2.17.57 Problems 5601 to 5700

Problem 5601

ODE	\[ \boxed {x y^{\prime \prime }+2 y^{\prime }+4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-2 x +\frac {4 x^{2}}{3}-\frac {4 x^{3}}{9}+\frac {4 x^{4}}{45}-\frac {8 x^{5}}{675}+O\left (x^{6}\right )\right )+c_{2} \left (\left (-4+8 x -\frac {16 x^{2}}{3}+\frac {16 x^{3}}{9}-\frac {16 x^{4}}{45}+\frac {32 x^{5}}{675}-4 O\left (x^{6}\right )\right ) \ln \left (x \right )+\frac {1-12 x^{2}+\frac {112 x^{3}}{9}-\frac {140 x^{4}}{27}+\frac {808 x^{5}}{675}+O\left (x^{6}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-2 x +\frac {4}{3} x^{2}-\frac {4}{9} x^{3}+\frac {4}{45} x^{4}-\frac {8}{675} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) x +c_{2} \left (\ln \left (x \right ) \left (\left (-4\right ) x +8 x^{2}-\frac {16}{3} x^{3}+\frac {16}{9} x^{4}-\frac {16}{45} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-12 x^{2}+\frac {112}{9} x^{3}-\frac {140}{27} x^{4}+\frac {808}{675} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]

Problem 5602

ODE	\[ \boxed {x y^{\prime \prime }+3 y^{\prime }+y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-\frac {x^{2}}{8}+\frac {x^{4}}{192}+O\left (x^{6}\right )\right )+c_{2} \left (\left (-\frac {1}{2}+\frac {x^{2}}{16}-\frac {x^{4}}{384}-\frac {O\left (x^{6}\right )}{2}\right ) \ln \left (x \right )+\frac {1-\frac {3 x^{4}}{64}+O\left (x^{6}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{2} \left (1-\frac {1}{8} x^{2}+\frac {1}{192} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (x^{2}-\frac {1}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+\frac {3}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]

Problem 5603

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

Problem 5604

ODE	\[ \boxed {x y^{\prime \prime }-5 y^{\prime }+y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{6} \left (1-\frac {x^{2}}{16}+\frac {x^{4}}{640}-\frac {x^{6}}{46080}+O\left (x^{7}\right )\right )+c_{2} \left (-\frac {x^{6} \left (1-\frac {x^{2}}{16}+\frac {x^{4}}{640}-\frac {x^{6}}{46080}+O\left (x^{7}\right )\right ) \ln \left (x \right )}{384}+1+\frac {x^{2}}{8}+\frac {x^{4}}{64}+O\left (x^{7}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{6} \left (1-\frac {1}{16} x^{2}+\frac {1}{640} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (-86400-10800 x^{2}-1350 x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 5605

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

Problem 5606

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

Problem 5607

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

Problem 5608

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

Problem 5609

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

Problem 5610

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

Problem 5611

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

Problem 5612

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

Problem 5613

ODE	\[ \boxed {16 x^{2} y^{\prime \prime }+32 y^{\prime } x +\left (x^{4}-12\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {x^{4}}{384}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {x^{4}}{128}+O\left (x^{6}\right )\right )}{x^{\frac {3}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{2} \left (1-\frac {1}{384} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (-2+\frac {1}{64} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {3}{2}}} \]

Problem 5614

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

Problem 5615

ODE	\[ \boxed {2 x y^{\prime \prime }+y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {x}{3}+\frac {x^{2}}{30}-\frac {x^{3}}{630}+\frac {x^{4}}{22680}-\frac {x^{5}}{1247400}+O\left (x^{6}\right )\right )+c_{2} \left (1-x +\frac {x^{2}}{6}-\frac {x^{3}}{90}+\frac {x^{4}}{2520}-\frac {x^{5}}{113400}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1-\frac {1}{3} x +\frac {1}{30} x^{2}-\frac {1}{630} x^{3}+\frac {1}{22680} x^{4}-\frac {1}{1247400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-x +\frac {1}{6} x^{2}-\frac {1}{90} x^{3}+\frac {1}{2520} x^{4}-\frac {1}{113400} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 5616

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

Problem 5617

ODE	\[ \boxed {\left (x -1\right ) y^{\prime \prime }+3 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {3}{2} x^{2}+\frac {1}{2} x^{3}+\frac {5}{8} x^{4}+\frac {9}{20} x^{5}+\frac {29}{80} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{3}+\frac {1}{4} x^{4}+\frac {9}{40} x^{5}+\frac {7}{40} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {3}{2} x^{2}+\frac {1}{2} x^{3}+\frac {5}{8} x^{4}+\frac {9}{20} x^{5}\right ) c_{1} +\left (x +\frac {1}{2} x^{3}+\frac {1}{4} x^{4}+\frac {9}{40} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {3}{2} x^{2}+\frac {1}{2} x^{3}+\frac {5}{8} x^{4}+\frac {9}{20} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{3}+\frac {1}{4} x^{4}+\frac {9}{40} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5618

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

Problem 5619

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

Problem 5620

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

Problem 5621

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

Problem 5622

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

Problem 5623

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

Problem 5624

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

Problem 5625

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

Problem 5626

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

Problem 5627

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

Problem 5628

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

Problem 5629

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

Problem 5630

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

Problem 5631

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

Problem 5632

ODE	\[ \boxed {4 y+y^{\prime }=1} \] With initial conditions \begin {align*} \left [y \left (0\right ) = {\frac {5}{4}}\right ] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {5}{4}-4 x +8 x^{2}-\frac {32 x^{3}}{3}+\frac {32 x^{4}}{3}-\frac {128 x^{5}}{15}+O\left (x^{6}\right ) \] Verified OK. \[ y = \frac {5}{4}-4 x +8 x^{2}-\frac {32 x^{3}}{3}+\frac {32 x^{4}}{3}-\frac {128 x^{5}}{15}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {5}{4}-4 x +8 x^{2}-\frac {32}{3} x^{3}+\frac {32}{3} x^{4}-\frac {128}{15} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 5633

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime } x +2 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 1] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = -x^{2}+x +1-\frac {5 x^{3}}{6}+\frac {2 x^{4}}{3}+\frac {11 x^{5}}{24}-\frac {14 x^{6}}{45}+O\left (x^{6}\right ) \] Verified OK. \[ y = 1-x^{2}+\frac {2 x^{4}}{3}+x -\frac {5 x^{3}}{6}+\frac {11 x^{5}}{24}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1+x -x^{2}-\frac {5}{6} x^{3}+\frac {2}{3} x^{4}+\frac {11}{24} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 5634

ODE	\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +30 y=0} \] With initial conditions \begin {align*} \left [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = {\frac {15}{8}}\right ] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {15 x}{8}-\frac {35 x^{3}}{4}+\frac {63 x^{5}}{8}+O\left (x^{6}\right ) \] Verified OK. \[ y = \frac {15 x}{8}-\frac {35 x^{3}}{4}+\frac {63 x^{5}}{8}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {15}{8} x -\frac {35}{4} x^{3}+\frac {63}{8} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 5635

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

Problem 5636

ODE	\[ \boxed {\left (x -2\right )^{2} y^{\prime \prime }+\left (x +2\right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{8} x^{2}+\frac {1}{48} x^{3}-\frac {1}{480} x^{5}-\frac {7}{5760} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{4} x^{2}-\frac {1}{24} x^{3}+\frac {1}{240} x^{5}+\frac {7}{2880} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{8} x^{2}+\frac {1}{48} x^{3}-\frac {1}{480} x^{5}\right ) c_{1} +\left (x -\frac {1}{4} x^{2}-\frac {1}{24} x^{3}+\frac {1}{240} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{8} x^{2}+\frac {1}{48} x^{3}-\frac {1}{480} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{4} x^{2}-\frac {1}{24} x^{3}+\frac {1}{240} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5637

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

Problem 5638

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

Problem 5639

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

Problem 5640

ODE	\[ \boxed {x y^{\prime \prime }+2 x^{3} y^{\prime }+\left (x^{2}-2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1+x +\frac {x^{2}}{3}-\frac {7 x^{3}}{36}-\frac {97 x^{4}}{360}-\frac {517 x^{5}}{5400}+O\left (x^{6}\right )\right )+c_{2} \left (2 x \left (1+x +\frac {x^{2}}{3}-\frac {7 x^{3}}{36}-\frac {97 x^{4}}{360}-\frac {517 x^{5}}{5400}+O\left (x^{6}\right )\right ) \ln \left (x \right )+1-3 x^{2}-\frac {31 x^{3}}{18}-\frac {85 x^{4}}{216}+\frac {4067 x^{5}}{5400}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1+x +\frac {1}{3} x^{2}-\frac {7}{36} x^{3}-\frac {97}{360} x^{4}-\frac {517}{5400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (2 x +2 x^{2}+\frac {2}{3} x^{3}-\frac {7}{18} x^{4}-\frac {97}{180} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-3 x^{2}-\frac {31}{18} x^{3}-\frac {85}{216} x^{4}+\frac {4067}{5400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

Problem 5641

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

Problem 5642

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

Problem 5643

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

Problem 5644

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

Problem 5645

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

Problem 5646

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

Problem 5647

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

Problem 5648

ODE	\[ \boxed {2 x \left (1-x \right ) y^{\prime \prime }-\left (1+6 x \right ) y^{\prime }-2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {3}{2}} \left (1+\frac {5 x}{2}+\frac {35 x^{2}}{8}+\frac {105 x^{3}}{16}+\frac {1155 x^{4}}{128}+\frac {3003 x^{5}}{256}+O\left (x^{6}\right )\right )+c_{2} \left (1-2 x -8 x^{2}-16 x^{3}-\frac {128 x^{4}}{5}-\frac {256 x^{5}}{7}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {3}{2}} \left (1+\frac {5}{2} x +\frac {35}{8} x^{2}+\frac {105}{16} x^{3}+\frac {1155}{128} x^{4}+\frac {3003}{256} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-2 x -8 x^{2}-16 x^{3}-\frac {128}{5} x^{4}-\frac {256}{7} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 5649

ODE	\[ \boxed {x \left (1-x \right ) y^{\prime \prime }+\left (\frac {1}{2}+2 x \right ) y^{\prime }-2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1+\frac {x}{2}-\frac {x^{2}}{40}-\frac {x^{3}}{560}-\frac {x^{4}}{2688}-\frac {x^{5}}{8448}+O\left (x^{6}\right )\right )+c_{2} \left (1+4 x +O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{2} x -\frac {1}{40} x^{2}-\frac {1}{560} x^{3}-\frac {1}{2688} x^{4}-\frac {1}{8448} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+4 x +\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 5650

ODE	\[ \boxed {4 x y^{\prime \prime }+y^{\prime }+8 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {3}{4}} \left (1-\frac {8 x}{7}+\frac {32 x^{2}}{77}-\frac {256 x^{3}}{3465}+\frac {512 x^{4}}{65835}-\frac {4096 x^{5}}{7571025}+O\left (x^{6}\right )\right )+c_{2} \left (1-8 x +\frac {32 x^{2}}{5}-\frac {256 x^{3}}{135}+\frac {512 x^{4}}{1755}-\frac {4096 x^{5}}{149175}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {3}{4}} \left (1-\frac {8}{7} x +\frac {32}{77} x^{2}-\frac {256}{3465} x^{3}+\frac {512}{65835} x^{4}-\frac {4096}{7571025} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-8 x +\frac {32}{5} x^{2}-\frac {256}{135} x^{3}+\frac {512}{1755} x^{4}-\frac {4096}{149175} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 5651

ODE	\[ \boxed {4 \left (t^{2}-3 t +2\right ) y^{\prime \prime }-2 y^{\prime }+y=0} \] With the expansion point for the power series method at \(t = 0\).
program solution	\[ y = \left (1-\frac {1}{16} t^{2}-\frac {7}{192} t^{3}-\frac {73}{3072} t^{4}-\frac {1037}{61440} t^{5}-\frac {6257}{491520} t^{6}\right ) y \left (0\right )+\left (t +\frac {1}{8} t^{2}+\frac {5}{96} t^{3}+\frac {47}{1536} t^{4}+\frac {643}{30720} t^{5}+\frac {3823}{245760} t^{6}\right ) y^{\prime }\left (0\right )+O\left (t^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{16} t^{2}-\frac {7}{192} t^{3}-\frac {73}{3072} t^{4}-\frac {1037}{61440} t^{5}\right ) c_{1} +\left (t +\frac {1}{8} t^{2}+\frac {5}{96} t^{3}+\frac {47}{1536} t^{4}+\frac {643}{30720} t^{5}\right ) c_{2} +O\left (t^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (1-\frac {1}{16} t^{2}-\frac {7}{192} t^{3}-\frac {73}{3072} t^{4}-\frac {1037}{61440} t^{5}\right ) y \left (0\right )+\left (t +\frac {1}{8} t^{2}+\frac {5}{96} t^{3}+\frac {47}{1536} t^{4}+\frac {643}{30720} t^{5}\right ) D\left (y \right )\left (0\right )+O\left (t^{6}\right ) \]

Problem 5652

ODE	\[ \boxed {2 \left (t^{2}-5 t +6\right ) y^{\prime \prime }+\left (2 t -3\right ) y^{\prime }-8 y=0} \] With the expansion point for the power series method at \(t = 0\).
program solution	\[ y = \left (1+\frac {1}{3} t^{2}+\frac {13}{108} t^{3}+\frac {299}{5184} t^{4}+\frac {923}{34560} t^{5}+\frac {30121}{2488320} t^{6}\right ) y \left (0\right )+\left (t +\frac {1}{8} t^{2}+\frac {37}{288} t^{3}+\frac {851}{13824} t^{4}+\frac {2627}{92160} t^{5}+\frac {85729}{6635520} t^{6}\right ) y^{\prime }\left (0\right )+O\left (t^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{3} t^{2}+\frac {13}{108} t^{3}+\frac {299}{5184} t^{4}+\frac {923}{34560} t^{5}\right ) c_{1} +\left (t +\frac {1}{8} t^{2}+\frac {37}{288} t^{3}+\frac {851}{13824} t^{4}+\frac {2627}{92160} t^{5}\right ) c_{2} +O\left (t^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (1+\frac {1}{3} t^{2}+\frac {13}{108} t^{3}+\frac {299}{5184} t^{4}+\frac {923}{34560} t^{5}\right ) y \left (0\right )+\left (t +\frac {1}{8} t^{2}+\frac {37}{288} t^{3}+\frac {851}{13824} t^{4}+\frac {2627}{92160} t^{5}\right ) D\left (y \right )\left (0\right )+O\left (t^{6}\right ) \]

Problem 5653

ODE	\[ \boxed {3 t \left (t +1\right ) y^{\prime \prime }+t y^{\prime }-y=0} \] With the expansion point for the power series method at \(t = 0\).
program solution	\[ y = c_{1} t \left (1+O\left (t^{6}\right )\right )+c_{2} \left (\frac {t \left (1+O\left (t^{6}\right )\right ) \ln \left (t \right )}{3}+1-\frac {2 t^{2}}{9}+\frac {7 t^{3}}{81}-\frac {35 t^{4}}{729}+\frac {91 t^{5}}{2916}+O\left (t^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} t \left (1+\operatorname {O}\left (t^{6}\right )\right )+\left (\frac {1}{3} t +\operatorname {O}\left (t^{6}\right )\right ) \ln \left (t \right ) c_{2} +\left (1-\frac {1}{3} t -\frac {2}{9} t^{2}+\frac {7}{81} t^{3}-\frac {35}{729} t^{4}+\frac {91}{2916} t^{5}+\operatorname {O}\left (t^{6}\right )\right ) c_{2} \]

Problem 5654

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (x^{2}-\frac {4}{49}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {2}{7}} \left (1-\frac {7 x^{2}}{36}+\frac {49 x^{4}}{4608}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {7 x^{2}}{20}+\frac {49 x^{4}}{1920}+O\left (x^{6}\right )\right )}{x^{\frac {2}{7}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {4}{7}} \left (1-\frac {7}{36} x^{2}+\frac {49}{4608} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1-\frac {7}{20} x^{2}+\frac {49}{1920} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {2}{7}}} \]

Problem 5655

ODE	\[ \boxed {x y^{\prime \prime }+y^{\prime }+\frac {y}{4}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-\frac {x}{4}+\frac {x^{2}}{64}-\frac {x^{3}}{2304}+\frac {x^{4}}{147456}-\frac {x^{5}}{14745600}+O\left (x^{6}\right )\right )+c_{2} \left (\left (1-\frac {x}{4}+\frac {x^{2}}{64}-\frac {x^{3}}{2304}+\frac {x^{4}}{147456}-\frac {x^{5}}{14745600}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\frac {x}{2}-\frac {3 x^{2}}{64}+\frac {11 x^{3}}{6912}-\frac {25 x^{4}}{884736}+\frac {137 x^{5}}{442368000}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {1}{4} x +\frac {1}{64} x^{2}-\frac {1}{2304} x^{3}+\frac {1}{147456} x^{4}-\frac {1}{14745600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{2} x -\frac {3}{64} x^{2}+\frac {11}{6912} x^{3}-\frac {25}{884736} x^{4}+\frac {137}{442368000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 5656

ODE	\[ \boxed {y^{\prime \prime }+\left ({\mathrm e}^{-2 x}-\frac {1}{9}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {4}{9} x^{2}+\frac {1}{3} x^{3}-\frac {65}{486} x^{4}+\frac {1}{135} x^{5}+\frac {1102}{32805} x^{6}\right ) y \left (0\right )+\left (x -\frac {4}{27} x^{3}+\frac {1}{6} x^{4}-\frac {227}{2430} x^{5}+\frac {4}{135} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {4}{9} x^{2}+\frac {1}{3} x^{3}-\frac {65}{486} x^{4}+\frac {1}{135} x^{5}\right ) c_{1} +\left (x -\frac {4}{27} x^{3}+\frac {1}{6} x^{4}-\frac {227}{2430} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {4}{9} x^{2}+\frac {1}{3} x^{3}-\frac {65}{486} x^{4}+\frac {1}{135} x^{5}\right ) y \left (0\right )+\left (x -\frac {4}{27} x^{3}+\frac {1}{6} x^{4}-\frac {227}{2430} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5657

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\frac {\left (x +\frac {3}{4}\right ) y}{4}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {3}{4}} \left (1-\frac {x}{6}+\frac {x^{2}}{120}-\frac {x^{3}}{5040}+\frac {x^{4}}{362880}-\frac {x^{5}}{39916800}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{4}} \left (1-\frac {x}{2}+\frac {x^{2}}{24}-\frac {x^{3}}{720}+\frac {x^{4}}{40320}-\frac {x^{5}}{3628800}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{4}} \left (1-\frac {1}{2} x +\frac {1}{24} x^{2}-\frac {1}{720} x^{3}+\frac {1}{40320} x^{4}-\frac {1}{3628800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{\frac {3}{4}} \left (1-\frac {1}{6} x +\frac {1}{120} x^{2}-\frac {1}{5040} x^{3}+\frac {1}{362880} x^{4}-\frac {1}{39916800} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 5658

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\frac {\left (x^{2}-1\right ) y}{4}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {x^{2}}{24}+\frac {x^{4}}{1920}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {x^{2}}{8}+\frac {x^{4}}{384}+O\left (x^{6}\right )\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x \left (1-\frac {1}{24} x^{2}+\frac {1}{1920} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-\frac {1}{8} x^{2}+\frac {1}{384} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

Problem 5659

ODE	\[ \boxed {\left (2 x +1\right )^{2} y^{\prime \prime }+2 \left (2 x +1\right ) y^{\prime }+16 x \left (1+x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {8}{3} x^{3}+\frac {16}{3} x^{4}-\frac {152}{15} x^{5}+\frac {184}{9} x^{6}\right ) y \left (0\right )+\left (x -x^{2}+\frac {4}{3} x^{3}-\frac {10}{3} x^{4}+\frac {104}{15} x^{5}-\frac {208}{15} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {8}{3} x^{3}+\frac {16}{3} x^{4}-\frac {152}{15} x^{5}\right ) c_{1} +\left (x -x^{2}+\frac {4}{3} x^{3}-\frac {10}{3} x^{4}+\frac {104}{15} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {8}{3} x^{3}+\frac {16}{3} x^{4}-\frac {152}{15} x^{5}\right ) y \left (0\right )+\left (x -x^{2}+\frac {4}{3} x^{3}-\frac {10}{3} x^{4}+\frac {104}{15} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5660

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

Problem 5661

ODE	\[ \boxed {x y^{\prime \prime }+5 y^{\prime }+y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-\frac {x^{2}}{12}+\frac {x^{4}}{384}+O\left (x^{6}\right )\right )+c_{2} \left (\left (-\frac {1}{16}+\frac {x^{2}}{192}-\frac {x^{4}}{6144}-\frac {O\left (x^{6}\right )}{16}\right ) \ln \left (x \right )+\frac {1+\frac {x^{2}}{4}+O\left (x^{6}\right )}{x^{4}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{4} \left (1-\frac {1}{12} x^{2}+\frac {1}{384} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (9 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-36 x^{2}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{4}} \]

Problem 5662

ODE	\[ \boxed {9 x^{2} y^{\prime \prime }+9 y^{\prime } x +\left (36 x^{4}-16\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {4}{3}} \left (1-\frac {3 x^{4}}{20}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {3 x^{4}}{4}+O\left (x^{6}\right )\right )}{x^{\frac {4}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {8}{3}} \left (1-\frac {3}{20} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1-\frac {3}{4} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {4}{3}}} \]

Problem 5663

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

Problem 5664

ODE	\[ \boxed {4 x y^{\prime \prime }+4 y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-\frac {x}{4}+\frac {x^{2}}{64}-\frac {x^{3}}{2304}+\frac {x^{4}}{147456}-\frac {x^{5}}{14745600}+O\left (x^{6}\right )\right )+c_{2} \left (\left (1-\frac {x}{4}+\frac {x^{2}}{64}-\frac {x^{3}}{2304}+\frac {x^{4}}{147456}-\frac {x^{5}}{14745600}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\frac {x}{2}-\frac {3 x^{2}}{64}+\frac {11 x^{3}}{6912}-\frac {25 x^{4}}{884736}+\frac {137 x^{5}}{442368000}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {1}{4} x +\frac {1}{64} x^{2}-\frac {1}{2304} x^{3}+\frac {1}{147456} x^{4}-\frac {1}{14745600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{2} x -\frac {3}{64} x^{2}+\frac {11}{6912} x^{3}-\frac {25}{884736} x^{4}+\frac {137}{442368000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 5665

ODE	\[ \boxed {x y^{\prime \prime }+y^{\prime }+36 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (2916 x^{4}-1296 x^{3}+324 x^{2}-36 x +1-\frac {104976 x^{5}}{25}+O\left (x^{6}\right )\right )+c_{2} \left (\left (2916 x^{4}-1296 x^{3}+324 x^{2}-36 x +1-\frac {104976 x^{5}}{25}+O\left (x^{6}\right )\right ) \ln \left (x \right )-12150 x^{4}+4752 x^{3}-972 x^{2}+72 x +\frac {2396952 x^{5}}{125}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-36 x +324 x^{2}-1296 x^{3}+2916 x^{4}-\frac {104976}{25} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (72 x -972 x^{2}+4752 x^{3}-12150 x^{4}+\frac {2396952}{125} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 5666

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

Problem 5667

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

Problem 5668

ODE	\[ \boxed {x y^{\prime \prime }-5 y^{\prime }+y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{6} \left (1-\frac {x^{2}}{16}+\frac {x^{4}}{640}-\frac {x^{6}}{46080}+O\left (x^{7}\right )\right )+c_{2} \left (-\frac {x^{6} \left (1-\frac {x^{2}}{16}+\frac {x^{4}}{640}-\frac {x^{6}}{46080}+O\left (x^{7}\right )\right ) \ln \left (x \right )}{384}+1+\frac {x^{2}}{8}+\frac {x^{4}}{64}+O\left (x^{7}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{6} \left (1-\frac {1}{16} x^{2}+\frac {1}{640} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (-86400-10800 x^{2}-1350 x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 5669

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

Problem 5670

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

Problem 5671

ODE	\[ \boxed {\left (x -1\right )^{2} y^{\prime \prime }-\left (x -1\right ) y^{\prime }-35 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {35}{2} x^{2}+\frac {35}{6} x^{3}+\frac {665}{12} x^{4}+\frac {259}{4} x^{5}+\frac {1505}{12} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {35}{6} x^{3}+\frac {35}{12} x^{4}+\frac {49}{4} x^{5}+\frac {203}{12} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {35}{2} x^{2}+\frac {35}{6} x^{3}+\frac {665}{12} x^{4}+\frac {259}{4} x^{5}\right ) c_{1} +\left (x -\frac {1}{2} x^{2}+\frac {35}{6} x^{3}+\frac {35}{12} x^{4}+\frac {49}{4} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {35}{2} x^{2}+\frac {35}{6} x^{3}+\frac {665}{12} x^{4}+\frac {259}{4} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {35}{6} x^{3}+\frac {35}{12} x^{4}+\frac {49}{4} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5672

ODE	\[ \boxed {16 \left (1+x \right )^{2} y^{\prime \prime }+3 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {3}{32} x^{2}+\frac {1}{16} x^{3}-\frac {93}{2048} x^{4}+\frac {9}{256} x^{5}-\frac {1863}{65536} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{32} x^{3}+\frac {1}{32} x^{4}-\frac {57}{2048} x^{5}+\frac {25}{1024} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {3}{32} x^{2}+\frac {1}{16} x^{3}-\frac {93}{2048} x^{4}+\frac {9}{256} x^{5}\right ) c_{1} +\left (x -\frac {1}{32} x^{3}+\frac {1}{32} x^{4}-\frac {57}{2048} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {3}{32} x^{2}+\frac {1}{16} x^{3}-\frac {93}{2048} x^{4}+\frac {9}{256} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{32} x^{3}+\frac {1}{32} x^{4}-\frac {57}{2048} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5673

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

Problem 5674

ODE	\[ \boxed {x^{2} y^{\prime \prime }+2 x^{3} y^{\prime }+\left (x^{2}-2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-\frac {x^{2}}{2}+\frac {9 x^{4}}{56}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {x^{2}}{2}+\frac {3 x^{4}}{8}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \left (1-\frac {1}{2} x^{2}+\frac {9}{56} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (12-6 x^{2}+\frac {9}{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 5675

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

Problem 5676

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

Problem 5677

ODE	\[ \boxed {y^{\prime \prime }+\frac {y}{4 x}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {x}{8}+\frac {x^{2}}{192}-\frac {x^{3}}{9216}+\frac {x^{4}}{737280}-\frac {x^{5}}{88473600}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {x \left (1-\frac {x}{8}+\frac {x^{2}}{192}-\frac {x^{3}}{9216}+\frac {x^{4}}{737280}-\frac {x^{5}}{88473600}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{4}+1-\frac {3 x^{2}}{64}+\frac {7 x^{3}}{2304}-\frac {35 x^{4}}{442368}+\frac {101 x^{5}}{88473600}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1-\frac {1}{8} x +\frac {1}{192} x^{2}-\frac {1}{9216} x^{3}+\frac {1}{737280} x^{4}-\frac {1}{88473600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-\frac {1}{4} x +\frac {1}{32} x^{2}-\frac {1}{768} x^{3}+\frac {1}{36864} x^{4}-\frac {1}{2949120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-\frac {3}{64} x^{2}+\frac {7}{2304} x^{3}-\frac {35}{442368} x^{4}+\frac {101}{88473600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

Problem 5678

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

Problem 5679

ODE	\[ \boxed {y^{\prime }+\frac {26 y}{5}=\frac {97 \sin \left (2 t \right )}{5}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {5 \,{\mathrm e}^{-\frac {26 t}{5}}}{4}-\frac {5 \cos \left (2 t \right )}{4}+\frac {13 \sin \left (2 t \right )}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {5 \,{\mathrm e}^{-\frac {26 t}{5}}}{4}-\frac {5 \cos \left (2 t \right )}{4}+\frac {13 \sin \left (2 t \right )}{4} \]

Problem 5680

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

Problem 5681

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }-6 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 11, y^{\prime }\left (0\right ) = 28] \end {align*}
program solution	\[ y = \left (10 \,{\mathrm e}^{5 t}+1\right ) {\mathrm e}^{-2 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (10 \,{\mathrm e}^{5 t}+1\right ) {\mathrm e}^{-2 t} \]

Problem 5682

ODE	\[ \boxed {y^{\prime \prime }+9 y=10 \,{\mathrm e}^{-t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = {\mathrm e}^{-t}-\cos \left (3 t \right )+\frac {\sin \left (3 t \right )}{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\cos \left (3 t \right )+\frac {\sin \left (3 t \right )}{3}+{\mathrm e}^{-t} \]

Problem 5683

ODE	\[ \boxed {y^{\prime \prime }-\frac {y}{4}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 12, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = 12 \cosh \left (\frac {t}{2}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = 12 \cosh \left (\frac {t}{2}\right ) \]

Problem 5684

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+5 y=29 \cos \left (2 t \right )} \] With initial conditions \begin {align*} \left [y \left (0\right ) = {\frac {16}{5}}, y^{\prime }\left (0\right ) = {\frac {31}{5}}\right ] \end {align*}
program solution	\[ y = \frac {\cos \left (2 t \right )}{5}-\frac {12 \sin \left (2 t \right )}{5}+2 \,{\mathrm e}^{5 t}+{\mathrm e}^{t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\cos \left (2 t \right )}{5}-\frac {12 \sin \left (2 t \right )}{5}+{\mathrm e}^{t}+2 \,{\mathrm e}^{5 t} \]

Problem 5685

ODE	\[ \boxed {y^{\prime \prime }+7 y^{\prime }+12 y=21 \,{\mathrm e}^{3 t}} \] With initial conditions \begin {align*} \left [y \left (0\right ) = {\frac {7}{2}}, y^{\prime }\left (0\right ) = -10\right ] \end {align*}
program solution	\[ y = \cosh \left (3 t \right )+\frac {5 \,{\mathrm e}^{-4 t}}{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {5 \,{\mathrm e}^{-4 t}}{2}+\cosh \left (3 t \right ) \]

Problem 5686

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+4 y=0} \] With initial conditions \begin {align*} \left [y \left (0\right ) = {\frac {81}{10}}, y^{\prime }\left (0\right ) = {\frac {39}{10}}\right ] \end {align*}
program solution	\[ y = -\frac {3 \left (41 t -27\right ) {\mathrm e}^{2 t}}{10} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {3 \left (41 t -27\right ) {\mathrm e}^{2 t}}{10} \]

Problem 5687

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+3 y=6 t -8} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = 2 t -2 \,{\mathrm e}^{2 t} \sinh \left (t \right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = 2 t -2 \,{\mathrm e}^{2 t} \sinh \left (t \right ) \]

Problem 5688

ODE	\[ \boxed {y^{\prime \prime }+\frac {y}{25}=\frac {t^{2}}{50}} \] With initial conditions \begin {align*} [y \left (0\right ) = -25, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = -25+\frac {t^{2}}{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {t^{2}}{2}-25 \]

Problem 5689

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }+\frac {9 y}{4}=9 t^{3}+64} \] With initial conditions \begin {align*} \left [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = {\frac {63}{2}}\right ] \end {align*}
program solution	\[ y = 4 t^{3}+t \,{\mathrm e}^{-\frac {3 t}{2}}-16 t^{2}+{\mathrm e}^{-\frac {3 t}{2}}+32 t \] Verified OK.
Maple solution	\[ y \left (t \right ) = 4 t^{3}+{\mathrm e}^{-\frac {3 t}{2}} t -16 t^{2}+{\mathrm e}^{-\frac {3 t}{2}}+32 t \]

Problem 5690

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }-3 y=0} \] With initial conditions \begin {align*} [y \left (4\right ) = -3, y^{\prime }\left (4\right ) = -17] \end {align*}
program solution	\[ y = -3 \left (\left (\cosh \left (8\right )-\frac {7 \sinh \left (8\right )}{3}\right ) \cosh \left (2 t \right )+\frac {7 \left (\cosh \left (8\right )-\frac {3 \sinh \left (8\right )}{7}\right ) \sinh \left (2 t \right )}{3}\right ) {\mathrm e}^{t -4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -5 \,{\mathrm e}^{3 t -12}+2 \,{\mathrm e}^{-t +4} \]

Problem 5691

ODE	\[ \boxed {y^{\prime }-6 y=0} \] With initial conditions \begin {align*} [y \left (-1\right ) = 4] \end {align*}
program solution	\[ y = 4 \,{\mathrm e}^{6} {\mathrm e}^{6 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = 4 \,{\mathrm e}^{6 t +6} \]

Problem 5692

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+5 y=50 t -100} \] With initial conditions \begin {align*} [y \left (2\right ) = -4, y^{\prime }\left (2\right ) = 14] \end {align*}
program solution	\[ y = 2 \cos \left (4\right ) {\mathrm e}^{2-t} \sin \left (2 t \right )-2 \,{\mathrm e}^{2-t} \sin \left (4\right ) \cos \left (2 t \right )+10 t -24 \] Verified OK.
Maple solution	\[ y \left (t \right ) = 2 \sin \left (2 t -4\right ) {\mathrm e}^{-t +2}-24+10 t \]

Problem 5693

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }-4 y=6 \,{\mathrm e}^{2 t -3}} \] With initial conditions \begin {align*} \left [y \left (\frac {3}{2}\right ) = 4, y^{\prime }\left (\frac {3}{2}\right ) = 5\right ] \end {align*}
program solution	\[ y = {\mathrm e}^{2 t -3}+3 \,{\mathrm e}^{t -\frac {3}{2}} \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{2 t -3}+3 \,{\mathrm e}^{t -\frac {3}{2}} \]

Problem 5694

ODE	\[ \boxed {9 y^{\prime \prime }-6 y^{\prime }+y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = 3 \,{\mathrm e}^{\frac {t}{3}} \] Verified OK.
Maple solution	\[ y \left (t \right ) = 3 \,{\mathrm e}^{\frac {t}{3}} \]

Problem 5695

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+8 y={\mathrm e}^{-3 t}-{\mathrm e}^{-5 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = -\frac {{\mathrm e}^{-5 t}}{3}+\frac {{\mathrm e}^{-2 t}}{3}-{\mathrm e}^{-3 t}+{\mathrm e}^{-4 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{-4 t}-{\mathrm e}^{-3 t}-\frac {{\mathrm e}^{-5 t}}{3}+\frac {{\mathrm e}^{-2 t}}{3} \]

Problem 5696

ODE	\[ \boxed {y^{\prime \prime }+10 y^{\prime }+24 y=144 t^{2}} \] With initial conditions \begin {align*} \left [y \left (0\right ) = {\frac {19}{12}}, y^{\prime }\left (0\right ) = -5\right ] \end {align*}
program solution	\[ y = 6 t^{2}-5 t +\frac {19}{12} \] Verified OK.
Maple solution	\[ y \left (t \right ) = 6 t^{2}-5 t +\frac {19}{12} \]

Problem 5697

ODE	\[ \boxed {y^{\prime \prime }+9 y=\left \{\begin {array}{cc} 8 \sin \left (t \right ) & 0
program solution	\[ y = 4 \left (\left \{\begin {array}{cc} \sin \left (t \right ) \cos \left (t \right )^{2} & t <\pi \\ \frac {\sin \left (3 t \right )}{3} & \pi \le t \end {array}\right .\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = 4 \left (\left \{\begin {array}{cc} \sin \left (t \right ) \cos \left (t \right )^{2} & t <\pi \\ \frac {\sin \left (3 t \right )}{3} & \pi \le t \end {array}\right .\right ) \]

Problem 5698

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }+2 y=\left \{\begin {array}{cc} 4 t & 0
program solution	\[ y = \left \{\begin {array}{cc} -3+4 \,{\mathrm e}^{-t}-{\mathrm e}^{-2 t}+2 t & t <1 \\ -6+4 \,{\mathrm e}^{-1}-{\mathrm e}^{-2} & t =1 \\ 4+4 \,{\mathrm e}^{-t}-{\mathrm e}^{-2 t}+3 \,{\mathrm e}^{-2 t +2}-8 \,{\mathrm e}^{-t +1} & 1
Maple solution	\[ y \left (t \right ) = \left \{\begin {array}{cc} 2 t -{\mathrm e}^{-2 t}-3+4 \,{\mathrm e}^{-t} & t <1 \\ -{\mathrm e}^{-2}+1+4 \,{\mathrm e}^{-1} & t =1 \\ 3 \,{\mathrm e}^{-2 t +2}-8 \,{\mathrm e}^{1-t}-{\mathrm e}^{-2 t}+4+4 \,{\mathrm e}^{-t} & 1

Problem 5699

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }-2 y=\left \{\begin {array}{cc} 3 \sin \left (t \right )-\cos \left (t \right ) & 0
program solution	\[ y = {\mathrm e}^{t}-\left (\left \{\begin {array}{cc} \sin \left (t \right ) & t <2 \pi \\ \frac {\sin \left (2 t \right )}{2} & 2 \pi \le t \end {array}\right .\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{t}-\left (\left \{\begin {array}{cc} \sin \left (t \right ) & t <2 \pi \\ \frac {\sin \left (2 t \right )}{2} & 2 \pi \le t \end {array}\right .\right ) \]

Problem 5700

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }+2 y=\left \{\begin {array}{cc} 1 & 0
program solution	\[ y = -\frac {\left (\left \{\begin {array}{cc} -1+2 \,{\mathrm e}^{-t}-{\mathrm e}^{-2 t} & t <1 \\ 2 \,{\mathrm e}^{-1}-{\mathrm e}^{-2}-2 & t =1 \\ 2 \,{\mathrm e}^{-t}-{\mathrm e}^{-2 t}-2 \,{\mathrm e}^{-t +1}+{\mathrm e}^{-2 t +2} & 1
Maple solution	\[ y \left (t \right ) = \frac {\left (\left \{\begin {array}{cc} 1-2 \,{\mathrm e}^{-t}+{\mathrm e}^{-2 t} & t <1 \\ -2 \,{\mathrm e}^{-1}+{\mathrm e}^{-2}+2 & t =1 \\ 2 \,{\mathrm e}^{1-t}-{\mathrm e}^{-2 t +2}-2 \,{\mathrm e}^{-t}+{\mathrm e}^{-2 t} & 1