2.17.30 Problems 2901 to 3000

Problem 2901

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

Problem 2902

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

Problem 2903

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

Problem 2904

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

Problem 2905

ODE	\[ \boxed {\left (x^{2}-3\right ) y^{\prime \prime }-3 y^{\prime } x -5 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {5}{6} x^{2}+\frac {5}{24} x^{4}-\frac {5}{432} x^{6}\right ) y \left (0\right )+\left (x -\frac {4}{9} x^{3}+\frac {8}{135} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {5}{6} x^{2}+\frac {5}{24} x^{4}\right ) c_{1} +\left (x -\frac {4}{9} x^{3}+\frac {8}{135} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {5}{6} x^{2}+\frac {5}{24} x^{4}\right ) y \left (0\right )+\left (x -\frac {4}{9} x^{3}+\frac {8}{135} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 2906

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

Problem 2907

ODE	\[ \boxed {\left (-4 x^{2}+1\right ) y^{\prime \prime }-20 y^{\prime } x -16 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+8 x^{2}+\frac {128}{3} x^{4}+\frac {1024}{5} x^{6}\right ) y \left (0\right )+\left (30 x^{5}+6 x^{3}+x \right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+8 x^{2}+\frac {128}{3} x^{4}\right ) c_{1} +\left (30 x^{5}+6 x^{3}+x \right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+8 x^{2}+\frac {128}{3} x^{4}\right ) y \left (0\right )+\left (30 x^{5}+6 x^{3}+x \right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 2908

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

Problem 2909

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+4 y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {2}{3} x^{3}+\frac {1}{3} x^{4}-\frac {2}{15} x^{5}+\frac {2}{15} x^{6}\right ) y \left (0\right )+\left (x -x^{2}+\frac {2}{3} x^{3}-\frac {2}{3} x^{4}+\frac {7}{15} x^{5}-\frac {11}{45} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {2}{3} x^{3}+\frac {1}{3} x^{4}-\frac {2}{15} x^{5}\right ) c_{1} +\left (x -x^{2}+\frac {2}{3} x^{3}-\frac {2}{3} x^{4}+\frac {7}{15} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {2}{3} x^{3}+\frac {1}{3} x^{4}-\frac {2}{15} x^{5}\right ) y \left (0\right )+\left (x -x^{2}+\frac {2}{3} x^{3}-\frac {2}{3} x^{4}+\frac {7}{15} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 2910

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime } x +\left (x +2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-x^{2}-\frac {1}{6} x^{3}+\frac {1}{3} x^{4}+\frac {11}{120} x^{5}-\frac {11}{180} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{3}-\frac {1}{12} x^{4}+\frac {1}{8} x^{5}+\frac {1}{30} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-x^{2}-\frac {1}{6} x^{3}+\frac {1}{3} x^{4}+\frac {11}{120} x^{5}\right ) c_{1} +\left (x -\frac {1}{2} x^{3}-\frac {1}{12} x^{4}+\frac {1}{8} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-x^{2}-\frac {1}{6} x^{3}+\frac {1}{3} x^{4}+\frac {11}{120} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{3}-\frac {1}{12} x^{4}+\frac {1}{8} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 2911

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

Problem 2912

ODE	\[ \boxed {x y^{\prime \prime }-\left (x -1\right ) y^{\prime }-y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (\frac {x^{2}}{4}+1+\frac {x^{3}}{18}+\frac {5 x^{4}}{192}+\frac {23 x^{5}}{3600}+O\left (x^{6}\right )\right )+c_{2} \left (\left (\frac {x^{2}}{4}+1+\frac {x^{3}}{18}+\frac {5 x^{4}}{192}+\frac {23 x^{5}}{3600}+O\left (x^{6}\right )\right ) \ln \left (x \right )+x +\frac {11 x^{3}}{108}+\frac {11 x^{4}}{1152}+\frac {883 x^{5}}{216000}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+\frac {1}{4} x^{2}+\frac {1}{18} x^{3}+\frac {5}{192} x^{4}+\frac {23}{3600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (x +\frac {11}{108} x^{3}+\frac {11}{1152} x^{4}+\frac {883}{216000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 2913

ODE	\[ \boxed {\left (2 x^{2}+1\right ) y^{\prime \prime }+7 y^{\prime } x +2 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = x -\frac {3 x^{3}}{2}+\frac {21 x^{5}}{8}+O\left (x^{6}\right ) \] Verified OK. \[ y = x -\frac {3 x^{3}}{2}+\frac {21 x^{5}}{8}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x -\frac {3}{2} x^{3}+\frac {21}{8} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 2914

ODE	\[ \boxed {4 y^{\prime \prime }+y^{\prime } x +4 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 1-\frac {x^{2}}{2}+\frac {x^{4}}{16}-\frac {x^{6}}{240}+O\left (x^{6}\right ) \] Verified OK. \[ y = 1-\frac {x^{2}}{2}+\frac {x^{4}}{16}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1-\frac {1}{2} x^{2}+\frac {1}{16} x^{4}+\operatorname {O}\left (x^{6}\right ) \]

Problem 2915

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime } x^{2}+y x=2 \cos \left (x \right )} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{6} x^{3}+\frac {7}{180} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{4} x^{4}\right ) y^{\prime }\left (0\right )+x^{2}-\frac {x^{4}}{12}-\frac {x^{5}}{4}+\frac {x^{6}}{360}+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {x^{3}}{6}\right ) c_{1} +\left (x -\frac {1}{4} x^{4}\right ) c_{2} +x^{2}-\frac {x^{4}}{12}-\frac {x^{5}}{4}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {x^{3}}{6}\right ) y \left (0\right )+\left (x -\frac {1}{4} x^{4}\right ) D\left (y \right )\left (0\right )+x^{2}-\frac {x^{4}}{12}-\frac {x^{5}}{4}+O\left (x^{6}\right ) \]

Problem 2916

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime } x -4 y=6 \,{\mathrm e}^{x}} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+2 x^{2}+\frac {1}{3} x^{4}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{3}+\frac {1}{40} x^{5}\right ) y^{\prime }\left (0\right )+3 x^{2}+x^{3}+\frac {3 x^{4}}{4}+\frac {x^{5}}{10}+\frac {x^{6}}{120}+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+2 x^{2}+\frac {1}{3} x^{4}\right ) c_{1} +\left (x +\frac {1}{2} x^{3}+\frac {1}{40} x^{5}\right ) c_{2} +3 x^{2}+x^{3}+\frac {3 x^{4}}{4}+\frac {x^{5}}{10}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+2 x^{2}+\frac {1}{3} x^{4}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{3}+\frac {1}{40} x^{5}\right ) D\left (y \right )\left (0\right )+3 x^{2}+x^{3}+\frac {3 x^{4}}{4}+\frac {x^{5}}{10}+O\left (x^{6}\right ) \]

Problem 2917

ODE	\[ \boxed {y^{\prime \prime }+\frac {y^{\prime }}{1-x}+y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{60} x^{5}+\frac {1}{72} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}-\frac {1}{12} x^{4}+\frac {1}{24} x^{5}+\frac {1}{240} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{60} x^{5}\right ) c_{1} +\left (x -\frac {1}{2} x^{2}-\frac {1}{12} x^{4}+\frac {1}{24} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{60} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}-\frac {1}{12} x^{4}+\frac {1}{24} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 2918

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

Problem 2919

ODE	\[ \boxed {\left (-2+x \right )^{2} y^{\prime \prime }+\left (-2+x \right ) {\mathrm e}^{x} y^{\prime }+\frac {4 y}{x}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {x}{4}-\frac {x^{2}}{24}-\frac {13 x^{3}}{576}-\frac {35 x^{4}}{2304}-\frac {1297 x^{5}}{138240}+O\left (x^{6}\right )\right )+c_{2} \left (-x \left (1-\frac {x}{4}-\frac {x^{2}}{24}-\frac {13 x^{3}}{576}-\frac {35 x^{4}}{2304}-\frac {1297 x^{5}}{138240}+O\left (x^{6}\right )\right ) \ln \left (x \right )+1-\frac {9 x^{2}}{8}-\frac {19 x^{3}}{72}-\frac {1019 x^{4}}{6912}-\frac {5827 x^{5}}{69120}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1-\frac {1}{4} x -\frac {1}{24} x^{2}-\frac {13}{576} x^{3}-\frac {35}{2304} x^{4}-\frac {1297}{138240} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-x +\frac {1}{4} x^{2}+\frac {1}{24} x^{3}+\frac {13}{576} x^{4}+\frac {35}{2304} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1+\frac {1}{2} x -\frac {5}{4} x^{2}-\frac {41}{144} x^{3}-\frac {1097}{6912} x^{4}-\frac {397}{4320} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

Problem 2920

ODE	\[ \boxed {y^{\prime \prime }+\frac {2 y^{\prime }}{x \left (x -3\right )}-\frac {y}{x^{3} \left (x +3\right )}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 2921

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x \left (1-x \right ) y^{\prime }-7 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\sqrt {7}} \left (1+\frac {\sqrt {7}\, x}{1+2 \sqrt {7}}+\frac {\sqrt {7}\, x^{2}}{4+8 \sqrt {7}}+\frac {\left (2+\sqrt {7}\right ) \sqrt {7}\, x^{3}}{372+96 \sqrt {7}}+\frac {\sqrt {7}\, \left (3+\sqrt {7}\right ) x^{4}}{2976+768 \sqrt {7}}+\frac {\sqrt {7}\, \left (3+\sqrt {7}\right ) \left (4+\sqrt {7}\right ) x^{5}}{48960 \sqrt {7}+128160}+O\left (x^{6}\right )\right )+c_{2} x^{-\sqrt {7}} \left (1+\frac {\sqrt {7}\, x}{-1+2 \sqrt {7}}+\frac {\sqrt {7}\, x^{2}}{-4+8 \sqrt {7}}+\frac {\sqrt {7}\, \left (-2+\sqrt {7}\right ) x^{3}}{372-96 \sqrt {7}}+\frac {\sqrt {7}\, \left (-3+\sqrt {7}\right ) x^{4}}{2976-768 \sqrt {7}}+\frac {\sqrt {7}\, \left (-3+\sqrt {7}\right ) \left (-4+\sqrt {7}\right ) x^{5}}{48960 \sqrt {7}-128160}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{-\sqrt {7}} \left (1+\frac {\sqrt {7}}{-1+2 \sqrt {7}} x +\frac {\sqrt {7}}{-4+8 \sqrt {7}} x^{2}+\frac {\sqrt {7}\, \left (\sqrt {7}-2\right )}{372-96 \sqrt {7}} x^{3}+\frac {\sqrt {7}\, \left (\sqrt {7}-3\right )}{2976-768 \sqrt {7}} x^{4}+\frac {\left (\sqrt {7}-4\right ) \left (\sqrt {7}-3\right ) \sqrt {7}}{48960 \sqrt {7}-128160} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{\sqrt {7}} \left (1+\frac {\sqrt {7}}{1+2 \sqrt {7}} x +\frac {\sqrt {7}}{4+8 \sqrt {7}} x^{2}+\frac {\sqrt {7}\, \left (\sqrt {7}+2\right )}{372+96 \sqrt {7}} x^{3}+\frac {\left (\sqrt {7}+3\right ) \sqrt {7}}{2976+768 \sqrt {7}} x^{4}+\frac {\left (\sqrt {7}+4\right ) \left (\sqrt {7}+3\right ) \sqrt {7}}{48960 \sqrt {7}+128160} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2922

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+y^{\prime } x \,{\mathrm e}^{x}-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {x}{9}-\frac {5 x^{2}}{468}-\frac {11 x^{3}}{23868}+\frac {79 x^{4}}{501228}+\frac {16043 x^{5}}{313267500}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {x}{4}+\frac {5 x^{2}}{96}+\frac {17 x^{3}}{8064}-\frac {313 x^{4}}{1419264}-\frac {69703 x^{5}}{709632000}+O\left (x^{6}\right )\right )}{x^{\frac {1}{4}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-\frac {1}{4} x +\frac {5}{96} x^{2}+\frac {17}{8064} x^{3}-\frac {313}{1419264} x^{4}-\frac {69703}{709632000} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {1}{4}}}+c_{2} x \left (1-\frac {1}{9} x -\frac {5}{468} x^{2}-\frac {11}{23868} x^{3}+\frac {79}{501228} x^{4}+\frac {16043}{313267500} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2923

ODE	\[ \boxed {4 x y^{\prime \prime }-y^{\prime } x +2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {x}{8}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {x \left (1-\frac {x}{8}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{2}+1-\frac {5 x^{2}}{32}+\frac {x^{3}}{384}+\frac {x^{4}}{18432}+\frac {x^{5}}{737280}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \ln \left (x \right ) \left (-\frac {1}{2} x +\frac {1}{16} x^{2}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} +c_{1} x \left (1-\frac {1}{8} x +\operatorname {O}\left (x^{6}\right )\right )+\left (1+\frac {1}{4} x -\frac {3}{16} x^{2}+\frac {1}{384} x^{3}+\frac {1}{18432} x^{4}+\frac {1}{737280} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 2924

ODE	\[ \boxed {x^{2} y^{\prime \prime }-x \cos \left (x \right ) y^{\prime }+5 y \,{\mathrm e}^{2 x}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{1+2 i} \left (1+\left (-\frac {10}{17}+\frac {40 i}{17}\right ) x +\left (-\frac {365}{136}-\frac {13 i}{17}\right ) x^{2}+\left (\frac {223}{1020}-\frac {1723 i}{765}\right ) x^{3}+\left (\frac {114911}{78336}-\frac {24835 i}{78336}\right ) x^{4}+\left (\frac {4041077}{8029440}+\frac {1112267 i}{1605888}\right ) x^{5}+O\left (x^{6}\right )\right )+c_{2} x^{1-2 i} \left (1+\left (-\frac {10}{17}-\frac {40 i}{17}\right ) x +\left (-\frac {365}{136}+\frac {13 i}{17}\right ) x^{2}+\left (\frac {223}{1020}+\frac {1723 i}{765}\right ) x^{3}+\left (\frac {114911}{78336}+\frac {24835 i}{78336}\right ) x^{4}+\left (\frac {4041077}{8029440}-\frac {1112267 i}{1605888}\right ) x^{5}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{1-2 i} \left (1+\left (-\frac {10}{17}-\frac {40 i}{17}\right ) x +\left (-\frac {365}{136}+\frac {13 i}{17}\right ) x^{2}+\left (\frac {223}{1020}+\frac {1723 i}{765}\right ) x^{3}+\left (\frac {114911}{78336}+\frac {24835 i}{78336}\right ) x^{4}+\left (\frac {4041077}{8029440}-\frac {1112267 i}{1605888}\right ) x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{1+2 i} \left (1+\left (-\frac {10}{17}+\frac {40 i}{17}\right ) x +\left (-\frac {365}{136}-\frac {13 i}{17}\right ) x^{2}+\left (\frac {223}{1020}-\frac {1723 i}{765}\right ) x^{3}+\left (\frac {114911}{78336}-\frac {24835 i}{78336}\right ) x^{4}+\left (\frac {4041077}{8029440}+\frac {1112267 i}{1605888}\right ) x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2925

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+3 y^{\prime } x +y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{4}} \left (1-\frac {x}{5}+\frac {x^{2}}{90}-\frac {x^{3}}{3510}+\frac {x^{4}}{238680}-\frac {x^{5}}{25061400}+O\left (x^{6}\right )\right )+c_{2} \left (1-\frac {x}{3}+\frac {x^{2}}{42}-\frac {x^{3}}{1386}+\frac {x^{4}}{83160}-\frac {x^{5}}{7900200}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{4}} \left (1-\frac {1}{5} x +\frac {1}{90} x^{2}-\frac {1}{3510} x^{3}+\frac {1}{238680} x^{4}-\frac {1}{25061400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-\frac {1}{3} x +\frac {1}{42} x^{2}-\frac {1}{1386} x^{3}+\frac {1}{83160} x^{4}-\frac {1}{7900200} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2926

ODE	\[ \boxed {6 x^{2} y^{\prime \prime }+x \left (1+18 x \right ) y^{\prime }+\left (1+12 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-3 x +\frac {9 x^{2}}{2}-\frac {9 x^{3}}{2}+\frac {27 x^{4}}{8}-\frac {81 x^{5}}{40}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{3}} \left (1-\frac {18 x}{5}+\frac {324 x^{2}}{55}-\frac {5832 x^{3}}{935}+\frac {104976 x^{4}}{21505}-\frac {1889568 x^{5}}{623645}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{3}} \left (1-\frac {18}{5} x +\frac {324}{55} x^{2}-\frac {5832}{935} x^{3}+\frac {104976}{21505} x^{4}-\frac {1889568}{623645} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1-3 x +\frac {9}{2} x^{2}-\frac {9}{2} x^{3}+\frac {27}{8} x^{4}-\frac {81}{40} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2927

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x -\left (x +2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\sqrt {2}} \left (1+\frac {x}{1+2 \sqrt {2}}+\frac {x^{2}}{20+12 \sqrt {2}}+\frac {x^{3}}{228 \sqrt {2}+324}+\frac {x^{4}}{8832+6240 \sqrt {2}}+\frac {x^{5}}{244320 \sqrt {2}+345600}+O\left (x^{6}\right )\right )+c_{2} x^{-\sqrt {2}} \left (1+\frac {x}{1-2 \sqrt {2}}+\frac {x^{2}}{20-12 \sqrt {2}}-\frac {x^{3}}{228 \sqrt {2}-324}+\frac {x^{4}}{8832-6240 \sqrt {2}}-\frac {x^{5}}{480 \left (-1+2 \sqrt {2}\right ) \left (\sqrt {2}-1\right ) \left (-3+2 \sqrt {2}\right ) \left (\sqrt {2}-2\right ) \left (-5+2 \sqrt {2}\right )}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{-\sqrt {2}} \left (1+\frac {1}{1-2 \sqrt {2}} x +\frac {1}{20-12 \sqrt {2}} x^{2}-\frac {1}{228 \sqrt {2}-324} x^{3}+\frac {1}{8832-6240 \sqrt {2}} x^{4}-\frac {1}{480} \frac {1}{\left (-1+2 \sqrt {2}\right ) \left (\sqrt {2}-1\right ) \left (-3+2 \sqrt {2}\right ) \left (\sqrt {2}-2\right ) \left (-5+2 \sqrt {2}\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{\sqrt {2}} \left (1+\frac {1}{1+2 \sqrt {2}} x +\frac {1}{20+12 \sqrt {2}} x^{2}+\frac {1}{228 \sqrt {2}+324} x^{3}+\frac {1}{8832+6240 \sqrt {2}} x^{4}+\frac {1}{244320 \sqrt {2}+345600} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2928

ODE	\[ \boxed {2 x y^{\prime \prime }+y^{\prime }-2 y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1+\frac {x^{2}}{5}+\frac {x^{4}}{90}+O\left (x^{6}\right )\right )+c_{2} \left (1+\frac {x^{2}}{3}+\frac {x^{4}}{42}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{5} x^{2}+\frac {1}{90} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+\frac {1}{3} x^{2}+\frac {1}{42} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2929

ODE	\[ \boxed {3 x^{2} y^{\prime \prime }-x \left (x +8\right ) y^{\prime }+6 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{3} \left (1+\frac {3 x}{10}+\frac {3 x^{2}}{65}+\frac {x^{3}}{208}+\frac {3 x^{4}}{7904}+\frac {21 x^{5}}{869440}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {2}{3}} \left (1-\frac {x}{6}+\frac {5 x^{2}}{36}+\frac {5 x^{3}}{81}+\frac {11 x^{4}}{972}+\frac {77 x^{5}}{58320}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {2}{3}} \left (1-\frac {1}{6} x +\frac {5}{36} x^{2}+\frac {5}{81} x^{3}+\frac {11}{972} x^{4}+\frac {77}{58320} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{3} \left (1+\frac {3}{10} x +\frac {3}{65} x^{2}+\frac {1}{208} x^{3}+\frac {3}{7904} x^{4}+\frac {21}{869440} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2930

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }-x \left (1+2 x \right ) y^{\prime }+2 \left (-1+4 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-\frac {4 x}{7}+\frac {4 x^{2}}{63}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+3 x +\frac {21 x^{2}}{2}-\frac {35 x^{3}}{2}+\frac {35 x^{4}}{8}-\frac {7 x^{5}}{40}+O\left (x^{6}\right )\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1+3 x +\frac {21}{2} x^{2}-\frac {35}{2} x^{3}+\frac {35}{8} x^{4}-\frac {7}{40} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+c_{2} x^{2} \left (1-\frac {4}{7} x +\frac {4}{63} x^{2}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2931

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x \left (1-x \right ) y^{\prime }-\left (x +5\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\sqrt {5}} \left (1+\frac {\left (\sqrt {5}+1\right ) x}{1+2 \sqrt {5}}+\frac {\left (2+\sqrt {5}\right ) x^{2}}{4+8 \sqrt {5}}+\frac {\left (2+\sqrt {5}\right ) \left (\sqrt {5}+3\right ) x^{3}}{276+96 \sqrt {5}}+\frac {\left (4+\sqrt {5}\right ) \left (\sqrt {5}+3\right ) x^{4}}{2208+768 \sqrt {5}}+\frac {\left (\sqrt {5}+3\right ) \left (4+\sqrt {5}\right ) \left (5+\sqrt {5}\right ) x^{5}}{41280 \sqrt {5}+93600}+O\left (x^{6}\right )\right )+c_{2} x^{-\sqrt {5}} \left (1+\frac {\left (\sqrt {5}-1\right ) x}{-1+2 \sqrt {5}}+\frac {\left (-2+\sqrt {5}\right ) x^{2}}{-4+8 \sqrt {5}}+\frac {\left (\sqrt {5}-3\right ) \left (-2+\sqrt {5}\right ) x^{3}}{276-96 \sqrt {5}}+\frac {\left (\sqrt {5}-3\right ) \left (-4+\sqrt {5}\right ) x^{4}}{2208-768 \sqrt {5}}+\frac {\left (\sqrt {5}-3\right ) \left (-4+\sqrt {5}\right ) \left (-5+\sqrt {5}\right ) x^{5}}{41280 \sqrt {5}-93600}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{-\sqrt {5}} \left (1+\frac {\sqrt {5}-1}{-1+2 \sqrt {5}} x +\frac {-2+\sqrt {5}}{-4+8 \sqrt {5}} x^{2}+\frac {\left (-2+\sqrt {5}\right ) \left (\sqrt {5}-3\right )}{276-96 \sqrt {5}} x^{3}+\frac {\left (\sqrt {5}-3\right ) \left (\sqrt {5}-4\right )}{2208-768 \sqrt {5}} x^{4}+\frac {\left (\sqrt {5}-3\right ) \left (\sqrt {5}-4\right ) \left (-5+\sqrt {5}\right )}{41280 \sqrt {5}-93600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{\sqrt {5}} \left (1+\frac {\sqrt {5}+1}{1+2 \sqrt {5}} x +\frac {\sqrt {5}+2}{4+8 \sqrt {5}} x^{2}+\frac {\left (\sqrt {5}+2\right ) \left (3+\sqrt {5}\right )}{276+96 \sqrt {5}} x^{3}+\frac {\left (3+\sqrt {5}\right ) \left (\sqrt {5}+4\right )}{2208+768 \sqrt {5}} x^{4}+\frac {\left (3+\sqrt {5}\right ) \left (\sqrt {5}+4\right ) \left (5+\sqrt {5}\right )}{41280 \sqrt {5}+93600} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2932

ODE	\[ \boxed {3 x^{2} y^{\prime \prime }+x \left (3 x +7\right ) y^{\prime }+\left (1+6 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1-x +\frac {x^{2}}{2}-\frac {x^{3}}{6}+\frac {x^{4}}{24}-\frac {x^{5}}{120}+O\left (x^{6}\right )\right )}{x^{\frac {1}{3}}}+\frac {c_{2} \left (1-3 x +\frac {9 x^{2}}{4}-\frac {27 x^{3}}{28}+\frac {81 x^{4}}{280}-\frac {243 x^{5}}{3640}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-3 x +\frac {9}{4} x^{2}-\frac {27}{28} x^{3}+\frac {81}{280} x^{4}-\frac {243}{3640} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x}+\frac {c_{2} \left (1-x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {1}{3}}} \]

Problem 2933

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (1-x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{i} \left (1+\left (\frac {1}{5}-\frac {2 i}{5}\right ) x +\left (-\frac {1}{40}-\frac {3 i}{40}\right ) x^{2}+\left (-\frac {3}{520}-\frac {7 i}{1560}\right ) x^{3}+\left (-\frac {1}{2496}-\frac {i}{12480}\right ) x^{4}+\left (-\frac {9}{603200}+\frac {i}{361920}\right ) x^{5}+O\left (x^{6}\right )\right )+c_{2} x^{-i} \left (1+\left (\frac {1}{5}+\frac {2 i}{5}\right ) x +\left (-\frac {1}{40}+\frac {3 i}{40}\right ) x^{2}+\left (-\frac {3}{520}+\frac {7 i}{1560}\right ) x^{3}+\left (-\frac {1}{2496}+\frac {i}{12480}\right ) x^{4}+\left (-\frac {9}{603200}-\frac {i}{361920}\right ) x^{5}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{-i} \left (1+\left (\frac {1}{5}+\frac {2 i}{5}\right ) x +\left (-\frac {1}{40}+\frac {3 i}{40}\right ) x^{2}+\left (-\frac {3}{520}+\frac {7 i}{1560}\right ) x^{3}+\left (-\frac {1}{2496}+\frac {i}{12480}\right ) x^{4}+\left (-\frac {9}{603200}-\frac {i}{361920}\right ) x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{i} \left (1+\left (\frac {1}{5}-\frac {2 i}{5}\right ) x +\left (-\frac {1}{40}-\frac {3 i}{40}\right ) x^{2}+\left (-\frac {3}{520}-\frac {7 i}{1560}\right ) x^{3}+\left (-\frac {1}{2496}-\frac {i}{12480}\right ) x^{4}+\left (-\frac {9}{603200}+\frac {i}{361920}\right ) x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2934

ODE	\[ \boxed {3 x^{2} y^{\prime \prime }+x \left (3 x^{2}+1\right ) y^{\prime }-2 y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {2}{3}} \left (1+\frac {2 x}{5}-\frac {3 x^{2}}{40}-\frac {43 x^{3}}{660}+\frac {31 x^{4}}{3696}+\frac {2259 x^{5}}{261800}+O\left (x^{6}\right )\right )+c_{2} \left (1+2 x +\frac {x^{2}}{2}-\frac {5 x^{3}}{21}-\frac {73 x^{4}}{840}+\frac {827 x^{5}}{27300}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {2}{3}} \left (1+\frac {2}{5} x -\frac {3}{40} x^{2}-\frac {43}{660} x^{3}+\frac {31}{3696} x^{4}+\frac {2259}{261800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+2 x +\frac {1}{2} x^{2}-\frac {5}{21} x^{3}-\frac {73}{840} x^{4}+\frac {827}{27300} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2935

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

Problem 2936

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x \left (3-2 x \right ) y^{\prime }+\left (-2 x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1+O\left (x^{6}\right )\right )}{x}+c_{2} \left (\frac {\left (1+O\left (x^{6}\right )\right ) \ln \left (x \right )}{x}+\frac {x^{2}+2 x +\frac {4 x^{3}}{9}+\frac {x^{4}}{6}+\frac {4 x^{5}}{75}+O\left (x^{6}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (2 x +x^{2}+\frac {4}{9} x^{3}+\frac {1}{6} x^{4}+\frac {4}{75} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} +\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 2937

ODE	\[ \boxed {x^{2} y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+\left (-x +4\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (3 x^{2}+3 x +1+\frac {5 x^{3}}{3}+\frac {5 x^{4}}{8}+\frac {7 x^{5}}{40}+O\left (x^{6}\right )\right )+c_{2} \left (x^{2} \left (3 x^{2}+3 x +1+\frac {5 x^{3}}{3}+\frac {5 x^{4}}{8}+\frac {7 x^{5}}{40}+O\left (x^{6}\right )\right ) \ln \left (x \right )+x^{2} \left (-5 x -\frac {29 x^{2}}{4}-\frac {173 x^{3}}{36}-\frac {193 x^{4}}{96}-\frac {1459 x^{5}}{2400}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (\left (-5\right ) x -\frac {29}{4} x^{2}-\frac {173}{36} x^{3}-\frac {193}{96} x^{4}-\frac {1459}{2400} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} +\left (1+3 x +3 x^{2}+\frac {5}{3} x^{3}+\frac {5}{8} x^{4}+\frac {7}{40} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \left (c_{2} \ln \left (x \right )+c_{1} \right )\right ) x^{2} \]

Problem 2938

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x \left (-x +3\right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1-x +O\left (x^{6}\right )\right )}{x}+c_{2} \left (\frac {\left (1-x +O\left (x^{6}\right )\right ) \ln \left (x \right )}{x}+\frac {3 x -\frac {x^{2}}{4}-\frac {x^{3}}{36}-\frac {x^{4}}{288}-\frac {x^{5}}{2400}+O\left (x^{6}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (3 x -\frac {1}{4} x^{2}-\frac {1}{36} x^{3}-\frac {1}{288} x^{4}-\frac {1}{2400} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} +\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-x +\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 2939

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x -\left (4+x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1+\frac {x}{5}+\frac {x^{2}}{60}+\frac {x^{3}}{1260}+\frac {x^{4}}{40320}+\frac {x^{5}}{1814400}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {x^{2} \left (1+\frac {x}{5}+\frac {x^{2}}{60}+\frac {x^{3}}{1260}+\frac {x^{4}}{40320}+\frac {x^{5}}{1814400}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{144}+\frac {1-\frac {x}{3}+\frac {x^{2}}{12}-\frac {x^{3}}{36}+\frac {x^{5}}{600}+O\left (x^{6}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{4} \left (1+\frac {1}{5} x +\frac {1}{60} x^{2}+\frac {1}{1260} x^{3}+\frac {1}{40320} x^{4}+\frac {1}{1814400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (x^{4}+\frac {1}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144+48 x -12 x^{2}+4 x^{3}-\frac {6}{25} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]

Problem 2940

ODE	\[ \boxed {x^{2} y^{\prime \prime }-\left (-x^{2}+x \right ) y^{\prime }+\left (x^{3}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-x +\frac {x^{2}}{2}-\frac {5 x^{3}}{18}+\frac {19 x^{4}}{144}-\frac {167 x^{5}}{3600}+O\left (x^{6}\right )\right )+c_{2} \left (x \left (1-x +\frac {x^{2}}{2}-\frac {5 x^{3}}{18}+\frac {19 x^{4}}{144}-\frac {167 x^{5}}{3600}+O\left (x^{6}\right )\right ) \ln \left (x \right )+x \left (x -\frac {3 x^{2}}{4}+\frac {41 x^{3}}{108}-\frac {89 x^{4}}{432}+\frac {2281 x^{5}}{27000}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (x -\frac {3}{4} x^{2}+\frac {41}{108} x^{3}-\frac {89}{432} x^{4}+\frac {2281}{27000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} +\left (1-x +\frac {1}{2} x^{2}-\frac {5}{18} x^{3}+\frac {19}{144} x^{4}-\frac {167}{3600} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \left (c_{2} \ln \left (x \right )+c_{1} \right )\right ) x \]

Problem 2941

ODE	\[ \boxed {x^{2} y^{\prime \prime }-\left (-1+2 \sqrt {5}\right ) x y^{\prime }+\left (\frac {19}{4}-3 x^{2}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{2}+\sqrt {5}} \left (1+\frac {x^{2}}{2}+\frac {3 x^{4}}{40}+O\left (x^{6}\right )\right )+c_{2} x^{-\frac {1}{2}+\sqrt {5}} \left (1-\frac {12 x^{2}}{-27+8 \left (-\frac {1}{2}+\sqrt {5}\right ) \sqrt {5}-4 \left (-\frac {1}{2}+\sqrt {5}\right )^{2}}-\frac {144 x^{4}}{-1304 \left (-\frac {1}{2}+\sqrt {5}\right )^{2}+64 \sqrt {5}\, \left (-\frac {1}{2}+\sqrt {5}\right )^{3}-16 \left (-\frac {1}{2}+\sqrt {5}\right )^{4}-3281-1920 \sqrt {5}+576 \sqrt {5}\, \left (-\frac {1}{2}+\sqrt {5}\right )^{2}-192 \left (-\frac {1}{2}+\sqrt {5}\right )^{3}+1968 \left (-\frac {1}{2}+\sqrt {5}\right ) \sqrt {5}}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{-\frac {1}{2}+\sqrt {5}} \left (\left (1+\frac {3}{2} x^{2}+\frac {3}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{1} +c_{2} x \left (\left (1+\frac {1}{2} x^{2}+\frac {3}{40} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right )+\left (-\frac {5}{12} x^{2}-\frac {77}{800} x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right )\right ) \]

Problem 2942

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\left (-2 x^{5}+9 x \right ) y^{\prime }+\left (10 x^{4}+5 x^{2}+25\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{-4+3 i} \left (1+\left (-\frac {1}{8}+\frac {3 i}{8}\right ) x^{2}+\left (-\frac {179}{832}+\frac {483 i}{832}\right ) x^{4}+\left (-\frac {433}{3744}-\frac {3943 i}{29952}\right ) x^{6}+O\left (x^{7}\right )\right )+c_{2} x^{-4-3 i} \left (1+\left (-\frac {1}{8}-\frac {3 i}{8}\right ) x^{2}+\left (-\frac {179}{832}-\frac {483 i}{832}\right ) x^{4}+\left (-\frac {433}{3744}+\frac {3943 i}{29952}\right ) x^{6}+O\left (x^{7}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{-4-3 i} \left (1+\left (-\frac {1}{8}-\frac {3 i}{8}\right ) x^{2}+\left (-\frac {179}{832}-\frac {483 i}{832}\right ) x^{4}+\left (-\frac {433}{3744}+\frac {3943 i}{29952}\right ) x^{6}+\operatorname {O}\left (x^{7}\right )\right )+c_{2} x^{-4+3 i} \left (1+\left (-\frac {1}{8}+\frac {3 i}{8}\right ) x^{2}+\left (-\frac {179}{832}+\frac {483 i}{832}\right ) x^{4}+\left (-\frac {433}{3744}-\frac {3943 i}{29952}\right ) x^{6}+\operatorname {O}\left (x^{7}\right )\right ) \]

Problem 2943

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\left (4 x +\frac {1}{2} x^{2}-\frac {1}{3} x^{3}\right ) y^{\prime }-\frac {7 y}{4}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {x}{20}+\frac {49 x^{2}}{2880}-\frac {533 x^{3}}{241920}+\frac {277 x^{4}}{491520}-\frac {203759 x^{5}}{2388787200}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {8491 \sqrt {x}\, \left (1-\frac {x}{20}+\frac {49 x^{2}}{2880}-\frac {533 x^{3}}{241920}+\frac {277 x^{4}}{491520}-\frac {203759 x^{5}}{2388787200}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{110592}+\frac {1-\frac {7 x}{12}+\frac {91 x^{2}}{192}-\frac {1939 x^{3}}{6912}+\frac {103033 x^{5}}{8294400}+O\left (x^{6}\right )}{x^{\frac {7}{2}}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{4} \left (1-\frac {1}{20} x +\frac {49}{2880} x^{2}-\frac {533}{241920} x^{3}+\frac {277}{491520} x^{4}-\frac {203759}{2388787200} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (\frac {8491}{768} x^{4}-\frac {8491}{15360} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144+84 x -\frac {273}{4} x^{2}+\frac {1939}{48} x^{3}-\frac {221}{12} x^{4}-\frac {49993}{57600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{\frac {7}{2}}} \]

Problem 2944

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

Problem 2945

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

Problem 2946

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+2 y^{\prime } x^{2}+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {x}{4}+\frac {3 x^{2}}{64}-\frac {5 x^{3}}{768}+\frac {35 x^{4}}{49152}-\frac {21 x^{5}}{327680}+O\left (x^{6}\right )\right )+c_{2} \left (\sqrt {x}\, \left (1-\frac {x}{4}+\frac {3 x^{2}}{64}-\frac {5 x^{3}}{768}+\frac {35 x^{4}}{49152}-\frac {21 x^{5}}{327680}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\sqrt {x}\, \left (-\frac {x^{2}}{64}+\frac {x^{3}}{256}-\frac {19 x^{4}}{32768}+\frac {25 x^{5}}{393216}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {1}{4} x +\frac {3}{64} x^{2}-\frac {5}{768} x^{3}+\frac {35}{49152} x^{4}-\frac {21}{327680} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {1}{64} x^{2}+\frac {1}{256} x^{3}-\frac {19}{32768} x^{4}+\frac {25}{393216} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]

Problem 2947

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x \cos \left (x \right ) y^{\prime }-2 \,{\mathrm e}^{x} y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\sqrt {2}} \left (1+\frac {2 x}{1+2 \sqrt {2}}+\frac {\left (5 \sqrt {2}+14\right ) x^{2}}{40+24 \sqrt {2}}+\frac {\left (122+75 \sqrt {2}\right ) x^{3}}{684 \sqrt {2}+972}+\frac {\left (1626 \sqrt {2}+2375\right ) x^{4}}{52992+37440 \sqrt {2}}+\frac {\left (75763+52810 \sqrt {2}\right ) x^{5}}{7200 \left (1+2 \sqrt {2}\right ) \left (1+\sqrt {2}\right ) \left (3+2 \sqrt {2}\right ) \left (2+\sqrt {2}\right ) \left (5+2 \sqrt {2}\right )}+O\left (x^{6}\right )\right )+c_{2} x^{-\sqrt {2}} \left (1-\frac {2 x}{-1+2 \sqrt {2}}+\frac {\left (-5 \sqrt {2}+14\right ) x^{2}}{40-24 \sqrt {2}}+\frac {\left (-122+75 \sqrt {2}\right ) x^{3}}{684 \sqrt {2}-972}+\frac {\left (-1626 \sqrt {2}+2375\right ) x^{4}}{52992-37440 \sqrt {2}}+\frac {\left (-75763+52810 \sqrt {2}\right ) x^{5}}{7200 \left (-1+2 \sqrt {2}\right ) \left (\sqrt {2}-1\right ) \left (-3+2 \sqrt {2}\right ) \left (\sqrt {2}-2\right ) \left (-5+2 \sqrt {2}\right )}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{-\sqrt {2}} \left (1-2 \frac {1}{-1+2 \sqrt {2}} x +\frac {-5 \sqrt {2}+14}{40-24 \sqrt {2}} x^{2}+\frac {-122+75 \sqrt {2}}{684 \sqrt {2}-972} x^{3}+\frac {-1626 \sqrt {2}+2375}{52992-37440 \sqrt {2}} x^{4}+\frac {1}{7200} \frac {-75763+52810 \sqrt {2}}{\left (-1+2 \sqrt {2}\right ) \left (\sqrt {2}-1\right ) \left (-3+2 \sqrt {2}\right ) \left (\sqrt {2}-2\right ) \left (-5+2 \sqrt {2}\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{\sqrt {2}} \left (1+2 \frac {1}{1+2 \sqrt {2}} x +\frac {5 \sqrt {2}+14}{40+24 \sqrt {2}} x^{2}+\frac {122+75 \sqrt {2}}{684 \sqrt {2}+972} x^{3}+\frac {1626 \sqrt {2}+2375}{52992+37440 \sqrt {2}} x^{4}+\frac {1}{7200} \frac {75763+52810 \sqrt {2}}{\left (1+2 \sqrt {2}\right ) \left (1+\sqrt {2}\right ) \left (3+2 \sqrt {2}\right ) \left (2+\sqrt {2}\right ) \left (5+2 \sqrt {2}\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2948

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x^{2}-\left (x +2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-\frac {x}{4}+\frac {x^{2}}{20}-\frac {x^{3}}{120}+\frac {x^{4}}{840}-\frac {x^{5}}{6720}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-x +\frac {x^{2}}{2}-\frac {x^{3}}{6}+\frac {x^{4}}{24}-\frac {x^{5}}{120}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \left (1-\frac {1}{4} x +\frac {1}{20} x^{2}-\frac {1}{120} x^{3}+\frac {1}{840} x^{4}-\frac {1}{6720} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (12-12 x +6 x^{2}-2 x^{3}+\frac {1}{2} x^{4}-\frac {1}{10} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 2949

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

Problem 2950

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (2 x -1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {2 x}{3}+\frac {x^{2}}{6}-\frac {x^{3}}{45}+\frac {x^{4}}{540}-\frac {x^{5}}{9450}+O\left (x^{6}\right )\right )+c_{2} \left (-2 x \left (1-\frac {2 x}{3}+\frac {x^{2}}{6}-\frac {x^{3}}{45}+\frac {x^{4}}{540}-\frac {x^{5}}{9450}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\frac {1+2 x -\frac {16 x^{3}}{9}+\frac {25 x^{4}}{36}-\frac {157 x^{5}}{1350}+O\left (x^{6}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{2} \left (1-\frac {2}{3} x +\frac {1}{6} x^{2}-\frac {1}{45} x^{3}+\frac {1}{540} x^{4}-\frac {1}{9450} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (4 x^{2}-\frac {8}{3} x^{3}+\frac {2}{3} x^{4}-\frac {4}{45} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2-4 x +\frac {32}{9} x^{3}-\frac {25}{18} x^{4}+\frac {157}{675} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]

Problem 2951

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x^{3}-\left (x +2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1+\frac {x}{4}-\frac {7 x^{2}}{40}-\frac {37 x^{3}}{720}+\frac {467 x^{4}}{20160}+\frac {5647 x^{5}}{806400}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {x^{2} \left (1+\frac {x}{4}-\frac {7 x^{2}}{40}-\frac {37 x^{3}}{720}+\frac {467 x^{4}}{20160}+\frac {5647 x^{5}}{806400}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{12}+\frac {1-\frac {x}{2}-\frac {x^{2}}{4}+\frac {17 x^{4}}{192}+\frac {67 x^{5}}{9600}+O\left (x^{6}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{3} \left (1+\frac {1}{4} x -\frac {7}{40} x^{2}-\frac {37}{720} x^{3}+\frac {467}{20160} x^{4}+\frac {5647}{806400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-x^{3}-\frac {1}{4} x^{4}+\frac {7}{40} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (12-6 x -3 x^{2}+3 x^{3}+\frac {29}{16} x^{4}-\frac {353}{800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]

Problem 2952

ODE	\[ \boxed {x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+7 y^{\prime } x \,{\mathrm e}^{x}+9 \left (1+\tan \left (x \right )\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1+12 x +\frac {117 x^{2}}{8}-\frac {67 x^{3}}{36}+\frac {505 x^{4}}{256}-\frac {262 x^{5}}{125}+\frac {2443637 x^{6}}{2304000}+O\left (x^{7}\right )\right )}{x^{3}}+c_{2} \left (\frac {\left (1+12 x +\frac {117 x^{2}}{8}-\frac {67 x^{3}}{36}+\frac {505 x^{4}}{256}-\frac {262 x^{5}}{125}+\frac {2443637 x^{6}}{2304000}+O\left (x^{7}\right )\right ) \ln \left (x \right )}{x^{3}}+\frac {-31 x -\frac {147 x^{2}}{2}+\frac {37 x^{3}}{8}-\frac {44803 x^{4}}{4608}+\frac {5057587 x^{5}}{480000}-\frac {3797765581 x^{6}}{622080000}+O\left (x^{7}\right )}{x^{3}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+12 x +\frac {117}{8} x^{2}-\frac {67}{36} x^{3}+\frac {505}{256} x^{4}-\frac {262}{125} x^{5}+\frac {2443637}{2304000} x^{6}+\operatorname {O}\left (x^{7}\right )\right )+\left (\left (-31\right ) x -\frac {147}{2} x^{2}+\frac {37}{8} x^{3}-\frac {44803}{4608} x^{4}+\frac {5057587}{480000} x^{5}-\frac {3797765581}{622080000} x^{6}+\operatorname {O}\left (x^{7}\right )\right ) c_{2}}{x^{3}} \]

Problem 2953

ODE	\[ \boxed {x^{2} \left (x +1\right ) y^{\prime \prime }+y^{\prime } x^{2}-2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-x +\frac {9 x^{2}}{10}-\frac {4 x^{3}}{5}+\frac {5 x^{4}}{7}-\frac {9 x^{5}}{14}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+\frac {x}{2}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \left (1-x +\frac {9}{10} x^{2}-\frac {4}{5} x^{3}+\frac {5}{7} x^{4}-\frac {9}{14} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (12+6 x +\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 2954

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

Problem 2955

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

Problem 2956

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

Problem 2957

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	\[ y = c_{1} x \left (1+\frac {x}{3}+\frac {x^{2}}{12}+\frac {x^{3}}{60}+\frac {x^{4}}{360}+\frac {x^{5}}{2520}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+x +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1+\frac {1}{3} x +\frac {1}{12} x^{2}+\frac {1}{60} x^{3}+\frac {1}{360} x^{4}+\frac {1}{2520} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (-2-2 x -x^{2}-\frac {1}{3} x^{3}-\frac {1}{12} x^{4}-\frac {1}{60} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 2958

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	\[ y \left (x \right ) = \sqrt {x}\, \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+x +\frac {1}{4} x^{2}+\frac {1}{36} x^{3}+\frac {1}{576} x^{4}+\frac {1}{14400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-2\right ) x -\frac {3}{4} x^{2}-\frac {11}{108} x^{3}-\frac {25}{3456} x^{4}-\frac {137}{432000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]

Problem 2959

ODE	\[ \boxed {x y^{\prime \prime }+y^{\prime }-2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (x^{2}+2 x +1+\frac {2 x^{3}}{9}+\frac {x^{4}}{36}+\frac {x^{5}}{450}+O\left (x^{6}\right )\right )+c_{2} \left (\left (x^{2}+2 x +1+\frac {2 x^{3}}{9}+\frac {x^{4}}{36}+\frac {x^{5}}{450}+O\left (x^{6}\right )\right ) \ln \left (x \right )-3 x^{2}-4 x -\frac {22 x^{3}}{27}-\frac {25 x^{4}}{216}-\frac {137 x^{5}}{13500}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+2 x +x^{2}+\frac {2}{9} x^{3}+\frac {1}{36} x^{4}+\frac {1}{450} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-4\right ) x -3 x^{2}-\frac {22}{27} x^{3}-\frac {25}{216} x^{4}-\frac {137}{13500} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 2960

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x -y \left (x +1\right )=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1+\frac {x}{3}+\frac {x^{2}}{24}+\frac {x^{3}}{360}+\frac {x^{4}}{8640}+\frac {x^{5}}{302400}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {x \left (1+\frac {x}{3}+\frac {x^{2}}{24}+\frac {x^{3}}{360}+\frac {x^{4}}{8640}+\frac {x^{5}}{302400}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{2}+\frac {1-x +\frac {2 x^{3}}{9}+\frac {25 x^{4}}{576}+\frac {157 x^{5}}{43200}+O\left (x^{6}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{2} \left (1+\frac {1}{3} x +\frac {1}{24} x^{2}+\frac {1}{360} x^{3}+\frac {1}{8640} x^{4}+\frac {1}{302400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (x^{2}+\frac {1}{3} x^{3}+\frac {1}{24} x^{4}+\frac {1}{360} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+2 x -\frac {4}{9} x^{3}-\frac {25}{288} x^{4}-\frac {157}{21600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]

Problem 2961

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	\[ y = c_{1} x^{2} \left (1+2 x +\frac {3 x^{2}}{2}+\frac {2 x^{3}}{3}+\frac {5 x^{4}}{24}+\frac {x^{5}}{20}+O\left (x^{6}\right )\right )+c_{2} \left (x^{2} \left (1+2 x +\frac {3 x^{2}}{2}+\frac {2 x^{3}}{3}+\frac {5 x^{4}}{24}+\frac {x^{5}}{20}+O\left (x^{6}\right )\right ) \ln \left (x \right )+x^{2} \left (-3 x -\frac {13 x^{2}}{4}-\frac {31 x^{3}}{18}-\frac {173 x^{4}}{288}-\frac {187 x^{5}}{1200}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+2 x +\frac {3}{2} x^{2}+\frac {2}{3} x^{3}+\frac {5}{24} x^{4}+\frac {1}{20} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-3\right ) x -\frac {13}{4} x^{2}-\frac {31}{18} x^{3}-\frac {173}{288} x^{4}-\frac {187}{1200} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) x^{2} \]

Problem 2962

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	\[ y = c_{1} x^{2} \left (1+\frac {x}{2}+\frac {3 x^{2}}{20}+\frac {x^{3}}{30}+\frac {x^{4}}{168}+\frac {x^{5}}{1120}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+\frac {x}{2}-\frac {x^{3}}{12}-\frac {x^{4}}{24}-\frac {x^{5}}{80}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \left (1+\frac {1}{2} x +\frac {3}{20} x^{2}+\frac {1}{30} x^{3}+\frac {1}{168} x^{4}+\frac {1}{1120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (12+6 x -x^{3}-\frac {1}{2} x^{4}-\frac {3}{20} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 2963

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

Problem 2964

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x \left (5-x \right ) y^{\prime }+4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (-2 x +1+\frac {x^{2}}{2}+O\left (x^{6}\right )\right )}{x^{2}}+c_{2} \left (\frac {\left (-2 x +1+\frac {x^{2}}{2}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{x^{2}}+\frac {5 x -\frac {9 x^{2}}{4}+\frac {x^{3}}{18}+\frac {x^{4}}{288}+\frac {x^{5}}{3600}+O\left (x^{6}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-2 x +\frac {1}{2} x^{2}+\operatorname {O}\left (x^{6}\right )\right )+\left (5 x -\frac {9}{4} x^{2}+\frac {1}{18} x^{3}+\frac {1}{288} x^{4}+\frac {1}{3600} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2}}{x^{2}} \]

Problem 2965

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+4 x \left (1-x \right ) y^{\prime }+\left (2 x -9\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {3}{2}} \left (1+\frac {x}{4}+\frac {x^{2}}{20}+\frac {x^{3}}{120}+\frac {x^{4}}{840}+\frac {x^{5}}{6720}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+x +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+O\left (x^{6}\right )\right )}{x^{\frac {3}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{3} \left (1+\frac {1}{4} x +\frac {1}{20} x^{2}+\frac {1}{120} x^{3}+\frac {1}{840} x^{4}+\frac {1}{6720} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (12+12 x +6 x^{2}+2 x^{3}+\frac {1}{2} x^{4}+\frac {1}{10} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {3}{2}}} \]

Problem 2966

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

Problem 2967

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

Problem 2968

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

Problem 2969

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }-\left (4 x +3\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {3}{2}} \left (1+\frac {x}{3}+\frac {x^{2}}{24}+\frac {x^{3}}{360}+\frac {x^{4}}{8640}+\frac {x^{5}}{302400}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {x^{\frac {3}{2}} \left (1+\frac {x}{3}+\frac {x^{2}}{24}+\frac {x^{3}}{360}+\frac {x^{4}}{8640}+\frac {x^{5}}{302400}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{2}+\frac {1-x +\frac {2 x^{3}}{9}+\frac {25 x^{4}}{576}+\frac {157 x^{5}}{43200}+O\left (x^{6}\right )}{\sqrt {x}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{2} \left (1+\frac {1}{3} x +\frac {1}{24} x^{2}+\frac {1}{360} x^{3}+\frac {1}{8640} x^{4}+\frac {1}{302400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (x^{2}+\frac {1}{3} x^{3}+\frac {1}{24} x^{4}+\frac {1}{360} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+2 x -\frac {4}{9} x^{3}-\frac {25}{288} x^{4}-\frac {157}{21600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{\sqrt {x}} \]

Problem 2970

ODE	\[ \boxed {x y^{\prime \prime }-y^{\prime } x +y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1+O\left (x^{6}\right )\right )+c_{2} \left (-x \left (1+O\left (x^{6}\right )\right ) \ln \left (x \right )+1-\frac {x^{2}}{2}-\frac {x^{3}}{12}-\frac {x^{4}}{72}-\frac {x^{5}}{480}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \ln \left (x \right ) \left (-x +\operatorname {O}\left (x^{6}\right )\right ) c_{2} +c_{1} x \left (1+\operatorname {O}\left (x^{6}\right )\right )+\left (1+x -\frac {1}{2} x^{2}-\frac {1}{12} x^{3}-\frac {1}{72} x^{4}-\frac {1}{480} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 2971

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x \left (4+x \right ) y^{\prime }+\left (x +2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1+O\left (x^{6}\right )\right )}{x}+c_{2} \left (\frac {\left (1+O\left (x^{6}\right )\right ) \ln \left (x \right )}{x}+\frac {1-\frac {x^{2}}{2}+\frac {x^{3}}{12}-\frac {x^{4}}{72}+\frac {x^{5}}{480}+O\left (x^{6}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\ln \left (x \right ) \left (x +\operatorname {O}\left (x^{6}\right )\right ) c_{2} +c_{1} x \left (1+\operatorname {O}\left (x^{6}\right )\right )+\left (1-x -\frac {1}{2} x^{2}+\frac {1}{12} x^{3}-\frac {1}{72} x^{4}+\frac {1}{480} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2}}{x^{2}} \]

Problem 2972

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (x^{2}-\frac {9}{4}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {3}{2}} \left (1-\frac {x^{2}}{10}+\frac {x^{4}}{280}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+\frac {x^{2}}{2}-\frac {x^{4}}{8}+O\left (x^{6}\right )\right )}{x^{\frac {3}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{3} \left (1-\frac {1}{10} x^{2}+\frac {1}{280} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (12+6 x^{2}-\frac {3}{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {3}{2}}} \]

Problem 2973

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	\[ y = c_{1} x^{2} \left (1-\frac {x^{2}}{8}+\frac {x^{4}}{192}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {x^{2} \left (1-\frac {x^{2}}{8}+\frac {x^{4}}{192}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{2}+1-\frac {3 x^{4}}{64}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \left (1-\frac {1}{8} x^{2}+\frac {1}{192} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (x^{2}-\frac {1}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+\frac {3}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

Problem 2974

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

Problem 2975

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

Problem 2976

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

Problem 2977

ODE	\[ \boxed {x y^{\prime \prime }+y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (x^{2}-2 x +1-\frac {2 x^{3}}{9}+\frac {x^{4}}{36}-\frac {x^{5}}{450}+O\left (x^{6}\right )\right )+c_{2} \left (\left (x^{2}-2 x +1-\frac {2 x^{3}}{9}+\frac {x^{4}}{36}-\frac {x^{5}}{450}+O\left (x^{6}\right )\right ) \ln \left (x \right )-3 x^{2}+4 x +\frac {22 x^{3}}{27}-\frac {25 x^{4}}{216}+\frac {137 x^{5}}{13500}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-2 x +x^{2}-\frac {2}{9} x^{3}+\frac {1}{36} x^{4}-\frac {1}{450} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (4 x -3 x^{2}+\frac {22}{27} x^{3}-\frac {25}{216} x^{4}+\frac {137}{13500} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 2978

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

Problem 2979

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

Problem 2980

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	\[ y = c_{1} \left (1-\frac {x^{2}}{4}+\frac {x^{4}}{64}+O\left (x^{6}\right )\right )+c_{2} \left (\left (1-\frac {x^{2}}{4}+\frac {x^{4}}{64}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\frac {x^{2}}{4}-\frac {3 x^{4}}{128}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {1}{4} x^{2}+\frac {1}{64} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{4} x^{2}-\frac {3}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 2981

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

Problem 2982

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

Problem 2983

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	\[ y = c_{1} x^{\frac {1}{4}} \left (1-\frac {3 x}{5}+\frac {x^{2}}{10}-\frac {x^{3}}{130}+\frac {3 x^{4}}{8840}-\frac {3 x^{5}}{309400}+O\left (x^{6}\right )\right )+c_{2} \left (1-x +\frac {3 x^{2}}{14}-\frac {3 x^{3}}{154}+\frac {3 x^{4}}{3080}-\frac {9 x^{5}}{292600}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{4}} \left (1-\frac {3}{5} x +\frac {1}{10} x^{2}-\frac {1}{130} x^{3}+\frac {3}{8840} x^{4}-\frac {3}{309400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-x +\frac {3}{14} x^{2}-\frac {3}{154} x^{3}+\frac {3}{3080} x^{4}-\frac {9}{292600} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2984

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\frac {3 y^{\prime } x}{2}-\frac {y \left (x +1\right )}{2}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1+\frac {x}{5}+\frac {x^{2}}{70}+\frac {x^{3}}{1890}+\frac {x^{4}}{83160}+\frac {x^{5}}{5405400}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-x -\frac {x^{2}}{2}-\frac {x^{3}}{18}-\frac {x^{4}}{360}-\frac {x^{5}}{12600}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {3}{2}} \left (1+\frac {1}{5} x +\frac {1}{70} x^{2}+\frac {1}{1890} x^{3}+\frac {1}{83160} x^{4}+\frac {1}{5405400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1-x -\frac {1}{2} x^{2}-\frac {1}{18} x^{3}-\frac {1}{360} x^{4}-\frac {1}{12600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 2985

ODE	\[ \boxed {x^{2} y^{\prime \prime }-x \left (-x +2\right ) y^{\prime }+\left (x^{2}+2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-x +\frac {x^{2}}{3}-\frac {x^{3}}{36}-\frac {7 x^{4}}{720}+\frac {31 x^{5}}{10800}+O\left (x^{6}\right )\right )+c_{2} \left (-x^{2} \left (1-x +\frac {x^{2}}{3}-\frac {x^{3}}{36}-\frac {7 x^{4}}{720}+\frac {31 x^{5}}{10800}+O\left (x^{6}\right )\right ) \ln \left (x \right )+x \left (1-\frac {3 x^{2}}{2}+\frac {31 x^{3}}{36}-\frac {65 x^{4}}{432}-\frac {121 x^{5}}{10800}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} x \left (1-x +\frac {1}{3} x^{2}-\frac {1}{36} x^{3}-\frac {7}{720} x^{4}+\frac {31}{10800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-x +x^{2}-\frac {1}{3} x^{3}+\frac {1}{36} x^{4}+\frac {7}{720} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-x -\frac {1}{2} x^{2}+\frac {19}{36} x^{3}-\frac {53}{432} x^{4}-\frac {1}{675} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )\right ) x \]

Problem 2986

ODE	\[ \boxed {x^{2} y^{\prime \prime }-3 y^{\prime } x +4 y \left (x +1\right )=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (4 x^{2}-4 x +1-\frac {16 x^{3}}{9}+\frac {4 x^{4}}{9}-\frac {16 x^{5}}{225}+O\left (x^{6}\right )\right )+c_{2} \left (x^{2} \left (4 x^{2}-4 x +1-\frac {16 x^{3}}{9}+\frac {4 x^{4}}{9}-\frac {16 x^{5}}{225}+O\left (x^{6}\right )\right ) \ln \left (x \right )+x^{2} \left (-12 x^{2}+8 x +\frac {176 x^{3}}{27}-\frac {50 x^{4}}{27}+\frac {1096 x^{5}}{3375}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-4 x +4 x^{2}-\frac {16}{9} x^{3}+\frac {4}{9} x^{4}-\frac {16}{225} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (8 x -12 x^{2}+\frac {176}{27} x^{3}-\frac {50}{27} x^{4}+\frac {1096}{3375} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) x^{2} \]

Problem 2987

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

Problem 2988

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

Problem 2989

ODE	\[ \boxed {2 x \tan \left (y\right )+\left (x -x^{2} \tan \left (y\right )\right ) y^{\prime }=0} \]
program solution	\[ -2 \,{\mathrm e}^{-\frac {y}{2}} x +\int _{0}^{y}-{\mathrm e}^{-\frac {\textit {\_a}}{2}} \cot \left (\textit {\_a} \right )d \textit {\_a} = c_{1} \] Verified OK.
Maple solution	\[ \frac {{\mathrm e}^{\frac {y \left (x \right )}{2}} \left (\int _{}^{y \left (x \right )}\cot \left (\textit {\_a} \right ) {\mathrm e}^{-\frac {\textit {\_a}}{2}}d \textit {\_a} \right )}{2}-{\mathrm e}^{\frac {y \left (x \right )}{2}} c_{1} +x = 0 \]

Problem 2990

ODE	\[ \boxed {y^{2} \left (x^{2}+1\right )+y+\left (2 y x +1\right ) y^{\prime }=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 2991

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

Problem 2992

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

Problem 2993

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

Problem 2994

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

Problem 2995

ODE	\[ \boxed {4 y+\left (4 x +2 y+2\right ) y^{\prime }=-1-6 x} \] With initial conditions \begin {align*} \left [y \left (\frac {1}{2}\right ) = 3\right ] \end {align*}
program solution	\[ y^{2}+\left (4 x +2\right ) y+3 x^{2}+x = {\frac {89}{4}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -2 x -1+\frac {\sqrt {4 x^{2}+12 x +93}}{2} \]

Problem 2996

ODE	\[ \boxed {-y+\left (x +y+2\right ) y^{\prime }=-3 x +6} \] With initial conditions \begin {align*} [y \left (2\right ) = -2] \end {align*}
program solution	\[ \frac {\ln \left (3 x^{2}+y^{2}-6 x +6 y+12\right )}{2}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (y+3\right ) \sqrt {3}}{3 x -3}\right )}{3} = \ln \left (2\right )+\frac {\sqrt {3}\, \pi }{18} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -3-\sqrt {3}\, \tan \left (\operatorname {RootOf}\left (-3 \sqrt {3}\, \ln \left (3\right )+6 \sqrt {3}\, \ln \left (2\right )-3 \sqrt {3}\, \ln \left (\sec \left (\textit {\_Z} \right )^{2} \left (x -1\right )^{2}\right )+\pi +6 \textit {\_Z} \right )\right ) \left (x -1\right ) \]

Problem 2997

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

Problem 2998

ODE	\[ \boxed {y-y^{\prime }-\frac {{y^{\prime }}^{2}}{2}=0} \]
program solution	\[ \sqrt {1+2 y}+\ln \left (-1+\sqrt {1+2 y}\right ) = x +c_{1} \] Verified OK. \[ -\sqrt {1+2 y}+\ln \left (1+\sqrt {1+2 y}\right ) = x +c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {{\mathrm e}^{2 \operatorname {RootOf}\left (-\textit {\_Z} -2 x -2 \,{\mathrm e}^{\textit {\_Z}}-2+2 c_{1} -\ln \left (2\right )+\ln \left ({\mathrm e}^{\textit {\_Z}} \left ({\mathrm e}^{\textit {\_Z}}+2\right )^{2}\right )\right )}}{2}+{\mathrm e}^{\operatorname {RootOf}\left (-\textit {\_Z} -2 x -2 \,{\mathrm e}^{\textit {\_Z}}-2+2 c_{1} -\ln \left (2\right )+\ln \left ({\mathrm e}^{\textit {\_Z}} \left ({\mathrm e}^{\textit {\_Z}}+2\right )^{2}\right )\right )} \\ y \left (x \right ) &= \frac {\operatorname {LambertW}\left (\sqrt {2}\, {\mathrm e}^{-c_{1} +x -1}\right ) \left (\operatorname {LambertW}\left (\sqrt {2}\, {\mathrm e}^{-c_{1} +x -1}\right )+2\right )}{2} \\ \end{align*}

Problem 2999

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

Problem 3000

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