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{3} \left (2 x^{2}+2 x +1+\frac {4 x^{3}}{3}+\frac {2 x^{4}}{3}+\frac {4 x^{5}}{15}+O\left (x^{6}\right )\right )+c_{2} \left (x^{3} \left (2 x^{2}+2 x +1+\frac {4 x^{3}}{3}+\frac {2 x^{4}}{3}+\frac {4 x^{5}}{15}+O\left (x^{6}\right )\right ) \ln \left (x \right )+x^{3} \left (-3 x^{2}-2 x -\frac {22 x^{3}}{9}-\frac {25 x^{4}}{18}-\frac {137 x^{5}}{225}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{1} +c_{2} \ln \left (x \right )\right ) \left (1+2 x +2 x^{2}+\frac {4}{3} x^{3}+\frac {2}{3} x^{4}+\frac {4}{15} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-2\right ) x -3 x^{2}-\frac {22}{9} x^{3}-\frac {25}{18} x^{4}-\frac {137}{225} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) x^{3} \]

Problem 13995

ODE	\[ \boxed {\left (-3 x^{3}+3 x^{2}\right ) y^{\prime \prime }-\left (5 x^{2}+4 x \right ) y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1+2 x +3 x^{2}+4 x^{3}+5 x^{4}+6 x^{5}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{3}} \left (1-\frac {x}{2}-2 x^{2}-\frac {7 x^{3}}{2}-5 x^{4}-\frac {13 x^{5}}{2}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{3}} \left (1-\frac {1}{2} x -2 x^{2}-\frac {7}{2} x^{3}-5 x^{4}-\frac {13}{2} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{2} \left (1+2 x +3 x^{2}+4 x^{3}+5 x^{4}+6 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 13996

ODE	\[ \boxed {x^{2} y^{\prime \prime }-\left (x^{2}+x \right ) y^{\prime }+4 y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-\frac {2 x}{3}+\frac {x^{2}}{12}+O\left (x^{6}\right )\right )+c_{2} \left (-6 x^{2} \left (1-\frac {2 x}{3}+\frac {x^{2}}{12}+O\left (x^{6}\right )\right ) \ln \left (x \right )+1+4 x -\frac {22 x^{3}}{3}+\frac {43 x^{4}}{24}-\frac {x^{5}}{30}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \left (1-\frac {2}{3} x +\frac {1}{12} x^{2}+\operatorname {O}\left (x^{6}\right )\right )+\ln \left (x \right ) \left (12 x^{2}-8 x^{3}+x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} +\left (-2-8 x -7 x^{2}+\frac {58}{3} x^{3}-\frac {25}{6} x^{4}+\frac {1}{15} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 13997

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+8 x^{2} y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-x +\frac {3 x^{2}}{4}-\frac {5 x^{3}}{12}+\frac {35 x^{4}}{192}-\frac {21 x^{5}}{320}+O\left (x^{6}\right )\right )+c_{2} \left (\sqrt {x}\, \left (1-x +\frac {3 x^{2}}{4}-\frac {5 x^{3}}{12}+\frac {35 x^{4}}{192}-\frac {21 x^{5}}{320}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\sqrt {x}\, \left (-\frac {x^{2}}{4}+\frac {x^{3}}{4}-\frac {19 x^{4}}{128}+\frac {25 x^{5}}{384}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{1} +c_{2} \ln \left (x \right )\right ) \left (1-x +\frac {3}{4} x^{2}-\frac {5}{12} x^{3}+\frac {35}{192} x^{4}-\frac {21}{320} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {1}{4} x^{2}+\frac {1}{4} x^{3}-\frac {19}{128} x^{4}+\frac {25}{384} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \sqrt {x} \]

Problem 13998

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

Problem 13999

ODE	\[ \boxed {\left (9 x^{3}+9 x^{2}\right ) y^{\prime \prime }+\left (27 x^{2}+9 x \right ) y^{\prime }+\left (8 x -1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1-x +x^{2}-x^{3}+x^{4}-x^{5}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-x +x^{2}-x^{3}+x^{4}-x^{5}+O\left (x^{6}\right )\right )}{x^{\frac {1}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-x^{5}+x^{4}-x^{3}+x^{2}-x +1\right ) \left (x^{\frac {2}{3}} c_{2} +c_{1} \right )}{x^{\frac {1}{3}}}+O\left (x^{6}\right ) \]

Problem 14000

ODE	\[ \boxed {\left (x -3\right ) y^{\prime \prime }+\left (x -3\right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 3\).
program solution	\[ y = c_{1} \left (x -3\right ) \left (4-x +\frac {\left (x -3\right )^{2}}{2}-\frac {\left (x -3\right )^{3}}{6}+\frac {\left (x -3\right )^{4}}{24}-\frac {\left (x -3\right )^{5}}{120}+O\left (\left (x -3\right )^{6}\right )\right )+c_{2} \left (-\left (x -3\right ) \left (4-x +\frac {\left (x -3\right )^{2}}{2}-\frac {\left (x -3\right )^{3}}{6}+\frac {\left (x -3\right )^{4}}{24}-\frac {\left (x -3\right )^{5}}{120}+O\left (\left (x -3\right )^{6}\right )\right ) \ln \left (x -3\right )+1-\left (x -3\right )^{2}+\frac {3 \left (x -3\right )^{3}}{4}-\frac {11 \left (x -3\right )^{4}}{36}+\frac {25 \left (x -3\right )^{5}}{288}+O\left (\left (x -3\right )^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (-3+x \right ) \left (1-\left (-3+x \right )+\frac {1}{2} \left (-3+x \right )^{2}-\frac {1}{6} \left (-3+x \right )^{3}+\frac {1}{24} \left (-3+x \right )^{4}-\frac {1}{120} \left (-3+x \right )^{5}+\operatorname {O}\left (\left (-3+x \right )^{6}\right )\right )+c_{2} \left (\ln \left (-3+x \right ) \left (-\left (-3+x \right )+\left (-3+x \right )^{2}-\frac {1}{2} \left (-3+x \right )^{3}+\frac {1}{6} \left (-3+x \right )^{4}-\frac {1}{24} \left (-3+x \right )^{5}+\operatorname {O}\left (\left (-3+x \right )^{6}\right )\right )+\left (1-\left (-3+x \right )+\frac {1}{4} \left (-3+x \right )^{3}-\frac {5}{36} \left (-3+x \right )^{4}+\frac {13}{288} \left (-3+x \right )^{5}+\operatorname {O}\left (\left (-3+x \right )^{6}\right )\right )\right ) \]