2.17.65 Problems 6401 to 6500

Problem 6401

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

Problem 6402

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

Problem 6403

ODE	\[ \boxed {y^{\prime \prime }+\sin \left (y\right )=0} \]
program solution	\[ \frac {2 \sqrt {\frac {\cos \left (y\right )+c_{1}}{1+c_{1}}}\, \operatorname {InverseJacobiAM}\left (\frac {y}{2}, \frac {2}{\sqrt {2+2 c_{1}}}\right )}{\sqrt {2 \cos \left (y\right )+2 c_{1}}} = x +c_{2} \] Verified OK. \[ -\frac {2 \sqrt {\frac {\cos \left (y\right )+c_{1}}{1+c_{1}}}\, \operatorname {InverseJacobiAM}\left (\frac {y}{2}, \frac {2}{\sqrt {2+2 c_{1}}}\right )}{\sqrt {2 \cos \left (y\right )+2 c_{1}}} = x +c_{3} \] Verified OK.
Maple solution	\begin{align*} \int _{}^{y \left (x \right )}\frac {1}{\sqrt {2 \cos \left (\textit {\_a} \right )+c_{1}}}d \textit {\_a} -x -c_{2} &= 0 \\ -\left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {2 \cos \left (\textit {\_a} \right )+c_{1}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*}

Problem 6404

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

Problem 6405

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

Problem 6406

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

Problem 6407

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

Problem 6408

ODE	\[ \boxed {y^{\prime }-y=2} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}\right ) y \left (0\right )+2 x +x^{2}+\frac {x^{3}}{3}+\frac {x^{4}}{12}+\frac {x^{5}}{60}+\frac {x^{6}}{360}+\frac {x^{7}}{2520}+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}\right ) c_{1} +2 x +x^{2}+\frac {x^{3}}{3}+\frac {x^{4}}{12}+\frac {x^{5}}{60}+\frac {x^{6}}{360}+\frac {x^{7}}{2520}+O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}\right ) y \left (0\right )+2 x +x^{2}+\frac {x^{3}}{3}+\frac {x^{4}}{12}+\frac {x^{5}}{60}+\frac {x^{6}}{360}+\frac {x^{7}}{2520}+O\left (x^{8}\right ) \]

Problem 6409

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

Problem 6410

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

Problem 6411

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

Problem 6412

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

Problem 6413

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

Problem 6414

ODE	\[ \boxed {y^{\prime }-y=x^{2}} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}\right ) y \left (0\right )+\frac {x^{3}}{3}+\frac {x^{4}}{12}+\frac {x^{5}}{60}+\frac {x^{6}}{360}+\frac {x^{7}}{2520}+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}\right ) c_{1} +\frac {x^{3}}{3}+\frac {x^{4}}{12}+\frac {x^{5}}{60}+\frac {x^{6}}{360}+\frac {x^{7}}{2520}+O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}\right ) y \left (0\right )+\frac {x^{3}}{3}+\frac {x^{4}}{12}+\frac {x^{5}}{60}+\frac {x^{6}}{360}+\frac {x^{7}}{2520}+O\left (x^{8}\right ) \]

Problem 6415

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

Problem 6416

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

Problem 6417

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

Problem 6418

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

Problem 6419

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

Problem 6420

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

Problem 6421

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

Problem 6422

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

Problem 6423

ODE	\[ \boxed {y^{\prime }=\frac {1}{\sqrt {-x^{2}+1}}} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = y \left (0\right )+x +\frac {x^{3}}{6}+\frac {3 x^{5}}{40}+\frac {5 x^{7}}{112}+O\left (x^{8}\right ) \] Verified OK. \[ y = c_{1} +x +\frac {x^{3}}{6}+\frac {3 x^{5}}{40}+\frac {5 x^{7}}{112}+O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = y \left (0\right )+x +\frac {x^{3}}{6}+\frac {3 x^{5}}{40}+\frac {5 x^{7}}{112}+O\left (x^{8}\right ) \]

Problem 6424

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

Problem 6425

ODE	\[ \boxed {y+y^{\prime }=x} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {x^{2}}{2}-\frac {x^{3}}{6}+\frac {x^{4}}{24}-\frac {x^{5}}{120}+\frac {x^{6}}{720}-\frac {x^{7}}{5040}+O\left (x^{8}\right ) \] Verified OK. \[ y = \frac {x^{2}}{2}-\frac {x^{3}}{6}+\frac {x^{4}}{24}-\frac {x^{5}}{120}+\frac {x^{6}}{720}-\frac {x^{7}}{5040}+O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\frac {1}{720} x^{6}-\frac {1}{5040} x^{7}+\operatorname {O}\left (x^{8}\right ) \]

Problem 6426

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

Problem 6427

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime } x +y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}-\frac {1}{48} x^{6}+\frac {1}{384} x^{8}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{15} x^{5}-\frac {1}{105} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}-\frac {1}{48} x^{6}\right ) c_{1} +\left (x -\frac {1}{3} x^{3}+\frac {1}{15} x^{5}-\frac {1}{105} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}-\frac {1}{48} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{15} x^{5}-\frac {1}{105} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6428

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }+y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{6} x^{3}-\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\frac {1}{240} x^{6}+\frac {1}{630} x^{7}+\frac {1}{2880} x^{8}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{24} x^{4}-\frac {1}{30} x^{5}-\frac {1}{90} x^{6}-\frac {1}{1680} x^{7}+\frac {1}{1920} x^{8}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{6} x^{3}-\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\frac {1}{240} x^{6}+\frac {1}{630} x^{7}\right ) c_{1} +\left (x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{24} x^{4}-\frac {1}{30} x^{5}-\frac {1}{90} x^{6}-\frac {1}{1680} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{6} x^{3}-\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\frac {1}{240} x^{6}+\frac {1}{630} x^{7}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{24} x^{4}-\frac {1}{30} x^{5}-\frac {1}{90} x^{6}-\frac {1}{1680} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6429

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime } x -y=x} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{2} x^{2}-\frac {1}{8} x^{4}+\frac {7}{240} x^{6}-\frac {11}{1920} x^{8}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {1}{24} x^{5}-\frac {1}{112} x^{7}\right ) y^{\prime }\left (0\right )+\frac {x^{3}}{6}-\frac {x^{5}}{24}+\frac {x^{7}}{112}+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{2} x^{2}-\frac {1}{8} x^{4}+\frac {7}{240} x^{6}\right ) c_{1} +\left (x -\frac {1}{6} x^{3}+\frac {1}{24} x^{5}-\frac {1}{112} x^{7}\right ) c_{2} +\frac {x^{3}}{6}-\frac {x^{5}}{24}+\frac {x^{7}}{112}+O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{2} x^{2}-\frac {1}{8} x^{4}+\frac {7}{240} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {1}{24} x^{5}-\frac {1}{112} x^{7}\right ) D\left (y \right )\left (0\right )+\frac {x^{3}}{6}-\frac {x^{5}}{24}+\frac {x^{7}}{112}+O\left (x^{8}\right ) \]

Problem 6430

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }-y x^{2}=1} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{12} x^{4}-\frac {1}{60} x^{5}+\frac {1}{360} x^{6}-\frac {1}{2520} x^{7}+\frac {31}{20160} x^{8}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{24} x^{4}+\frac {7}{120} x^{5}-\frac {19}{720} x^{6}+\frac {13}{1680} x^{7}-\frac {23}{13440} x^{8}\right ) y^{\prime }\left (0\right )+\frac {x^{2}}{2}-\frac {x^{3}}{6}+\frac {x^{4}}{24}-\frac {x^{5}}{120}+\frac {13 x^{6}}{720}-\frac {11 x^{7}}{1680}+\frac {x^{8}}{640}+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{12} x^{4}-\frac {1}{60} x^{5}+\frac {1}{360} x^{6}-\frac {1}{2520} x^{7}\right ) c_{1} +\left (x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{24} x^{4}+\frac {7}{120} x^{5}-\frac {19}{720} x^{6}+\frac {13}{1680} x^{7}\right ) c_{2} +\frac {x^{2}}{2}-\frac {x^{3}}{6}+\frac {x^{4}}{24}-\frac {x^{5}}{120}+\frac {13 x^{6}}{720}-\frac {11 x^{7}}{1680}+O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{12} x^{4}-\frac {1}{60} x^{5}+\frac {1}{360} x^{6}-\frac {1}{2520} x^{7}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{24} x^{4}+\frac {7}{120} x^{5}-\frac {19}{720} x^{6}+\frac {13}{1680} x^{7}\right ) D\left (y \right )\left (0\right )+\frac {x^{2}}{2}-\frac {x^{3}}{6}+\frac {x^{4}}{24}-\frac {x^{5}}{120}+\frac {13 x^{6}}{720}-\frac {11 x^{7}}{1680}+O\left (x^{8}\right ) \]

Problem 6431

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	\[ y = \left (1-\frac {1}{2} x^{2}+\frac {5}{24} x^{4}-\frac {17}{144} x^{6}+\frac {629}{8064} x^{8}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{6} x^{5}-\frac {13}{126} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}+\frac {5}{24} x^{4}-\frac {17}{144} x^{6}\right ) c_{1} +\left (x -\frac {1}{3} x^{3}+\frac {1}{6} x^{5}-\frac {13}{126} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}+\frac {5}{24} x^{4}-\frac {17}{144} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{6} x^{5}-\frac {13}{126} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6432

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	\[ y = \left (1+\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{60} x^{5}-\frac {1}{360} x^{6}-\frac {1}{840} x^{7}+\frac {1}{2520} x^{8}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{60} x^{5}+\frac {1}{360} x^{6}+\frac {1}{840} x^{7}-\frac {1}{2520} x^{8}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{60} x^{5}-\frac {1}{360} x^{6}-\frac {1}{840} x^{7}\right ) c_{1} +\left (x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{60} x^{5}+\frac {1}{360} x^{6}+\frac {1}{840} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{60} x^{5}-\frac {1}{360} x^{6}-\frac {1}{840} x^{7}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{60} x^{5}+\frac {1}{360} x^{6}+\frac {1}{840} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6433

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	\[ y = \left (1+x^{2}-\frac {1}{3} x^{4}+\frac {1}{5} x^{6}-\frac {1}{7} x^{8}\right ) y \left (0\right )+y^{\prime }\left (0\right ) x +O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1+x^{2}-\frac {1}{3} x^{4}+\frac {1}{5} x^{6}\right ) c_{1} +c_{2} x +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+x^{2}-\frac {1}{3} x^{4}+\frac {1}{5} x^{6}\right ) y \left (0\right )+D\left (y \right )\left (0\right ) x +O\left (x^{8}\right ) \]

Problem 6434

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime } x +y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}-\frac {1}{48} x^{6}+\frac {1}{384} x^{8}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{15} x^{5}-\frac {1}{105} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}-\frac {1}{48} x^{6}\right ) c_{1} +\left (x -\frac {1}{3} x^{3}+\frac {1}{15} x^{5}-\frac {1}{105} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}-\frac {1}{48} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{15} x^{5}-\frac {1}{105} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6435

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }-y x=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 1+\frac {x^{3}}{6}-\frac {x^{4}}{24}+\frac {x^{5}}{120}+\frac {x^{6}}{240}-\frac {x^{7}}{630}+\frac {x^{8}}{2880}+O\left (x^{8}\right ) \] Verified OK. \[ y = 1+\frac {x^{3}}{6}-\frac {x^{4}}{24}+\frac {x^{5}}{120}+\frac {x^{6}}{240}-\frac {x^{7}}{630}+O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1+\frac {1}{6} x^{3}-\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{240} x^{6}-\frac {1}{630} x^{7}+\operatorname {O}\left (x^{8}\right ) \]

Problem 6436

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }-y x=0} \] 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	\[ y = x -\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}-\frac {x^{5}}{30}+\frac {x^{6}}{90}-\frac {x^{7}}{1680}-\frac {x^{8}}{1920}+O\left (x^{8}\right ) \] Verified OK. \[ y = x -\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}-\frac {x^{5}}{30}+\frac {x^{6}}{90}-\frac {x^{7}}{1680}+O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{30} x^{5}+\frac {1}{90} x^{6}-\frac {1}{1680} x^{7}+\operatorname {O}\left (x^{8}\right ) \]

Problem 6437

ODE	\[ \boxed {y^{\prime \prime }+\left (p +\frac {1}{2}-\frac {x^{2}}{4}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{4} x^{2}+\frac {1}{32} x^{4}-\frac {1}{384} x^{6}+\frac {1}{6144} x^{8}+\frac {1}{40320} x^{8} p^{4}+\frac {1}{20160} x^{8} p^{3}+\frac {5}{16128} x^{8} p^{2}+\frac {23}{80640} x^{8} p -\frac {1}{2} x^{2} p +\frac {1}{24} p^{2} x^{4}+\frac {1}{24} p \,x^{4}-\frac {1}{720} x^{6} p^{3}-\frac {1}{480} x^{6} p^{2}-\frac {17}{2880} p \,x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{12} x^{3}+\frac {7}{480} x^{5}-\frac {3}{4480} x^{7}-\frac {1}{3360} x^{7} p^{2}-\frac {29}{20160} x^{7} p -\frac {1}{6} x^{3} p +\frac {1}{120} x^{5} p^{2}+\frac {1}{120} x^{5} p -\frac {1}{5040} x^{7} p^{3}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1+\left (-\frac {p}{2}-\frac {1}{4}\right ) x^{2}+\left (\frac {1}{32}+\frac {1}{24} p^{2}+\frac {1}{24} p \right ) x^{4}+\left (-\frac {17}{2880} p -\frac {1}{384}-\frac {1}{720} p^{3}-\frac {1}{480} p^{2}\right ) x^{6}\right ) c_{1} +\left (x +\left (-\frac {p}{6}-\frac {1}{12}\right ) x^{3}+\left (\frac {7}{480}+\frac {1}{120} p^{2}+\frac {1}{120} p \right ) x^{5}+\left (-\frac {29}{20160} p -\frac {3}{4480}-\frac {1}{5040} p^{3}-\frac {1}{3360} p^{2}\right ) x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {\left (1+2 p \right ) x^{2}}{4}+\frac {\left (4 p^{2}+4 p +3\right ) x^{4}}{96}-\frac {\left (8 p^{3}+12 p^{2}+34 p +15\right ) x^{6}}{5760}\right ) y \left (0\right )+\left (x -\frac {\left (1+2 p \right ) x^{3}}{12}+\frac {\left (4 p^{2}+4 p +7\right ) x^{5}}{480}-\frac {\left (8 p^{3}+12 p^{2}+58 p +27\right ) x^{7}}{40320}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6438

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

Problem 6439

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	\[ y = \left (1-\frac {1}{2} x^{2} p^{2}+\frac {1}{24} p^{4} x^{4}-\frac {1}{6} p^{2} x^{4}-\frac {1}{720} p^{6} x^{6}+\frac {1}{36} x^{6} p^{4}-\frac {4}{45} x^{6} p^{2}+\frac {1}{40320} x^{8} p^{8}-\frac {1}{720} x^{8} p^{6}+\frac {7}{360} x^{8} p^{4}-\frac {2}{35} x^{8} p^{2}\right ) y \left (0\right )+\left (x -\frac {1}{6} p^{2} x^{3}+\frac {1}{6} x^{3}+\frac {1}{120} p^{4} x^{5}-\frac {1}{12} x^{5} p^{2}+\frac {3}{40} x^{5}-\frac {1}{5040} x^{7} p^{6}+\frac {1}{144} x^{7} p^{4}-\frac {37}{720} x^{7} p^{2}+\frac {5}{112} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {x^{2} p^{2}}{2}+\left (\frac {1}{24} p^{4}-\frac {1}{6} p^{2}\right ) x^{4}+\left (-\frac {1}{720} p^{6}+\frac {1}{36} p^{4}-\frac {4}{45} p^{2}\right ) x^{6}\right ) c_{1} +\left (x +\left (-\frac {p^{2}}{6}+\frac {1}{6}\right ) x^{3}+\left (\frac {1}{120} p^{4}-\frac {1}{12} p^{2}+\frac {3}{40}\right ) x^{5}+\left (-\frac {1}{5040} p^{6}+\frac {1}{144} p^{4}-\frac {37}{720} p^{2}+\frac {5}{112}\right ) x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {p^{2} x^{2}}{2}+\frac {p^{2} \left (p^{2}-4\right ) x^{4}}{24}-\frac {p^{2} \left (p^{4}-20 p^{2}+64\right ) x^{6}}{720}\right ) y \left (0\right )+\left (x -\frac {\left (p^{2}-1\right ) x^{3}}{6}+\frac {\left (p^{4}-10 p^{2}+9\right ) x^{5}}{120}-\frac {\left (p^{6}-35 p^{4}+259 p^{2}-225\right ) x^{7}}{5040}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6440

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime } x +2 p y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-x^{2} p +\frac {1}{6} p^{2} x^{4}-\frac {1}{3} p \,x^{4}-\frac {1}{90} x^{6} p^{3}+\frac {1}{15} x^{6} p^{2}-\frac {4}{45} p \,x^{6}+\frac {1}{2520} x^{8} p^{4}-\frac {1}{210} x^{8} p^{3}+\frac {11}{630} x^{8} p^{2}-\frac {2}{105} x^{8} p \right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3} p +\frac {1}{3} x^{3}+\frac {1}{30} x^{5} p^{2}-\frac {2}{15} x^{5} p +\frac {1}{10} x^{5}-\frac {1}{630} x^{7} p^{3}+\frac {1}{70} x^{7} p^{2}-\frac {23}{630} x^{7} p +\frac {1}{42} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-x^{2} p +\left (\frac {1}{6} p^{2}-\frac {1}{3} p \right ) x^{4}+\left (-\frac {1}{90} p^{3}+\frac {1}{15} p^{2}-\frac {4}{45} p \right ) x^{6}\right ) c_{1} +\left (x +\left (-\frac {p}{3}+\frac {1}{3}\right ) x^{3}+\left (\frac {1}{30} p^{2}-\frac {2}{15} p +\frac {1}{10}\right ) x^{5}+\left (-\frac {1}{630} p^{3}+\frac {1}{70} p^{2}-\frac {23}{630} p +\frac {1}{42}\right ) x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-p \,x^{2}+\frac {p \left (p -2\right ) x^{4}}{6}-\frac {p \left (p -2\right ) \left (p -4\right ) x^{6}}{90}\right ) y \left (0\right )+\left (x -\frac {\left (p -1\right ) x^{3}}{3}+\frac {\left (p^{2}-4 p +3\right ) x^{5}}{30}-\frac {\left (p^{3}-9 p^{2}+23 p -15\right ) x^{7}}{630}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6441

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

Problem 6442

ODE	\[ \boxed {x^{2} \left (x^{2}-1\right ) y^{\prime \prime }-x \left (1-x \right ) y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{-\sqrt {2}} \left (1+\frac {\sqrt {2}\, x}{-1+2 \sqrt {2}}+\frac {\sqrt {2}\, x^{2}}{-5+3 \sqrt {2}}-\frac {2 \sqrt {2}\, x^{3}}{15-9 \sqrt {2}}+\frac {\left (-69+49 \sqrt {2}\right ) x^{4}}{-1104+780 \sqrt {2}}+\frac {\left (-414+293 \sqrt {2}\right ) x^{5}}{6108 \sqrt {2}-8640}+\frac {\left (3898-2757 \sqrt {2}\right ) x^{6}}{114408-80892 \sqrt {2}}-\frac {\sqrt {2}\, \left (-77567+54843 \sqrt {2}\right ) x^{7}}{126 \left (-1+2 \sqrt {2}\right ) \left (\sqrt {2}-1\right ) \left (-3+2 \sqrt {2}\right ) \left (\sqrt {2}-2\right ) \left (-5+2 \sqrt {2}\right ) \left (-3+\sqrt {2}\right ) \left (-7+2 \sqrt {2}\right )}+O\left (x^{8}\right )\right )+c_{2} x^{\sqrt {2}} \left (1+\frac {\sqrt {2}\, x}{1+2 \sqrt {2}}+\frac {\sqrt {2}\, x^{2}}{5+3 \sqrt {2}}+\frac {2 \sqrt {2}\, x^{3}}{15+9 \sqrt {2}}+\frac {\left (69+49 \sqrt {2}\right ) x^{4}}{1104+780 \sqrt {2}}+\frac {\left (414+293 \sqrt {2}\right ) x^{5}}{6108 \sqrt {2}+8640}+\frac {\left (3898+2757 \sqrt {2}\right ) x^{6}}{114408+80892 \sqrt {2}}+\frac {\sqrt {2}\, \left (77567+54843 \sqrt {2}\right ) x^{7}}{126 \left (1+2 \sqrt {2}\right ) \left (1+\sqrt {2}\right ) \left (3+2 \sqrt {2}\right ) \left (2+\sqrt {2}\right ) \left (5+2 \sqrt {2}\right ) \left (3+\sqrt {2}\right ) \left (7+2 \sqrt {2}\right )}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{-\sqrt {2}} \left (1+\frac {\sqrt {2}}{-1+2 \sqrt {2}} x +\frac {\sqrt {2}}{\left (1-2 \sqrt {2}\right ) \left (\sqrt {2}-1\right )} x^{2}+\frac {6 \sqrt {2}-8}{57 \sqrt {2}-81} x^{3}+\frac {-49 \sqrt {2}+69}{1104-780 \sqrt {2}} x^{4}+\frac {293 \sqrt {2}-414}{6108 \sqrt {2}-8640} x^{5}+\frac {-2757 \sqrt {2}+3898}{114408-80892 \sqrt {2}} x^{6}+\frac {1}{126} \frac {77567 \sqrt {2}-109686}{\left (-1+2 \sqrt {2}\right ) \left (\sqrt {2}-1\right ) \left (2 \sqrt {2}-3\right ) \left (-2+\sqrt {2}\right ) \left (-5+2 \sqrt {2}\right ) \left (-3+\sqrt {2}\right ) \left (-7+2 \sqrt {2}\right )} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x^{\sqrt {2}} \left (1+\frac {\sqrt {2}}{1+2 \sqrt {2}} x +\frac {\sqrt {2}}{5+3 \sqrt {2}} x^{2}+\frac {6 \sqrt {2}+8}{57 \sqrt {2}+81} x^{3}+\frac {49 \sqrt {2}+69}{1104+780 \sqrt {2}} x^{4}+\frac {293 \sqrt {2}+414}{6108 \sqrt {2}+8640} x^{5}+\frac {2757 \sqrt {2}+3898}{114408+80892 \sqrt {2}} x^{6}+\frac {1}{126} \frac {77567 \sqrt {2}+109686}{\left (1+2 \sqrt {2}\right ) \left (1+\sqrt {2}\right ) \left (3+2 \sqrt {2}\right ) \left (2+\sqrt {2}\right ) \left (5+2 \sqrt {2}\right ) \left (3+\sqrt {2}\right ) \left (7+2 \sqrt {2}\right )} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6443

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

Problem 6444

ODE	\[ \boxed {\left (3 x +1\right ) x y^{\prime \prime }-\left (1+x \right ) y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-2 x +\frac {17 x^{2}}{4}-\frac {289 x^{3}}{30}+\frac {5491 x^{4}}{240}-\frac {236113 x^{5}}{4200}+\frac {28569673 x^{6}}{201600}-\frac {28569673 x^{7}}{78400}+O\left (x^{8}\right )\right )+c_{2} \left (-x^{2} \left (1-2 x +\frac {17 x^{2}}{4}-\frac {289 x^{3}}{30}+\frac {5491 x^{4}}{240}-\frac {236113 x^{5}}{4200}+\frac {28569673 x^{6}}{201600}-\frac {28569673 x^{7}}{78400}+O\left (x^{8}\right )\right ) \ln \left (x \right )+1+2 x -\frac {5 x^{4}}{16}+\frac {2227 x^{5}}{1800}-\frac {27659 x^{6}}{7200}+\frac {9774983 x^{7}}{882000}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \left (1-2 x +\frac {17}{4} x^{2}-\frac {289}{30} x^{3}+\frac {5491}{240} x^{4}-\frac {236113}{4200} x^{5}+\frac {28569673}{201600} x^{6}-\frac {28569673}{78400} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (2 x^{2}-4 x^{3}+\frac {17}{2} x^{4}-\frac {289}{15} x^{5}+\frac {5491}{120} x^{6}-\frac {236113}{2100} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-2-4 x +6 x^{2}-12 x^{3}+\frac {209}{8} x^{4}-\frac {54247}{900} x^{5}+\frac {521849}{3600} x^{6}-\frac {158526173}{441000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right ) \]

Problem 6445

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	\[ y = \left (1-\frac {1}{6} x^{3}+\frac {1}{120} x^{5}+\frac {1}{180} x^{6}-\frac {1}{5040} x^{7}-\frac {13}{20160} x^{8}\right ) y \left (0\right )+\left (x -\frac {1}{12} x^{4}+\frac {1}{180} x^{6}+\frac {1}{504} x^{7}-\frac {1}{6720} x^{8}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{6} x^{3}+\frac {1}{120} x^{5}+\frac {1}{180} x^{6}-\frac {1}{5040} x^{7}\right ) c_{1} +\left (x -\frac {1}{12} x^{4}+\frac {1}{180} x^{6}+\frac {1}{504} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{6} x^{3}+\frac {1}{120} x^{5}+\frac {1}{180} x^{6}-\frac {1}{5040} x^{7}\right ) y \left (0\right )+\left (x -\frac {1}{12} x^{4}+\frac {1}{180} x^{6}+\frac {1}{504} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6446

ODE	\[ \boxed {x y^{\prime \prime }+\sin \left (x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{18} x^{4}-\frac {53}{10800} x^{6}+\frac {467}{1411200} x^{8}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {1}{60} x^{5}-\frac {19}{15120} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{18} x^{4}-\frac {53}{10800} x^{6}\right ) c_{1} +\left (x -\frac {1}{6} x^{3}+\frac {1}{60} x^{5}-\frac {19}{15120} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{18} x^{4}-\frac {53}{10800} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {1}{60} x^{5}-\frac {19}{15120} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6447

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\sin \left (x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {x}{2}+\frac {x^{2}}{12}+\frac {x^{3}}{144}-\frac {13 x^{4}}{2880}+\frac {29 x^{5}}{86400}+\frac {431 x^{6}}{3628800}-\frac {4961 x^{7}}{203212800}-\frac {5197 x^{8}}{4877107200}+O\left (x^{8}\right )\right )+c_{2} \left (-x \left (1-\frac {x}{2}+\frac {x^{2}}{12}+\frac {x^{3}}{144}-\frac {13 x^{4}}{2880}+\frac {29 x^{5}}{86400}+\frac {431 x^{6}}{3628800}-\frac {4961 x^{7}}{203212800}-\frac {5197 x^{8}}{4877107200}+O\left (x^{8}\right )\right ) \ln \left (x \right )+1-\frac {3 x^{2}}{4}+\frac {2 x^{3}}{9}-\frac {25 x^{4}}{1728}-\frac {689 x^{5}}{86400}+\frac {263 x^{6}}{162000}+\frac {71809 x^{7}}{762048000}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1-\frac {1}{2} x +\frac {1}{12} x^{2}+\frac {1}{144} x^{3}-\frac {13}{2880} x^{4}+\frac {29}{86400} x^{5}+\frac {431}{3628800} x^{6}-\frac {4961}{203212800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-x +\frac {1}{2} x^{2}-\frac {1}{12} x^{3}-\frac {1}{144} x^{4}+\frac {13}{2880} x^{5}-\frac {29}{86400} x^{6}-\frac {431}{3628800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (1-\frac {3}{4} x^{2}+\frac {2}{9} x^{3}-\frac {25}{1728} x^{4}-\frac {689}{86400} x^{5}+\frac {263}{162000} x^{6}+\frac {71809}{762048000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right ) \]

Problem 6448

ODE	\[ \boxed {x^{3} y^{\prime \prime }+\sin \left (x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{2}+\frac {i \sqrt {3}}{2}} \left (1+\frac {x^{2}}{12 i \sqrt {3}+24}+\frac {\left (-3 i \sqrt {3}-1\right ) x^{4}}{1440 \left (i \sqrt {3}+2\right ) \left (i \sqrt {3}+4\right )}+\frac {\left (9 i \sqrt {3}-115\right ) x^{6}}{362880 \left (i \sqrt {3}+2\right ) \left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+6\right )}+O\left (x^{8}\right )\right )+c_{2} x^{\frac {1}{2}-\frac {i \sqrt {3}}{2}} \left (1+\frac {x^{2}}{-12 i \sqrt {3}+24}+\frac {\left (3 i \sqrt {3}-1\right ) x^{4}}{1440 \left (2-i \sqrt {3}\right ) \left (-i \sqrt {3}+4\right )}+\frac {\left (-9 i \sqrt {3}-115\right ) x^{6}}{362880 \left (2-i \sqrt {3}\right ) \left (-i \sqrt {3}+4\right ) \left (-i \sqrt {3}+6\right )}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (c_{2} x^{\frac {i \sqrt {3}}{2}} \left (1+\frac {1}{12 i \sqrt {3}+24} x^{2}+\frac {1}{1440} \frac {-3 i \sqrt {3}-1}{\left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+2\right )} x^{4}+\frac {1}{362880} \frac {9 i \sqrt {3}-115}{\left (i \sqrt {3}+6\right ) \left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+2\right )} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_{1} x^{-\frac {i \sqrt {3}}{2}} \left (1-\frac {1}{12 i \sqrt {3}-24} x^{2}+\frac {-3 \sqrt {3}-i}{7200 i+8640 \sqrt {3}} x^{4}+\frac {9 \sqrt {3}-115 i}{4354560 i+14878080 \sqrt {3}} x^{6}+\operatorname {O}\left (x^{8}\right )\right )\right ) \]

Problem 6449

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

Problem 6450

ODE	\[ \boxed {x^{3} y^{\prime \prime }+\left (-1+\cos \left (2 x \right )\right ) y^{\prime }+2 y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-\frac {2 x^{2}}{9}+\frac {26 x^{4}}{675}-\frac {1742 x^{6}}{297675}+O\left (x^{8}\right )\right )+c_{2} x \left (1-\frac {x^{2}}{3}+\frac {17 x^{4}}{270}-\frac {173 x^{6}}{17010}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \left (1-\frac {2}{9} x^{2}+\frac {26}{675} x^{4}-\frac {1742}{297675} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x \left (1-\frac {1}{3} x^{2}+\frac {17}{270} x^{4}-\frac {173}{17010} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6451

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+\left (2 x^{4}-5 x \right ) y^{\prime }+\left (3 x^{2}+2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-\frac {x^{2}}{10}-\frac {4 x^{3}}{57}+\frac {3 x^{4}}{920}+\frac {32 x^{5}}{4275}+\frac {36287 x^{6}}{9753840}-\frac {4037 x^{7}}{16059750}+O\left (x^{8}\right )\right )+c_{2} x^{\frac {1}{4}} \left (1-\frac {3 x^{2}}{2}-\frac {x^{3}}{30}+\frac {x^{4}}{8}+\frac {137 x^{5}}{1300}-\frac {19 x^{6}}{12240}-\frac {7169 x^{7}}{764400}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{4}} \left (1-\frac {3}{2} x^{2}-\frac {1}{30} x^{3}+\frac {1}{8} x^{4}+\frac {137}{1300} x^{5}-\frac {19}{12240} x^{6}-\frac {7169}{764400} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x^{2} \left (1-\frac {1}{10} x^{2}-\frac {4}{57} x^{3}+\frac {3}{920} x^{4}+\frac {32}{4275} x^{5}+\frac {36287}{9753840} x^{6}-\frac {4037}{16059750} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6452

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

Problem 6453

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

Problem 6454

ODE	\[ \boxed {4 x y^{\prime \prime }+3 y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{4}} \left (1-\frac {x}{5}+\frac {x^{2}}{90}-\frac {x^{3}}{3510}+\frac {x^{4}}{238680}-\frac {x^{5}}{25061400}+\frac {x^{6}}{3759210000}-\frac {x^{7}}{763119630000}+O\left (x^{8}\right )\right )+c_{2} \left (1-\frac {x}{3}+\frac {x^{2}}{42}-\frac {x^{3}}{1386}+\frac {x^{4}}{83160}-\frac {x^{5}}{7900200}+\frac {x^{6}}{1090227600}-\frac {x^{7}}{206053016400}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{4}} \left (1-\frac {1}{5} x +\frac {1}{90} x^{2}-\frac {1}{3510} x^{3}+\frac {1}{238680} x^{4}-\frac {1}{25061400} x^{5}+\frac {1}{3759210000} x^{6}-\frac {1}{763119630000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1-\frac {1}{3} x +\frac {1}{42} x^{2}-\frac {1}{1386} x^{3}+\frac {1}{83160} x^{4}-\frac {1}{7900200} x^{5}+\frac {1}{1090227600} x^{6}-\frac {1}{206053016400} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6455

ODE	\[ \boxed {2 x y^{\prime \prime }+\left (-x +3\right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1+\frac {x}{3}+\frac {x^{2}}{15}+\frac {x^{3}}{105}+\frac {x^{4}}{945}+\frac {x^{5}}{10395}+\frac {x^{6}}{135135}+\frac {x^{7}}{2027025}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1+\frac {x}{2}+\frac {x^{2}}{8}+\frac {x^{3}}{48}+\frac {x^{4}}{384}+\frac {x^{5}}{3840}+\frac {x^{6}}{46080}+\frac {x^{7}}{645120}+O\left (x^{8}\right )\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1+\frac {1}{2} x +\frac {1}{8} x^{2}+\frac {1}{48} x^{3}+\frac {1}{384} x^{4}+\frac {1}{3840} x^{5}+\frac {1}{46080} x^{6}+\frac {1}{645120} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{\sqrt {x}}+c_{2} \left (1+\frac {1}{3} x +\frac {1}{15} x^{2}+\frac {1}{105} x^{3}+\frac {1}{945} x^{4}+\frac {1}{10395} x^{5}+\frac {1}{135135} x^{6}+\frac {1}{2027025} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6456

ODE	\[ \boxed {2 x y^{\prime \prime }+\left (1+x \right ) y^{\prime }+3 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {7 x}{6}+\frac {21 x^{2}}{40}-\frac {11 x^{3}}{80}+\frac {143 x^{4}}{5760}-\frac {13 x^{5}}{3840}+\frac {17 x^{6}}{46080}-\frac {323 x^{7}}{9676800}+O\left (x^{8}\right )\right )+c_{2} \left (1-3 x +2 x^{2}-\frac {2 x^{3}}{3}+\frac {x^{4}}{7}-\frac {x^{5}}{45}+\frac {4 x^{6}}{1485}-\frac {4 x^{7}}{15015}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1-\frac {7}{6} x +\frac {21}{40} x^{2}-\frac {11}{80} x^{3}+\frac {143}{5760} x^{4}-\frac {13}{3840} x^{5}+\frac {17}{46080} x^{6}-\frac {323}{9676800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1-3 x +2 x^{2}-\frac {2}{3} x^{3}+\frac {1}{7} x^{4}-\frac {1}{45} x^{5}+\frac {4}{1485} x^{6}-\frac {4}{15015} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6457

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }+y^{\prime } x -y \left (1+x \right )=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1+\frac {x}{5}+\frac {x^{2}}{70}+\frac {x^{3}}{1890}+\frac {x^{4}}{83160}+\frac {x^{5}}{5405400}+\frac {x^{6}}{486486000}+\frac {x^{7}}{57891834000}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1-x -\frac {x^{2}}{2}-\frac {x^{3}}{18}-\frac {x^{4}}{360}-\frac {x^{5}}{12600}-\frac {x^{6}}{680400}-\frac {x^{7}}{52390800}+O\left (x^{8}\right )\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-x -\frac {1}{2} x^{2}-\frac {1}{18} x^{3}-\frac {1}{360} x^{4}-\frac {1}{12600} x^{5}-\frac {1}{680400} x^{6}-\frac {1}{52390800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{\sqrt {x}}+c_{2} x \left (1+\frac {1}{5} x +\frac {1}{70} x^{2}+\frac {1}{1890} x^{3}+\frac {1}{83160} x^{4}+\frac {1}{5405400} x^{5}+\frac {1}{486486000} x^{6}+\frac {1}{57891834000} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6458

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

Problem 6459

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

Problem 6460

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 6461

ODE	\[ \boxed {x^{2} y^{\prime \prime }-3 y^{\prime } x +\left (4 x +4\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (4 x^{2}-4 x +1-\frac {16 x^{3}}{9}+\frac {4 x^{4}}{9}-\frac {16 x^{5}}{225}+\frac {16 x^{6}}{2025}-\frac {64 x^{7}}{99225}+O\left (x^{8}\right )\right )+c_{2} \left (x^{2} \left (4 x^{2}-4 x +1-\frac {16 x^{3}}{9}+\frac {4 x^{4}}{9}-\frac {16 x^{5}}{225}+\frac {16 x^{6}}{2025}-\frac {64 x^{7}}{99225}+O\left (x^{8}\right )\right ) \ln \left (x \right )+x^{2} \left (-12 x^{2}+8 x +\frac {176 x^{3}}{27}-\frac {50 x^{4}}{27}+\frac {1096 x^{5}}{3375}-\frac {392 x^{6}}{10125}+\frac {3872 x^{7}}{1157625}+O\left (x^{8}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-4 x +4 x^{2}-\frac {16}{9} x^{3}+\frac {4}{9} x^{4}-\frac {16}{225} x^{5}+\frac {16}{2025} x^{6}-\frac {64}{99225} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (8 x -12 x^{2}+\frac {176}{27} x^{3}-\frac {50}{27} x^{4}+\frac {1096}{3375} x^{5}-\frac {392}{10125} x^{6}+\frac {3872}{1157625} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \right ) x^{2} \]

Problem 6462

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }-8 x^{2} y^{\prime }+\left (4 x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1+x +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+\frac {x^{6}}{720}+\frac {x^{7}}{5040}+O\left (x^{8}\right )\right )+c_{2} \left (\sqrt {x}\, \left (1+x +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+\frac {x^{6}}{720}+\frac {x^{7}}{5040}+O\left (x^{8}\right )\right ) \ln \left (x \right )+\sqrt {x}\, O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}\right ) \left (c_{2} \ln \left (x \right )+c_{1} \right )+O\left (x^{8}\right ) \]

Problem 6463

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	\[ y = c_{1} \left (1-\frac {x^{2}}{6}+\frac {x^{4}}{120}-\frac {x^{6}}{5040}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1-\frac {x^{2}}{2}+\frac {x^{4}}{24}-\frac {x^{6}}{720}+O\left (x^{8}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (1-\frac {1}{6} x^{2}+\frac {1}{120} x^{4}-\frac {1}{5040} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}-\frac {1}{720} x^{6}+\operatorname {O}\left (x^{8}\right )\right )}{x} \]

Problem 6464

ODE	\[ \boxed {x^{2} y^{\prime \prime }-x^{2} y^{\prime }+\left (x^{2}-2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1+\frac {x}{2}+\frac {x^{2}}{20}-\frac {x^{3}}{60}-\frac {x^{4}}{210}-\frac {x^{5}}{3360}+\frac {x^{6}}{20160}+\frac {x^{7}}{100800}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1+\frac {x}{2}+\frac {x^{2}}{2}+\frac {5 x^{3}}{12}+\frac {x^{4}}{12}-\frac {x^{5}}{60}-\frac {x^{6}}{120}-\frac {x^{7}}{1120}+O\left (x^{8}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \left (1+\frac {1}{2} x +\frac {1}{20} x^{2}-\frac {1}{60} x^{3}-\frac {1}{210} x^{4}-\frac {1}{3360} x^{5}+\frac {1}{20160} x^{6}+\frac {1}{100800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (12+6 x +6 x^{2}+5 x^{3}+x^{4}-\frac {1}{5} x^{5}-\frac {1}{10} x^{6}-\frac {3}{280} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x} \]

Problem 6465

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

Problem 6466

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

Problem 6467

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

Problem 6468

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	\[ y = c_{1} x \left (1-\frac {x^{2}}{8}+\frac {x^{4}}{192}-\frac {x^{6}}{9216}+O\left (x^{8}\right )\right )+c_{2} \left (-\frac {x \left (1-\frac {x^{2}}{8}+\frac {x^{4}}{192}-\frac {x^{6}}{9216}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{2}+\frac {1-\frac {3 x^{4}}{64}+\frac {7 x^{6}}{2304}+O\left (x^{8}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{2} \left (1-\frac {1}{8} x^{2}+\frac {1}{192} x^{4}-\frac {1}{9216} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\left (x^{2}-\frac {1}{8} x^{4}+\frac {1}{192} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) \ln \left (x \right )+\left (-2+\frac {3}{32} x^{4}-\frac {7}{1152} x^{6}+\operatorname {O}\left (x^{8}\right )\right )\right )}{x} \]

Problem 6469

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	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {x^{2}}{6}+\frac {x^{4}}{120}-\frac {x^{6}}{5040}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1-\frac {x^{2}}{2}+\frac {x^{4}}{24}-\frac {x^{6}}{720}+O\left (x^{8}\right )\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-\frac {1}{6} x^{2}+\frac {1}{120} x^{4}-\frac {1}{5040} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) x +c_{2} \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}-\frac {1}{720} x^{6}+\operatorname {O}\left (x^{8}\right )\right )}{\sqrt {x}} \]

Problem 6470

ODE	\[ \boxed {x \left (1-x \right ) y^{\prime \prime }+\left (\frac {3}{2}-2 x \right ) y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-\frac {4 x}{3}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1-\frac {9 x}{2}+\frac {15 x^{2}}{8}+\frac {7 x^{3}}{16}+\frac {27 x^{4}}{128}+\frac {33 x^{5}}{256}+\frac {91 x^{6}}{1024}+\frac {135 x^{7}}{2048}+O\left (x^{8}\right )\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-\frac {9}{2} x +\frac {15}{8} x^{2}+\frac {7}{16} x^{3}+\frac {27}{128} x^{4}+\frac {33}{256} x^{5}+\frac {91}{1024} x^{6}+\frac {135}{2048} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{\sqrt {x}}+c_{2} \left (1-\frac {4}{3} x +\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6471

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

Problem 6472

ODE	\[ \boxed {\left (x^{2}-1\right ) y^{\prime \prime }+\left (5 x +4\right ) y^{\prime }+4 y=0} \] With the expansion point for the power series method at \(x = -1\).
program solution	\[ y = c_{1} \sqrt {1+x}\, \left (\frac {37}{12}+\frac {25 x}{12}+\frac {245 \left (1+x \right )^{2}}{96}+\frac {315 \left (1+x \right )^{3}}{128}+\frac {4235 \left (1+x \right )^{4}}{2048}+\frac {13013 \left (1+x \right )^{5}}{8192}+\frac {75075 \left (1+x \right )^{6}}{65536}+\frac {206635 \left (1+x \right )^{7}}{262144}+O\left (\left (1+x \right )^{8}\right )\right )+c_{2} \left (5+4 x +6 \left (1+x \right )^{2}+\frac {32 \left (1+x \right )^{3}}{5}+\frac {40 \left (1+x \right )^{4}}{7}+\frac {32 \left (1+x \right )^{5}}{7}+\frac {112 \left (1+x \right )^{6}}{33}+\frac {1024 \left (1+x \right )^{7}}{429}+O\left (\left (1+x \right )^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x +1}\, \left (1+\frac {25}{12} \left (x +1\right )+\frac {245}{96} \left (x +1\right )^{2}+\frac {315}{128} \left (x +1\right )^{3}+\frac {4235}{2048} \left (x +1\right )^{4}+\frac {13013}{8192} \left (x +1\right )^{5}+\frac {75075}{65536} \left (x +1\right )^{6}+\frac {206635}{262144} \left (x +1\right )^{7}+\operatorname {O}\left (\left (x +1\right )^{8}\right )\right )+c_{2} \left (1+4 \left (x +1\right )+6 \left (x +1\right )^{2}+\frac {32}{5} \left (x +1\right )^{3}+\frac {40}{7} \left (x +1\right )^{4}+\frac {32}{7} \left (x +1\right )^{5}+\frac {112}{33} \left (x +1\right )^{6}+\frac {1024}{429} \left (x +1\right )^{7}+\operatorname {O}\left (\left (x +1\right )^{8}\right )\right ) \]

Problem 6473

ODE	\[ \boxed {\left (x^{2}-x -6\right ) y^{\prime \prime }+\left (3 x +5\right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 3\).
program solution	\[ y = c_{1} \left (\frac {17}{14}-\frac {x}{14}+\frac {\left (x -3\right )^{2}}{133}-\frac {\left (x -3\right )^{3}}{1064}+\frac {\left (x -3\right )^{4}}{7714}-\frac {5 \left (x -3\right )^{5}}{262276}+\frac {5 \left (x -3\right )^{6}}{1704794}-\frac {5 \left (x -3\right )^{7}}{10715848}+O\left (\left (x -3\right )^{8}\right )\right )+\frac {c_{2} \left (\frac {13}{25}+\frac {4 x}{25}-\frac {2 \left (x -3\right )^{2}}{625}+\frac {4 \left (x -3\right )^{3}}{15625}-\frac {11 \left (x -3\right )^{4}}{390625}+\frac {176 \left (x -3\right )^{5}}{48828125}-\frac {616 \left (x -3\right )^{6}}{1220703125}+\frac {2288 \left (x -3\right )^{7}}{30517578125}+O\left (\left (x -3\right )^{8}\right )\right )}{\left (x -3\right )^{\frac {9}{5}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1+\frac {4}{25} \left (x -3\right )-\frac {2}{625} \left (x -3\right )^{2}+\frac {4}{15625} \left (x -3\right )^{3}-\frac {11}{390625} \left (x -3\right )^{4}+\frac {176}{48828125} \left (x -3\right )^{5}-\frac {616}{1220703125} \left (x -3\right )^{6}+\frac {2288}{30517578125} \left (x -3\right )^{7}+\operatorname {O}\left (\left (x -3\right )^{8}\right )\right )}{\left (x -3\right )^{\frac {9}{5}}}+c_{2} \left (1-\frac {1}{14} \left (x -3\right )+\frac {1}{133} \left (x -3\right )^{2}-\frac {1}{1064} \left (x -3\right )^{3}+\frac {1}{7714} \left (x -3\right )^{4}-\frac {5}{262276} \left (x -3\right )^{5}+\frac {5}{1704794} \left (x -3\right )^{6}-\frac {5}{10715848} \left (x -3\right )^{7}+\operatorname {O}\left (\left (x -3\right )^{8}\right )\right ) \]

Problem 6474

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 \(x = 1\).
program solution	\[ y = c_{1} \sqrt {x -1}\, \left (1+\left (\frac {p^{2}}{3}-\frac {1}{12}\right ) \left (x -1\right )+\left (\frac {1}{30} p^{4}-\frac {1}{12} p^{2}+\frac {3}{160}\right ) \left (x -1\right )^{2}+\left (\frac {1}{630} p^{6}-\frac {1}{72} p^{4}+\frac {37}{1440} p^{2}-\frac {5}{896}\right ) \left (x -1\right )^{3}+\frac {\left (4 p^{2}-49\right ) \left (4 p^{2}-25\right ) \left (4 p^{2}-9\right ) \left (4 p^{2}-1\right ) \left (x -1\right )^{4}}{5806080}+\frac {\left (4 p^{2}-81\right ) \left (4 p^{2}-49\right ) \left (4 p^{2}-25\right ) \left (4 p^{2}-9\right ) \left (4 p^{2}-1\right ) \left (x -1\right )^{5}}{1277337600}+\frac {\left (4 p^{2}-121\right ) \left (4 p^{2}-81\right ) \left (4 p^{2}-49\right ) \left (4 p^{2}-25\right ) \left (4 p^{2}-9\right ) \left (4 p^{2}-1\right ) \left (x -1\right )^{6}}{398529331200}+\frac {\left (4 p^{2}-169\right ) \left (4 p^{2}-121\right ) \left (4 p^{2}-81\right ) \left (4 p^{2}-49\right ) \left (4 p^{2}-25\right ) \left (4 p^{2}-9\right ) \left (4 p^{2}-1\right ) \left (x -1\right )^{7}}{167382319104000}+O\left (\left (x -1\right )^{8}\right )\right )+c_{2} \left (1+p^{2} \left (x -1\right )+\left (\frac {1}{6} p^{4}-\frac {1}{6} p^{2}\right ) \left (x -1\right )^{2}+\frac {p^{2} \left (p^{4}-5 p^{2}+4\right ) \left (x -1\right )^{3}}{90}+\frac {p^{2} \left (p^{6}-14 p^{4}+49 p^{2}-36\right ) \left (x -1\right )^{4}}{2520}+\frac {p^{2} \left (p^{8}-30 p^{6}+273 p^{4}-820 p^{2}+576\right ) \left (x -1\right )^{5}}{113400}+\frac {\left (p^{2}-25\right ) \left (p^{2}-16\right ) \left (p^{2}-9\right ) \left (p^{2}-4\right ) \left (p^{2}-1\right ) p^{2} \left (x -1\right )^{6}}{7484400}+\frac {\left (p^{2}-36\right ) \left (p^{2}-25\right ) \left (p^{2}-16\right ) \left (p^{2}-9\right ) \left (p^{2}-4\right ) \left (p^{2}-1\right ) p^{2} \left (x -1\right )^{7}}{681080400}+O\left (\left (x -1\right )^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x -1}\, \left (1+\left (\frac {p^{2}}{3}-\frac {1}{12}\right ) \left (x -1\right )+\left (\frac {1}{30} p^{4}-\frac {1}{12} p^{2}+\frac {3}{160}\right ) \left (x -1\right )^{2}+\left (\frac {1}{630} p^{6}-\frac {1}{72} p^{4}+\frac {37}{1440} p^{2}-\frac {5}{896}\right ) \left (x -1\right )^{3}+\left (\frac {1}{22680} p^{8}-\frac {1}{1080} p^{6}+\frac {47}{8640} p^{4}-\frac {3229}{362880} p^{2}+\frac {35}{18432}\right ) \left (x -1\right )^{4}+\left (\frac {1}{1247400} p^{10}-\frac {1}{30240} p^{8}+\frac {19}{43200} p^{6}-\frac {1571}{725760} p^{4}+\frac {10679}{3225600} p^{2}-\frac {63}{90112}\right ) \left (x -1\right )^{5}+\left (\frac {1}{97297200} p^{12}-\frac {1}{1360800} p^{10}+\frac {67}{3628800} p^{8}-\frac {2159}{10886400} p^{6}+\frac {153617}{174182400} p^{4}-\frac {550499}{425779200} p^{2}+\frac {231}{851968}\right ) \left (x -1\right )^{6}+\left (\frac {1}{10216206000} p^{14}-\frac {1}{89812800} p^{12}+\frac {11}{23328000} p^{10}-\frac {8521}{914457600} p^{8}+\frac {230443}{2612736000} p^{6}-\frac {1206053}{3284582400} p^{4}+\frac {2430898831}{4649508864000} p^{2}-\frac {143}{1310720}\right ) \left (x -1\right )^{7}+\operatorname {O}\left (\left (x -1\right )^{8}\right )\right )+c_{2} \left (1+p^{2} \left (x -1\right )+\left (\frac {1}{6} p^{4}-\frac {1}{6} p^{2}\right ) \left (x -1\right )^{2}+\left (\frac {1}{90} p^{6}-\frac {1}{18} p^{4}+\frac {2}{45} p^{2}\right ) \left (x -1\right )^{3}+\left (\frac {1}{2520} p^{8}-\frac {1}{180} p^{6}+\frac {7}{360} p^{4}-\frac {1}{70} p^{2}\right ) \left (x -1\right )^{4}+\left (\frac {1}{113400} p^{10}-\frac {1}{3780} p^{8}+\frac {13}{5400} p^{6}-\frac {41}{5670} p^{4}+\frac {8}{1575} p^{2}\right ) \left (x -1\right )^{5}+\left (\frac {1}{7484400} p^{12}-\frac {1}{136080} p^{10}+\frac {31}{226800} p^{8}-\frac {139}{136080} p^{6}+\frac {479}{170100} p^{4}-\frac {4}{2079} p^{2}\right ) \left (x -1\right )^{6}+\left (\frac {1}{681080400} p^{14}-\frac {1}{7484400} p^{12}+\frac {1}{226800} p^{10}-\frac {311}{4762800} p^{8}+\frac {37}{85050} p^{6}-\frac {59}{51975} p^{4}+\frac {16}{21021} p^{2}\right ) \left (x -1\right )^{7}+\operatorname {O}\left (\left (x -1\right )^{8}\right )\right ) \]

Problem 6475

ODE	\[ \boxed {\left (1-{\mathrm e}^{x}\right ) y^{\prime \prime }+\frac {y^{\prime }}{2}+{\mathrm e}^{x} y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {3}{2}} \left (1+\frac {x}{4}+\frac {3 x^{2}}{32}+\frac {7 x^{3}}{384}+\frac {109 x^{4}}{30720}+\frac {13 x^{5}}{24576}+\frac {4439 x^{6}}{61931520}+\frac {2069 x^{7}}{247726080}+\frac {685613 x^{8}}{753087283200}+O\left (x^{8}\right )\right )+c_{2} \left (1-2 x -x^{2}-\frac {x^{3}}{3}-\frac {x^{4}}{12}-\frac {x^{5}}{60}-\frac {x^{6}}{360}-\frac {x^{7}}{2520}-\frac {x^{8}}{20160}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {3}{2}} \left (1+\frac {1}{4} x +\frac {3}{32} x^{2}+\frac {7}{384} x^{3}+\frac {109}{30720} x^{4}+\frac {13}{24576} x^{5}+\frac {4439}{61931520} x^{6}+\frac {2069}{247726080} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1-2 x -x^{2}-\frac {1}{3} x^{3}-\frac {1}{12} x^{4}-\frac {1}{60} x^{5}-\frac {1}{360} x^{6}-\frac {1}{2520} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6476

ODE	\[ \boxed {y^{\prime \prime }+2 y x=x^{2}} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{3} x^{3}+\frac {1}{45} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{4}+\frac {1}{126} x^{7}\right ) y^{\prime }\left (0\right )+\frac {x^{4}}{12}-\frac {x^{7}}{252}+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{3} x^{3}+\frac {1}{45} x^{6}\right ) c_{1} +\left (x -\frac {1}{6} x^{4}+\frac {1}{126} x^{7}\right ) c_{2} +\frac {x^{4}}{12}-\frac {x^{7}}{252}+O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{3} x^{3}+\frac {1}{45} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{4}+\frac {1}{126} x^{7}\right ) D\left (y \right )\left (0\right )+\frac {x^{4}}{12}-\frac {x^{7}}{252}+O\left (x^{8}\right ) \]

Problem 6477

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime } x +y=x} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{24} x^{4}-\frac {1}{240} x^{6}-\frac {1}{2688} x^{8}\right ) y \left (0\right )+y^{\prime }\left (0\right ) x +\frac {x^{3}}{6}+\frac {x^{5}}{60}+\frac {x^{7}}{630}+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{24} x^{4}-\frac {1}{240} x^{6}\right ) c_{1} +c_{2} x +\frac {x^{3}}{6}+\frac {x^{5}}{60}+\frac {x^{7}}{630}+O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}-\frac {1}{24} x^{4}-\frac {1}{240} x^{6}\right ) y \left (0\right )+D\left (y \right )\left (0\right ) x +\frac {x^{3}}{6}+\frac {x^{5}}{60}+\frac {x^{7}}{630}+O\left (x^{8}\right ) \]

Problem 6478

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }+y=x^{3}-x} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{120} x^{5}+\frac {1}{720} x^{6}-\frac {1}{40320} x^{8}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\frac {1}{5040} x^{7}-\frac {1}{40320} x^{8}\right ) y^{\prime }\left (0\right )-\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{20}-\frac {7 x^{6}}{720}+\frac {x^{7}}{5040}+\frac {x^{8}}{6720}+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{120} x^{5}+\frac {1}{720} x^{6}\right ) c_{1} +\left (x -\frac {1}{2} x^{2}+\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\frac {1}{5040} x^{7}\right ) c_{2} -\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{20}-\frac {7 x^{6}}{720}+\frac {x^{7}}{5040}+O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{120} x^{5}+\frac {1}{720} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\frac {1}{5040} x^{7}\right ) D\left (y \right )\left (0\right )-\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{20}-\frac {7 x^{6}}{720}+\frac {x^{7}}{5040}+O\left (x^{8}\right ) \]

Problem 6479

ODE	\[ \boxed {2 y^{\prime \prime }+y^{\prime } x +y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{4} x^{2}+\frac {1}{32} x^{4}-\frac {1}{384} x^{6}+\frac {1}{6144} x^{8}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {1}{60} x^{5}-\frac {1}{840} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{4} x^{2}+\frac {1}{32} x^{4}-\frac {1}{384} x^{6}\right ) c_{1} +\left (x -\frac {1}{6} x^{3}+\frac {1}{60} x^{5}-\frac {1}{840} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{4} x^{2}+\frac {1}{32} x^{4}-\frac {1}{384} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {1}{60} x^{5}-\frac {1}{840} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6480

ODE	\[ \boxed {\left (x^{2}+4\right ) y^{\prime \prime }-y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{8} x^{2}-\frac {1}{96} x^{3}+\frac {11}{1536} x^{4}+\frac {13}{10240} x^{5}-\frac {533}{737280} x^{6}-\frac {3809}{20643840} x^{7}+\frac {20761}{220200960} x^{8}\right ) y \left (0\right )+\left (x +\frac {1}{8} x^{2}-\frac {1}{32} x^{3}-\frac {5}{512} x^{4}+\frac {23}{10240} x^{5}+\frac {283}{245760} x^{6}-\frac {1649}{6881280} x^{7}-\frac {12247}{73400320} x^{8}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{8} x^{2}-\frac {1}{96} x^{3}+\frac {11}{1536} x^{4}+\frac {13}{10240} x^{5}-\frac {533}{737280} x^{6}-\frac {3809}{20643840} x^{7}\right ) c_{1} +\left (x +\frac {1}{8} x^{2}-\frac {1}{32} x^{3}-\frac {5}{512} x^{4}+\frac {23}{10240} x^{5}+\frac {283}{245760} x^{6}-\frac {1649}{6881280} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{8} x^{2}-\frac {1}{96} x^{3}+\frac {11}{1536} x^{4}+\frac {13}{10240} x^{5}-\frac {533}{737280} x^{6}-\frac {3809}{20643840} x^{7}\right ) y \left (0\right )+\left (x +\frac {1}{8} x^{2}-\frac {1}{32} x^{3}-\frac {5}{512} x^{4}+\frac {23}{10240} x^{5}+\frac {283}{245760} x^{6}-\frac {1649}{6881280} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6481

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	\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}-\frac {1}{80} x^{6}+\frac {5}{896} x^{8}\right ) y \left (0\right )+y^{\prime }\left (0\right ) x +O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}-\frac {1}{80} x^{6}\right ) c_{1} +c_{2} x +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}-\frac {1}{80} x^{6}\right ) y \left (0\right )+D\left (y \right )\left (0\right ) x +O\left (x^{8}\right ) \]

Problem 6482

ODE	\[ \boxed {y^{\prime \prime }-\left (1+x \right ) y^{\prime }-y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{30} x^{5}+\frac {1}{60} x^{6}+\frac {37}{5040} x^{7}+\frac {19}{5760} x^{8}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{3} x^{3}+\frac {1}{4} x^{4}+\frac {1}{8} x^{5}+\frac {47}{720} x^{6}+\frac {19}{630} x^{7}+\frac {131}{10080} x^{8}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{30} x^{5}+\frac {1}{60} x^{6}+\frac {37}{5040} x^{7}\right ) c_{1} +\left (x +\frac {1}{2} x^{2}+\frac {1}{3} x^{3}+\frac {1}{4} x^{4}+\frac {1}{8} x^{5}+\frac {47}{720} x^{6}+\frac {19}{630} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{30} x^{5}+\frac {1}{60} x^{6}+\frac {37}{5040} x^{7}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{3} x^{3}+\frac {1}{4} x^{4}+\frac {1}{8} x^{5}+\frac {47}{720} x^{6}+\frac {19}{630} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6483

ODE	\[ \boxed {\left (x -1\right ) y^{\prime \prime }+\left (1+x \right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{3} x^{3}+\frac {3}{8} x^{4}+\frac {11}{30} x^{5}+\frac {53}{144} x^{6}+\frac {103}{280} x^{7}+\frac {2119}{5760} x^{8}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {2}{3} x^{3}+\frac {5}{8} x^{4}+\frac {19}{30} x^{5}+\frac {91}{144} x^{6}+\frac {177}{280} x^{7}+\frac {3641}{5760} x^{8}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{3} x^{3}+\frac {3}{8} x^{4}+\frac {11}{30} x^{5}+\frac {53}{144} x^{6}+\frac {103}{280} x^{7}\right ) c_{1} +\left (x +\frac {1}{2} x^{2}+\frac {2}{3} x^{3}+\frac {5}{8} x^{4}+\frac {19}{30} x^{5}+\frac {91}{144} x^{6}+\frac {177}{280} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{2} x^{2}+\frac {1}{3} x^{3}+\frac {3}{8} x^{4}+\frac {11}{30} x^{5}+\frac {53}{144} x^{6}+\frac {103}{280} x^{7}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {2}{3} x^{3}+\frac {5}{8} x^{4}+\frac {19}{30} x^{5}+\frac {91}{144} x^{6}+\frac {177}{280} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6484

ODE	\[ \boxed {\left (x^{2}+1\right ) x^{2} y^{\prime \prime }-y^{\prime } x +\left (x +2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{1+i} \left (1+\left (-\frac {1}{5}+\frac {2 i}{5}\right ) x +\left (-\frac {1}{40}-\frac {13 i}{40}\right ) x^{2}+\left (\frac {71}{520}-\frac {17 i}{520}\right ) x^{3}+\left (-\frac {31}{832}+\frac {541 i}{4160}\right ) x^{4}+\left (-\frac {1423}{20800}-\frac {7 i}{4160}\right ) x^{5}+\left (\frac {12849}{416000}-\frac {10853 i}{156000}\right ) x^{6}+\left (\frac {209609}{5088000}+\frac {106907 i}{17808000}\right ) x^{7}+O\left (x^{8}\right )\right )+c_{2} x^{1-i} \left (1+\left (-\frac {1}{5}-\frac {2 i}{5}\right ) x +\left (-\frac {1}{40}+\frac {13 i}{40}\right ) x^{2}+\left (\frac {71}{520}+\frac {17 i}{520}\right ) x^{3}+\left (-\frac {31}{832}-\frac {541 i}{4160}\right ) x^{4}+\left (-\frac {1423}{20800}+\frac {7 i}{4160}\right ) x^{5}+\left (\frac {12849}{416000}+\frac {10853 i}{156000}\right ) x^{6}+\left (\frac {209609}{5088000}-\frac {106907 i}{17808000}\right ) x^{7}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{1-i} \left (1+\left (-\frac {1}{5}-\frac {2 i}{5}\right ) x +\left (-\frac {1}{40}+\frac {13 i}{40}\right ) x^{2}+\left (\frac {71}{520}+\frac {17 i}{520}\right ) x^{3}+\left (-\frac {31}{832}-\frac {541 i}{4160}\right ) x^{4}+\left (-\frac {1423}{20800}+\frac {7 i}{4160}\right ) x^{5}+\left (\frac {12849}{416000}+\frac {10853 i}{156000}\right ) x^{6}+\left (\frac {209609}{5088000}-\frac {106907 i}{17808000}\right ) x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x^{1+i} \left (1+\left (-\frac {1}{5}+\frac {2 i}{5}\right ) x +\left (-\frac {1}{40}-\frac {13 i}{40}\right ) x^{2}+\left (\frac {71}{520}-\frac {17 i}{520}\right ) x^{3}+\left (-\frac {31}{832}+\frac {541 i}{4160}\right ) x^{4}+\left (-\frac {1423}{20800}-\frac {7 i}{4160}\right ) x^{5}+\left (\frac {12849}{416000}-\frac {10853 i}{156000}\right ) x^{6}+\left (\frac {209609}{5088000}+\frac {106907 i}{17808000}\right ) x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6485

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	\[ y = c_{1} x^{i} \left (1+\left (-\frac {1}{5}+\frac {2 i}{5}\right ) x +\left (-\frac {1}{40}-\frac {3 i}{40}\right ) x^{2}+\left (\frac {3}{520}+\frac {7 i}{1560}\right ) x^{3}+\left (-\frac {1}{2496}-\frac {i}{12480}\right ) x^{4}+\left (\frac {9}{603200}-\frac {i}{361920}\right ) x^{5}+\left (-\frac {19}{54288000}+\frac {7 i}{36192000}\right ) x^{6}+\left (\frac {1}{179829000}-\frac {223 i}{40281696000}\right ) x^{7}+O\left (x^{8}\right )\right )+c_{2} x^{-i} \left (1+\left (-\frac {1}{5}-\frac {2 i}{5}\right ) x +\left (-\frac {1}{40}+\frac {3 i}{40}\right ) x^{2}+\left (\frac {3}{520}-\frac {7 i}{1560}\right ) x^{3}+\left (-\frac {1}{2496}+\frac {i}{12480}\right ) x^{4}+\left (\frac {9}{603200}+\frac {i}{361920}\right ) x^{5}+\left (-\frac {19}{54288000}-\frac {7 i}{36192000}\right ) x^{6}+\left (\frac {1}{179829000}+\frac {223 i}{40281696000}\right ) x^{7}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{-i} \left (1+\left (-\frac {1}{5}-\frac {2 i}{5}\right ) x +\left (-\frac {1}{40}+\frac {3 i}{40}\right ) x^{2}+\left (\frac {3}{520}-\frac {7 i}{1560}\right ) x^{3}+\left (-\frac {1}{2496}+\frac {i}{12480}\right ) x^{4}+\left (\frac {9}{603200}+\frac {i}{361920}\right ) x^{5}+\left (-\frac {19}{54288000}-\frac {7 i}{36192000}\right ) x^{6}+\left (\frac {1}{179829000}+\frac {223 i}{40281696000}\right ) x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x^{i} \left (1+\left (-\frac {1}{5}+\frac {2 i}{5}\right ) x +\left (-\frac {1}{40}-\frac {3 i}{40}\right ) x^{2}+\left (\frac {3}{520}+\frac {7 i}{1560}\right ) x^{3}+\left (-\frac {1}{2496}-\frac {i}{12480}\right ) x^{4}+\left (\frac {9}{603200}-\frac {i}{361920}\right ) x^{5}+\left (-\frac {19}{54288000}+\frac {7 i}{36192000}\right ) x^{6}+\left (\frac {1}{179829000}-\frac {223 i}{40281696000}\right ) x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6486

ODE	\[ \boxed {x y^{\prime \prime }-4 y^{\prime }+y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{5} \left (1-\frac {x^{2}}{14}+\frac {x^{4}}{504}-\frac {x^{6}}{33264}+O\left (x^{8}\right )\right )+c_{2} \left (1+\frac {x^{2}}{6}+\frac {x^{4}}{24}-\frac {x^{6}}{144}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{5} \left (1-\frac {1}{14} x^{2}+\frac {1}{504} x^{4}-\frac {1}{33264} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (2880+480 x^{2}+120 x^{4}-20 x^{6}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6487

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+4 x^{2} y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{2}+\frac {i}{2}} \left (1-\frac {x}{2}+\left (\frac {7}{40}-\frac {i}{40}\right ) x^{2}+\left (-\frac {11}{240}+\frac {i}{80}\right ) x^{3}+\left (\frac {31}{3264}-\frac {i}{272}\right ) x^{4}+\left (-\frac {53}{32640}+\frac {13 i}{16320}\right ) x^{5}+\left (\frac {3421}{14492160}-\frac {223 i}{1610240}\right ) x^{6}+\left (-\frac {30269}{1014451200}+\frac {977 i}{48307200}\right ) x^{7}+O\left (x^{8}\right )\right )+c_{2} x^{\frac {1}{2}-\frac {i}{2}} \left (1-\frac {x}{2}+\left (\frac {7}{40}+\frac {i}{40}\right ) x^{2}+\left (-\frac {11}{240}-\frac {i}{80}\right ) x^{3}+\left (\frac {31}{3264}+\frac {i}{272}\right ) x^{4}+\left (-\frac {53}{32640}-\frac {13 i}{16320}\right ) x^{5}+\left (\frac {3421}{14492160}+\frac {223 i}{1610240}\right ) x^{6}+\left (-\frac {30269}{1014451200}-\frac {977 i}{48307200}\right ) x^{7}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{2}-\frac {i}{2}} \left (1-\frac {1}{2} x +\left (\frac {7}{40}+\frac {i}{40}\right ) x^{2}+\left (-\frac {11}{240}-\frac {i}{80}\right ) x^{3}+\left (\frac {31}{3264}+\frac {i}{272}\right ) x^{4}+\left (-\frac {53}{32640}-\frac {13 i}{16320}\right ) x^{5}+\left (\frac {3421}{14492160}+\frac {223 i}{1610240}\right ) x^{6}+\left (-\frac {30269}{1014451200}-\frac {977 i}{48307200}\right ) x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x^{\frac {1}{2}+\frac {i}{2}} \left (1-\frac {1}{2} x +\left (\frac {7}{40}-\frac {i}{40}\right ) x^{2}+\left (-\frac {11}{240}+\frac {i}{80}\right ) x^{3}+\left (\frac {31}{3264}-\frac {i}{272}\right ) x^{4}+\left (-\frac {53}{32640}+\frac {13 i}{16320}\right ) x^{5}+\left (\frac {3421}{14492160}-\frac {223 i}{1610240}\right ) x^{6}+\left (-\frac {30269}{1014451200}+\frac {977 i}{48307200}\right ) x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6488

ODE	\[ \boxed {2 x y^{\prime \prime }+\left (1-x \right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {x}{6}-\frac {x^{2}}{120}-\frac {x^{3}}{1680}-\frac {x^{4}}{24192}-\frac {x^{5}}{380160}-\frac {x^{6}}{6589440}-\frac {x^{7}}{125798400}+O\left (x^{8}\right )\right )+c_{2} \left (1-x +O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1-\frac {1}{6} x -\frac {1}{120} x^{2}-\frac {1}{1680} x^{3}-\frac {1}{24192} x^{4}-\frac {1}{380160} x^{5}-\frac {1}{6589440} x^{6}-\frac {1}{125798400} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1-x +\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6489

ODE	\[ \boxed {x y^{\prime \prime }-\left (x -1\right ) y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (-2 x +1+\frac {x^{2}}{2}+O\left (x^{8}\right )\right )+c_{2} \left (\left (-2 x +1+\frac {x^{2}}{2}+O\left (x^{8}\right )\right ) \ln \left (x \right )+5 x -\frac {9 x^{2}}{4}+\frac {x^{3}}{18}+\frac {x^{4}}{288}+\frac {x^{5}}{3600}+\frac {x^{6}}{43200}+\frac {x^{7}}{529200}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-2 x +\frac {1}{2} x^{2}+\operatorname {O}\left (x^{8}\right )\right )+\left (5 x -\frac {9}{4} x^{2}+\frac {1}{18} x^{3}+\frac {1}{288} x^{4}+\frac {1}{3600} x^{5}+\frac {1}{43200} x^{6}+\frac {1}{529200} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \]

Problem 6490

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x \left (1-x \right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{i} \left (1+\left (\frac {2}{5}+\frac {i}{5}\right ) x +\left (\frac {1}{10}+\frac {i}{20}\right ) x^{2}+\left (\frac {17}{780}+\frac {i}{130}\right ) x^{3}+\left (\frac {5}{1248}+\frac {i}{1248}\right ) x^{4}+\left (\frac {113}{180960}+\frac {7 i}{180960}\right ) x^{5}+\left (\frac {911}{10857600}-\frac {19 i}{3619200}\right ) x^{6}+\left (\frac {39799}{4028169600}-\frac {1009 i}{575452800}\right ) x^{7}+O\left (x^{8}\right )\right )+c_{2} x^{-i} \left (1+\left (\frac {2}{5}-\frac {i}{5}\right ) x +\left (\frac {1}{10}-\frac {i}{20}\right ) x^{2}+\left (\frac {17}{780}-\frac {i}{130}\right ) x^{3}+\left (\frac {5}{1248}-\frac {i}{1248}\right ) x^{4}+\left (\frac {113}{180960}-\frac {7 i}{180960}\right ) x^{5}+\left (\frac {911}{10857600}+\frac {19 i}{3619200}\right ) x^{6}+\left (\frac {39799}{4028169600}+\frac {1009 i}{575452800}\right ) x^{7}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{-i} \left (1+\left (\frac {2}{5}-\frac {i}{5}\right ) x +\left (\frac {1}{10}-\frac {i}{20}\right ) x^{2}+\left (\frac {17}{780}-\frac {i}{130}\right ) x^{3}+\left (\frac {5}{1248}-\frac {i}{1248}\right ) x^{4}+\left (\frac {113}{180960}-\frac {7 i}{180960}\right ) x^{5}+\left (\frac {911}{10857600}+\frac {19 i}{3619200}\right ) x^{6}+\left (\frac {39799}{4028169600}+\frac {1009 i}{575452800}\right ) x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x^{i} \left (1+\left (\frac {2}{5}+\frac {i}{5}\right ) x +\left (\frac {1}{10}+\frac {i}{20}\right ) x^{2}+\left (\frac {17}{780}+\frac {i}{130}\right ) x^{3}+\left (\frac {5}{1248}+\frac {i}{1248}\right ) x^{4}+\left (\frac {113}{180960}+\frac {7 i}{180960}\right ) x^{5}+\left (\frac {911}{10857600}-\frac {19 i}{3619200}\right ) x^{6}+\left (\frac {39799}{4028169600}-\frac {1009 i}{575452800}\right ) x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6491

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

Problem 6492

ODE	Solve \begin {gather*} \boxed {x^{3} y^{\prime \prime \prime }+2 x^{2} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }+y x=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
program solution	N/A
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, x^{-\frac {i \sqrt {3}}{2}} \left (1+\frac {1}{i \sqrt {3}-1} x -\frac {1}{2+6 i \sqrt {3}} x^{2}-\frac {1}{18} \frac {1}{\left (\frac {\sqrt {3}}{3}+i\right ) \left (-2+i \sqrt {3}\right ) \left (\sqrt {3}+i\right )} x^{3}+\frac {1}{-1728-480 i \sqrt {3}} x^{4}+\frac {1}{3360 i \sqrt {3}+50400} x^{5}-\frac {1}{720} \frac {1}{\left (\sqrt {3}+6 i\right ) \left (\sqrt {3}+5 i\right ) \left (\sqrt {3}+4 i\right ) \left (\sqrt {3}+3 i\right ) \left (2 i+\sqrt {3}\right ) \left (\sqrt {3}+i\right )} x^{6}-\frac {1}{5040} \frac {1}{\left (-2+i \sqrt {3}\right ) \left (\sqrt {3}+7 i\right ) \left (\sqrt {3}+6 i\right ) \left (\sqrt {3}+5 i\right ) \left (\sqrt {3}+4 i\right ) \left (\sqrt {3}+3 i\right ) \left (\sqrt {3}+i\right )} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \sqrt {x}\, x^{\frac {i \sqrt {3}}{2}} \left (1+\frac {1}{-i \sqrt {3}-1} x +\frac {1}{6 i \sqrt {3}-2} x^{2}+\frac {1}{6} \frac {1}{\left (\sqrt {3}-2 i\right ) \left (-i+\sqrt {3}\right ) \left (i \sqrt {3}+3\right )} x^{3}+\frac {1}{24} \frac {1}{\left (-\sqrt {3}+2 i\right ) \left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+3\right ) \left (-i+\sqrt {3}\right )} x^{4}+\frac {1}{120} \frac {1}{\left (\sqrt {3}-2 i\right ) \left (-i+\sqrt {3}\right ) \left (i \sqrt {3}+5\right ) \left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+3\right )} x^{5}+\frac {1}{720} \frac {1}{\left (-\sqrt {3}+2 i\right ) \left (i \sqrt {3}+6\right ) \left (i \sqrt {3}+5\right ) \left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+3\right ) \left (-i+\sqrt {3}\right )} x^{6}+\frac {1}{5040} \frac {1}{\left (\sqrt {3}-2 i\right ) \left (-i+\sqrt {3}\right ) \left (i \sqrt {3}+7\right ) \left (i \sqrt {3}+6\right ) \left (i \sqrt {3}+5\right ) \left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+3\right )} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{3} \left (1-x +\frac {1}{3} x^{2}-\frac {1}{21} x^{3}+\frac {1}{273} x^{4}-\frac {1}{5733} x^{5}+\frac {1}{177723} x^{6}-\frac {1}{7642089} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6493

ODE	Solve \begin {gather*} \boxed {x^{3} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }-3 x y^{\prime }+y \left (x -1\right )=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y \left (x \right ) = c_{1} x^{\left \{\lambda =\lambda , x =-\lambda ^{3}+2 \lambda ^{2}+2 \lambda +1\right \}} \] Warning, solution could not be verified
Maple solution	\[ \text {Expression too large to display} \]

Problem 6494

ODE	Solve \begin {gather*} \boxed {x^{3} y^{\prime \prime \prime }-2 x^{2} y^{\prime \prime }+\left (x^{2}+2 x \right ) y^{\prime }-y x=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
program solution	N/A
Maple solution	\[ y \left (x \right ) = c_{1} x^{3} \left (1-\frac {1}{4} x +\frac {1}{40} x^{2}-\frac {1}{720} x^{3}+\frac {1}{20160} x^{4}-\frac {1}{806400} x^{5}+\frac {1}{43545600} x^{6}-\frac {1}{3048192000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x^{2} \left (\ln \left (x \right ) \left (\left (-240\right ) x +60 x^{2}-6 x^{3}+\frac {1}{3} x^{4}-\frac {1}{84} x^{5}+\frac {1}{3360} x^{6}-\frac {1}{181440} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (720-908 x +152 x^{2}-11 x^{3}+\frac {4}{9} x^{4}-\frac {79}{7056} x^{5}+\frac {517}{2822400} x^{6}-\frac {851}{457228800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )+c_{3} \left (2 \ln \left (x \right ) \left (x^{3}-\frac {1}{4} x^{4}+\frac {1}{40} x^{5}-\frac {1}{720} x^{6}+\frac {1}{20160} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-24-12 x -6 x^{2}+\frac {5}{8} x^{4}-\frac {39}{400} x^{5}+\frac {49}{7200} x^{6}-\frac {199}{705600} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right ) \]

Problem 6495

ODE	Solve \begin {gather*} \boxed {x^{3} y^{\prime \prime \prime }+\left (2 x^{3}-x^{2}\right ) y^{\prime \prime }-y^{\prime } x +y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
program solution	N/A
Maple solution	\[ y \left (x \right ) = c_{3} x \left (1+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x^{\frac {3}{2}-\frac {\sqrt {13}}{2}} \left (1-x +\frac {-3+\sqrt {13}}{-4+2 \sqrt {13}} x^{2}+\frac {5-\sqrt {13}}{6 \sqrt {13}-12} x^{3}+\frac {1}{24} \frac {\left (-5+\sqrt {13}\right ) \left (-7+\sqrt {13}\right )}{\left (-2+\sqrt {13}\right ) \left (-4+\sqrt {13}\right )} x^{4}+\frac {1}{30} \frac {-19+4 \sqrt {13}}{\left (-2+\sqrt {13}\right ) \left (-4+\sqrt {13}\right )} x^{5}+\frac {1}{20} \frac {-29+7 \sqrt {13}}{\left (-2+\sqrt {13}\right ) \left (-4+\sqrt {13}\right ) \left (-6+\sqrt {13}\right )} x^{6}+\frac {-\frac {117}{35}+\frac {6 \sqrt {13}}{7}}{\left (-2+\sqrt {13}\right ) \left (-4+\sqrt {13}\right ) \left (-6+\sqrt {13}\right ) \left (-7+\sqrt {13}\right )} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{1} x^{\frac {3}{2}+\frac {\sqrt {13}}{2}} \left (1-x +\frac {3+\sqrt {13}}{4+2 \sqrt {13}} x^{2}+\frac {-5-\sqrt {13}}{6 \sqrt {13}+12} x^{3}+\frac {1}{24} \frac {\left (5+\sqrt {13}\right ) \left (7+\sqrt {13}\right )}{\left (2+\sqrt {13}\right ) \left (4+\sqrt {13}\right )} x^{4}-\frac {1}{30} \frac {19+4 \sqrt {13}}{\left (2+\sqrt {13}\right ) \left (4+\sqrt {13}\right )} x^{5}+\frac {1}{20} \frac {29+7 \sqrt {13}}{\left (2+\sqrt {13}\right ) \left (4+\sqrt {13}\right ) \left (6+\sqrt {13}\right )} x^{6}+\frac {-\frac {117}{35}-\frac {6 \sqrt {13}}{7}}{\left (2+\sqrt {13}\right ) \left (4+\sqrt {13}\right ) \left (6+\sqrt {13}\right ) \left (7+\sqrt {13}\right )} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6496

ODE	\[ \boxed {x^{3} y^{\prime \prime }+x^{2} y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = \infty \).
program solution	\[ y = c_{1} \left (-\frac {1}{x}+1+\frac {1}{4 x^{2}}-\frac {1}{36 x^{3}}+\frac {1}{576 x^{4}}-\frac {1}{14400 x^{5}}+\frac {1}{518400 x^{6}}-\frac {1}{25401600 x^{7}}+O\left (\frac {1}{x^{8}}\right )\right )+c_{2} \left (\left (-\frac {1}{x}+1+\frac {1}{4 x^{2}}-\frac {1}{36 x^{3}}+\frac {1}{576 x^{4}}-\frac {1}{14400 x^{5}}+\frac {1}{518400 x^{6}}-\frac {1}{25401600 x^{7}}+O\left (\frac {1}{x^{8}}\right )\right ) \ln \left (\frac {1}{x}\right )+\frac {2}{x}-\frac {3}{4 x^{2}}+\frac {11}{108 x^{3}}-\frac {25}{3456 x^{4}}+\frac {137}{432000 x^{5}}-\frac {49}{5184000 x^{6}}+\frac {121}{592704000 x^{7}}+O\left (\frac {1}{x^{8}}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {\left (x -\operatorname {Infinity} \right )^{2}}{2 \operatorname {Infinity}^{3}}+\frac {2 \left (x -\operatorname {Infinity} \right )^{3}}{3 \operatorname {Infinity}^{4}}+\frac {\left (-18 \operatorname {Infinity} +1\right ) \left (x -\operatorname {Infinity} \right )^{4}}{24 \operatorname {Infinity}^{6}}+\frac {\left (96 \operatorname {Infinity} -14\right ) \left (x -\operatorname {Infinity} \right )^{5}}{120 \operatorname {Infinity}^{7}}+\frac {\left (-600 \operatorname {Infinity}^{2}+156 \operatorname {Infinity} -1\right ) \left (x -\operatorname {Infinity} \right )^{6}}{720 \operatorname {Infinity}^{9}}+\frac {\left (4320 \operatorname {Infinity}^{2}-1692 \operatorname {Infinity} +30\right ) \left (x -\operatorname {Infinity} \right )^{7}}{5040 \operatorname {Infinity}^{10}}\right ) y \left (\operatorname {Infinity} \right )+\left (x -\operatorname {Infinity} -\frac {\left (x -\operatorname {Infinity} \right )^{2}}{2 \operatorname {Infinity}}+\frac {\left (2 \operatorname {Infinity}^{2}-\operatorname {Infinity} \right ) \left (x -\operatorname {Infinity} \right )^{3}}{6 \operatorname {Infinity}^{4}}-\frac {\left (\operatorname {Infinity} -\frac {4}{3}\right ) \left (x -\operatorname {Infinity} \right )^{4}}{4 \operatorname {Infinity}^{4}}+\frac {\left (24 \operatorname {Infinity}^{3}-58 \operatorname {Infinity}^{2}+\operatorname {Infinity} \right ) \left (x -\operatorname {Infinity} \right )^{5}}{120 \operatorname {Infinity}^{7}}+\frac {\left (-120 \operatorname {Infinity}^{4}+444 \operatorname {Infinity}^{3}-21 \operatorname {Infinity}^{2}\right ) \left (x -\operatorname {Infinity} \right )^{6}}{720 \operatorname {Infinity}^{9}}+\frac {\left (720 \operatorname {Infinity}^{4}-3708 \operatorname {Infinity}^{3}+324 \operatorname {Infinity}^{2}-\operatorname {Infinity} \right ) \left (x -\operatorname {Infinity} \right )^{7}}{5040 \operatorname {Infinity}^{10}}\right ) D\left (y \right )\left (\operatorname {Infinity} \right )+O\left (x^{8}\right ) \]

Problem 6497

ODE	\[ \boxed {9 \left (x -2\right )^{2} \left (x -3\right ) y^{\prime \prime }+6 x \left (x -2\right ) y^{\prime }+16 y=0} \] With the expansion point for the power series method at \(x = \infty \).
program solution	\[ y = c_{1} \left (1-\frac {4}{3 x}-\frac {28}{9 x^{2}}-\frac {3004}{405 x^{3}}-\frac {285704}{15795 x^{4}}-\frac {822592}{18225 x^{5}}-\frac {4666732192}{40514175 x^{6}}-\frac {401483448544}{1336967775 x^{7}}+O\left (\frac {1}{x^{8}}\right )\right )+\frac {c_{2} \left (1-\frac {13}{3 x}-\frac {251}{45 x^{2}}-\frac {7781}{810 x^{3}}-\frac {22151}{1215 x^{4}}-\frac {669229}{18225 x^{5}}-\frac {216463313}{2788425 x^{6}}-\frac {7179886604}{41826375 x^{7}}+O\left (\frac {1}{x^{8}}\right )\right )}{\left (\frac {1}{x}\right )^{\frac {1}{3}}} \] Verified OK.
Maple solution	\[ \text {Expression too large to display} \]

Problem 6498

ODE	\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +p \left (p +1\right ) y=0} \] With the expansion point for the power series method at \(x = \infty \).
program solution	\[ y = c_{1} \left (\frac {1}{x}\right )^{-p} \left (1-\frac {p \left (p -1\right )}{\left (4 p -2\right ) x^{2}}+\frac {\left (p -1\right ) p \left (-3+p \right ) \left (p -2\right )}{\left (32 p^{2}-64 p +24\right ) x^{4}}-\frac {\left (p -1\right ) p \left (-3+p \right ) \left (p -2\right ) \left (p -5\right ) \left (p -4\right )}{\left (384 p^{3}-1728 p^{2}+2208 p -720\right ) x^{6}}+O\left (\frac {1}{x^{8}}\right )\right )+c_{2} \left (\frac {1}{x}\right )^{p +1} \left (1+\frac {\left (p +2\right ) \left (p +1\right )}{\left (6+4 p \right ) x^{2}}+\frac {\left (p +3\right ) \left (p +1\right ) \left (p +4\right ) \left (p +2\right )}{\left (32 p^{2}+128 p +120\right ) x^{4}}+\frac {\left (p +6\right ) \left (p +5\right ) \left (p +4\right ) \left (p +3\right ) \left (p +2\right ) \left (p +1\right )}{\left (384 p^{3}+2880 p^{2}+6816 p +5040\right ) x^{6}}+O\left (\frac {1}{x^{8}}\right )\right ) \] Verified OK.
Maple solution	\[ \text {Expression too large to display} \]

Problem 6499

ODE	\[ \boxed {y^{\prime \prime }+5 y^{\prime }+6 y=5 \,{\mathrm e}^{3 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = -{\mathrm e}^{-2 t}+\cosh \left (3 t \right )-\frac {2 \sinh \left (3 t \right )}{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \cosh \left (3 t \right )-\frac {2 \sinh \left (3 t \right )}{3}-{\mathrm e}^{-2 t} \]

Problem 6500

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }-6 y=t} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = -\frac {\left (-9 \,{\mathrm e}^{5 t}+30 t \,{\mathrm e}^{3 t}+5 \,{\mathrm e}^{3 t}+4\right ) {\mathrm e}^{-3 t}}{180} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (9 \,{\mathrm e}^{5 t}-30 t \,{\mathrm e}^{3 t}-5 \,{\mathrm e}^{3 t}-4\right ) {\mathrm e}^{-3 t}}{180} \]