2.17.56 Problems 5501 to 5600

Problem 5501

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

Problem 5502

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

Problem 5503

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

Problem 5504

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

Problem 5505

ODE	\[ \boxed {y^{\prime \prime }-y x=0} \] With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = \left (1+\frac {\left (x -1\right )^{2}}{2}+\frac {\left (x -1\right )^{3}}{6}+\frac {\left (x -1\right )^{4}}{24}+\frac {\left (x -1\right )^{5}}{30}+\frac {\left (x -1\right )^{6}}{144}\right ) y \left (1\right )+\left (x -1+\frac {\left (x -1\right )^{3}}{6}+\frac {\left (x -1\right )^{4}}{12}+\frac {\left (x -1\right )^{5}}{120}+\frac {\left (x -1\right )^{6}}{120}\right ) y^{\prime }\left (1\right )+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {\left (x -1\right )^{2}}{2}+\frac {\left (x -1\right )^{3}}{6}+\frac {\left (x -1\right )^{4}}{24}+\frac {\left (x -1\right )^{5}}{30}\right ) y \left (1\right )+\left (x -1+\frac {\left (x -1\right )^{3}}{6}+\frac {\left (x -1\right )^{4}}{12}+\frac {\left (x -1\right )^{5}}{120}\right ) D\left (y \right )\left (1\right )+O\left (x^{6}\right ) \]

Problem 5506

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 5507

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

Problem 5508

ODE	\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x +p^{2} y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2} p^{2}+\frac {1}{24} p^{4} x^{4}-\frac {1}{6} p^{2} x^{4}-\frac {1}{720} x^{6} p^{6}+\frac {1}{36} x^{6} p^{4}-\frac {4}{45} x^{6} p^{2}\right ) y \left (0\right )+\left (x -\frac {1}{6} p^{2} x^{3}+\frac {1}{6} x^{3}+\frac {1}{120} p^{4} x^{5}-\frac {1}{12} p^{2} x^{5}+\frac {3}{40} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {x^{2} p^{2}}{2}+\left (\frac {1}{24} p^{4}-\frac {1}{6} p^{2}\right ) x^{4}\right ) c_{1} +\left (x +\left (-\frac {p^{2}}{6}+\frac {1}{6}\right ) x^{3}+\left (\frac {1}{120} p^{4}-\frac {1}{12} p^{2}+\frac {3}{40}\right ) x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {p^{2} x^{2}}{2}+\frac {p^{2} \left (p^{2}-4\right ) x^{4}}{24}\right ) y \left (0\right )+\left (x -\frac {\left (p^{2}-1\right ) x^{3}}{6}+\frac {\left (p^{4}-10 p^{2}+9\right ) x^{5}}{120}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5509

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

Problem 5510

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 5511

ODE	\[ \boxed {x y^{\prime \prime }+y=0} \] With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = \left (1-\frac {\left (x -1\right )^{2}}{2}+\frac {\left (x -1\right )^{3}}{6}-\frac {\left (x -1\right )^{4}}{24}+\frac {\left (x -1\right )^{5}}{60}-\frac {7 \left (x -1\right )^{6}}{720}\right ) y \left (1\right )+\left (x -1-\frac {\left (x -1\right )^{3}}{6}+\frac {\left (x -1\right )^{4}}{12}-\frac {\left (x -1\right )^{5}}{24}+\frac {\left (x -1\right )^{6}}{40}\right ) y^{\prime }\left (1\right )+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {\left (x -1\right )^{2}}{2}+\frac {\left (x -1\right )^{3}}{6}-\frac {\left (x -1\right )^{4}}{24}+\frac {\left (x -1\right )^{5}}{60}\right ) y \left (1\right )+\left (x -1-\frac {\left (x -1\right )^{3}}{6}+\frac {\left (x -1\right )^{4}}{12}-\frac {\left (x -1\right )^{5}}{24}\right ) D\left (y \right )\left (1\right )+O\left (x^{6}\right ) \]

Problem 5512

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

Problem 5513

ODE	\[ \boxed {y^{\prime \prime }-y x=\frac {1}{1-x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{12}+\frac {3 x^{5}}{40}+\frac {7 x^{6}}{180}+O\left (x^{6}\right ) \] Verified OK. \[ y = \frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{12}+\frac {3 x^{5}}{40}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {3}{40} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 5514

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

Problem 5515

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +y \left (1+x \right )=0} \] 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 5516

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

Problem 5517

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

Problem 5518

ODE	\[ \boxed {2 x y^{\prime \prime }+y^{\prime }-y x^{2}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1+\frac {x^{3}}{21}+O\left (x^{6}\right )\right )+c_{2} \left (1+\frac {x^{3}}{15}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{21} x^{3}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+\frac {1}{15} x^{3}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 5519

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

Problem 5520

ODE	\[ \boxed {x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {x}{2}+\frac {x^{2}}{12}+\frac {11 x^{3}}{144}-\frac {83 x^{4}}{2880}-\frac {2557 x^{5}}{86400}+O\left (x^{6}\right )\right )+c_{2} \left (-x \left (1-\frac {x}{2}+\frac {x^{2}}{12}+\frac {11 x^{3}}{144}-\frac {83 x^{4}}{2880}-\frac {2557 x^{5}}{86400}+O\left (x^{6}\right )\right ) \ln \left (x \right )+1-\frac {3 x^{2}}{4}+\frac {13 x^{3}}{36}+\frac {25 x^{4}}{1728}-\frac {8743 x^{5}}{86400}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1-\frac {1}{2} x +\frac {1}{12} x^{2}+\frac {11}{144} x^{3}-\frac {83}{2880} x^{4}-\frac {2557}{86400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-x +\frac {1}{2} x^{2}-\frac {1}{12} x^{3}-\frac {11}{144} x^{4}+\frac {83}{2880} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-\frac {3}{4} x^{2}+\frac {13}{36} x^{3}+\frac {25}{1728} x^{4}-\frac {8743}{86400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

Problem 5521

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

Problem 5522

ODE	\[ \boxed {x y^{\prime \prime }+x^{3} y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {x}{2}+\frac {x^{2}}{12}-\frac {13 x^{3}}{144}+\frac {157 x^{4}}{2880}-\frac {877 x^{5}}{86400}+O\left (x^{6}\right )\right )+c_{2} \left (-x \left (1-\frac {x}{2}+\frac {x^{2}}{12}-\frac {13 x^{3}}{144}+\frac {157 x^{4}}{2880}-\frac {877 x^{5}}{86400}+O\left (x^{6}\right )\right ) \ln \left (x \right )+1-\frac {3 x^{2}}{4}+\frac {7 x^{3}}{36}+\frac {25 x^{4}}{1728}+\frac {6377 x^{5}}{86400}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1-\frac {1}{2} x +\frac {1}{12} x^{2}-\frac {13}{144} x^{3}+\frac {157}{2880} x^{4}-\frac {877}{86400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-x +\frac {1}{2} x^{2}-\frac {1}{12} x^{3}+\frac {13}{144} x^{4}-\frac {157}{2880} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-\frac {3}{4} x^{2}+\frac {7}{36} x^{3}+\frac {25}{1728} x^{4}+\frac {6377}{86400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

Problem 5523

ODE	\[ \boxed {x y^{\prime \prime }+y^{\prime } x -{\mathrm e}^{x} y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1+\frac {x^{2}}{6}+\frac {x^{3}}{72}+\frac {7 x^{4}}{480}+\frac {29 x^{5}}{10800}+\frac {191 x^{6}}{181440}+O\left (x^{6}\right )\right )+c_{2} \left (x \left (1+\frac {x^{2}}{6}+\frac {x^{3}}{72}+\frac {7 x^{4}}{480}+\frac {29 x^{5}}{10800}+\frac {191 x^{6}}{181440}+O\left (x^{6}\right )\right ) \ln \left (x \right )+1-\frac {x^{3}}{18}+\frac {x^{4}}{864}-\frac {13 x^{5}}{1600}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1+\frac {1}{6} x^{2}+\frac {1}{72} x^{3}+\frac {7}{480} x^{4}+\frac {29}{10800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (x +\frac {1}{6} x^{3}+\frac {1}{72} x^{4}+\frac {7}{480} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-x -\frac {2}{9} x^{3}-\frac {11}{864} x^{4}-\frac {109}{4800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

Problem 5524

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

Problem 5525

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

Problem 5526

ODE	\[ \boxed {y^{\prime \prime } x^{3}+y \left (1+x \right )=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 5527

ODE	\[ \boxed {x y^{\prime \prime }+y^{\prime } x^{5}+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {x}{2}+\frac {x^{2}}{12}-\frac {x^{3}}{144}+\frac {x^{4}}{2880}-\frac {2881 x^{5}}{86400}+O\left (x^{6}\right )\right )+c_{2} \left (-x \left (1-\frac {x}{2}+\frac {x^{2}}{12}-\frac {x^{3}}{144}+\frac {x^{4}}{2880}-\frac {2881 x^{5}}{86400}+O\left (x^{6}\right )\right ) \ln \left (x \right )+1-\frac {3 x^{2}}{4}+\frac {7 x^{3}}{36}-\frac {35 x^{4}}{1728}+\frac {101 x^{5}}{86400}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1-\frac {1}{2} x +\frac {1}{12} x^{2}-\frac {1}{144} x^{3}+\frac {1}{2880} x^{4}-\frac {2881}{86400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-x +\frac {1}{2} x^{2}-\frac {1}{12} x^{3}+\frac {1}{144} x^{4}-\frac {1}{2880} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-\frac {3}{4} x^{2}+\frac {7}{36} x^{3}-\frac {35}{1728} x^{4}+\frac {101}{86400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

Problem 5528

ODE	\[ \boxed {\sin \left (x \right ) y^{\prime \prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1+\frac {x}{2}+\frac {x^{2}}{12}+\frac {x^{3}}{48}+\frac {x^{4}}{192}+\frac {37 x^{5}}{28800}+O\left (x^{6}\right )\right )+c_{2} \left (x \left (1+\frac {x}{2}+\frac {x^{2}}{12}+\frac {x^{3}}{48}+\frac {x^{4}}{192}+\frac {37 x^{5}}{28800}+O\left (x^{6}\right )\right ) \ln \left (x \right )+1-\frac {3 x^{2}}{4}-\frac {x^{3}}{6}-\frac {5 x^{4}}{192}-\frac {257 x^{5}}{28800}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1+\frac {1}{2} x +\frac {1}{12} x^{2}+\frac {1}{48} x^{3}+\frac {1}{192} x^{4}+\frac {37}{28800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (x +\frac {1}{2} x^{2}+\frac {1}{12} x^{3}+\frac {1}{48} x^{4}+\frac {1}{192} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-\frac {3}{4} x^{2}-\frac {1}{6} x^{3}-\frac {5}{192} x^{4}-\frac {257}{28800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

Problem 5529

ODE	\[ \boxed {\cos \left (x \right ) y^{\prime \prime }-\sin \left (x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{6} x^{3}+\frac {1}{60} x^{5}+\frac {1}{180} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{12} x^{4}+\frac {1}{90} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{6} x^{3}+\frac {1}{60} x^{5}\right ) c_{1} +\left (x +\frac {1}{12} x^{4}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{6} x^{3}+\frac {1}{60} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{12} x^{4}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5530

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

Problem 5531

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\left (x -\frac {3}{4}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {3}{2}} \left (1-\frac {x}{3}+\frac {x^{2}}{24}-\frac {x^{3}}{360}+\frac {x^{4}}{8640}-\frac {x^{5}}{302400}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {x^{\frac {3}{2}} \left (1-\frac {x}{3}+\frac {x^{2}}{24}-\frac {x^{3}}{360}+\frac {x^{4}}{8640}-\frac {x^{5}}{302400}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{2}+\frac {1+x -\frac {2 x^{3}}{9}+\frac {25 x^{4}}{576}-\frac {157 x^{5}}{43200}+O\left (x^{6}\right )}{\sqrt {x}}\right ) \] 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 5532

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

Problem 5533

ODE	\[ \boxed {\left (x^{2}-25\right ) y^{\prime \prime }+2 y^{\prime } x +y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{50} x^{2}+\frac {7}{15000} x^{4}+\frac {49}{3750000} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{50} x^{3}+\frac {13}{25000} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{50} x^{2}+\frac {7}{15000} x^{4}\right ) c_{1} +\left (x +\frac {1}{50} x^{3}+\frac {13}{25000} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{50} x^{2}+\frac {7}{15000} x^{4}\right ) y \left (0\right )+\left (x +\frac {1}{50} x^{3}+\frac {13}{25000} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5534

ODE	\[ \boxed {\left (x^{2}-25\right ) y^{\prime \prime }+2 y^{\prime } x +y=0} \] With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = \left (1+\frac {\left (x -1\right )^{2}}{48}+\frac {\left (x -1\right )^{3}}{864}+\frac {\left (x -1\right )^{4}}{1728}+\frac {29 \left (x -1\right )^{5}}{414720}+\frac {649 \left (x -1\right )^{6}}{29859840}\right ) y \left (1\right )+\left (x -1+\frac {\left (x -1\right )^{2}}{24}+\frac {5 \left (x -1\right )^{3}}{216}+\frac {17 \left (x -1\right )^{4}}{6912}+\frac {41 \left (x -1\right )^{5}}{51840}+\frac {1891 \left (x -1\right )^{6}}{14929920}\right ) y^{\prime }\left (1\right )+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {\left (x -1\right )^{2}}{48}+\frac {\left (x -1\right )^{3}}{864}+\frac {\left (x -1\right )^{4}}{1728}+\frac {29 \left (x -1\right )^{5}}{414720}\right ) y \left (1\right )+\left (x -1+\frac {\left (x -1\right )^{2}}{24}+\frac {5 \left (x -1\right )^{3}}{216}+\frac {17 \left (x -1\right )^{4}}{6912}+\frac {41 \left (x -1\right )^{5}}{51840}\right ) D\left (y \right )\left (1\right )+O\left (x^{6}\right ) \]

Problem 5535

ODE	\[ \boxed {\left (x^{2}-2 x +10\right ) y^{\prime \prime }+y^{\prime } x -4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{5} x^{2}+\frac {1}{75} x^{3}+\frac {1}{750} x^{4}-\frac {13}{75000} x^{5}-\frac {43}{562500} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{20} x^{3}+\frac {1}{200} x^{4}-\frac {13}{20000} x^{5}-\frac {43}{150000} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{5} x^{2}+\frac {1}{75} x^{3}+\frac {1}{750} x^{4}-\frac {13}{75000} x^{5}\right ) c_{1} +\left (x +\frac {1}{20} x^{3}+\frac {1}{200} x^{4}-\frac {13}{20000} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{5} x^{2}+\frac {1}{75} x^{3}+\frac {1}{750} x^{4}-\frac {13}{75000} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{20} x^{3}+\frac {1}{200} x^{4}-\frac {13}{20000} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5536

ODE	\[ \boxed {\left (x^{2}-2 x +10\right ) y^{\prime \prime }+y^{\prime } x -4 y=0} \] With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = \left (1+\frac {2 \left (x -1\right )^{2}}{9}-\frac {2 \left (x -1\right )^{3}}{243}+\frac {\left (x -1\right )^{4}}{4374}+\frac {22 \left (x -1\right )^{5}}{98415}-\frac {38 \left (x -1\right )^{6}}{2657205}\right ) y \left (1\right )+\left (x -1-\frac {\left (x -1\right )^{2}}{18}+\frac {14 \left (x -1\right )^{3}}{243}-\frac {7 \left (x -1\right )^{4}}{4374}-\frac {154 \left (x -1\right )^{5}}{98415}+\frac {266 \left (x -1\right )^{6}}{2657205}\right ) y^{\prime }\left (1\right )+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {2 \left (x -1\right )^{2}}{9}-\frac {2 \left (x -1\right )^{3}}{243}+\frac {\left (x -1\right )^{4}}{4374}+\frac {22 \left (x -1\right )^{5}}{98415}\right ) y \left (1\right )+\left (x -1-\frac {\left (x -1\right )^{2}}{18}+\frac {14 \left (x -1\right )^{3}}{243}-\frac {7 \left (x -1\right )^{4}}{4374}-\frac {154 \left (x -1\right )^{5}}{98415}\right ) D\left (y \right )\left (1\right )+O\left (x^{6}\right ) \]

Problem 5537

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 5538

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 5539

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime } x +y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{8} x^{4}-\frac {7}{240} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{24} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{8} x^{4}\right ) c_{1} +\left (x +\frac {1}{6} x^{3}+\frac {1}{24} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}-\frac {1}{8} x^{4}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{24} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5540

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

Problem 5541

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

Problem 5542

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

Problem 5543

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

Problem 5544

ODE	\[ \boxed {\left (x +2\right ) y^{\prime \prime }+y^{\prime } x -y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{4} x^{2}-\frac {1}{24} x^{3}+\frac {1}{480} x^{5}-\frac {1}{1440} x^{6}\right ) y \left (0\right )+y^{\prime }\left (0\right ) x +O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{4} x^{2}-\frac {1}{24} x^{3}+\frac {1}{480} x^{5}\right ) c_{1} +c_{2} x +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{4} x^{2}-\frac {1}{24} x^{3}+\frac {1}{480} x^{5}\right ) y \left (0\right )+D\left (y \right )\left (0\right ) x +O\left (x^{6}\right ) \]

Problem 5545

ODE	\[ \boxed {y^{\prime \prime }-\left (1+x \right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{6} x^{4}+\frac {1}{15} x^{5}+\frac {7}{180} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{2} x^{3}+\frac {1}{4} x^{4}+\frac {3}{20} x^{5}+\frac {1}{15} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{6} x^{4}+\frac {1}{15} x^{5}\right ) c_{1} +\left (x +\frac {1}{2} x^{2}+\frac {1}{2} x^{3}+\frac {1}{4} x^{4}+\frac {3}{20} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{6} x^{4}+\frac {1}{15} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{2} x^{3}+\frac {1}{4} x^{4}+\frac {3}{20} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5546

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

Problem 5547

ODE	\[ \boxed {\left (x^{2}+2\right ) y^{\prime \prime }+3 y^{\prime } x -y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{4} x^{2}-\frac {7}{96} x^{4}+\frac {161}{5760} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {7}{120} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{4} x^{2}-\frac {7}{96} x^{4}\right ) c_{1} +\left (x -\frac {1}{6} x^{3}+\frac {7}{120} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{4} x^{2}-\frac {7}{96} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {7}{120} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5548

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

Problem 5549

ODE	\[ \boxed {\left (x -1\right ) y^{\prime \prime }-y^{\prime } x +y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -2, y^{\prime }\left (0\right ) = 6] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = -x^{2}+6 x -2-\frac {x^{3}}{3}-\frac {x^{4}}{12}-\frac {x^{5}}{60}-\frac {x^{6}}{360}+O\left (x^{6}\right ) \] Verified OK. \[ y = -2-x^{2}-\frac {x^{3}}{3}-\frac {x^{4}}{12}-\frac {x^{5}}{60}+6 x +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -2+6 x -x^{2}-\frac {1}{3} x^{3}-\frac {1}{12} x^{4}-\frac {1}{60} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 5550

ODE	\[ \boxed {\left (1+x \right ) y^{\prime \prime }-\left (-x +2\right ) y^{\prime }+y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = -1] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = -2 x^{2}-x +2-\frac {x^{3}}{3}+\frac {x^{4}}{2}-\frac {x^{5}}{30}-\frac {13 x^{6}}{180}+O\left (x^{6}\right ) \] Verified OK. \[ y = 2-2 x^{2}-\frac {x^{3}}{3}+\frac {x^{4}}{2}-\frac {x^{5}}{30}-x +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2-x -2 x^{2}-\frac {1}{3} x^{3}+\frac {1}{2} x^{4}-\frac {1}{30} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 5551

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

Problem 5552

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

Problem 5553

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

Problem 5554

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime } {\mathrm e}^{x}-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{2} x^{2}-\frac {1}{6} x^{3}-\frac {1}{120} x^{5}+\frac {1}{240} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{24} x^{4}+\frac {1}{120} x^{5}-\frac {1}{720} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{2} x^{2}-\frac {1}{6} x^{3}-\frac {1}{120} x^{5}\right ) c_{1} +\left (x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{24} x^{4}+\frac {1}{120} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{2} x^{2}-\frac {1}{6} x^{3}-\frac {1}{120} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{24} x^{4}+\frac {1}{120} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5555

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

Problem 5556

ODE	\[ \boxed {x^{3} y^{\prime \prime }+4 x^{2} y^{\prime }+3 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 5557

ODE	\[ \boxed {x \left (x +3\right )^{2} y^{\prime \prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1+\frac {x}{18}-\frac {11 x^{2}}{972}+\frac {277 x^{3}}{104976}-\frac {12539 x^{4}}{18895680}+\frac {893821 x^{5}}{5101833600}+O\left (x^{6}\right )\right )+c_{2} \left (\frac {x \left (1+\frac {x}{18}-\frac {11 x^{2}}{972}+\frac {277 x^{3}}{104976}-\frac {12539 x^{4}}{18895680}+\frac {893821 x^{5}}{5101833600}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{9}+1-\frac {5 x^{2}}{108}+\frac {167 x^{3}}{26244}-\frac {13583 x^{4}}{11337408}+\frac {1327279 x^{5}}{5101833600}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1+\frac {1}{18} x -\frac {11}{972} x^{2}+\frac {277}{104976} x^{3}-\frac {12539}{18895680} x^{4}+\frac {893821}{5101833600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (\frac {1}{9} x +\frac {1}{162} x^{2}-\frac {11}{8748} x^{3}+\frac {277}{944784} x^{4}-\frac {12539}{170061120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-\frac {5}{108} x^{2}+\frac {167}{26244} x^{3}-\frac {13583}{11337408} x^{4}+\frac {1327279}{5101833600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

Problem 5558

ODE	\[ \boxed {\left (x^{2}-9\right )^{2} y^{\prime \prime }+\left (x +3\right ) y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{81} x^{2}+\frac {1}{6561} x^{3}-\frac {289}{708588} x^{4}+\frac {304}{23914845} x^{5}-\frac {194981}{7748409780} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{54} x^{2}-\frac {13}{2187} x^{3}-\frac {131}{236196} x^{4}-\frac {596}{1594323} x^{5}-\frac {78469}{2582803260} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{81} x^{2}+\frac {1}{6561} x^{3}-\frac {289}{708588} x^{4}+\frac {304}{23914845} x^{5}\right ) c_{1} +\left (x -\frac {1}{54} x^{2}-\frac {13}{2187} x^{3}-\frac {131}{236196} x^{4}-\frac {596}{1594323} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{81} x^{2}+\frac {1}{6561} x^{3}-\frac {289}{708588} x^{4}+\frac {304}{23914845} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{54} x^{2}-\frac {13}{2187} x^{3}-\frac {131}{236196} x^{4}-\frac {596}{1594323} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5559

ODE	\[ \boxed {y^{\prime \prime }-\frac {y^{\prime }}{x}+\frac {y}{\left (x -1\right )^{3}}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1+\frac {x^{2}}{8}+\frac {x^{3}}{5}+\frac {49 x^{4}}{192}+\frac {423 x^{5}}{1400}+O\left (x^{6}\right )\right )+c_{2} \left (\frac {x^{2} \left (1+\frac {x^{2}}{8}+\frac {x^{3}}{5}+\frac {49 x^{4}}{192}+\frac {423 x^{5}}{1400}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{2}+1+x^{3}+\frac {45 x^{4}}{64}+\frac {17 x^{5}}{25}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \left (1+\frac {1}{8} x^{2}+\frac {1}{5} x^{3}+\frac {49}{192} x^{4}+\frac {423}{1400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-x^{2}-\frac {1}{8} x^{4}-\frac {1}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2-2 x^{3}-\frac {45}{32} x^{4}-\frac {34}{25} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

Problem 5560

ODE	\[ \boxed {\left (x^{3}+4 x \right ) y^{\prime \prime }-2 y^{\prime } x +6 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {x}{2}+\frac {x^{2}}{24}+\frac {x^{3}}{48}-\frac {x^{4}}{384}-\frac {5 x^{5}}{2304}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {3 x \left (1-\frac {x}{2}+\frac {x^{2}}{24}+\frac {x^{3}}{48}-\frac {x^{4}}{384}-\frac {5 x^{5}}{2304}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{2}+1-\frac {3 x^{2}}{2}+\frac {29 x^{3}}{96}+\frac {x^{4}}{32}-\frac {21 x^{5}}{1024}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1-\frac {1}{2} x +\frac {1}{24} x^{2}+\frac {1}{48} x^{3}-\frac {1}{384} x^{4}-\frac {5}{2304} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-\frac {3}{2} x +\frac {3}{4} x^{2}-\frac {1}{16} x^{3}-\frac {1}{32} x^{4}+\frac {1}{256} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1+\frac {1}{2} x -\frac {7}{4} x^{2}+\frac {31}{96} x^{3}+\frac {1}{24} x^{4}-\frac {67}{3072} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

Problem 5561

ODE	\[ \boxed {x^{2} \left (x -5\right )^{2} y^{\prime \prime }+4 y^{\prime } x +\left (x^{2}-25\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {21}{50}+\frac {\sqrt {2941}}{50}} \left (1+\frac {\left (21+\sqrt {2941}\right ) \left (-29+\sqrt {2941}\right ) x}{6250+250 \sqrt {2941}}+\frac {9 \left (79709+879 \sqrt {2941}\right ) x^{2}}{15625 \left (25+\sqrt {2941}\right ) \left (50+\sqrt {2941}\right )}+\frac {12 \left (75561897+1274257 \sqrt {2941}\right ) x^{3}}{1953125 \left (25+\sqrt {2941}\right ) \left (50+\sqrt {2941}\right ) \left (75+\sqrt {2941}\right )}+\frac {12 \left (122814219551+2200649681 \sqrt {2941}\right ) x^{4}}{244140625 \left (25+\sqrt {2941}\right ) \left (50+\sqrt {2941}\right ) \left (75+\sqrt {2941}\right ) \left (100+\sqrt {2941}\right )}+\frac {1152 \left (8688311436917+157371578127 \sqrt {2941}\right ) x^{5}}{152587890625 \left (25+\sqrt {2941}\right ) \left (50+\sqrt {2941}\right ) \left (75+\sqrt {2941}\right ) \left (100+\sqrt {2941}\right ) \left (125+\sqrt {2941}\right )}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {21}{50}-\frac {\sqrt {2941}}{50}} \left (1-\frac {\left (-21+\sqrt {2941}\right ) \left (29+\sqrt {2941}\right ) x}{-6250+250 \sqrt {2941}}+\frac {9 \left (79709-879 \sqrt {2941}\right ) x^{2}}{15625 \left (-25+\sqrt {2941}\right ) \left (-50+\sqrt {2941}\right )}+\frac {12 \left (-75561897+1274257 \sqrt {2941}\right ) x^{3}}{1953125 \left (-25+\sqrt {2941}\right ) \left (-50+\sqrt {2941}\right ) \left (-75+\sqrt {2941}\right )}+\frac {12 \left (122814219551-2200649681 \sqrt {2941}\right ) x^{4}}{244140625 \left (-25+\sqrt {2941}\right ) \left (-50+\sqrt {2941}\right ) \left (-75+\sqrt {2941}\right ) \left (-100+\sqrt {2941}\right )}+\frac {1152 \left (-8688311436917+157371578127 \sqrt {2941}\right ) x^{5}}{152587890625 \left (-25+\sqrt {2941}\right ) \left (-50+\sqrt {2941}\right ) \left (-75+\sqrt {2941}\right ) \left (-100+\sqrt {2941}\right ) \left (-125+\sqrt {2941}\right )}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{\frac {21}{50}} \left (c_{1} x^{-\frac {\sqrt {2941}}{50}} \left (1+\frac {-1166-4 \sqrt {2941}}{-3125+125 \sqrt {2941}} x -\frac {9}{15625} \frac {879 \sqrt {2941}-79709}{\left (-25+\sqrt {2941}\right ) \left (-50+\sqrt {2941}\right )} x^{2}+\frac {\frac {15291084 \sqrt {2941}}{1953125}-\frac {906742764}{1953125}}{\left (-25+\sqrt {2941}\right ) \left (-50+\sqrt {2941}\right ) \left (-75+\sqrt {2941}\right )} x^{3}-\frac {12}{244140625} \frac {2200649681 \sqrt {2941}-122814219551}{\left (-25+\sqrt {2941}\right ) \left (-50+\sqrt {2941}\right ) \left (-75+\sqrt {2941}\right ) \left (-100+\sqrt {2941}\right )} x^{4}+\frac {\frac {181292058002304 \sqrt {2941}}{152587890625}-\frac {10008934775328384}{152587890625}}{\left (-25+\sqrt {2941}\right ) \left (-50+\sqrt {2941}\right ) \left (-75+\sqrt {2941}\right ) \left (-100+\sqrt {2941}\right ) \left (-125+\sqrt {2941}\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{\frac {\sqrt {2941}}{50}} \left (1+\frac {1166-4 \sqrt {2941}}{125 \sqrt {2941}+3125} x +\frac {\frac {7911 \sqrt {2941}}{15625}+\frac {717381}{15625}}{\left (\sqrt {2941}+25\right ) \left (50+\sqrt {2941}\right )} x^{2}+\frac {\frac {15291084 \sqrt {2941}}{1953125}+\frac {906742764}{1953125}}{\left (\sqrt {2941}+25\right ) \left (50+\sqrt {2941}\right ) \left (\sqrt {2941}+75\right )} x^{3}+\frac {\frac {26407796172 \sqrt {2941}}{244140625}+\frac {1473770634612}{244140625}}{\left (\sqrt {2941}+25\right ) \left (50+\sqrt {2941}\right ) \left (\sqrt {2941}+75\right ) \left (100+\sqrt {2941}\right )} x^{4}+\frac {\frac {181292058002304 \sqrt {2941}}{152587890625}+\frac {10008934775328384}{152587890625}}{\left (\sqrt {2941}+25\right ) \left (50+\sqrt {2941}\right ) \left (\sqrt {2941}+75\right ) \left (100+\sqrt {2941}\right ) \left (125+\sqrt {2941}\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

Problem 5562

ODE	\[ \boxed {\left (x^{2}+x -6\right ) y^{\prime \prime }+\left (x +3\right ) y^{\prime }+y \left (x -2\right )=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{6} x^{2}-\frac {1}{108} x^{3}-\frac {17}{2592} x^{4}-\frac {7}{2160} x^{5}-\frac {139}{116640} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{4} x^{2}+\frac {1}{36} x^{3}+\frac {23}{864} x^{4}+\frac {13}{1440} x^{5}+\frac {619}{155520} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{6} x^{2}-\frac {1}{108} x^{3}-\frac {17}{2592} x^{4}-\frac {7}{2160} x^{5}\right ) c_{1} +\left (x +\frac {1}{4} x^{2}+\frac {1}{36} x^{3}+\frac {23}{864} x^{4}+\frac {13}{1440} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{6} x^{2}-\frac {1}{108} x^{3}-\frac {17}{2592} x^{4}-\frac {7}{2160} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{4} x^{2}+\frac {1}{36} x^{3}+\frac {23}{864} x^{4}+\frac {13}{1440} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5563

ODE	\[ \boxed {x \left (x^{2}+1\right )^{2} y^{\prime \prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {x}{2}+\frac {x^{2}}{12}+\frac {23 x^{3}}{144}-\frac {167 x^{4}}{2880}-\frac {7993 x^{5}}{86400}+O\left (x^{6}\right )\right )+c_{2} \left (-x \left (1-\frac {x}{2}+\frac {x^{2}}{12}+\frac {23 x^{3}}{144}-\frac {167 x^{4}}{2880}-\frac {7993 x^{5}}{86400}+O\left (x^{6}\right )\right ) \ln \left (x \right )+1-\frac {3 x^{2}}{4}+\frac {19 x^{3}}{36}+\frac {85 x^{4}}{1728}-\frac {21907 x^{5}}{86400}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1-\frac {1}{2} x +\frac {1}{12} x^{2}+\frac {23}{144} x^{3}-\frac {167}{2880} x^{4}-\frac {7993}{86400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-x +\frac {1}{2} x^{2}-\frac {1}{12} x^{3}-\frac {23}{144} x^{4}+\frac {167}{2880} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-\frac {3}{4} x^{2}+\frac {19}{36} x^{3}+\frac {85}{1728} x^{4}-\frac {21907}{86400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

Problem 5564

ODE	\[ \boxed {x^{3} \left (x^{2}-25\right ) \left (x -2\right )^{2} y^{\prime \prime }+3 x \left (x -2\right ) y^{\prime }+7 \left (x +5\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 5565

ODE	\[ \boxed {\left (x^{3}-2 x^{2}+3 x \right )^{2} y^{\prime \prime }+x \left (x -3\right )^{2} y^{\prime }-y \left (1+x \right )=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1+\frac {x}{45}+\frac {149 x^{2}}{3240}+\frac {2701 x^{3}}{192456}+\frac {236933 x^{4}}{121247280}-\frac {67092967 x^{5}}{92754169200}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+\frac {13 x}{9}-\frac {5 x^{2}}{162}+\frac {1591 x^{3}}{30618}+\frac {106583 x^{4}}{5511240}+\frac {7435523 x^{5}}{3224075400}+O\left (x^{6}\right )\right )}{x^{\frac {1}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {2}{3}} \left (1+\frac {1}{45} x +\frac {149}{3240} x^{2}+\frac {2701}{192456} x^{3}+\frac {236933}{121247280} x^{4}-\frac {67092967}{92754169200} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1+\frac {13}{9} x -\frac {5}{162} x^{2}+\frac {1591}{30618} x^{3}+\frac {106583}{5511240} x^{4}+\frac {7435523}{3224075400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {1}{3}}} \]

Problem 5566

ODE	\[ \boxed {\left (x^{2}-1\right ) y^{\prime \prime }+5 \left (1+x \right ) y^{\prime }+\left (x^{2}-x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{6} x^{3}-\frac {1}{8} x^{4}-\frac {3}{10} x^{5}-\frac {17}{45} x^{6}\right ) y \left (0\right )+\left (x +\frac {5}{2} x^{2}+5 x^{3}+\frac {26}{3} x^{4}+\frac {1661}{120} x^{5}+\frac {4967}{240} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{6} x^{3}-\frac {1}{8} x^{4}-\frac {3}{10} x^{5}\right ) c_{1} +\left (x +\frac {5}{2} x^{2}+5 x^{3}+\frac {26}{3} x^{4}+\frac {1661}{120} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{6} x^{3}-\frac {1}{8} x^{4}-\frac {3}{10} x^{5}\right ) y \left (0\right )+\left (x +\frac {5}{2} x^{2}+5 x^{3}+\frac {26}{3} x^{4}+\frac {1661}{120} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5567

ODE	\[ \boxed {x y^{\prime \prime }+\left (x +3\right ) y^{\prime }+7 y x^{2}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-\frac {7 x^{3}}{15}+\frac {7 x^{4}}{120}-\frac {x^{5}}{150}+O\left (x^{6}\right )\right )+c_{2} \left (\left (-1+\frac {7 x^{3}}{15}-\frac {7 x^{4}}{120}+\frac {x^{5}}{150}-O\left (x^{6}\right )\right ) \ln \left (x \right )+\frac {1-2 x -2 x^{3}+2 x^{4}-\frac {116 x^{5}}{225}+O\left (x^{6}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (1-\frac {7}{15} x^{3}+\frac {7}{120} x^{4}-\frac {1}{150} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (\ln \left (x \right ) \left (2 x^{2}-\frac {14}{15} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+4 x -3 x^{2}+4 x^{3}-4 x^{4}+\frac {547}{225} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]

Problem 5568

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\left (\frac {5}{3} x +x^{2}\right ) y^{\prime }-\frac {y}{3}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1-\frac {x}{7}+\frac {x^{2}}{35}-\frac {x^{3}}{195}+\frac {x^{4}}{1248}-\frac {x^{5}}{9120}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-3 x +O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {4}{3}} \left (1-\frac {1}{7} x +\frac {1}{35} x^{2}-\frac {1}{195} x^{3}+\frac {1}{1248} x^{4}-\frac {1}{9120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1-3 x +\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 5569

ODE	\[ \boxed {x y^{\prime \prime }+y^{\prime }+10 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (25 x^{2}-10 x +1-\frac {250 x^{3}}{9}+\frac {625 x^{4}}{36}-\frac {125 x^{5}}{18}+O\left (x^{6}\right )\right )+c_{2} \left (\left (25 x^{2}-10 x +1-\frac {250 x^{3}}{9}+\frac {625 x^{4}}{36}-\frac {125 x^{5}}{18}+O\left (x^{6}\right )\right ) \ln \left (x \right )-75 x^{2}+20 x +\frac {2750 x^{3}}{27}-\frac {15625 x^{4}}{216}+\frac {3425 x^{5}}{108}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-10 x +25 x^{2}-\frac {250}{9} x^{3}+\frac {625}{36} x^{4}-\frac {125}{18} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (20 x -75 x^{2}+\frac {2750}{27} x^{3}-\frac {15625}{216} x^{4}+\frac {3425}{108} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 5570

ODE	\[ \boxed {2 x y^{\prime \prime }-y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {3}{2}} \left (1-\frac {2 x}{5}+\frac {2 x^{2}}{35}-\frac {4 x^{3}}{945}+\frac {2 x^{4}}{10395}-\frac {4 x^{5}}{675675}+O\left (x^{6}\right )\right )+c_{2} \left (1+2 x -2 x^{2}+\frac {4 x^{3}}{9}-\frac {2 x^{4}}{45}+\frac {4 x^{5}}{1575}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {3}{2}} \left (1-\frac {2}{5} x +\frac {2}{35} x^{2}-\frac {4}{945} x^{3}+\frac {2}{10395} x^{4}-\frac {4}{675675} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+2 x -2 x^{2}+\frac {4}{9} x^{3}-\frac {2}{45} x^{4}+\frac {4}{1575} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 5571

ODE	\[ \boxed {2 x y^{\prime \prime }+5 y^{\prime }+y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-\frac {x^{2}}{14}+\frac {x^{4}}{616}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {x^{2}}{2}+\frac {x^{4}}{40}+O\left (x^{6}\right )\right )}{x^{\frac {3}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-\frac {1}{2} x^{2}+\frac {1}{40} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {3}{2}}}+c_{2} \left (1-\frac {1}{14} x^{2}+\frac {1}{616} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 5572

ODE	\[ \boxed {4 x y^{\prime \prime }+\frac {y^{\prime }}{2}+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {7}{8}} \left (1-\frac {2 x}{15}+\frac {2 x^{2}}{345}-\frac {4 x^{3}}{32085}+\frac {2 x^{4}}{1251315}-\frac {4 x^{5}}{294059025}+O\left (x^{6}\right )\right )+c_{2} \left (1-2 x +\frac {2 x^{2}}{9}-\frac {4 x^{3}}{459}+\frac {2 x^{4}}{11475}-\frac {4 x^{5}}{1893375}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {7}{8}} \left (1-\frac {2}{15} x +\frac {2}{345} x^{2}-\frac {4}{32085} x^{3}+\frac {2}{1251315} x^{4}-\frac {4}{294059025} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-2 x +\frac {2}{9} x^{2}-\frac {4}{459} x^{3}+\frac {2}{11475} x^{4}-\frac {4}{1893375} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 5573

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }-y^{\prime } x +y \left (x^{2}+1\right )=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {x^{2}}{10}+\frac {x^{4}}{360}+O\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1-\frac {x^{2}}{6}+\frac {x^{4}}{168}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1-\frac {1}{6} x^{2}+\frac {1}{168} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x \left (1-\frac {1}{10} x^{2}+\frac {1}{360} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 5574

ODE	\[ \boxed {3 x y^{\prime \prime }+\left (-x +2\right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1+\frac {x}{3}+\frac {x^{2}}{18}+\frac {x^{3}}{162}+\frac {x^{4}}{1944}+\frac {x^{5}}{29160}+O\left (x^{6}\right )\right )+c_{2} \left (1+\frac {x}{2}+\frac {x^{2}}{10}+\frac {x^{3}}{80}+\frac {x^{4}}{880}+\frac {x^{5}}{12320}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{3}} \left (1+\frac {1}{3} x +\frac {1}{18} x^{2}+\frac {1}{162} x^{3}+\frac {1}{1944} x^{4}+\frac {1}{29160} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+\frac {1}{2} x +\frac {1}{10} x^{2}+\frac {1}{80} x^{3}+\frac {1}{880} x^{4}+\frac {1}{12320} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 5575

ODE	\[ \boxed {x^{2} y^{\prime \prime }-\left (x -\frac {2}{9}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {2}{3}} \left (1+\frac {3 x}{4}+\frac {9 x^{2}}{56}+\frac {9 x^{3}}{560}+\frac {27 x^{4}}{29120}+\frac {81 x^{5}}{2329600}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{3}} \left (1+\frac {3 x}{2}+\frac {9 x^{2}}{20}+\frac {9 x^{3}}{160}+\frac {27 x^{4}}{7040}+\frac {81 x^{5}}{492800}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{3}} \left (1+\frac {3}{2} x +\frac {9}{20} x^{2}+\frac {9}{160} x^{3}+\frac {27}{7040} x^{4}+\frac {81}{492800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{\frac {2}{3}} \left (1+\frac {3}{4} x +\frac {9}{56} x^{2}+\frac {9}{560} x^{3}+\frac {27}{29120} x^{4}+\frac {81}{2329600} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 5576

ODE	\[ \boxed {2 x y^{\prime \prime }-\left (2 x +3\right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {5}{2}} \left (1+\frac {4 x}{7}+\frac {4 x^{2}}{21}+\frac {32 x^{3}}{693}+\frac {80 x^{4}}{9009}+\frac {64 x^{5}}{45045}+O\left (x^{6}\right )\right )+c_{2} \left (1+\frac {x}{3}-\frac {x^{2}}{6}-\frac {x^{3}}{6}-\frac {5 x^{4}}{72}-\frac {7 x^{5}}{360}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {5}{2}} \left (1+\frac {4}{7} x +\frac {4}{21} x^{2}+\frac {32}{693} x^{3}+\frac {80}{9009} x^{4}+\frac {64}{45045} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+\frac {1}{3} x -\frac {1}{6} x^{2}-\frac {1}{6} x^{3}-\frac {5}{72} x^{4}-\frac {7}{360} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 5577

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (x^{2}-\frac {4}{9}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {2}{3}} \left (1-\frac {3 x^{2}}{20}+\frac {9 x^{4}}{1280}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {3 x^{2}}{4}+\frac {9 x^{4}}{128}+O\left (x^{6}\right )\right )}{x^{\frac {2}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {4}{3}} \left (1-\frac {3}{20} x^{2}+\frac {9}{1280} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1-\frac {3}{4} x^{2}+\frac {9}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {2}{3}}} \]

Problem 5578

ODE	\[ \boxed {9 x^{2} y^{\prime \prime }+9 x^{2} y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {2}{3}} \left (1-\frac {x}{2}+\frac {5 x^{2}}{28}-\frac {x^{3}}{21}+\frac {11 x^{4}}{1092}-\frac {11 x^{5}}{6240}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{3}} \left (1-\frac {x}{2}+\frac {x^{2}}{5}-\frac {7 x^{3}}{120}+\frac {7 x^{4}}{528}-\frac {13 x^{5}}{5280}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{3}} \left (1-\frac {1}{2} x +\frac {1}{5} x^{2}-\frac {7}{120} x^{3}+\frac {7}{528} x^{4}-\frac {13}{5280} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{\frac {2}{3}} \left (1-\frac {1}{2} x +\frac {5}{28} x^{2}-\frac {1}{21} x^{3}+\frac {11}{1092} x^{4}-\frac {11}{6240} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 5579

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }+3 y^{\prime } x +\left (2 x -1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {2 x}{5}+\frac {2 x^{2}}{35}-\frac {4 x^{3}}{945}+\frac {2 x^{4}}{10395}-\frac {4 x^{5}}{675675}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+2 x -2 x^{2}+\frac {4 x^{3}}{9}-\frac {2 x^{4}}{45}+\frac {4 x^{5}}{1575}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {3}{2}} \left (1-\frac {2}{5} x +\frac {2}{35} x^{2}-\frac {4}{945} x^{3}+\frac {2}{10395} x^{4}-\frac {4}{675675} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1+2 x -2 x^{2}+\frac {4}{9} x^{3}-\frac {2}{45} x^{4}+\frac {4}{1575} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 5580

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 5581

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

Problem 5582

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 5583

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

Problem 5584

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

Problem 5585

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

Problem 5586

ODE	\[ \boxed {x y^{\prime \prime }+\left (x -6\right ) y^{\prime }-3 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{7} \left (1-\frac {x}{2}+\frac {5 x^{2}}{36}-\frac {x^{3}}{36}+\frac {7 x^{4}}{1584}-\frac {7 x^{5}}{11880}+\frac {7 x^{6}}{102960}-\frac {x^{7}}{144144}+O\left (x^{8}\right )\right )+c_{2} \left (1-\frac {x}{2}+\frac {x^{2}}{10}-\frac {x^{3}}{120}+\frac {x^{7}}{100800}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{7} \left (1-\frac {1}{2} x +\frac {5}{36} x^{2}-\frac {1}{36} x^{3}+\frac {7}{1584} x^{4}-\frac {7}{11880} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (3628800-1814400 x +362880 x^{2}-30240 x^{3}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 5587

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

Problem 5588

ODE	\[ \boxed {x^{4} y^{\prime \prime }+\lambda y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 5589

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

Problem 5590

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\left (3 x -1\right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 5591

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (x^{2}-\frac {1}{9}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1-\frac {3 x^{2}}{16}+\frac {9 x^{4}}{896}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {3 x^{2}}{8}+\frac {9 x^{4}}{320}+O\left (x^{6}\right )\right )}{x^{\frac {1}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {2}{3}} \left (1-\frac {3}{16} x^{2}+\frac {9}{896} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1-\frac {3}{8} x^{2}+\frac {9}{320} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {1}{3}}} \]

Problem 5592

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

Problem 5593

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+4 y^{\prime } x +\left (4 x^{2}-25\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {5}{2}} \left (1-\frac {x^{2}}{14}+\frac {x^{4}}{504}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+\frac {x^{2}}{6}+\frac {x^{4}}{24}+O\left (x^{6}\right )\right )}{x^{\frac {5}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{5} \left (1-\frac {1}{14} x^{2}+\frac {1}{504} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (2880+480 x^{2}+120 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {5}{2}}} \]

Problem 5594

ODE	\[ \boxed {16 x^{2} y^{\prime \prime }+16 y^{\prime } x +\left (16 x^{2}-1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{4}} \left (1-\frac {x^{2}}{5}+\frac {x^{4}}{90}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {x^{2}}{3}+\frac {x^{4}}{42}+O\left (x^{6}\right )\right )}{x^{\frac {1}{4}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} \sqrt {x}\, \left (1-\frac {1}{5} x^{2}+\frac {1}{90} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1-\frac {1}{3} x^{2}+\frac {1}{42} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {1}{4}}} \]

Problem 5595

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 5596

ODE	\[ \boxed {x y^{\prime \prime }+y^{\prime }+\left (x -\frac {4}{x}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-\frac {x^{2}}{12}+\frac {x^{4}}{384}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {x^{2} \left (1-\frac {x^{2}}{12}+\frac {x^{4}}{384}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{16}+\frac {1+\frac {x^{2}}{4}+O\left (x^{6}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{4} \left (1-\frac {1}{12} x^{2}+\frac {1}{384} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (9 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-36 x^{2}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]

Problem 5597

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (9 x^{2}-4\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-\frac {3 x^{2}}{4}+\frac {27 x^{4}}{128}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {81 x^{2} \left (1-\frac {3 x^{2}}{4}+\frac {27 x^{4}}{128}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{16}+\frac {1+\frac {9 x^{2}}{4}+O\left (x^{6}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{4} \left (1-\frac {3}{4} x^{2}+\frac {27}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (729 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-324 x^{2}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]

Problem 5598

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (36 x^{2}-\frac {1}{4}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-6 x^{2}+\frac {54 x^{4}}{5}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-18 x^{2}+54 x^{4}+O\left (x^{6}\right )\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x \left (1-6 x^{2}+\frac {54}{5} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-18 x^{2}+54 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

Problem 5599

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (25 x^{2}-\frac {4}{9}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {2}{3}} \left (1-\frac {15 x^{2}}{4}+\frac {1125 x^{4}}{256}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {75 x^{2}}{4}+\frac {5625 x^{4}}{128}+O\left (x^{6}\right )\right )}{x^{\frac {2}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {4}{3}} \left (1-\frac {15}{4} x^{2}+\frac {1125}{256} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1-\frac {75}{4} x^{2}+\frac {5625}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {2}{3}}} \]

Problem 5600

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (2 x^{2}-64\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{8} \left (1-\frac {x^{2}}{18}+\frac {x^{4}}{720}-\frac {x^{6}}{47520}+\frac {x^{8}}{4561920}-\frac {x^{10}}{593049600}+\frac {x^{12}}{99632332800}-\frac {x^{14}}{20922789888000}+\frac {x^{16}}{5356234211328000}+O\left (x^{17}\right )\right )+c_{2} \left (-\frac {x^{8} \left (1-\frac {x^{2}}{18}+\frac {x^{4}}{720}-\frac {x^{6}}{47520}+\frac {x^{8}}{4561920}-\frac {x^{10}}{593049600}+\frac {x^{12}}{99632332800}-\frac {x^{14}}{20922789888000}+\frac {x^{16}}{5356234211328000}+O\left (x^{17}\right )\right ) \ln \left (x \right )}{26011238400}+\frac {1+\frac {x^{2}}{14}+\frac {x^{4}}{336}+\frac {x^{6}}{10080}+\frac {x^{8}}{322560}+\frac {x^{10}}{9676800}+\frac {x^{12}}{232243200}+\frac {x^{14}}{3251404800}+O\left (x^{17}\right )}{x^{8}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{8} \left (1-\frac {1}{18} x^{2}+\frac {1}{720} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (-27360196043587190784000000-1954299717399085056000000 x^{2}-81429154891628544000000 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{8}} \]