2.17.51 Problems 5001 to 5100

Problem 5001

ODE	\[ \boxed {t^{3} y^{2}+\frac {t^{4} y^{\prime }}{y^{6}}=0} \]
program solution	\[ \frac {1}{7 y^{7}}-\ln \left (t \right ) = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (t \right ) &= \frac {1}{\left (c_{1} +7 \ln \left (t \right )\right )^{\frac {1}{7}}} \\ y \left (t \right ) &= -\frac {\left (-1\right )^{\frac {1}{7}}}{\left (c_{1} +7 \ln \left (t \right )\right )^{\frac {1}{7}}} \\ y \left (t \right ) &= \frac {\left (-1\right )^{\frac {6}{7}}}{\left (c_{1} +7 \ln \left (t \right )\right )^{\frac {1}{7}}} \\ y \left (t \right ) &= -\frac {\left (-1\right )^{\frac {5}{7}}}{\left (c_{1} +7 \ln \left (t \right )\right )^{\frac {1}{7}}} \\ y \left (t \right ) &= \frac {\left (-1\right )^{\frac {2}{7}}}{\left (c_{1} +7 \ln \left (t \right )\right )^{\frac {1}{7}}} \\ y \left (t \right ) &= -\frac {\left (-1\right )^{\frac {3}{7}}}{\left (c_{1} +7 \ln \left (t \right )\right )^{\frac {1}{7}}} \\ y \left (t \right ) &= \frac {\left (-1\right )^{\frac {4}{7}}}{\left (c_{1} +7 \ln \left (t \right )\right )^{\frac {1}{7}}} \\ \end{align*}

Problem 5002

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

Problem 5003

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

Problem 5004

ODE	\[ \boxed {\left (x^{2}-2\right ) y^{\prime \prime }+2 y^{\prime }+\sin \left (x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{12} x^{3}+\frac {1}{48} x^{4}+\frac {1}{80} x^{5}+\frac {11}{1440} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{8} x^{4}+\frac {1}{16} x^{5}+\frac {17}{480} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{12} x^{3}+\frac {1}{48} x^{4}+\frac {1}{80} x^{5}\right ) c_{1} +\left (x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{8} x^{4}+\frac {1}{16} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{12} x^{3}+\frac {1}{48} x^{4}+\frac {1}{80} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{8} x^{4}+\frac {1}{16} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5005

ODE	\[ \boxed {\left (x^{2}+x \right ) y^{\prime \prime }+3 y^{\prime }-6 y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1+\frac {3 x^{2}}{4}-\frac {x^{3}}{10}+\frac {17 x^{4}}{80}-\frac {9 x^{5}}{100}+O\left (x^{6}\right )\right )+c_{2} \left (\left (-3-\frac {9 x^{2}}{4}+\frac {3 x^{3}}{10}-\frac {51 x^{4}}{80}+\frac {27 x^{5}}{100}-3 O\left (x^{6}\right )\right ) \ln \left (x \right )+\frac {1+6 x +11 x^{3}+\frac {27 x^{4}}{16}+\frac {893 x^{5}}{200}+O\left (x^{6}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1+\frac {3}{4} x^{2}-\frac {1}{10} x^{3}+\frac {17}{80} x^{4}-\frac {9}{100} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) x^{2}+c_{2} \left (\ln \left (x \right ) \left (6 x^{2}+\frac {9}{2} x^{4}-\frac {3}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2-12 x -24 x^{2}-22 x^{3}-\frac {171}{8} x^{4}-\frac {653}{100} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]

Problem 5006

ODE	\[ \boxed {\left (t^{2}-t -2\right ) x^{\prime \prime }+\left (t +1\right ) x^{\prime }-\left (-2+t \right ) x=0} \] With the expansion point for the power series method at \(t = 0\).
program solution	\[ x = \left (1+\frac {1}{2} t^{2}-\frac {1}{12} t^{3}+\frac {13}{96} t^{4}-\frac {1}{16} t^{5}+\frac {83}{1440} t^{6}\right ) x \left (0\right )+\left (t +\frac {1}{4} t^{2}+\frac {1}{4} t^{3}-\frac {1}{96} t^{4}+\frac {31}{480} t^{5}-\frac {3}{128} t^{6}\right ) x^{\prime }\left (0\right )+O\left (t^{6}\right ) \] Verified OK. \[ x = \left (1+\frac {1}{2} t^{2}-\frac {1}{12} t^{3}+\frac {13}{96} t^{4}-\frac {1}{16} t^{5}\right ) c_{1} +\left (t +\frac {1}{4} t^{2}+\frac {1}{4} t^{3}-\frac {1}{96} t^{4}+\frac {31}{480} t^{5}\right ) c_{2} +O\left (t^{6}\right ) \] Verified OK.
Maple solution	\[ x \left (t \right ) = \left (1+\frac {1}{2} t^{2}-\frac {1}{12} t^{3}+\frac {13}{96} t^{4}-\frac {1}{16} t^{5}\right ) x \left (0\right )+\left (t +\frac {1}{4} t^{2}+\frac {1}{4} t^{3}-\frac {1}{96} t^{4}+\frac {31}{480} t^{5}\right ) D\left (x \right )\left (0\right )+O\left (t^{6}\right ) \]

Problem 5007

ODE	\[ \boxed {\left (x^{2}-1\right ) y^{\prime \prime }+\left (1-x \right ) y^{\prime }+\left (x^{2}-2 x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{12} x^{4}-\frac {1}{15} x^{5}+\frac {1}{24} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{12} x^{4}+\frac {1}{60} x^{5}-\frac {1}{60} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{12} x^{4}-\frac {1}{15} x^{5}\right ) c_{1} +\left (x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{12} x^{4}+\frac {1}{60} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{12} x^{4}-\frac {1}{15} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{12} x^{4}+\frac {1}{60} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5008

ODE	\[ \boxed {\sin \left (x \right ) y^{\prime \prime }+y \cos \left (x \right )=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {x}{2}+\frac {x^{2}}{12}+\frac {x^{3}}{48}-\frac {3 x^{4}}{320}+\frac {19 x^{5}}{9600}-\frac {59 x^{6}}{403200}+O\left (x^{6}\right )\right )+c_{2} \left (-x \left (1-\frac {x}{2}+\frac {x^{2}}{12}+\frac {x^{3}}{48}-\frac {3 x^{4}}{320}+\frac {19 x^{5}}{9600}-\frac {59 x^{6}}{403200}+O\left (x^{6}\right )\right ) \ln \left (x \right )+1-\frac {3 x^{2}}{4}+\frac {x^{3}}{4}-\frac {5 x^{4}}{576}-\frac {437 x^{5}}{28800}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1-\frac {1}{2} x +\frac {1}{12} x^{2}+\frac {1}{48} x^{3}-\frac {3}{320} x^{4}+\frac {19}{9600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-x +\frac {1}{2} x^{2}-\frac {1}{12} x^{3}-\frac {1}{48} x^{4}+\frac {3}{320} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-\frac {3}{4} x^{2}+\frac {1}{4} x^{3}-\frac {5}{576} x^{4}-\frac {437}{28800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

Problem 5009

ODE	\[ \boxed {{\mathrm e}^{x} y^{\prime \prime }-\left (x^{2}-1\right ) y^{\prime }+2 y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{3} x^{3}+\frac {1}{4} x^{4}-\frac {3}{20} x^{5}+\frac {3}{40} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{3} x^{3}-\frac {7}{24} x^{4}+\frac {23}{120} x^{5}-\frac {1}{10} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{3} x^{3}+\frac {1}{4} x^{4}-\frac {3}{20} x^{5}\right ) c_{1} +\left (x -\frac {1}{2} x^{2}+\frac {1}{3} x^{3}-\frac {7}{24} x^{4}+\frac {23}{120} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{3} x^{3}+\frac {1}{4} x^{4}-\frac {3}{20} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{3} x^{3}-\frac {7}{24} x^{4}+\frac {23}{120} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5010

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

Problem 5011

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

Problem 5012

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

Problem 5013

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

Problem 5014

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}-\frac {13}{240} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {7}{120} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}\right ) c_{1} +\left (x -\frac {1}{6} x^{3}+\frac {7}{120} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {7}{120} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5015

ODE	\[ \boxed {y^{\prime \prime }+\left (x -1\right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{20} x^{5}-\frac {1}{180} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}-\frac {1}{6} x^{4}+\frac {1}{36} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{20} x^{5}\right ) c_{1} +\left (x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}-\frac {1}{6} x^{4}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{20} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}-\frac {1}{6} x^{4}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5016

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{3} x^{3}-\frac {1}{8} x^{4}-\frac {1}{30} x^{5}-\frac {1}{144} x^{6}\right ) y \left (0\right )+\left (x +x^{2}+\frac {1}{2} x^{3}+\frac {1}{6} x^{4}+\frac {1}{24} x^{5}+\frac {1}{120} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{3} x^{3}-\frac {1}{8} x^{4}-\frac {1}{30} x^{5}\right ) c_{1} +\left (x +x^{2}+\frac {1}{2} x^{3}+\frac {1}{6} x^{4}+\frac {1}{24} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}-\frac {1}{3} x^{3}-\frac {1}{8} x^{4}-\frac {1}{30} x^{5}\right ) y \left (0\right )+\left (x +x^{2}+\frac {1}{2} x^{3}+\frac {1}{6} x^{4}+\frac {1}{24} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5017

ODE	\[ \boxed {w^{\prime \prime }-x^{2} w^{\prime }+w=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ w = \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}-\frac {1}{20} x^{5}-\frac {1}{720} x^{6}\right ) w \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{120} x^{5}-\frac {7}{360} x^{6}\right ) w^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ w = \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}-\frac {1}{20} x^{5}\right ) c_{1} +\left (x -\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{120} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ w \left (x \right ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}-\frac {1}{20} x^{5}\right ) w \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{120} x^{5}\right ) D\left (w \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5018

ODE	\[ \boxed {\left (2 x -3\right ) y^{\prime \prime }-y^{\prime } x +y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{6} x^{2}+\frac {1}{27} x^{3}+\frac {5}{648} x^{4}+\frac {1}{540} x^{5}+\frac {11}{19440} x^{6}\right ) y \left (0\right )+y^{\prime }\left (0\right ) x +O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{6} x^{2}+\frac {1}{27} x^{3}+\frac {5}{648} x^{4}+\frac {1}{540} x^{5}\right ) c_{1} +c_{2} x +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{6} x^{2}+\frac {1}{27} x^{3}+\frac {5}{648} x^{4}+\frac {1}{540} x^{5}\right ) y \left (0\right )+D\left (y \right )\left (0\right ) x +O\left (x^{6}\right ) \]

Problem 5019

ODE	\[ \boxed {\left (x +1\right ) y^{\prime \prime }-3 y^{\prime } x +2 y=0} \] With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = \left (1-\frac {\left (x -1\right )^{2}}{2}-\frac {\left (x -1\right )^{3}}{6}-\frac {5 \left (x -1\right )^{4}}{48}-\frac {7 \left (x -1\right )^{5}}{240}-\frac {43 \left (x -1\right )^{6}}{2880}\right ) y \left (1\right )+\left (x -1+\frac {3 \left (x -1\right )^{2}}{4}+\frac {\left (x -1\right )^{3}}{3}+\frac {\left (x -1\right )^{4}}{6}+\frac {7 \left (x -1\right )^{5}}{120}+\frac {11 \left (x -1\right )^{6}}{480}\right ) y^{\prime }\left (1\right )+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {\left (x -1\right )^{2}}{2}-\frac {\left (x -1\right )^{3}}{6}-\frac {5 \left (x -1\right )^{4}}{48}-\frac {7 \left (x -1\right )^{5}}{240}\right ) y \left (1\right )+\left (x -1+\frac {3 \left (x -1\right )^{2}}{4}+\frac {\left (x -1\right )^{3}}{3}+\frac {\left (x -1\right )^{4}}{6}+\frac {7 \left (x -1\right )^{5}}{120}\right ) D\left (y \right )\left (1\right )+O\left (x^{6}\right ) \]

Problem 5020

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime } x -3 y=0} \] With the expansion point for the power series method at \(x = 2\).
program solution	\[ y = \left (1+\frac {3 \left (x -2\right )^{2}}{2}+\left (x -2\right )^{3}+\frac {9 \left (x -2\right )^{4}}{8}+\frac {3 \left (x -2\right )^{5}}{4}+\frac {41 \left (x -2\right )^{6}}{80}\right ) y \left (2\right )+\left (x -2+\left (x -2\right )^{2}+\frac {4 \left (x -2\right )^{3}}{3}+\frac {13 \left (x -2\right )^{4}}{12}+\frac {5 \left (x -2\right )^{5}}{6}+\frac {191 \left (x -2\right )^{6}}{360}\right ) y^{\prime }\left (2\right )+O\left (\left (x -2\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {3 \left (-2+x \right )^{2}}{2}+\left (-2+x \right )^{3}+\frac {9 \left (-2+x \right )^{4}}{8}+\frac {3 \left (-2+x \right )^{5}}{4}\right ) y \left (2\right )+\left (-2+x +\left (-2+x \right )^{2}+\frac {4 \left (-2+x \right )^{3}}{3}+\frac {13 \left (-2+x \right )^{4}}{12}+\frac {5 \left (-2+x \right )^{5}}{6}\right ) D\left (y \right )\left (2\right )+O\left (x^{6}\right ) \]

Problem 5021

ODE	\[ \boxed {\left (x^{2}+x +1\right ) y^{\prime \prime }-3 y=0} \] With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = \left (1+\frac {\left (x -1\right )^{2}}{2}-\frac {\left (x -1\right )^{3}}{6}+\frac {7 \left (x -1\right )^{4}}{72}-\frac {\left (x -1\right )^{5}}{20}+\frac {17 \left (x -1\right )^{6}}{720}\right ) y \left (1\right )+\left (x -1+\frac {\left (x -1\right )^{3}}{6}-\frac {\left (x -1\right )^{4}}{12}+\frac {\left (x -1\right )^{5}}{24}-\frac {7 \left (x -1\right )^{6}}{360}\right ) y^{\prime }\left (1\right )+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {\left (x -1\right )^{2}}{2}-\frac {\left (x -1\right )^{3}}{6}+\frac {7 \left (x -1\right )^{4}}{72}-\frac {\left (x -1\right )^{5}}{20}\right ) y \left (1\right )+\left (x -1+\frac {\left (x -1\right )^{3}}{6}-\frac {\left (x -1\right )^{4}}{12}+\frac {\left (x -1\right )^{5}}{24}\right ) D\left (y \right )\left (1\right )+O\left (x^{6}\right ) \]

Problem 5022

ODE	\[ \boxed {\left (x^{2}-5 x +6\right ) y^{\prime \prime }-3 y^{\prime } x -y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{12} x^{2}+\frac {5}{216} x^{3}+\frac {5}{324} x^{4}+\frac {11}{1296} x^{5}+\frac {7}{1458} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{9} x^{3}+\frac {5}{108} x^{4}+\frac {29}{1080} x^{5}+\frac {59}{3888} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{12} x^{2}+\frac {5}{216} x^{3}+\frac {5}{324} x^{4}+\frac {11}{1296} x^{5}\right ) c_{1} +\left (x +\frac {1}{9} x^{3}+\frac {5}{108} x^{4}+\frac {29}{1080} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{12} x^{2}+\frac {5}{216} x^{3}+\frac {5}{324} x^{4}+\frac {11}{1296} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{9} x^{3}+\frac {5}{108} x^{4}+\frac {29}{1080} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5023

ODE	\[ \boxed {y^{\prime \prime }-\tan \left (x \right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = \left (1-\frac {\left (x -1\right )^{2}}{2}-\frac {\left (x -1\right )^{3} \tan \left (1\right )}{6}-\frac {\left (x -1\right )^{4} \sec \left (1\right )^{2}}{8}+\frac {\left (x -1\right )^{4}}{12}-\frac {\left (x -1\right )^{5} \tan \left (1\right ) \sec \left (1\right )^{2}}{10}+\frac {\left (x -1\right )^{5} \tan \left (1\right )}{40}-\frac {\left (x -1\right )^{6} \sec \left (1\right )^{4}}{12}+\frac {3 \left (x -1\right )^{6} \sec \left (1\right )^{2}}{40}-\frac {\left (x -1\right )^{6}}{144}\right ) y \left (1\right )+\left (x -1+\frac {\left (x -1\right )^{2} \tan \left (1\right )}{2}+\frac {\left (x -1\right )^{3} \tan \left (1\right )^{2}}{3}+\frac {\left (x -1\right )^{4} \tan \left (1\right ) \sec \left (1\right )^{2}}{4}-\frac {\left (x -1\right )^{4} \tan \left (1\right )}{8}+\frac {\left (x -1\right )^{5} \sec \left (1\right )^{4}}{5}-\frac {9 \left (x -1\right )^{5} \sec \left (1\right )^{2}}{40}+\frac {\left (x -1\right )^{5}}{24}+\frac {\left (x -1\right )^{6} \tan \left (1\right )}{90}-\frac {31 \left (x -1\right )^{6} \tan \left (1\right ) \sec \left (1\right )^{2}}{240}+\frac {\left (x -1\right )^{6} \tan \left (1\right ) \sec \left (1\right )^{4}}{6}\right ) y^{\prime }\left (1\right )+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {\left (x -1\right )^{2}}{2}-\frac {\tan \left (1\right ) \left (x -1\right )^{3}}{6}+\left (\frac {1}{12}-\frac {\sec \left (1\right )^{2}}{8}\right ) \left (x -1\right )^{4}+\frac {\tan \left (1\right ) \left (1-4 \sec \left (1\right )^{2}\right ) \left (x -1\right )^{5}}{40}\right ) y \left (1\right )+\left (x -1+\frac {\tan \left (1\right ) \left (x -1\right )^{2}}{2}+\frac {\tan \left (1\right )^{2} \left (x -1\right )^{3}}{3}+\frac {\tan \left (1\right ) \left (2 \sec \left (1\right )^{2}-1\right ) \left (x -1\right )^{4}}{8}+\frac {\left (5-27 \sec \left (1\right )^{2}+24 \sec \left (1\right )^{4}\right ) \left (x -1\right )^{5}}{120}\right ) D\left (y \right )\left (1\right )+O\left (x^{6}\right ) \]

Problem 5024

ODE	\[ \boxed {\left (x^{3}+1\right ) y^{\prime \prime }-y^{\prime } x +2 y x^{2}=0} \] With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = \left (1-\frac {\left (x -1\right )^{2}}{2}-\frac {\left (x -1\right )^{3}}{6}+\frac {7 \left (x -1\right )^{4}}{48}+\frac {7 \left (x -1\right )^{5}}{240}-\frac {187 \left (x -1\right )^{6}}{2880}\right ) y \left (1\right )+\left (x -1+\frac {\left (x -1\right )^{2}}{4}-\frac {\left (x -1\right )^{3}}{6}-\frac {\left (x -1\right )^{4}}{8}+\frac {\left (x -1\right )^{5}}{12}+\frac {\left (x -1\right )^{6}}{72}\right ) y^{\prime }\left (1\right )+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {\left (x -1\right )^{2}}{2}-\frac {\left (x -1\right )^{3}}{6}+\frac {7 \left (x -1\right )^{4}}{48}+\frac {7 \left (x -1\right )^{5}}{240}\right ) y \left (1\right )+\left (x -1+\frac {\left (x -1\right )^{2}}{4}-\frac {\left (x -1\right )^{3}}{6}-\frac {\left (x -1\right )^{4}}{8}+\frac {\left (x -1\right )^{5}}{12}\right ) D\left (y \right )\left (1\right )+O\left (x^{6}\right ) \]

Problem 5025

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

Problem 5026

ODE	\[ \boxed {y^{\prime }-2 y x=0} \] With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = \left (-1+2 x +3 \left (x -1\right )^{2}+\frac {10 \left (x -1\right )^{3}}{3}+\frac {19 \left (x -1\right )^{4}}{6}+\frac {13 \left (x -1\right )^{5}}{5}\right ) y \left (1\right )+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (-1+2 x +3 \left (x -1\right )^{2}+\frac {10 \left (x -1\right )^{3}}{3}+\frac {19 \left (x -1\right )^{4}}{6}+\frac {13 \left (x -1\right )^{5}}{5}\right ) y \left (1\right )+O\left (x^{6}\right ) \]

Problem 5027

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

Problem 5028

ODE	\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x +2 y=0} \] With the expansion point for the power series method at \(x = 2\).
program solution	\[ y = \left (1-\frac {\left (x -2\right )^{2}}{4}+\frac {\left (x -2\right )^{3}}{24}-\frac {\left (x -2\right )^{4}}{192}+\frac {\left (x -2\right )^{6}}{2304}\right ) y \left (2\right )+\left (x -2+\frac {\left (x -2\right )^{2}}{4}-\frac {\left (x -2\right )^{3}}{12}+\frac {\left (x -2\right )^{4}}{48}-\frac {\left (x -2\right )^{5}}{192}+\frac {\left (x -2\right )^{6}}{768}\right ) y^{\prime }\left (2\right )+O\left (\left (x -2\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {\left (-2+x \right )^{2}}{4}+\frac {\left (-2+x \right )^{3}}{24}-\frac {\left (-2+x \right )^{4}}{192}\right ) y \left (2\right )+\left (-2+x +\frac {\left (-2+x \right )^{2}}{4}-\frac {\left (-2+x \right )^{3}}{12}+\frac {\left (-2+x \right )^{4}}{48}-\frac {\left (-2+x \right )^{5}}{192}\right ) D\left (y \right )\left (2\right )+O\left (x^{6}\right ) \]

Problem 5029

ODE	\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 2\).
program solution	\[ y = \left (1-\frac {\left (x -2\right )^{2}}{8}+\frac {\left (x -2\right )^{3}}{32}-\frac {3 \left (x -2\right )^{4}}{512}+\frac {\left (x -2\right )^{5}}{2048}+\frac {27 \left (x -2\right )^{6}}{81920}\right ) y \left (2\right )+\left (x -2+\frac {\left (x -2\right )^{2}}{8}-\frac {7 \left (x -2\right )^{3}}{96}+\frac {37 \left (x -2\right )^{4}}{1536}-\frac {211 \left (x -2\right )^{5}}{30720}+\frac {1241 \left (x -2\right )^{6}}{737280}\right ) y^{\prime }\left (2\right )+O\left (\left (x -2\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {\left (-2+x \right )^{2}}{8}+\frac {\left (-2+x \right )^{3}}{32}-\frac {3 \left (-2+x \right )^{4}}{512}+\frac {\left (-2+x \right )^{5}}{2048}\right ) y \left (2\right )+\left (-2+x +\frac {\left (-2+x \right )^{2}}{8}-\frac {7 \left (-2+x \right )^{3}}{96}+\frac {37 \left (-2+x \right )^{4}}{1536}-\frac {211 \left (-2+x \right )^{5}}{30720}\right ) D\left (y \right )\left (2\right )+O\left (x^{6}\right ) \]

Problem 5030

ODE	\[ \boxed {y^{\prime \prime }+\left (3 x -1\right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = -1\).
program solution	\[ y = \left (1+\frac {\left (x +1\right )^{2}}{2}+\frac {2 \left (x +1\right )^{3}}{3}+\frac {11 \left (x +1\right )^{4}}{24}+\frac {\left (x +1\right )^{5}}{10}-\frac {73 \left (x +1\right )^{6}}{720}\right ) y \left (-1\right )+\left (x +1+2 \left (x +1\right )^{2}+\frac {7 \left (x +1\right )^{3}}{3}+\frac {3 \left (x +1\right )^{4}}{2}+\frac {4 \left (x +1\right )^{5}}{15}-\frac {67 \left (x +1\right )^{6}}{180}\right ) y^{\prime }\left (-1\right )+O\left (\left (x +1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {\left (x +1\right )^{2}}{2}+\frac {2 \left (x +1\right )^{3}}{3}+\frac {11 \left (x +1\right )^{4}}{24}+\frac {\left (x +1\right )^{5}}{10}\right ) y \left (-1\right )+\left (x +1+2 \left (x +1\right )^{2}+\frac {7 \left (x +1\right )^{3}}{3}+\frac {3 \left (x +1\right )^{4}}{2}+\frac {4 \left (x +1\right )^{5}}{15}\right ) D\left (y \right )\left (-1\right )+O\left (x^{6}\right ) \]

Problem 5031

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

Problem 5032

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

Problem 5033

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }-y^{\prime } {\mathrm e}^{x}+y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 1] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = x +1+\frac {x^{4}}{24}+\frac {x^{5}}{60}-\frac {x^{6}}{120}+O\left (x^{6}\right ) \] Verified OK. \[ y = 1+\frac {x^{5}}{60}+x +\frac {x^{4}}{24}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1+x +\frac {1}{24} x^{4}+\frac {1}{60} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 5034

ODE	\[ \boxed {y^{\prime \prime }+t y^{\prime }+{\mathrm e}^{t} y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = -1] \end {align*} With the expansion point for the power series method at \(t = 0\).
program solution	\[ y = -t +1-\frac {t^{2}}{2}+\frac {t^{3}}{6}+\frac {t^{4}}{6}+\frac {t^{5}}{120}-\frac {t^{6}}{48}+O\left (t^{6}\right ) \] Verified OK. \[ y = 1-\frac {t^{2}}{2}+\frac {t^{3}}{6}+\frac {t^{4}}{6}+\frac {t^{5}}{120}-t +O\left (t^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = 1-t -\frac {1}{2} t^{2}+\frac {1}{6} t^{3}+\frac {1}{6} t^{4}+\frac {1}{120} t^{5}+\operatorname {O}\left (t^{6}\right ) \]

Problem 5035

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime } {\mathrm e}^{2 x}+y \cos \left (x \right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = 1] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = x^{2}+x -1+\frac {x^{3}}{2}+\frac {x^{4}}{2}+\frac {31 x^{5}}{60}+\frac {311 x^{6}}{720}+O\left (x^{6}\right ) \] Verified OK. \[ y = -1+x^{2}+\frac {x^{3}}{2}+\frac {x^{4}}{2}+\frac {31 x^{5}}{60}+x +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -1+x +x^{2}+\frac {1}{2} x^{3}+\frac {1}{2} x^{4}+\frac {31}{60} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 5036

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

Problem 5037

ODE	\[ \boxed {w^{\prime }+w x={\mathrm e}^{x}} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ w = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}\right ) w \left (0\right )+x +\frac {x^{2}}{2}-\frac {x^{3}}{6}-\frac {x^{4}}{12}+\frac {x^{5}}{24}+O\left (x^{6}\right ) \] Verified OK. \[ w = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}\right ) c_{1} +x +\frac {x^{2}}{2}-\frac {x^{3}}{6}-\frac {x^{4}}{12}+\frac {x^{5}}{24}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ w \left (x \right ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}\right ) w \left (0\right )+x +\frac {x^{2}}{2}-\frac {x^{3}}{6}-\frac {x^{4}}{12}+\frac {x^{5}}{24}+O\left (x^{6}\right ) \]

Problem 5038

ODE	\[ \boxed {z^{\prime \prime }+x z^{\prime }+z=x^{2}+2 x +1} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ z = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}-\frac {1}{48} x^{6}\right ) z \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{15} x^{5}\right ) z^{\prime }\left (0\right )+\frac {x^{2}}{2}+\frac {x^{3}}{3}-\frac {x^{4}}{24}-\frac {x^{5}}{15}+\frac {x^{6}}{144}+O\left (x^{6}\right ) \] Verified OK. \[ z = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}\right ) c_{1} +\left (x -\frac {1}{3} x^{3}+\frac {1}{15} x^{5}\right ) c_{2} +\frac {x^{2}}{2}+\frac {x^{3}}{3}-\frac {x^{4}}{24}-\frac {x^{5}}{15}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ z \left (x \right ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}\right ) z \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{15} x^{5}\right ) D\left (z \right )\left (0\right )+\frac {x^{2}}{2}+\frac {x^{3}}{3}-\frac {x^{4}}{24}-\frac {x^{5}}{15}+O\left (x^{6}\right ) \]

Problem 5039

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime } x +3 y=x^{2}} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {3}{2} x^{2}-\frac {1}{8} x^{4}-\frac {1}{48} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}-\frac {1}{40} x^{5}\right ) y^{\prime }\left (0\right )+\frac {x^{4}}{12}+\frac {x^{6}}{72}+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {3}{2} x^{2}-\frac {1}{8} x^{4}\right ) c_{1} +\left (x -\frac {1}{6} x^{3}-\frac {1}{40} x^{5}\right ) c_{2} +\frac {x^{4}}{12}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {3}{2} x^{2}-\frac {1}{8} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}-\frac {1}{40} x^{5}\right ) D\left (y \right )\left (0\right )+\frac {x^{4}}{12}+O\left (x^{6}\right ) \]

Problem 5040

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x +y=\cos \left (x \right )} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}-\frac {1}{80} x^{6}\right ) y \left (0\right )+y^{\prime }\left (0\right ) x +\frac {x^{2}}{2}-\frac {x^{4}}{12}+\frac {19 x^{6}}{720}+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (\frac {1}{24} x^{4}-\frac {1}{2} x^{2}+1\right ) c_{1} +c_{2} x +\frac {x^{2}}{2}-\frac {x^{4}}{12}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}\right ) y \left (0\right )+D\left (y \right )\left (0\right ) x +\frac {x^{2}}{2}-\frac {x^{4}}{12}+O\left (x^{6}\right ) \]

Problem 5041

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

Problem 5042

ODE	\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime }+y=\tan \left (x \right )} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{6} x^{3}-\frac {1}{12} x^{4}-\frac {7}{120} x^{5}-\frac {29}{720} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{60} x^{6}\right ) y^{\prime }\left (0\right )+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{15}+\frac {19 x^{6}}{720}+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{6} x^{3}-\frac {1}{12} x^{4}-\frac {7}{120} x^{5}\right ) c_{1} +\left (x +\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}\right ) c_{2} +\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{15}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}-\frac {1}{6} x^{3}-\frac {1}{12} x^{4}-\frac {7}{120} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}\right ) D\left (y \right )\left (0\right )+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{15}+O\left (x^{6}\right ) \]

Problem 5043

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

Problem 5044

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

Problem 5045

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

Problem 5046

ODE	\[ \boxed {x^{\prime \prime \prime }-x^{\prime \prime }+x^{\prime }-x=0} \]
program solution	\[ x = c_{1} {\mathrm e}^{t}+{\mathrm e}^{i t} c_{2} +{\mathrm e}^{-i t} c_{3} \] Verified OK.
Maple solution	\[ x \left (t \right ) = c_{1} {\mathrm e}^{t}+c_{2} \sin \left (t \right )+c_{3} \cos \left (t \right ) \]

Problem 5047

ODE	\[ \boxed {x^{\prime \prime }+42 x^{\prime }+x=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 1, x^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ x = \frac {\left (220+21 \sqrt {110}\right ) {\mathrm e}^{\left (-21+2 \sqrt {110}\right ) t}}{440}+\frac {\left (220-21 \sqrt {110}\right ) {\mathrm e}^{\left (-21-2 \sqrt {110}\right ) t}}{440} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {\left (220+21 \sqrt {110}\right ) {\mathrm e}^{\left (-21+2 \sqrt {110}\right ) t}}{440}+\frac {\left (220-21 \sqrt {110}\right ) {\mathrm e}^{\left (-21-2 \sqrt {110}\right ) t}}{440} \]

Problem 5048

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

Problem 5049

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

Problem 5050

ODE	\[ \boxed {x^{\prime \prime }+2 \gamma x^{\prime }+\omega _{0} x=F \cos \left (\omega t \right )} \]
program solution	\[ x = c_{1} {\mathrm e}^{\left (-\gamma +\sqrt {\gamma ^{2}-\omega _{0}}\right ) t}-\frac {c_{2} {\mathrm e}^{-\left (\gamma +\sqrt {\gamma ^{2}-\omega _{0}}\right ) t}}{2 \sqrt {\gamma ^{2}-\omega _{0}}}-\frac {\left (\omega ^{2}-\omega _{0} \right ) F \cos \left (\omega t \right )}{\omega ^{4}+\left (4 \gamma ^{2}-2 \omega _{0} \right ) \omega ^{2}+\omega _{0}^{2}}+\frac {2 F \gamma \omega \sin \left (\omega t \right )}{\omega ^{4}+\left (4 \gamma ^{2}-2 \omega _{0} \right ) \omega ^{2}+\omega _{0}^{2}} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {-F \left (\omega ^{2}-\omega _{0} \right ) \cos \left (\omega t \right )+2 F \sin \left (\omega t \right ) \gamma \omega +4 \left (\frac {\omega ^{4}}{4}+\left (\gamma ^{2}-\frac {\omega _{0}}{2}\right ) \omega ^{2}+\frac {\omega _{0}^{2}}{4}\right ) \left ({\mathrm e}^{-\left (\gamma +\sqrt {\gamma ^{2}-\omega _{0}}\right ) t} c_{1} +{\mathrm e}^{\left (-\gamma +\sqrt {\gamma ^{2}-\omega _{0}}\right ) t} c_{2} \right )}{\omega ^{4}+\left (4 \gamma ^{2}-2 \omega _{0} \right ) \omega ^{2}+\omega _{0}^{2}} \]

Problem 5051

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

Problem 5052

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

Problem 5053

ODE	\[ \boxed {y^{\prime \prime }+16 y=16 \cos \left (4 x \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = 2 x \sin \left (4 x \right )+\cos \left (4 x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \cos \left (4 x \right )+2 \sin \left (4 x \right ) x \]

Problem 5054

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

Problem 5055

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

Problem 5056

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

Problem 5057

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

Problem 5058

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

Problem 5059

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

Problem 5060

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

Problem 5061

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

Problem 5062

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

Problem 5063

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

Problem 5064

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

Problem 5065

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

Problem 5066

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x -9 y=0} \]
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 5067

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

Problem 5068

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

Problem 5069

ODE	\[ \boxed {2 x y^{\prime \prime }-y^{\prime }+2 y=0} \]
program solution	\[ y = \frac {c_{1} \left (2 \sqrt {-x}-1\right ) {\mathrm e}^{2 \sqrt {-x}} x^{\frac {1}{4}}}{\left (-x \right )^{\frac {1}{4}}}+\frac {c_{2} x^{\frac {1}{4}} \left (2 \sqrt {-x}+1\right ) {\mathrm e}^{-2 \sqrt {-x}}}{8 \left (-x \right )^{\frac {1}{4}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (2 \sqrt {x}\, c_{1} +c_{2} \right ) \cos \left (2 \sqrt {x}\right )-\sin \left (2 \sqrt {x}\right ) \left (-2 \sqrt {x}\, c_{2} +c_{1} \right ) \]

Problem 5070

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

Problem 5071

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

Problem 5072

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

Problem 5073

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

Problem 5074

ODE	\[ \boxed {x y^{\prime \prime }+\left (1-x \right ) y^{\prime }+m y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {KummerM}\left (-m , 1, x\right )+c_{2} \operatorname {KummerU}\left (-m , 1, x\right ) \]

Problem 5075

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

Problem 5076

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

Problem 5077

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

Problem 5078

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

Problem 5079

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

Problem 5080

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

Problem 5081

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

Problem 5082

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

Problem 5083

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

Problem 5084

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

Problem 5085

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

Problem 5086

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

Problem 5087

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

Problem 5088

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

Problem 5089

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

Problem 5090

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

Problem 5091

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

Problem 5092

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

Problem 5093

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

Problem 5094

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

Problem 5095

ODE	\[ \boxed {y+\left (4 y+3 x \right ) y^{\prime }=3 x} \]
program solution	\[ y = \frac {x \operatorname {RootOf}\left (x^{8} \textit {\_Z}^{8}+4 \textit {\_Z}^{7} x^{8}-8 x^{8} \textit {\_Z}^{6}-28 \textit {\_Z}^{5} x^{8}+50 x^{8} \textit {\_Z}^{4}+44 \textit {\_Z}^{3} x^{8}-144 x^{8} \textit {\_Z}^{2}-256 c_{3} {\mathrm e}^{8 c_{2}}+108 \textit {\_Z} \,x^{8}-27 x^{8}\right )}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {-3 x^{8} c_{1} \operatorname {RootOf}\left (\textit {\_Z}^{64} c_{1} x^{8}+12 \textit {\_Z}^{56} c_{1} x^{8}+48 \textit {\_Z}^{48} c_{1} x^{8}+64 \textit {\_Z}^{40} c_{1} x^{8}-1\right )^{56}-24 x^{8} c_{1} \operatorname {RootOf}\left (\textit {\_Z}^{64} c_{1} x^{8}+12 \textit {\_Z}^{56} c_{1} x^{8}+48 \textit {\_Z}^{48} c_{1} x^{8}+64 \textit {\_Z}^{40} c_{1} x^{8}-1\right )^{48}-48 x^{8} c_{1} \operatorname {RootOf}\left (\textit {\_Z}^{64} c_{1} x^{8}+12 \textit {\_Z}^{56} c_{1} x^{8}+48 \textit {\_Z}^{48} c_{1} x^{8}+64 \textit {\_Z}^{40} c_{1} x^{8}-1\right )^{40}+1}{2 c_{1} x^{7} \operatorname {RootOf}\left (\textit {\_Z}^{64} c_{1} x^{8}+12 \textit {\_Z}^{56} c_{1} x^{8}+48 \textit {\_Z}^{48} c_{1} x^{8}+64 \textit {\_Z}^{40} c_{1} x^{8}-1\right )^{40} \left (\operatorname {RootOf}\left (\textit {\_Z}^{64} c_{1} x^{8}+12 \textit {\_Z}^{56} c_{1} x^{8}+48 \textit {\_Z}^{48} c_{1} x^{8}+64 \textit {\_Z}^{40} c_{1} x^{8}-1\right )^{16}+8 \operatorname {RootOf}\left (\textit {\_Z}^{64} c_{1} x^{8}+12 \textit {\_Z}^{56} c_{1} x^{8}+48 \textit {\_Z}^{48} c_{1} x^{8}+64 \textit {\_Z}^{40} c_{1} x^{8}-1\right )^{8}+16\right )} \]

Problem 5096

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

Problem 5097

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

Problem 5098

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

Problem 5099

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

Problem 5100

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