2.17.70 Problems 6901 to 7000

Problem 6901

ODE	\[ \boxed {y^{\prime \prime }+y x^{2}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{12} x^{4}+\frac {1}{672} x^{8}\right ) y \left (0\right )+\left (x -\frac {1}{20} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {x^{4}}{12}\right ) c_{1} +\left (x -\frac {1}{20} x^{5}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {x^{4}}{12}\right ) y \left (0\right )+\left (x -\frac {1}{20} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6902

ODE	\[ \boxed {\left (-4 x^{2}+1\right ) y^{\prime \prime }+6 y^{\prime } x -4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (2 x^{2}+1\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}-\frac {1}{6} x^{5}-\frac {3}{14} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (2 x^{2}+1\right ) c_{1} +\left (x -\frac {1}{3} x^{3}-\frac {1}{6} x^{5}-\frac {3}{14} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (2 x^{2}+1\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}-\frac {1}{6} x^{5}-\frac {3}{14} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6903

ODE	\[ \boxed {\left (2 x^{2}+1\right ) y^{\prime \prime }+3 x y^{\prime }-3 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {3}{2} x^{2}-\frac {7}{8} x^{4}+\frac {77}{80} x^{6}-\frac {165}{128} x^{8}\right ) y \left (0\right )+x y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1+\frac {3}{2} x^{2}-\frac {7}{8} x^{4}+\frac {77}{80} x^{6}\right ) c_{1} +c_{2} x +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {3}{2} x^{2}-\frac {7}{8} x^{4}+\frac {77}{80} x^{6}\right ) y \left (0\right )+D\left (y \right )\left (0\right ) x +O\left (x^{8}\right ) \]

Problem 6904

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

Problem 6905

ODE	\[ \boxed {y^{\prime \prime }+x y^{\prime }+3 y=x^{2}} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {3}{2} x^{2}+\frac {5}{8} x^{4}-\frac {7}{48} x^{6}+\frac {3}{128} x^{8}\right ) y \left (0\right )+\left (x -\frac {2}{3} x^{3}+\frac {1}{5} x^{5}-\frac {4}{105} x^{7}\right ) y^{\prime }\left (0\right )+\frac {x^{4}}{12}-\frac {7 x^{6}}{360}+\frac {x^{8}}{320}+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {3}{2} x^{2}+\frac {5}{8} x^{4}-\frac {7}{48} x^{6}\right ) c_{1} +\left (x -\frac {2}{3} x^{3}+\frac {1}{5} x^{5}-\frac {4}{105} x^{7}\right ) c_{2} +\frac {x^{4}}{12}-\frac {7 x^{6}}{360}+O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {3}{2} x^{2}+\frac {5}{8} x^{4}-\frac {7}{48} x^{6}\right ) y \left (0\right )+\left (x -\frac {2}{3} x^{3}+\frac {1}{5} x^{5}-\frac {4}{105} x^{7}\right ) D\left (y \right )\left (0\right )+\frac {x^{4}}{12}-\frac {7 x^{6}}{360}+O\left (x^{8}\right ) \]

Problem 6906

ODE	\[ \boxed {y^{\prime \prime }+2 x y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-x^{2}+\frac {1}{2} x^{4}-\frac {1}{6} x^{6}+\frac {1}{24} x^{8}\right ) y \left (0\right )+\left (x -\frac {2}{3} x^{3}+\frac {4}{15} x^{5}-\frac {8}{105} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-x^{2}+\frac {1}{2} x^{4}-\frac {1}{6} x^{6}\right ) c_{1} +\left (x -\frac {2}{3} x^{3}+\frac {4}{15} x^{5}-\frac {8}{105} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-x^{2}+\frac {1}{2} x^{4}-\frac {1}{6} x^{6}\right ) y \left (0\right )+\left (x -\frac {2}{3} x^{3}+\frac {4}{15} x^{5}-\frac {8}{105} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6907

ODE	\[ \boxed {y^{\prime \prime }+3 x y^{\prime }+7 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {7}{2} x^{2}+\frac {91}{24} x^{4}-\frac {1729}{720} x^{6}+\frac {1235}{1152} x^{8}\right ) y \left (0\right )+\left (x -\frac {5}{3} x^{3}+\frac {4}{3} x^{5}-\frac {44}{63} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {7}{2} x^{2}+\frac {91}{24} x^{4}-\frac {1729}{720} x^{6}\right ) c_{1} +\left (x -\frac {5}{3} x^{3}+\frac {4}{3} x^{5}-\frac {44}{63} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {7}{2} x^{2}+\frac {91}{24} x^{4}-\frac {1729}{720} x^{6}\right ) y \left (0\right )+\left (x -\frac {5}{3} x^{3}+\frac {4}{3} x^{5}-\frac {44}{63} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6908

ODE	\[ \boxed {2 y^{\prime \prime }+9 x y^{\prime }-36 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+9 x^{2}+\frac {27}{4} x^{4}\right ) y \left (0\right )+\left (x +\frac {9}{4} x^{3}+\frac {81}{160} x^{5}-\frac {243}{4480} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1+9 x^{2}+\frac {27}{4} x^{4}\right ) c_{1} +\left (x +\frac {9}{4} x^{3}+\frac {81}{160} x^{5}-\frac {243}{4480} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\frac {27}{4} x^{4}+9 x^{2}+1\right ) y \left (0\right )+\left (x +\frac {9}{4} x^{3}+\frac {81}{160} x^{5}-\frac {243}{4480} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6909

ODE	\[ \boxed {\left (x^{2}+4\right ) y^{\prime \prime }+x y^{\prime }-9 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {9}{8} x^{2}+\frac {15}{128} x^{4}-\frac {7}{1024} x^{6}+\frac {27}{32768} x^{8}\right ) y \left (0\right )+\left (x +\frac {1}{3} x^{3}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1+\frac {9}{8} x^{2}+\frac {15}{128} x^{4}-\frac {7}{1024} x^{6}\right ) c_{1} +\left (x +\frac {1}{3} x^{3}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {9}{8} x^{2}+\frac {15}{128} x^{4}-\frac {7}{1024} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{3} x^{3}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6910

ODE	\[ \boxed {\left (x^{2}+4\right ) y^{\prime \prime }+3 x y^{\prime }-8 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (x^{2}+1\right ) y \left (0\right )+\left (x +\frac {5}{24} x^{3}-\frac {7}{384} x^{5}+\frac {3}{1024} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (x^{2}+1\right ) c_{1} +\left (x +\frac {5}{24} x^{3}-\frac {7}{384} x^{5}+\frac {3}{1024} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (x^{2}+1\right ) y \left (0\right )+\left (x +\frac {5}{24} x^{3}-\frac {7}{384} x^{5}+\frac {3}{1024} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6911

ODE	\[ \boxed {\left (9 x^{2}+1\right ) y^{\prime \prime }-18 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (9 x^{2}+1\right ) y \left (0\right )+\left (x +3 x^{3}-\frac {27}{5} x^{5}+\frac {729}{35} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (9 x^{2}+1\right ) c_{1} +\left (x +3 x^{3}-\frac {27}{5} x^{5}+\frac {729}{35} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (9 x^{2}+1\right ) y \left (0\right )+\left (x +3 x^{3}-\frac {27}{5} x^{5}+\frac {729}{35} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6912

ODE	\[ \boxed {\left (3 x^{2}+1\right ) y^{\prime \prime }+13 x y^{\prime }+7 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {7}{2} x^{2}+\frac {91}{8} x^{4}-\frac {1729}{48} x^{6}+\frac {43225}{384} x^{8}\right ) y \left (0\right )+\left (x -\frac {10}{3} x^{3}+\frac {32}{3} x^{5}-\frac {704}{21} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {7}{2} x^{2}+\frac {91}{8} x^{4}-\frac {1729}{48} x^{6}\right ) c_{1} +\left (x -\frac {10}{3} x^{3}+\frac {32}{3} x^{5}-\frac {704}{21} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {7}{2} x^{2}+\frac {91}{8} x^{4}-\frac {1729}{48} x^{6}\right ) y \left (0\right )+\left (x -\frac {10}{3} x^{3}+\frac {32}{3} x^{5}-\frac {704}{21} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6913

ODE	\[ \boxed {\left (2 x^{2}+1\right ) y^{\prime \prime }+11 x y^{\prime }+9 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {9}{2} x^{2}+\frac {105}{8} x^{4}-\frac {539}{16} x^{6}+\frac {10395}{128} x^{8}\right ) y \left (0\right )+\left (x -\frac {10}{3} x^{3}+9 x^{5}-\frac {156}{7} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {9}{2} x^{2}+\frac {105}{8} x^{4}-\frac {539}{16} x^{6}\right ) c_{1} +\left (x -\frac {10}{3} x^{3}+9 x^{5}-\frac {156}{7} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {9}{2} x^{2}+\frac {105}{8} x^{4}-\frac {539}{16} x^{6}\right ) y \left (0\right )+\left (x -\frac {10}{3} x^{3}+9 x^{5}-\frac {156}{7} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6914

ODE	\[ \boxed {y^{\prime \prime }-2 \left (x +3\right ) y^{\prime }-3 y=0} \] With the expansion point for the power series method at \(x = -3\).
program solution	\[ y = \left (1+\frac {3 \left (x +3\right )^{2}}{2}+\frac {7 \left (x +3\right )^{4}}{8}+\frac {77 \left (x +3\right )^{6}}{240}+\frac {11 \left (x +3\right )^{8}}{128}\right ) y \left (-3\right )+\left (x +3+\frac {5 \left (x +3\right )^{3}}{6}+\frac {3 \left (x +3\right )^{5}}{8}+\frac {13 \left (x +3\right )^{7}}{112}\right ) y^{\prime }\left (-3\right )+O\left (\left (x +3\right )^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {3 \left (x +3\right )^{2}}{2}+\frac {7 \left (x +3\right )^{4}}{8}+\frac {77 \left (x +3\right )^{6}}{240}\right ) y \left (-3\right )+\left (x +3+\frac {5 \left (x +3\right )^{3}}{6}+\frac {3 \left (x +3\right )^{5}}{8}+\frac {13 \left (x +3\right )^{7}}{112}\right ) D\left (y \right )\left (-3\right )+O\left (x^{8}\right ) \]

Problem 6915

ODE	\[ \boxed {y^{\prime \prime }+\left (x -2\right ) y=0} \] With the expansion point for the power series method at \(x = 2\).
program solution	\[ y = \left (1-\frac {\left (x -2\right )^{3}}{6}+\frac {\left (x -2\right )^{6}}{180}\right ) y \left (2\right )+\left (x -2-\frac {\left (x -2\right )^{4}}{12}+\frac {\left (x -2\right )^{7}}{504}\right ) y^{\prime }\left (2\right )+O\left (\left (x -2\right )^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {\left (-2+x \right )^{3}}{6}+\frac {\left (-2+x \right )^{6}}{180}\right ) y \left (2\right )+\left (-2+x -\frac {\left (-2+x \right )^{4}}{12}+\frac {\left (-2+x \right )^{7}}{504}\right ) D\left (y \right )\left (2\right )+O\left (x^{8}\right ) \]

Problem 6916

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

Problem 6917

ODE	\[ \boxed {2 x \left (x +1\right ) y^{\prime \prime }+3 \left (x +1\right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1+\frac {x}{3}-\frac {x^{2}}{15}+\frac {x^{3}}{35}-\frac {x^{4}}{63}+\frac {x^{5}}{99}-\frac {x^{6}}{143}+\frac {x^{7}}{195}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1+x +O\left (x^{8}\right )\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1+x +\operatorname {O}\left (x^{8}\right )\right )}{\sqrt {x}}+c_{2} \left (1+\frac {1}{3} x -\frac {1}{15} x^{2}+\frac {1}{35} x^{3}-\frac {1}{63} x^{4}+\frac {1}{99} x^{5}-\frac {1}{143} x^{6}+\frac {1}{195} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6918

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}-1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {x^{2}}{6}+\frac {x^{4}}{120}-\frac {x^{6}}{5040}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1-\frac {x^{2}}{2}+\frac {x^{4}}{24}-\frac {x^{6}}{720}+O\left (x^{8}\right )\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x \left (1-\frac {1}{6} x^{2}+\frac {1}{120} x^{4}-\frac {1}{5040} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}-\frac {1}{720} x^{6}+\operatorname {O}\left (x^{8}\right )\right )}{\sqrt {x}} \]

Problem 6919

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+4 x y^{\prime }-\left (4 x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1+\frac {x^{2}}{6}+\frac {x^{4}}{120}+\frac {x^{6}}{5040}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1+\frac {x^{2}}{2}+\frac {x^{4}}{24}+\frac {x^{6}}{720}+O\left (x^{8}\right )\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x \left (1+\frac {1}{6} x^{2}+\frac {1}{120} x^{4}+\frac {1}{5040} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1+\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\frac {1}{720} x^{6}+\operatorname {O}\left (x^{8}\right )\right )}{\sqrt {x}} \]

Problem 6920

ODE	\[ \boxed {4 x y^{\prime \prime }+3 y^{\prime }+3 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{4}} \left (1-\frac {3 x}{5}+\frac {x^{2}}{10}-\frac {x^{3}}{130}+\frac {3 x^{4}}{8840}-\frac {3 x^{5}}{309400}+\frac {3 x^{6}}{15470000}-\frac {9 x^{7}}{3140410000}+O\left (x^{8}\right )\right )+c_{2} \left (1-x +\frac {3 x^{2}}{14}-\frac {3 x^{3}}{154}+\frac {3 x^{4}}{3080}-\frac {9 x^{5}}{292600}+\frac {9 x^{6}}{13459600}-\frac {x^{7}}{94217200}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{4}} \left (1-\frac {3}{5} x +\frac {1}{10} x^{2}-\frac {1}{130} x^{3}+\frac {3}{8840} x^{4}-\frac {3}{309400} x^{5}+\frac {3}{15470000} x^{6}-\frac {9}{3140410000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1-x +\frac {3}{14} x^{2}-\frac {3}{154} x^{3}+\frac {3}{3080} x^{4}-\frac {9}{292600} x^{5}+\frac {9}{13459600} x^{6}-\frac {1}{94217200} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6921

ODE	\[ \boxed {2 x^{2} \left (1-x \right ) y^{\prime \prime }-x \left (1+7 x \right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1+\frac {7 x}{3}+\frac {21 x^{2}}{5}+\frac {33 x^{3}}{5}+\frac {143 x^{4}}{15}+13 x^{5}+17 x^{6}+\frac {323 x^{7}}{15}+O\left (x^{8}\right )\right )+c_{2} \sqrt {x}\, \left (1+3 x +6 x^{2}+10 x^{3}+15 x^{4}+21 x^{5}+28 x^{6}+36 x^{7}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+3 x +6 x^{2}+10 x^{3}+15 x^{4}+21 x^{5}+28 x^{6}+36 x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x \left (1+\frac {7}{3} x +\frac {21}{5} x^{2}+\frac {33}{5} x^{3}+\frac {143}{15} x^{4}+13 x^{5}+17 x^{6}+\frac {323}{15} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6922

ODE	\[ \boxed {2 x y^{\prime \prime }+5 \left (1-2 x \right ) y^{\prime }-5 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1+x +\frac {15 x^{2}}{14}+\frac {125 x^{3}}{126}+\frac {625 x^{4}}{792}+\frac {625 x^{5}}{1144}+\frac {625 x^{6}}{1872}+\frac {3125 x^{7}}{17136}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1+10 x +O\left (x^{8}\right )\right )}{x^{\frac {3}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1+10 x +\operatorname {O}\left (x^{8}\right )\right )}{x^{\frac {3}{2}}}+c_{2} \left (1+x +\frac {15}{14} x^{2}+\frac {125}{126} x^{3}+\frac {625}{792} x^{4}+\frac {625}{1144} x^{5}+\frac {625}{1872} x^{6}+\frac {3125}{17136} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6923

ODE	\[ \boxed {8 x^{2} y^{\prime \prime }+10 x y^{\prime }-\left (1+x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{4}} \left (1+\frac {x}{14}+\frac {x^{2}}{616}+\frac {x^{3}}{55440}+\frac {x^{4}}{8426880}+\frac {x^{5}}{1938182400}+\frac {x^{6}}{627971097600}+\frac {x^{7}}{272539456358400}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1+\frac {x}{2}+\frac {x^{2}}{40}+\frac {x^{3}}{2160}+\frac {x^{4}}{224640}+\frac {x^{5}}{38188800}+\frac {x^{6}}{9623577600}+\frac {x^{7}}{3368252160000}+O\left (x^{8}\right )\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {3}{4}} \left (1+\frac {1}{14} x +\frac {1}{616} x^{2}+\frac {1}{55440} x^{3}+\frac {1}{8426880} x^{4}+\frac {1}{1938182400} x^{5}+\frac {1}{627971097600} x^{6}+\frac {1}{272539456358400} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{1} \left (1+\frac {1}{2} x +\frac {1}{40} x^{2}+\frac {1}{2160} x^{3}+\frac {1}{224640} x^{4}+\frac {1}{38188800} x^{5}+\frac {1}{9623577600} x^{6}+\frac {1}{3368252160000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{\sqrt {x}} \]

Problem 6924

ODE	\[ \boxed {2 x y^{\prime \prime }+\left (-x +2\right ) y^{\prime }-2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1+x +\frac {3 x^{2}}{8}+\frac {x^{3}}{12}+\frac {5 x^{4}}{384}+\frac {x^{5}}{640}+\frac {7 x^{6}}{46080}+\frac {x^{7}}{80640}+O\left (x^{8}\right )\right )+c_{2} \left (\left (1+x +\frac {3 x^{2}}{8}+\frac {x^{3}}{12}+\frac {5 x^{4}}{384}+\frac {x^{5}}{640}+\frac {7 x^{6}}{46080}+\frac {x^{7}}{80640}+O\left (x^{8}\right )\right ) \ln \left (x \right )-\frac {3 x}{2}-\frac {13 x^{2}}{16}-\frac {31 x^{3}}{144}-\frac {173 x^{4}}{4608}-\frac {187 x^{5}}{38400}-\frac {463 x^{6}}{921600}-\frac {971 x^{7}}{22579200}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+x +\frac {3}{8} x^{2}+\frac {1}{12} x^{3}+\frac {5}{384} x^{4}+\frac {1}{640} x^{5}+\frac {7}{46080} x^{6}+\frac {1}{80640} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-\frac {3}{2} x -\frac {13}{16} x^{2}-\frac {31}{144} x^{3}-\frac {173}{4608} x^{4}-\frac {187}{38400} x^{5}-\frac {463}{921600} x^{6}-\frac {971}{22579200} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \]

Problem 6925

ODE	\[ \boxed {2 x \left (x +3\right ) y^{\prime \prime }-3 \left (1+x \right ) y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {3}{2}} \left (1+\frac {x}{15}-\frac {x^{2}}{315}+\frac {x^{3}}{2835}-\frac {x^{4}}{18711}+\frac {x^{5}}{104247}-\frac {x^{6}}{521235}+\frac {x^{7}}{2416635}+O\left (x^{8}\right )\right )+c_{2} \left (1+\frac {2 x}{3}+\frac {x^{2}}{9}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {3}{2}} \left (1+\frac {1}{15} x -\frac {1}{315} x^{2}+\frac {1}{2835} x^{3}-\frac {1}{18711} x^{4}+\frac {1}{104247} x^{5}-\frac {1}{521235} x^{6}+\frac {1}{2416635} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1+\frac {2}{3} x +\frac {1}{9} x^{2}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6926

ODE	\[ \boxed {2 x y^{\prime \prime }+\left (-2 x^{2}+1\right ) y^{\prime }-4 y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1+\frac {x^{2}}{2}+\frac {x^{4}}{8}+\frac {x^{6}}{48}+O\left (x^{8}\right )\right )+c_{2} \left (1+\frac {2 x^{2}}{3}+\frac {4 x^{4}}{21}+\frac {8 x^{6}}{231}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{2} x^{2}+\frac {1}{8} x^{4}+\frac {1}{48} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1+\frac {2}{3} x^{2}+\frac {4}{21} x^{4}+\frac {8}{231} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6927

ODE	\[ \boxed {x \left (4-x \right ) y^{\prime \prime }+\left (-x +2\right ) y^{\prime }+4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {5 x}{8}+\frac {7 x^{2}}{128}+\frac {3 x^{3}}{1024}+\frac {11 x^{4}}{32768}+\frac {13 x^{5}}{262144}+\frac {35 x^{6}}{4194304}+\frac {51 x^{7}}{33554432}+O\left (x^{8}\right )\right )+c_{2} \left (1-2 x +\frac {x^{2}}{2}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1-\frac {5}{8} x +\frac {7}{128} x^{2}+\frac {3}{1024} x^{3}+\frac {11}{32768} x^{4}+\frac {13}{262144} x^{5}+\frac {35}{4194304} x^{6}+\frac {51}{33554432} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1-2 x +\frac {1}{2} x^{2}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6928

ODE	\[ \boxed {3 x^{2} y^{\prime \prime }+x y^{\prime }-\left (1+x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1+\frac {x}{7}+\frac {x^{2}}{140}+\frac {x^{3}}{5460}+\frac {x^{4}}{349440}+\frac {x^{5}}{33196800}+\frac {x^{6}}{4381977600}+\frac {x^{7}}{766846080000}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1-x -\frac {x^{2}}{4}-\frac {x^{3}}{60}-\frac {x^{4}}{1920}-\frac {x^{5}}{105600}-\frac {x^{6}}{8870400}-\frac {x^{7}}{1055577600}+O\left (x^{8}\right )\right )}{x^{\frac {1}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-x -\frac {1}{4} x^{2}-\frac {1}{60} x^{3}-\frac {1}{1920} x^{4}-\frac {1}{105600} x^{5}-\frac {1}{8870400} x^{6}-\frac {1}{1055577600} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x^{\frac {1}{3}}}+c_{2} x \left (1+\frac {1}{7} x +\frac {1}{140} x^{2}+\frac {1}{5460} x^{3}+\frac {1}{349440} x^{4}+\frac {1}{33196800} x^{5}+\frac {1}{4381977600} x^{6}+\frac {1}{766846080000} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6929

ODE	\[ \boxed {2 x y^{\prime \prime }+\left (1+2 x \right ) y^{\prime }+4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {5 x}{3}+\frac {7 x^{2}}{6}-\frac {x^{3}}{2}+\frac {11 x^{4}}{72}-\frac {13 x^{5}}{360}+\frac {x^{6}}{144}-\frac {17 x^{7}}{15120}+O\left (x^{8}\right )\right )+c_{2} \left (1-4 x +4 x^{2}-\frac {32 x^{3}}{15}+\frac {16 x^{4}}{21}-\frac {64 x^{5}}{315}+\frac {64 x^{6}}{1485}-\frac {1024 x^{7}}{135135}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1-\frac {5}{3} x +\frac {7}{6} x^{2}-\frac {1}{2} x^{3}+\frac {11}{72} x^{4}-\frac {13}{360} x^{5}+\frac {1}{144} x^{6}-\frac {17}{15120} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1-4 x +4 x^{2}-\frac {32}{15} x^{3}+\frac {16}{21} x^{4}-\frac {64}{315} x^{5}+\frac {64}{1485} x^{6}-\frac {1024}{135135} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6930

ODE	\[ \boxed {2 x y^{\prime \prime }+\left (1+2 x \right ) y^{\prime }-5 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1+\frac {4 x}{3}+\frac {4 x^{2}}{15}+O\left (x^{8}\right )\right )+c_{2} \left (1+5 x +\frac {5 x^{2}}{2}+\frac {x^{3}}{6}-\frac {x^{4}}{168}+\frac {x^{5}}{2520}-\frac {x^{6}}{33264}+\frac {x^{7}}{432432}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {4}{3} x +\frac {4}{15} x^{2}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1+5 x +\frac {5}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{168} x^{4}+\frac {1}{2520} x^{5}-\frac {1}{33264} x^{6}+\frac {1}{432432} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6931

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }-3 x \left (1-x \right ) y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-\frac {6 x}{5}+\frac {27 x^{2}}{35}-\frac {12 x^{3}}{35}+\frac {9 x^{4}}{77}-\frac {162 x^{5}}{5005}+\frac {27 x^{6}}{3575}-\frac {648 x^{7}}{425425}+O\left (x^{8}\right )\right )+c_{2} \sqrt {x}\, \left (1+\frac {3 x}{2}-\frac {27 x^{2}}{8}+\frac {45 x^{3}}{16}-\frac {189 x^{4}}{128}+\frac {729 x^{5}}{1280}-\frac {891 x^{6}}{5120}+\frac {3159 x^{7}}{71680}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {3}{2} x -\frac {27}{8} x^{2}+\frac {45}{16} x^{3}-\frac {189}{128} x^{4}+\frac {729}{1280} x^{5}-\frac {891}{5120} x^{6}+\frac {3159}{71680} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x^{2} \left (1-\frac {6}{5} x +\frac {27}{35} x^{2}-\frac {12}{35} x^{3}+\frac {9}{77} x^{4}-\frac {162}{5005} x^{5}+\frac {27}{3575} x^{6}-\frac {648}{425425} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6932

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }+x \left (4 x -1\right ) y^{\prime }+2 \left (3 x -1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-2 x +2 x^{2}-\frac {4 x^{3}}{3}+\frac {2 x^{4}}{3}-\frac {4 x^{5}}{15}+\frac {4 x^{6}}{45}-\frac {8 x^{7}}{315}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1+\frac {4 x}{3}+\frac {16 x^{2}}{3}-\frac {64 x^{3}}{3}+\frac {256 x^{4}}{9}-\frac {1024 x^{5}}{45}+\frac {4096 x^{6}}{315}-\frac {16384 x^{7}}{2835}+O\left (x^{8}\right )\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1+\frac {4}{3} x +\frac {16}{3} x^{2}-\frac {64}{3} x^{3}+\frac {256}{9} x^{4}-\frac {1024}{45} x^{5}+\frac {4096}{315} x^{6}-\frac {16384}{2835} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{\sqrt {x}}+c_{2} x^{2} \left (1-2 x +2 x^{2}-\frac {4}{3} x^{3}+\frac {2}{3} x^{4}-\frac {4}{15} x^{5}+\frac {4}{45} x^{6}-\frac {8}{315} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6933

ODE	\[ \boxed {2 x y^{\prime \prime }-\left (2 x^{2}+1\right ) y^{\prime }-y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {3}{2}} \left (1+\frac {2 x^{2}}{7}+\frac {4 x^{4}}{77}+\frac {8 x^{6}}{1155}+O\left (x^{8}\right )\right )+c_{2} \left (1+\frac {x^{2}}{2}+\frac {x^{4}}{8}+\frac {x^{6}}{48}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {3}{2}} \left (1+\frac {2}{7} x^{2}+\frac {4}{77} x^{4}+\frac {8}{1155} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1+\frac {1}{2} x^{2}+\frac {1}{8} x^{4}+\frac {1}{48} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6934

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

Problem 6935

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

Problem 6936

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

Problem 6937

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

Problem 6938

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

Problem 6939

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

Problem 6940

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

Problem 6941

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

Problem 6942

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

Problem 6943

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

Problem 6944

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

Problem 6945

ODE	\[ \boxed {x^{2} y^{\prime \prime }-5 x y^{\prime }+9 y=0} \]
program solution	\[ y = c_{1} x^{3}+c_{2} x^{3} \ln \left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{3} \left (c_{2} \ln \left (x \right )+c_{1} \right ) \]

Problem 6946

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

Problem 6947

ODE	\[ \boxed {x^{3} y^{\prime \prime \prime }+4 x^{2} y^{\prime \prime }-8 x y^{\prime }+8 y=0} \]
program solution	\[ y = c_{2} x^{2}+c_{1} x +\frac {c_{3}}{x^{4}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{6}+c_{3} x^{5}+c_{2}}{x^{4}} \]

Problem 6948

ODE	\[ \boxed {x^{2} y^{\prime \prime }-x \left (1+x \right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1+x +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+\frac {x^{6}}{720}+\frac {x^{7}}{5040}+O\left (x^{8}\right )\right )+c_{2} \left (x \left (1+x +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+\frac {x^{6}}{720}+\frac {x^{7}}{5040}+O\left (x^{8}\right )\right ) \ln \left (x \right )+x \left (-x -\frac {3 x^{2}}{4}-\frac {11 x^{3}}{36}-\frac {25 x^{4}}{288}-\frac {137 x^{5}}{7200}-\frac {49 x^{6}}{14400}-\frac {121 x^{7}}{235200}+O\left (x^{8}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-x -\frac {3}{4} x^{2}-\frac {11}{36} x^{3}-\frac {25}{288} x^{4}-\frac {137}{7200} x^{5}-\frac {49}{14400} x^{6}-\frac {121}{235200} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \right ) x \]

Problem 6949

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+\left (1-2 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1+\frac {x}{2}+\frac {x^{2}}{16}+\frac {x^{3}}{288}+\frac {x^{4}}{9216}+\frac {x^{5}}{460800}+\frac {x^{6}}{33177600}+\frac {x^{7}}{3251404800}+O\left (x^{8}\right )\right )+c_{2} \left (\sqrt {x}\, \left (1+\frac {x}{2}+\frac {x^{2}}{16}+\frac {x^{3}}{288}+\frac {x^{4}}{9216}+\frac {x^{5}}{460800}+\frac {x^{6}}{33177600}+\frac {x^{7}}{3251404800}+O\left (x^{8}\right )\right ) \ln \left (x \right )+\sqrt {x}\, \left (-x -\frac {3 x^{2}}{16}-\frac {11 x^{3}}{864}-\frac {25 x^{4}}{55296}-\frac {137 x^{5}}{13824000}-\frac {49 x^{6}}{331776000}-\frac {121 x^{7}}{75866112000}+O\left (x^{8}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+\frac {1}{2} x +\frac {1}{16} x^{2}+\frac {1}{288} x^{3}+\frac {1}{9216} x^{4}+\frac {1}{460800} x^{5}+\frac {1}{33177600} x^{6}+\frac {1}{3251404800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-x -\frac {3}{16} x^{2}-\frac {11}{864} x^{3}-\frac {25}{55296} x^{4}-\frac {137}{13824000} x^{5}-\frac {49}{331776000} x^{6}-\frac {121}{75866112000} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \right ) \sqrt {x} \]

Problem 6950

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x \left (x -3\right ) y^{\prime }+4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-2 x +\frac {3 x^{2}}{2}-\frac {2 x^{3}}{3}+\frac {5 x^{4}}{24}-\frac {x^{5}}{20}+\frac {7 x^{6}}{720}-\frac {x^{7}}{630}+O\left (x^{8}\right )\right )+c_{2} \left (x^{2} \left (1-2 x +\frac {3 x^{2}}{2}-\frac {2 x^{3}}{3}+\frac {5 x^{4}}{24}-\frac {x^{5}}{20}+\frac {7 x^{6}}{720}-\frac {x^{7}}{630}+O\left (x^{8}\right )\right ) \ln \left (x \right )+x^{2} \left (3 x -\frac {13 x^{2}}{4}+\frac {31 x^{3}}{18}-\frac {173 x^{4}}{288}+\frac {187 x^{5}}{1200}-\frac {463 x^{6}}{14400}+\frac {971 x^{7}}{176400}+O\left (x^{8}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-2 x +\frac {3}{2} x^{2}-\frac {2}{3} x^{3}+\frac {5}{24} x^{4}-\frac {1}{20} x^{5}+\frac {7}{720} x^{6}-\frac {1}{630} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (3 x -\frac {13}{4} x^{2}+\frac {31}{18} x^{3}-\frac {173}{288} x^{4}+\frac {187}{1200} x^{5}-\frac {463}{14400} x^{6}+\frac {971}{176400} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \right ) x^{2} \]

Problem 6951

ODE	\[ \boxed {x^{2} y^{\prime \prime }+3 x y^{\prime }+\left (4 x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (-x^{2}+1+\frac {x^{4}}{4}-\frac {x^{6}}{36}+O\left (x^{8}\right )\right )}{x}+c_{2} \left (\frac {\left (-x^{2}+1+\frac {x^{4}}{4}-\frac {x^{6}}{36}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{x}+\frac {x^{2}-\frac {3 x^{4}}{8}+\frac {11 x^{6}}{216}+O\left (x^{8}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-x^{2}+\frac {1}{4} x^{4}-\frac {1}{36} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+\left (x^{2}-\frac {3}{8} x^{4}+\frac {11}{216} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) c_{2}}{x} \]

Problem 6952

ODE	\[ \boxed {x \left (1+x \right ) y^{\prime \prime }+\left (5 x +1\right ) y^{\prime }+3 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (-36 x^{7}+28 x^{6}-21 x^{5}+15 x^{4}-10 x^{3}+6 x^{2}-3 x +1+O\left (x^{8}\right )\right )+c_{2} \left (\left (-36 x^{7}+28 x^{6}-21 x^{5}+15 x^{4}-10 x^{3}+6 x^{2}-3 x +1+O\left (x^{8}\right )\right ) \ln \left (x \right )+2 x -\frac {11 x^{2}}{2}+\frac {21 x^{3}}{2}-17 x^{4}+25 x^{5}-\frac {69 x^{6}}{2}+\frac {91 x^{7}}{2}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-3 x +6 x^{2}-10 x^{3}+15 x^{4}-21 x^{5}+28 x^{6}-36 x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (2 x -\frac {11}{2} x^{2}+\frac {21}{2} x^{3}-17 x^{4}+25 x^{5}-\frac {69}{2} x^{6}+\frac {91}{2} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \]

Problem 6953

ODE	\[ \boxed {x^{2} y^{\prime \prime }-x \left (1+3 x \right ) y^{\prime }+\left (1-6 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (45 x^{3}+27 x^{2}+9 x +1+\frac {405 x^{4}}{8}+\frac {1701 x^{5}}{40}+\frac {567 x^{6}}{20}+\frac {2187 x^{7}}{140}+O\left (x^{8}\right )\right )+c_{2} \left (x \left (45 x^{3}+27 x^{2}+9 x +1+\frac {405 x^{4}}{8}+\frac {1701 x^{5}}{40}+\frac {567 x^{6}}{20}+\frac {2187 x^{7}}{140}+O\left (x^{8}\right )\right ) \ln \left (x \right )+x \left (-15 x -\frac {261 x^{2}}{4}-\frac {519 x^{3}}{4}-\frac {5211 x^{4}}{32}-\frac {118179 x^{5}}{800}-\frac {83511 x^{6}}{800}-\frac {2361717 x^{7}}{39200}+O\left (x^{8}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+9 x +27 x^{2}+45 x^{3}+\frac {405}{8} x^{4}+\frac {1701}{40} x^{5}+\frac {567}{20} x^{6}+\frac {2187}{140} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (\left (-15\right ) x -\frac {261}{4} x^{2}-\frac {519}{4} x^{3}-\frac {5211}{32} x^{4}-\frac {118179}{800} x^{5}-\frac {83511}{800} x^{6}-\frac {2361717}{39200} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \right ) x \]

Problem 6954

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x \left (x -1\right ) y^{\prime }+\left (1-x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1+O\left (x^{8}\right )\right )+c_{2} \left (x \left (1+O\left (x^{8}\right )\right ) \ln \left (x \right )+x \left (-x +\frac {x^{2}}{4}-\frac {x^{3}}{18}+\frac {x^{4}}{96}-\frac {x^{5}}{600}+\frac {x^{6}}{4320}-\frac {x^{7}}{35280}+O\left (x^{8}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+\operatorname {O}\left (x^{8}\right )\right )+\left (-x +\frac {1}{4} x^{2}-\frac {1}{18} x^{3}+\frac {1}{96} x^{4}-\frac {1}{600} x^{5}+\frac {1}{4320} x^{6}-\frac {1}{35280} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \right ) x \]

Problem 6955

ODE	\[ \boxed {x \left (x -2\right ) y^{\prime \prime }+2 \left (x -1\right ) y^{\prime }-2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-x +O\left (x^{8}\right )\right )+c_{2} \left (\left (1-x +O\left (x^{8}\right )\right ) \ln \left (x \right )+\frac {5 x}{2}-\frac {3 x^{2}}{8}-\frac {x^{3}}{12}-\frac {5 x^{4}}{192}-\frac {3 x^{5}}{320}-\frac {7 x^{6}}{1920}-\frac {x^{7}}{672}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-x +\operatorname {O}\left (x^{8}\right )\right )+\left (\frac {5}{2} x -\frac {3}{8} x^{2}-\frac {1}{12} x^{3}-\frac {5}{192} x^{4}-\frac {3}{320} x^{5}-\frac {7}{1920} x^{6}-\frac {1}{672} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \]

Problem 6956

ODE	\[ \boxed {x \left (x -2\right ) y^{\prime \prime }+2 \left (x -1\right ) y^{\prime }-2 y=0} \] With the expansion point for the power series method at \(x = 2\).
program solution	\[ y = c_{1} \left (x -1+O\left (\left (x -2\right )^{8}\right )\right )+c_{2} \left (\left (x -1+O\left (\left (x -2\right )^{8}\right )\right ) \ln \left (x -2\right )-\frac {5 x}{2}+5-\frac {3 \left (x -2\right )^{2}}{8}+\frac {\left (x -2\right )^{3}}{12}-\frac {5 \left (x -2\right )^{4}}{192}+\frac {3 \left (x -2\right )^{5}}{320}-\frac {7 \left (x -2\right )^{6}}{1920}+\frac {\left (x -2\right )^{7}}{672}+O\left (\left (x -2\right )^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\ln \left (-2+x \right ) c_{2} +c_{1} \right ) \left (1+\left (-2+x \right )+\operatorname {O}\left (\left (-2+x \right )^{8}\right )\right )+\left (-\frac {5}{2} \left (-2+x \right )-\frac {3}{8} \left (-2+x \right )^{2}+\frac {1}{12} \left (-2+x \right )^{3}-\frac {5}{192} \left (-2+x \right )^{4}+\frac {3}{320} \left (-2+x \right )^{5}-\frac {7}{1920} \left (-2+x \right )^{6}+\frac {1}{672} \left (-2+x \right )^{7}+\operatorname {O}\left (\left (-2+x \right )^{8}\right )\right ) c_{2} \]

Problem 6957

ODE	\[ \boxed {4 \left (-4+x \right )^{2} y^{\prime \prime }+\left (-4+x \right ) \left (x -8\right ) y^{\prime }+y x=0} \] With the expansion point for the power series method at \(x = 4\).
program solution	\[ y = c_{1} \left (-4+x \right ) \left (3-\frac {x}{2}+\frac {3 \left (-4+x \right )^{2}}{32}-\frac {\left (-4+x \right )^{3}}{96}+\frac {5 \left (-4+x \right )^{4}}{6144}-\frac {\left (-4+x \right )^{5}}{20480}+\frac {7 \left (-4+x \right )^{6}}{2949120}-\frac {\left (-4+x \right )^{7}}{10321920}+O\left (\left (-4+x \right )^{8}\right )\right )+c_{2} \left (\left (-4+x \right ) \left (3-\frac {x}{2}+\frac {3 \left (-4+x \right )^{2}}{32}-\frac {\left (-4+x \right )^{3}}{96}+\frac {5 \left (-4+x \right )^{4}}{6144}-\frac {\left (-4+x \right )^{5}}{20480}+\frac {7 \left (-4+x \right )^{6}}{2949120}-\frac {\left (-4+x \right )^{7}}{10321920}+O\left (\left (-4+x \right )^{8}\right )\right ) \ln \left (-4+x \right )+\left (-4+x \right ) \left (-3+\frac {3 x}{4}-\frac {13 \left (-4+x \right )^{2}}{64}+\frac {31 \left (-4+x \right )^{3}}{1152}-\frac {173 \left (-4+x \right )^{4}}{73728}+\frac {187 \left (-4+x \right )^{5}}{1228800}-\frac {463 \left (-4+x \right )^{6}}{58982400}+\frac {971 \left (-4+x \right )^{7}}{2890137600}+O\left (\left (-4+x \right )^{8}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (\ln \left (x -4\right ) c_{2} +c_{1} \right ) \left (1-\frac {1}{2} \left (x -4\right )+\frac {3}{32} \left (x -4\right )^{2}-\frac {1}{96} \left (x -4\right )^{3}+\frac {5}{6144} \left (x -4\right )^{4}-\frac {1}{20480} \left (x -4\right )^{5}+\frac {7}{2949120} \left (x -4\right )^{6}-\frac {1}{10321920} \left (x -4\right )^{7}+\operatorname {O}\left (\left (x -4\right )^{8}\right )\right )+\left (\frac {3}{4} \left (x -4\right )-\frac {13}{64} \left (x -4\right )^{2}+\frac {31}{1152} \left (x -4\right )^{3}-\frac {173}{73728} \left (x -4\right )^{4}+\frac {187}{1228800} \left (x -4\right )^{5}-\frac {463}{58982400} \left (x -4\right )^{6}+\frac {971}{2890137600} \left (x -4\right )^{7}+\operatorname {O}\left (\left (x -4\right )^{8}\right )\right ) c_{2} \right ) \left (x -4\right ) \]

Problem 6958

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

Problem 6959

ODE	\[ \boxed {x y^{\prime \prime }+y^{\prime }-y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1+\frac {x^{2}}{4}+\frac {x^{4}}{64}+\frac {x^{6}}{2304}+O\left (x^{8}\right )\right )+c_{2} \left (\left (1+\frac {x^{2}}{4}+\frac {x^{4}}{64}+\frac {x^{6}}{2304}+O\left (x^{8}\right )\right ) \ln \left (x \right )-\frac {x^{2}}{4}-\frac {3 x^{4}}{128}-\frac {11 x^{6}}{13824}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+\frac {1}{4} x^{2}+\frac {1}{64} x^{4}+\frac {1}{2304} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+\left (-\frac {1}{4} x^{2}-\frac {3}{128} x^{4}-\frac {11}{13824} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \]

Problem 6960

ODE	\[ \boxed {x y^{\prime \prime }+\left (-x^{2}+1\right ) y^{\prime }-y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1+\frac {x^{2}}{4}+\frac {3 x^{4}}{64}+\frac {5 x^{6}}{768}+O\left (x^{8}\right )\right )+c_{2} \left (\left (1+\frac {x^{2}}{4}+\frac {3 x^{4}}{64}+\frac {5 x^{6}}{768}+O\left (x^{8}\right )\right ) \ln \left (x \right )-\frac {x^{4}}{128}-\frac {x^{6}}{512}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+\frac {1}{4} x^{2}+\frac {3}{64} x^{4}+\frac {5}{768} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+\left (-\frac {1}{128} x^{4}-\frac {1}{512} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \]

Problem 6961

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x \left (3+2 x \right ) y^{\prime }+\left (1+3 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1-x +\frac {3 x^{2}}{4}-\frac {5 x^{3}}{12}+\frac {35 x^{4}}{192}-\frac {21 x^{5}}{320}+\frac {77 x^{6}}{3840}-\frac {143 x^{7}}{26880}+O\left (x^{8}\right )\right )}{x}+c_{2} \left (\frac {\left (1-x +\frac {3 x^{2}}{4}-\frac {5 x^{3}}{12}+\frac {35 x^{4}}{192}-\frac {21 x^{5}}{320}+\frac {77 x^{6}}{3840}-\frac {143 x^{7}}{26880}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{x}+\frac {-\frac {x^{2}}{4}+\frac {x^{3}}{4}-\frac {19 x^{4}}{128}+\frac {25 x^{5}}{384}-\frac {317 x^{6}}{13824}+\frac {469 x^{7}}{69120}+O\left (x^{8}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-x +\frac {3}{4} x^{2}-\frac {5}{12} x^{3}+\frac {35}{192} x^{4}-\frac {21}{320} x^{5}+\frac {77}{3840} x^{6}-\frac {143}{26880} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-\frac {1}{4} x^{2}+\frac {1}{4} x^{3}-\frac {19}{128} x^{4}+\frac {25}{384} x^{5}-\frac {317}{13824} x^{6}+\frac {469}{69120} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2}}{x} \]

Problem 6962

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+8 x \left (1+x \right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1+x -\frac {x^{2}}{4}+\frac {x^{3}}{12}-\frac {5 x^{4}}{192}+\frac {7 x^{5}}{960}-\frac {7 x^{6}}{3840}+\frac {11 x^{7}}{26880}+O\left (x^{8}\right )\right )}{\sqrt {x}}+c_{2} \left (\frac {\left (1+x -\frac {x^{2}}{4}+\frac {x^{3}}{12}-\frac {5 x^{4}}{192}+\frac {7 x^{5}}{960}-\frac {7 x^{6}}{3840}+\frac {11 x^{7}}{26880}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{\sqrt {x}}+\frac {-4 x +\frac {3 x^{2}}{4}-\frac {x^{3}}{4}+\frac {31 x^{4}}{384}-\frac {3 x^{5}}{128}+\frac {419 x^{6}}{69120}-\frac {97 x^{7}}{69120}+O\left (x^{8}\right )}{\sqrt {x}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+x -\frac {1}{4} x^{2}+\frac {1}{12} x^{3}-\frac {5}{192} x^{4}+\frac {7}{960} x^{5}-\frac {7}{3840} x^{6}+\frac {11}{26880} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (\left (-4\right ) x +\frac {3}{4} x^{2}-\frac {1}{4} x^{3}+\frac {31}{384} x^{4}-\frac {3}{128} x^{5}+\frac {419}{69120} x^{6}-\frac {97}{69120} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2}}{\sqrt {x}} \]

Problem 6963

ODE	\[ \boxed {x^{2} y^{\prime \prime }+3 x \left (1+x \right ) y^{\prime }+\left (1-3 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1+6 x +\frac {9 x^{2}}{2}+O\left (x^{8}\right )\right )}{x}+c_{2} \left (\frac {\left (1+6 x +\frac {9 x^{2}}{2}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{x}+\frac {-15 x -\frac {81 x^{2}}{4}-\frac {3 x^{3}}{2}+\frac {9 x^{4}}{32}-\frac {27 x^{5}}{400}+\frac {27 x^{6}}{1600}-\frac {81 x^{7}}{19600}+O\left (x^{8}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+6 x +\frac {9}{2} x^{2}+\operatorname {O}\left (x^{8}\right )\right )+\left (\left (-15\right ) x -\frac {81}{4} x^{2}-\frac {3}{2} x^{3}+\frac {9}{32} x^{4}-\frac {27}{400} x^{5}+\frac {27}{1600} x^{6}-\frac {81}{19600} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2}}{x} \]

Problem 6964

ODE	\[ \boxed {x y^{\prime \prime }+\left (1-x \right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1+x +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+\frac {x^{6}}{720}+\frac {x^{7}}{5040}+O\left (x^{8}\right )\right )+c_{2} \left (\left (1+x +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+\frac {x^{6}}{720}+\frac {x^{7}}{5040}+O\left (x^{8}\right )\right ) \ln \left (x \right )-x -\frac {3 x^{2}}{4}-\frac {11 x^{3}}{36}-\frac {25 x^{4}}{288}-\frac {137 x^{5}}{7200}-\frac {49 x^{6}}{14400}-\frac {121 x^{7}}{235200}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-x -\frac {3}{4} x^{2}-\frac {11}{36} x^{3}-\frac {25}{288} x^{4}-\frac {137}{7200} x^{5}-\frac {49}{14400} x^{6}-\frac {121}{235200} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \]

Problem 6965

ODE	\[ \boxed {x^{2} y^{\prime \prime }+2 x \left (x -2\right ) y^{\prime }+2 \left (2-3 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{4} \left (1-\frac {x}{2}+\frac {x^{2}}{5}-\frac {x^{3}}{15}+\frac {2 x^{4}}{105}-\frac {x^{5}}{210}+\frac {x^{6}}{945}-\frac {x^{7}}{4725}+O\left (x^{8}\right )\right )+c_{2} x \left (1-2 x +2 x^{2}-\frac {4 x^{3}}{3}+\frac {2 x^{4}}{3}-\frac {4 x^{5}}{15}+\frac {4 x^{6}}{45}-\frac {8 x^{7}}{315}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{4} \left (1-\frac {1}{2} x +\frac {1}{5} x^{2}-\frac {1}{15} x^{3}+\frac {2}{105} x^{4}-\frac {1}{210} x^{5}+\frac {1}{945} x^{6}-\frac {1}{4725} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x \left (12-24 x +24 x^{2}-16 x^{3}+8 x^{4}-\frac {16}{5} x^{5}+\frac {16}{15} x^{6}-\frac {32}{105} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6966

ODE	\[ \boxed {x^{2} \left (1+2 x \right ) y^{\prime \prime }+2 x \left (1+6 x \right ) y^{\prime }-2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-3 x +\frac {42 x^{2}}{5}-\frac {112 x^{3}}{5}+\frac {288 x^{4}}{5}-144 x^{5}+352 x^{6}-\frac {4224 x^{7}}{5}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1-6 x +24 x^{2}-80 x^{3}+240 x^{4}-672 x^{5}+1792 x^{6}-4608 x^{7}+O\left (x^{8}\right )\right )}{x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1-3 x +\frac {42}{5} x^{2}-\frac {112}{5} x^{3}+\frac {288}{5} x^{4}-144 x^{5}+352 x^{6}-\frac {4224}{5} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (12-72 x +288 x^{2}-960 x^{3}+2880 x^{4}-8064 x^{5}+21504 x^{6}-55296 x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x^{2}} \]

Problem 6967

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x \left (3 x +2\right ) y^{\prime }-2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {3 x}{4}+\frac {9 x^{2}}{20}-\frac {9 x^{3}}{40}+\frac {27 x^{4}}{280}-\frac {81 x^{5}}{2240}+\frac {27 x^{6}}{2240}-\frac {81 x^{7}}{22400}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1-3 x +\frac {9 x^{2}}{2}-\frac {9 x^{3}}{2}+\frac {27 x^{4}}{8}-\frac {81 x^{5}}{40}+\frac {81 x^{6}}{80}-\frac {243 x^{7}}{560}+O\left (x^{8}\right )\right )}{x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1-\frac {3}{4} x +\frac {9}{20} x^{2}-\frac {9}{40} x^{3}+\frac {27}{280} x^{4}-\frac {81}{2240} x^{5}+\frac {27}{2240} x^{6}-\frac {81}{22400} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (12-36 x +54 x^{2}-54 x^{3}+\frac {81}{2} x^{4}-\frac {243}{10} x^{5}+\frac {243}{20} x^{6}-\frac {729}{140} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x^{2}} \]

Problem 6968

ODE	\[ \boxed {x y^{\prime \prime }-\left (x +3\right ) y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{4} \left (1+\frac {2 x}{5}+\frac {x^{2}}{10}+\frac {2 x^{3}}{105}+\frac {x^{4}}{336}+\frac {x^{5}}{2520}+\frac {x^{6}}{21600}+\frac {x^{7}}{207900}+O\left (x^{8}\right )\right )+c_{2} \left (1+\frac {2 x}{3}+\frac {x^{2}}{6}-\frac {x^{4}}{72}-\frac {x^{5}}{180}-\frac {x^{6}}{720}-\frac {x^{7}}{3780}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{4} \left (1+\frac {2}{5} x +\frac {1}{10} x^{2}+\frac {2}{105} x^{3}+\frac {1}{336} x^{4}+\frac {1}{2520} x^{5}+\frac {1}{21600} x^{6}+\frac {1}{207900} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (-144-96 x -24 x^{2}+2 x^{4}+\frac {4}{5} x^{5}+\frac {1}{5} x^{6}+\frac {4}{105} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6969

ODE	\[ \boxed {x \left (1+x \right ) y^{\prime \prime }+\left (x +5\right ) y^{\prime }-4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1+\frac {4 x}{5}+\frac {x^{2}}{5}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1+4 x +5 x^{2}-5 x^{4}-4 x^{5}-x^{6}+O\left (x^{8}\right )\right )}{x^{4}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (1+\frac {4}{5} x +\frac {1}{5} x^{2}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (-144-576 x -720 x^{2}+720 x^{4}+576 x^{5}+144 x^{6}+\operatorname {O}\left (x^{8}\right )\right )}{x^{4}} \]

Problem 6970

ODE	\[ \boxed {x \left (1+x \right ) y^{\prime \prime }+\left (x +5\right ) y^{\prime }-4 y=0} \] With the expansion point for the power series method at \(x = -1\).
program solution	\[ y = c_{1} \left (1+x \right )^{5} \left (\frac {9}{2}+\frac {7 x}{2}+8 \left (1+x \right )^{2}+15 \left (1+x \right )^{3}+25 \left (1+x \right )^{4}+\frac {77 \left (1+x \right )^{5}}{2}+56 \left (1+x \right )^{6}+78 \left (1+x \right )^{7}+O\left (\left (1+x \right )^{8}\right )\right )+c_{2} \left (2+x +\frac {\left (1+x \right )^{2}}{2}+\left (1+x \right )^{5}+\frac {7 \left (1+x \right )^{6}}{2}+8 \left (1+x \right )^{7}+O\left (\left (1+x \right )^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (x +1\right )^{5} \left (1+\frac {7}{2} \left (x +1\right )+8 \left (x +1\right )^{2}+15 \left (x +1\right )^{3}+25 \left (x +1\right )^{4}+\frac {77}{2} \left (x +1\right )^{5}+56 \left (x +1\right )^{6}+78 \left (x +1\right )^{7}+\operatorname {O}\left (\left (x +1\right )^{8}\right )\right )+c_{2} \left (2880+2880 \left (x +1\right )+1440 \left (x +1\right )^{2}+2880 \left (x +1\right )^{5}+10080 \left (x +1\right )^{6}+23040 \left (x +1\right )^{7}+\operatorname {O}\left (\left (x +1\right )^{8}\right )\right ) \]

Problem 6971

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x^{2}-2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-\frac {x}{2}+\frac {3 x^{2}}{20}-\frac {x^{3}}{30}+\frac {x^{4}}{168}-\frac {x^{5}}{1120}+\frac {x^{6}}{8640}-\frac {x^{7}}{75600}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1-\frac {x}{2}+\frac {x^{3}}{12}-\frac {x^{4}}{24}+\frac {x^{5}}{80}-\frac {x^{6}}{360}+\frac {x^{7}}{2016}+O\left (x^{8}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \left (1-\frac {1}{2} x +\frac {3}{20} x^{2}-\frac {1}{30} x^{3}+\frac {1}{168} x^{4}-\frac {1}{1120} x^{5}+\frac {1}{8640} x^{6}-\frac {1}{75600} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (12-6 x +x^{3}-\frac {1}{2} x^{4}+\frac {3}{20} x^{5}-\frac {1}{30} x^{6}+\frac {1}{168} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x} \]

Problem 6972

ODE	\[ \boxed {x \left (1-x \right ) y^{\prime \prime }-3 y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{4} \left (1+2 x +3 x^{2}+4 x^{3}+5 x^{4}+6 x^{5}+7 x^{6}+8 x^{7}+O\left (x^{8}\right )\right )+c_{2} \left (1+\frac {2 x}{3}+\frac {x^{2}}{3}-\frac {x^{4}}{3}-\frac {2 x^{5}}{3}-x^{6}-\frac {4 x^{7}}{3}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{4} \left (1+2 x +3 x^{2}+4 x^{3}+5 x^{4}+6 x^{5}+7 x^{6}+8 x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (-144-96 x -48 x^{2}+48 x^{4}+96 x^{5}+144 x^{6}+192 x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6973

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

Problem 6974

ODE	\[ \boxed {x y^{\prime \prime }+\left (4+3 x \right ) y^{\prime }+3 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-\frac {3 x}{4}+\frac {9 x^{2}}{20}-\frac {9 x^{3}}{40}+\frac {27 x^{4}}{280}-\frac {81 x^{5}}{2240}+\frac {27 x^{6}}{2240}-\frac {81 x^{7}}{22400}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1-3 x +\frac {9 x^{2}}{2}-\frac {9 x^{3}}{2}+\frac {27 x^{4}}{8}-\frac {81 x^{5}}{40}+\frac {81 x^{6}}{80}-\frac {243 x^{7}}{560}+O\left (x^{8}\right )\right )}{x^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (1-\frac {3}{4} x +\frac {9}{20} x^{2}-\frac {9}{40} x^{3}+\frac {27}{280} x^{4}-\frac {81}{2240} x^{5}+\frac {27}{2240} x^{6}-\frac {81}{22400} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (12-36 x +54 x^{2}-54 x^{3}+\frac {81}{2} x^{4}-\frac {243}{10} x^{5}+\frac {243}{20} x^{6}-\frac {729}{140} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x^{3}} \]

Problem 6975

ODE	\[ \boxed {x y^{\prime \prime }-2 \left (x +2\right ) y^{\prime }+4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{5} \left (1+x +\frac {4 x^{2}}{7}+\frac {5 x^{3}}{21}+\frac {5 x^{4}}{63}+\frac {x^{5}}{45}+\frac {8 x^{6}}{1485}+\frac {4 x^{7}}{3465}+O\left (x^{8}\right )\right )+c_{2} \left (1+x +\frac {x^{2}}{3}+\frac {2 x^{5}}{45}+\frac {2 x^{6}}{45}+\frac {8 x^{7}}{315}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{5} \left (1+x +\frac {4}{7} x^{2}+\frac {5}{21} x^{3}+\frac {5}{63} x^{4}+\frac {1}{45} x^{5}+\frac {8}{1485} x^{6}+\frac {4}{3465} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (2880+2880 x +960 x^{2}+128 x^{5}+128 x^{6}+\frac {512}{7} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6976

ODE	\[ \boxed {x y^{\prime \prime }+\left (3+2 x \right ) y^{\prime }+4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-\frac {4 x}{3}+x^{2}-\frac {8 x^{3}}{15}+\frac {2 x^{4}}{9}-\frac {8 x^{5}}{105}+\frac {x^{6}}{45}-\frac {16 x^{7}}{2835}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1-2 x^{2}+\frac {8 x^{3}}{3}-2 x^{4}+\frac {16 x^{5}}{15}-\frac {4 x^{6}}{9}+\frac {16 x^{7}}{105}+O\left (x^{8}\right )\right )}{x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (1-\frac {4}{3} x +x^{2}-\frac {8}{15} x^{3}+\frac {2}{9} x^{4}-\frac {8}{105} x^{5}+\frac {1}{45} x^{6}-\frac {16}{2835} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (-2+4 x^{2}-\frac {16}{3} x^{3}+4 x^{4}-\frac {32}{15} x^{5}+\frac {8}{9} x^{6}-\frac {32}{105} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x^{2}} \]

Problem 6977

ODE	\[ \boxed {x \left (x +3\right ) y^{\prime \prime }-9 y^{\prime }-6 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{4} \left (1-\frac {2 x}{5}+\frac {7 x^{2}}{45}-\frac {8 x^{3}}{135}+\frac {x^{4}}{45}-\frac {2 x^{5}}{243}+\frac {11 x^{6}}{3645}-\frac {4 x^{7}}{3645}+O\left (x^{8}\right )\right )+c_{2} \left (1-\frac {2 x}{3}+\frac {x^{2}}{3}-\frac {4 x^{3}}{27}+\frac {5 x^{4}}{81}-\frac {2 x^{5}}{81}+\frac {7 x^{6}}{729}-\frac {8 x^{7}}{2187}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{4} \left (1-\frac {2}{5} x +\frac {7}{45} x^{2}-\frac {8}{135} x^{3}+\frac {1}{45} x^{4}-\frac {2}{243} x^{5}+\frac {11}{3645} x^{6}-\frac {4}{3645} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (-144+96 x -48 x^{2}+\frac {64}{3} x^{3}-\frac {80}{9} x^{4}+\frac {32}{9} x^{5}-\frac {112}{81} x^{6}+\frac {128}{243} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6978

ODE	\[ \boxed {x \left (1-2 x \right ) y^{\prime \prime }-2 \left (x +2\right ) y^{\prime }+8 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{5} \left (1+7 x +32 x^{2}+120 x^{3}+400 x^{4}+1232 x^{5}+3584 x^{6}+9984 x^{7}+O\left (x^{8}\right )\right )+c_{2} \left (1+2 x +2 x^{2}+32 x^{5}+224 x^{6}+1024 x^{7}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{5} \left (1+7 x +32 x^{2}+120 x^{3}+400 x^{4}+1232 x^{5}+3584 x^{6}+9984 x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (2880+5760 x +5760 x^{2}+92160 x^{5}+645120 x^{6}+2949120 x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6979

ODE	\[ \boxed {x y^{\prime \prime }+\left (x^{3}-1\right ) y^{\prime }+y x^{2}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-\frac {x^{3}}{5}+\frac {x^{6}}{40}+O\left (x^{8}\right )\right )+c_{2} \left (1-\frac {x^{3}}{3}+\frac {x^{6}}{18}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \left (1-\frac {1}{5} x^{3}+\frac {1}{40} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (-2+\frac {2}{3} x^{3}-\frac {1}{9} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6980

ODE	\[ \boxed {x^{2} \left (4 x -1\right ) y^{\prime \prime }+x \left (5 x +1\right ) y^{\prime }+3 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{3} \left (1+\frac {39 x}{5}+\frac {221 x^{2}}{5}+221 x^{3}+\frac {16575 x^{4}}{16}+\frac {224315 x^{5}}{48}+\frac {493493 x^{6}}{24}+\frac {711399 x^{7}}{8}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1-x -\frac {15 x^{4}}{8}-\frac {117 x^{5}}{8}-\frac {663 x^{6}}{8}-\frac {3315 x^{7}}{8}+O\left (x^{8}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{3} \left (1+\frac {39}{5} x +\frac {221}{5} x^{2}+221 x^{3}+\frac {16575}{16} x^{4}+\frac {224315}{48} x^{5}+\frac {493493}{24} x^{6}+\frac {711399}{8} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (-144+144 x +270 x^{4}+2106 x^{5}+11934 x^{6}+59670 x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x} \]

Problem 6981

ODE	\[ \boxed {x y^{\prime \prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {x}{2}+\frac {x^{2}}{12}-\frac {x^{3}}{144}+\frac {x^{4}}{2880}-\frac {x^{5}}{86400}+\frac {x^{6}}{3628800}-\frac {x^{7}}{203212800}+O\left (x^{8}\right )\right )+c_{2} \left (-x \left (1-\frac {x}{2}+\frac {x^{2}}{12}-\frac {x^{3}}{144}+\frac {x^{4}}{2880}-\frac {x^{5}}{86400}+\frac {x^{6}}{3628800}-\frac {x^{7}}{203212800}+O\left (x^{8}\right )\right ) \ln \left (x \right )+1-\frac {3 x^{2}}{4}+\frac {7 x^{3}}{36}-\frac {35 x^{4}}{1728}+\frac {101 x^{5}}{86400}-\frac {7 x^{6}}{162000}+\frac {283 x^{7}}{254016000}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1-\frac {1}{2} x +\frac {1}{12} x^{2}-\frac {1}{144} x^{3}+\frac {1}{2880} x^{4}-\frac {1}{86400} x^{5}+\frac {1}{3628800} x^{6}-\frac {1}{203212800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-x +\frac {1}{2} x^{2}-\frac {1}{12} x^{3}+\frac {1}{144} x^{4}-\frac {1}{2880} x^{5}+\frac {1}{86400} x^{6}-\frac {1}{3628800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (1-\frac {3}{4} x^{2}+\frac {7}{36} x^{3}-\frac {35}{1728} x^{4}+\frac {101}{86400} x^{5}-\frac {7}{162000} x^{6}+\frac {283}{254016000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right ) \]

Problem 6982

ODE	\[ \boxed {x^{2} y^{\prime \prime }-3 x y^{\prime }+\left (4 x +3\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{3} \left (1-\frac {4 x}{3}+\frac {2 x^{2}}{3}-\frac {8 x^{3}}{45}+\frac {4 x^{4}}{135}-\frac {16 x^{5}}{4725}+\frac {4 x^{6}}{14175}-\frac {16 x^{7}}{893025}+O\left (x^{8}\right )\right )+c_{2} \left (-8 x^{3} \left (1-\frac {4 x}{3}+\frac {2 x^{2}}{3}-\frac {8 x^{3}}{45}+\frac {4 x^{4}}{135}-\frac {16 x^{5}}{4725}+\frac {4 x^{6}}{14175}-\frac {16 x^{7}}{893025}+O\left (x^{8}\right )\right ) \ln \left (x \right )+x \left (1+4 x -\frac {128 x^{3}}{9}+\frac {100 x^{4}}{9}-\frac {2512 x^{5}}{675}+\frac {1456 x^{6}}{2025}-\frac {45376 x^{7}}{496125}+O\left (x^{8}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x \left (c_{1} x^{2} \left (1-\frac {4}{3} x +\frac {2}{3} x^{2}-\frac {8}{45} x^{3}+\frac {4}{135} x^{4}-\frac {16}{4725} x^{5}+\frac {4}{14175} x^{6}-\frac {16}{893025} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (16 x^{2}-\frac {64}{3} x^{3}+\frac {32}{3} x^{4}-\frac {128}{45} x^{5}+\frac {64}{135} x^{6}-\frac {256}{4725} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-2-8 x +\frac {256}{9} x^{3}-\frac {200}{9} x^{4}+\frac {5024}{675} x^{5}-\frac {2912}{2025} x^{6}+\frac {90752}{496125} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )\right ) \]

Problem 6983

ODE	\[ \boxed {2 x y^{\prime \prime }+6 y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-\frac {x}{6}+\frac {x^{2}}{96}-\frac {x^{3}}{2880}+\frac {x^{4}}{138240}-\frac {x^{5}}{9676800}+\frac {x^{6}}{928972800}-\frac {x^{7}}{117050572800}+O\left (x^{8}\right )\right )+c_{2} \left (\left (-\frac {1}{8}+\frac {x}{48}-\frac {x^{2}}{768}+\frac {x^{3}}{23040}-\frac {x^{4}}{1105920}+\frac {x^{5}}{77414400}-\frac {x^{6}}{7431782400}+\frac {x^{7}}{936404582400}-\frac {O\left (x^{8}\right )}{8}\right ) \ln \left (x \right )+\frac {1+\frac {x}{2}-\frac {x^{3}}{36}+\frac {25 x^{4}}{9216}-\frac {157 x^{5}}{1382400}+\frac {91 x^{6}}{33177600}-\frac {709 x^{7}}{16257024000}+O\left (x^{8}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-\frac {1}{6} x +\frac {1}{96} x^{2}-\frac {1}{2880} x^{3}+\frac {1}{138240} x^{4}-\frac {1}{9676800} x^{5}+\frac {1}{928972800} x^{6}-\frac {1}{117050572800} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) x^{2}+c_{2} \left (\ln \left (x \right ) \left (\frac {1}{4} x^{2}-\frac {1}{24} x^{3}+\frac {1}{384} x^{4}-\frac {1}{11520} x^{5}+\frac {1}{552960} x^{6}-\frac {1}{38707200} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-2-x +\frac {1}{18} x^{3}-\frac {25}{4608} x^{4}+\frac {157}{691200} x^{5}-\frac {91}{16588800} x^{6}+\frac {709}{8128512000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )}{x^{2}} \]

Problem 6984

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+2 x \left (-x +2\right ) y^{\prime }-\left (1+3 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1+\frac {x}{2}+\frac {x^{2}}{8}+\frac {x^{3}}{48}+\frac {x^{4}}{384}+\frac {x^{5}}{3840}+\frac {x^{6}}{46080}+\frac {x^{7}}{645120}+O\left (x^{8}\right )\right )+c_{2} \left (\frac {\sqrt {x}\, \left (1+\frac {x}{2}+\frac {x^{2}}{8}+\frac {x^{3}}{48}+\frac {x^{4}}{384}+\frac {x^{5}}{3840}+\frac {x^{6}}{46080}+\frac {x^{7}}{645120}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{2}+\frac {1-\frac {x^{2}}{4}-\frac {3 x^{3}}{32}-\frac {11 x^{4}}{576}-\frac {25 x^{5}}{9216}-\frac {137 x^{6}}{460800}-\frac {49 x^{7}}{1843200}+O\left (x^{8}\right )}{\sqrt {x}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x \left (1+\frac {1}{2} x +\frac {1}{8} x^{2}+\frac {1}{48} x^{3}+\frac {1}{384} x^{4}+\frac {1}{3840} x^{5}+\frac {1}{46080} x^{6}+\frac {1}{645120} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (\frac {1}{2} x +\frac {1}{4} x^{2}+\frac {1}{16} x^{3}+\frac {1}{96} x^{4}+\frac {1}{768} x^{5}+\frac {1}{7680} x^{6}+\frac {1}{92160} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (1+\frac {1}{2} x -\frac {1}{32} x^{3}-\frac {5}{576} x^{4}-\frac {13}{9216} x^{5}-\frac {77}{460800} x^{6}-\frac {29}{1843200} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )}{\sqrt {x}} \]

Problem 6985

ODE	\[ \boxed {x^{2} y^{\prime \prime }-x \left (x +6\right ) y^{\prime }+10 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{5} \left (1+\frac {5 x}{4}+\frac {3 x^{2}}{4}+\frac {7 x^{3}}{24}+\frac {x^{4}}{12}+\frac {3 x^{5}}{160}+\frac {x^{6}}{288}+\frac {11 x^{7}}{20160}+O\left (x^{8}\right )\right )+c_{2} \left (2 x^{5} \left (1+\frac {5 x}{4}+\frac {3 x^{2}}{4}+\frac {7 x^{3}}{24}+\frac {x^{4}}{12}+\frac {3 x^{5}}{160}+\frac {x^{6}}{288}+\frac {11 x^{7}}{20160}+O\left (x^{8}\right )\right ) \ln \left (x \right )+x^{2} \left (1-x +\frac {3 x^{2}}{2}-\frac {21 x^{4}}{8}-\frac {19 x^{5}}{8}-\frac {163 x^{6}}{144}-\frac {53 x^{7}}{144}+O\left (x^{8}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{2} \left (c_{1} x^{3} \left (1+\frac {5}{4} x +\frac {3}{4} x^{2}+\frac {7}{24} x^{3}+\frac {1}{12} x^{4}+\frac {3}{160} x^{5}+\frac {1}{288} x^{6}+\frac {11}{20160} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (24 x^{3}+30 x^{4}+18 x^{5}+7 x^{6}+2 x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (12-12 x +18 x^{2}+26 x^{3}+x^{4}-9 x^{5}-6 x^{6}-\frac {9}{4} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )\right ) \]

Problem 6986

ODE	\[ \boxed {x y^{\prime \prime }+\left (3+2 x \right ) y^{\prime }+8 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-\frac {8 x}{3}+\frac {10 x^{2}}{3}-\frac {8 x^{3}}{3}+\frac {14 x^{4}}{9}-\frac {32 x^{5}}{45}+\frac {4 x^{6}}{15}-\frac {16 x^{7}}{189}+O\left (x^{8}\right )\right )+c_{2} \left (\left (-12+32 x -40 x^{2}+32 x^{3}-\frac {56 x^{4}}{3}+\frac {128 x^{5}}{15}-\frac {16 x^{6}}{5}+\frac {64 x^{7}}{63}-12 O\left (x^{8}\right )\right ) \ln \left (x \right )+\frac {1+4 x -\frac {104 x^{3}}{3}+\frac {196 x^{4}}{3}-64 x^{5}+\frac {382 x^{6}}{9}-\frac {4784 x^{7}}{225}+O\left (x^{8}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-\frac {8}{3} x +\frac {10}{3} x^{2}-\frac {8}{3} x^{3}+\frac {14}{9} x^{4}-\frac {32}{45} x^{5}+\frac {4}{15} x^{6}-\frac {16}{189} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) x^{2}+c_{2} \left (\ln \left (x \right ) \left (24 x^{2}-64 x^{3}+80 x^{4}-64 x^{5}+\frac {112}{3} x^{6}-\frac {256}{15} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-2-8 x +20 x^{2}+16 x^{3}-64 x^{4}+\frac {224}{3} x^{5}-\frac {484}{9} x^{6}+\frac {6368}{225} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )}{x^{2}} \]

Problem 6987

ODE	\[ \boxed {x \left (1-x \right ) y^{\prime \prime }+2 \left (1-x \right ) y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-x +O\left (x^{8}\right )\right )+c_{2} \left (\left (-2+2 x -2 O\left (x^{8}\right )\right ) \ln \left (x \right )+\frac {1-4 x^{2}+x^{3}+\frac {x^{4}}{3}+\frac {x^{5}}{6}+\frac {x^{6}}{10}+\frac {x^{7}}{15}+O\left (x^{8}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\ln \left (x \right ) \left (\left (-2\right ) x +2 x^{2}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} +c_{1} \left (1-x +\operatorname {O}\left (x^{8}\right )\right ) x +\left (1-4 x^{2}+x^{3}+\frac {1}{3} x^{4}+\frac {1}{6} x^{5}+\frac {1}{10} x^{6}+\frac {1}{15} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2}}{x} \]

Problem 6988

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

Problem 6989

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

Problem 6990

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {x^{2}}{8}+\frac {x^{4}}{192}-\frac {x^{6}}{9216}+O\left (x^{8}\right )\right )+c_{2} \left (-\frac {x \left (1-\frac {x^{2}}{8}+\frac {x^{4}}{192}-\frac {x^{6}}{9216}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{2}+\frac {1-\frac {3 x^{4}}{64}+\frac {7 x^{6}}{2304}+O\left (x^{8}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{2} \left (1-\frac {1}{8} x^{2}+\frac {1}{192} x^{4}-\frac {1}{9216} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\left (x^{2}-\frac {1}{8} x^{4}+\frac {1}{192} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) \ln \left (x \right )+\left (-2+\frac {3}{32} x^{4}-\frac {7}{1152} x^{6}+\operatorname {O}\left (x^{8}\right )\right )\right )}{x} \]

Problem 6991

ODE	\[ \boxed {x^{2} y^{\prime \prime }-5 x y^{\prime }+\left (8+5 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{4} \left (1-\frac {5 x}{3}+\frac {25 x^{2}}{24}-\frac {25 x^{3}}{72}+\frac {125 x^{4}}{1728}-\frac {125 x^{5}}{12096}+\frac {625 x^{6}}{580608}-\frac {3125 x^{7}}{36578304}+O\left (x^{8}\right )\right )+c_{2} \left (-\frac {25 x^{4} \left (1-\frac {5 x}{3}+\frac {25 x^{2}}{24}-\frac {25 x^{3}}{72}+\frac {125 x^{4}}{1728}-\frac {125 x^{5}}{12096}+\frac {625 x^{6}}{580608}-\frac {3125 x^{7}}{36578304}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{2}+x^{2} \left (1+5 x -\frac {250 x^{3}}{9}+\frac {15625 x^{4}}{576}-\frac {19625 x^{5}}{1728}+\frac {56875 x^{6}}{20736}-\frac {443125 x^{7}}{1016064}+O\left (x^{8}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{2} \left (c_{1} x^{2} \left (1-\frac {5}{3} x +\frac {25}{24} x^{2}-\frac {25}{72} x^{3}+\frac {125}{1728} x^{4}-\frac {125}{12096} x^{5}+\frac {625}{580608} x^{6}-\frac {3125}{36578304} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (25 x^{2}-\frac {125}{3} x^{3}+\frac {625}{24} x^{4}-\frac {625}{72} x^{5}+\frac {3125}{1728} x^{6}-\frac {3125}{12096} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-2-10 x +\frac {500}{9} x^{3}-\frac {15625}{288} x^{4}+\frac {19625}{864} x^{5}-\frac {56875}{10368} x^{6}+\frac {443125}{508032} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )\right ) \]

Problem 6992

ODE	\[ \boxed {x y^{\prime \prime }+\left (3-x \right ) y^{\prime }-5 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1+\frac {5 x}{3}+\frac {5 x^{2}}{4}+\frac {7 x^{3}}{12}+\frac {7 x^{4}}{36}+\frac {x^{5}}{20}+\frac {x^{6}}{96}+\frac {11 x^{7}}{6048}+O\left (x^{8}\right )\right )+c_{2} \left (\left (-6-10 x -\frac {15 x^{2}}{2}-\frac {7 x^{3}}{2}-\frac {7 x^{4}}{6}-\frac {3 x^{5}}{10}-\frac {x^{6}}{16}-\frac {11 x^{7}}{1008}-6 O\left (x^{8}\right )\right ) \ln \left (x \right )+\frac {1-3 x +\frac {34 x^{3}}{3}+\frac {103 x^{4}}{8}+\frac {59 x^{5}}{8}+\frac {403 x^{6}}{144}+\frac {947 x^{7}}{1200}+O\left (x^{8}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1+\frac {5}{3} x +\frac {5}{4} x^{2}+\frac {7}{12} x^{3}+\frac {7}{36} x^{4}+\frac {1}{20} x^{5}+\frac {1}{96} x^{6}+\frac {11}{6048} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) x^{2}+c_{2} \left (\ln \left (x \right ) \left (12 x^{2}+20 x^{3}+15 x^{4}+7 x^{5}+\frac {7}{3} x^{6}+\frac {3}{5} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-2+6 x +7 x^{2}-11 x^{3}-17 x^{4}-\frac {32}{3} x^{5}-\frac {305}{72} x^{6}-\frac {737}{600} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )}{x^{2}} \]

Problem 6993

ODE	\[ \boxed {9 x^{2} y^{\prime \prime }-15 x y^{\prime }+7 \left (1+x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {7}{3}} \left (1-\frac {7 x}{27}+\frac {49 x^{2}}{1944}-\frac {343 x^{3}}{262440}+\frac {2401 x^{4}}{56687040}-\frac {2401 x^{5}}{2550916800}+\frac {16807 x^{6}}{1101996057600}-\frac {16807 x^{7}}{89261680665600}+O\left (x^{8}\right )\right )+c_{2} \left (-\frac {49 x^{\frac {7}{3}} \left (1-\frac {7 x}{27}+\frac {49 x^{2}}{1944}-\frac {343 x^{3}}{262440}+\frac {2401 x^{4}}{56687040}-\frac {2401 x^{5}}{2550916800}+\frac {16807 x^{6}}{1101996057600}-\frac {16807 x^{7}}{89261680665600}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{162}+x^{\frac {1}{3}} \left (1+\frac {7 x}{9}-\frac {686 x^{3}}{6561}+\frac {60025 x^{4}}{3779136}-\frac {2638699 x^{5}}{2550916800}+\frac {10706059 x^{6}}{275499014400}-\frac {11916163 x^{7}}{12397455648000}+O\left (x^{8}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{\frac {1}{3}} \left (x^{2} \left (1-\frac {7}{27} x +\frac {49}{1944} x^{2}-\frac {343}{262440} x^{3}+\frac {2401}{56687040} x^{4}-\frac {2401}{2550916800} x^{5}+\frac {16807}{1101996057600} x^{6}-\frac {16807}{89261680665600} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{1} +c_{2} \left (\ln \left (x \right ) \left (\frac {49}{81} x^{2}-\frac {343}{2187} x^{3}+\frac {2401}{157464} x^{4}-\frac {16807}{21257640} x^{5}+\frac {117649}{4591650240} x^{6}-\frac {117649}{206624260800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-2-\frac {14}{9} x +\frac {1372}{6561} x^{3}-\frac {60025}{1889568} x^{4}+\frac {2638699}{1275458400} x^{5}-\frac {10706059}{137749507200} x^{6}+\frac {11916163}{6198727824000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )\right ) \]

Problem 6994

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x \left (1-2 x \right ) y^{\prime }-\left (1+x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1+x +\frac {5 x^{2}}{8}+\frac {7 x^{3}}{24}+\frac {7 x^{4}}{64}+\frac {11 x^{5}}{320}+\frac {143 x^{6}}{15360}+\frac {143 x^{7}}{64512}+O\left (x^{8}\right )\right )+c_{2} \left (\frac {x \left (1+x +\frac {5 x^{2}}{8}+\frac {7 x^{3}}{24}+\frac {7 x^{4}}{64}+\frac {11 x^{5}}{320}+\frac {143 x^{6}}{15360}+\frac {143 x^{7}}{64512}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{2}+\frac {1+x -\frac {x^{3}}{3}-\frac {61 x^{4}}{192}-\frac {59 x^{5}}{320}-\frac {919 x^{6}}{11520}-\frac {449 x^{7}}{16128}+O\left (x^{8}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{2} \left (1+x +\frac {5}{8} x^{2}+\frac {7}{24} x^{3}+\frac {7}{64} x^{4}+\frac {11}{320} x^{5}+\frac {143}{15360} x^{6}+\frac {143}{64512} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-x^{2}-x^{3}-\frac {5}{8} x^{4}-\frac {7}{24} x^{5}-\frac {7}{64} x^{6}-\frac {11}{320} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-2-2 x +\frac {2}{3} x^{3}+\frac {61}{96} x^{4}+\frac {59}{160} x^{5}+\frac {919}{5760} x^{6}+\frac {449}{8064} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )}{x} \]

Problem 6995

ODE	\[ \boxed {x^{2} y^{\prime \prime }+3 x y^{\prime }+\left (x^{3}+x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1-x +\frac {x^{2}}{4}-\frac {5 x^{3}}{36}+\frac {41 x^{4}}{576}-\frac {37 x^{5}}{2880}+\frac {437 x^{6}}{103680}-\frac {7817 x^{7}}{5080320}+O\left (x^{8}\right )\right )}{x}+c_{2} \left (\frac {\left (1-x +\frac {x^{2}}{4}-\frac {5 x^{3}}{36}+\frac {41 x^{4}}{576}-\frac {37 x^{5}}{2880}+\frac {437 x^{6}}{103680}-\frac {7817 x^{7}}{5080320}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{x}+\frac {2 x -\frac {3 x^{2}}{4}+\frac {19 x^{3}}{108}-\frac {593 x^{4}}{3456}+\frac {3629 x^{5}}{86400}-\frac {7733 x^{6}}{1036800}+\frac {485257 x^{7}}{118540800}+O\left (x^{8}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-x +\frac {1}{4} x^{2}-\frac {5}{36} x^{3}+\frac {41}{576} x^{4}-\frac {37}{2880} x^{5}+\frac {437}{103680} x^{6}-\frac {7817}{5080320} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (2 x -\frac {3}{4} x^{2}+\frac {19}{108} x^{3}-\frac {593}{3456} x^{4}+\frac {3629}{86400} x^{5}-\frac {7733}{1036800} x^{6}+\frac {485257}{118540800} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2}}{x} \]

Problem 6996

ODE	\[ \boxed {2 x \left (1-x \right ) y^{\prime \prime }+\left (1-2 x \right ) y^{\prime }+\left (x +2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {x}{2}-\frac {9 x^{2}}{40}-\frac {149 x^{3}}{1680}-\frac {661 x^{4}}{13440}-\frac {16171 x^{5}}{492800}-\frac {5530601 x^{6}}{230630400}-\frac {299137703 x^{7}}{16144128000}+O\left (x^{8}\right )\right )+c_{2} \left (1-2 x -\frac {x^{2}}{6}+\frac {x^{3}}{15}+\frac {37 x^{4}}{840}+\frac {527 x^{5}}{18900}+\frac {16309 x^{6}}{831600}+\frac {14339 x^{7}}{970200}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1-\frac {1}{2} x -\frac {9}{40} x^{2}-\frac {149}{1680} x^{3}-\frac {661}{13440} x^{4}-\frac {16171}{492800} x^{5}-\frac {5530601}{230630400} x^{6}-\frac {299137703}{16144128000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1-2 x -\frac {1}{6} x^{2}+\frac {1}{15} x^{3}+\frac {37}{840} x^{4}+\frac {527}{18900} x^{5}+\frac {16309}{831600} x^{6}+\frac {14339}{970200} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6997

ODE	\[ \boxed {x y^{\prime \prime }+y^{\prime }+x \left (1+x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-\frac {x^{2}}{4}-\frac {x^{3}}{9}+\frac {x^{4}}{64}+\frac {13 x^{5}}{900}+\frac {55 x^{6}}{20736}-\frac {433 x^{7}}{705600}+O\left (x^{8}\right )\right )+c_{2} \left (\left (1-\frac {x^{2}}{4}-\frac {x^{3}}{9}+\frac {x^{4}}{64}+\frac {13 x^{5}}{900}+\frac {55 x^{6}}{20736}-\frac {433 x^{7}}{705600}+O\left (x^{8}\right )\right ) \ln \left (x \right )+\frac {x^{2}}{4}+\frac {2 x^{3}}{27}-\frac {3 x^{4}}{128}-\frac {253 x^{5}}{13500}-\frac {95 x^{6}}{41472}+\frac {153527 x^{7}}{148176000}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {1}{4} x^{2}-\frac {1}{9} x^{3}+\frac {1}{64} x^{4}+\frac {13}{900} x^{5}+\frac {55}{20736} x^{6}-\frac {433}{705600} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (\frac {1}{4} x^{2}+\frac {2}{27} x^{3}-\frac {3}{128} x^{4}-\frac {253}{13500} x^{5}-\frac {95}{41472} x^{6}+\frac {153527}{148176000} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \]

Problem 6998

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-\left (6 x^{2}-3 x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {4 x}{3}+\frac {19 x^{2}}{12}-\frac {7 x^{3}}{6}+\frac {53 x^{4}}{72}-\frac {116 x^{5}}{315}+\frac {3247 x^{6}}{20160}-\frac {5501 x^{7}}{90720}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1+2 x -\frac {5 x^{2}}{2}+\frac {22 x^{3}}{3}-\frac {155 x^{4}}{24}+\frac {331 x^{5}}{60}-\frac {2321 x^{6}}{720}+\frac {106 x^{7}}{63}+O\left (x^{8}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1-\frac {4}{3} x +\frac {19}{12} x^{2}-\frac {7}{6} x^{3}+\frac {53}{72} x^{4}-\frac {116}{315} x^{5}+\frac {3247}{20160} x^{6}-\frac {5501}{90720} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (-2-4 x +5 x^{2}-\frac {44}{3} x^{3}+\frac {155}{12} x^{4}-\frac {331}{30} x^{5}+\frac {2321}{360} x^{6}-\frac {212}{63} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x} \]

Problem 6999

ODE	\[ \boxed {x y^{\prime \prime }+x y^{\prime }+\left (x^{4}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-x +\frac {x^{2}}{2}-\frac {x^{3}}{6}+\frac {x^{4}}{24}-\frac {x^{5}}{24}+\frac {31 x^{6}}{1008}-\frac {47 x^{7}}{3528}+O\left (x^{8}\right )\right )+c_{2} \left (-x \left (1-x +\frac {x^{2}}{2}-\frac {x^{3}}{6}+\frac {x^{4}}{24}-\frac {x^{5}}{24}+\frac {31 x^{6}}{1008}-\frac {47 x^{7}}{3528}+O\left (x^{8}\right )\right ) \ln \left (x \right )+1-x^{2}+\frac {3 x^{3}}{4}-\frac {11 x^{4}}{36}+\frac {53 x^{5}}{1440}-\frac {17 x^{6}}{800}+\frac {75947 x^{7}}{2116800}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1-x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{24} x^{5}+\frac {31}{1008} x^{6}-\frac {47}{3528} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-x +x^{2}-\frac {1}{2} x^{3}+\frac {1}{6} x^{4}-\frac {1}{24} x^{5}+\frac {1}{24} x^{6}-\frac {31}{1008} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (1-x +\frac {1}{4} x^{3}-\frac {5}{36} x^{4}-\frac {7}{1440} x^{5}+\frac {49}{2400} x^{6}+\frac {10847}{2116800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right ) \]

Problem 7000

ODE	\[ \boxed {x \left (x -2\right )^{2} y^{\prime \prime }-2 \left (x -2\right ) y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-\frac {x}{2}+O\left (x^{8}\right )\right )+c_{2} \left (\left (1-\frac {x}{2}+O\left (x^{8}\right )\right ) \ln \left (x \right )+\frac {x}{2}-\frac {x^{2}}{8}-\frac {x^{3}}{48}-\frac {x^{4}}{192}-\frac {x^{5}}{640}-\frac {x^{6}}{1920}-\frac {x^{7}}{5376}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {1}{2} x +\operatorname {O}\left (x^{8}\right )\right )+\left (\frac {1}{2} x -\frac {1}{8} x^{2}-\frac {1}{48} x^{3}-\frac {1}{192} x^{4}-\frac {1}{640} x^{5}-\frac {1}{1920} x^{6}-\frac {1}{5376} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \]