2.17.15 Problems 1401 to 1500

Problem 1401

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

Problem 1402

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

Problem 1403

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

Problem 1404

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+2 x \left (-x^{2}+4\right ) y^{\prime }+\left (7 x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (-\frac {x^{2}}{2}+1+\frac {x^{4}}{32}+O\left (x^{6}\right )\right )}{\sqrt {x}}+c_{2} \left (\frac {\left (-\frac {x^{2}}{2}+1+\frac {x^{4}}{32}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{\sqrt {x}}+\frac {\frac {5 x^{2}}{8}-\frac {9 x^{4}}{128}+O\left (x^{6}\right )}{\sqrt {x}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {1}{2} x^{2}+\frac {1}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {5}{8} x^{2}-\frac {9}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2}}{\sqrt {x}} \]

Problem 1405

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

Problem 1406

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

Problem 1407

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

Problem 1408

ODE	\[ \boxed {16 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+8 x \left (9 x^{2}+1\right ) y^{\prime }+\left (49 x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{4}} \left (x^{4}-x^{2}+1+O\left (x^{6}\right )\right )+c_{2} \left (x^{\frac {1}{4}} \left (x^{4}-x^{2}+1+O\left (x^{6}\right )\right ) \ln \left (x \right )+x^{\frac {1}{4}} O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{\frac {1}{4}} \left (x^{4}-x^{2}+1\right ) \left (c_{2} \ln \left (x \right )+c_{1} \right )+O\left (x^{6}\right ) \]

Problem 1409

ODE	\[ \boxed {x^{2} \left (3 x +4\right ) y^{\prime \prime }-x \left (4-3 x \right ) y^{\prime }+4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {3 x}{4}+\frac {9 x^{2}}{16}-\frac {27 x^{3}}{64}+\frac {81 x^{4}}{256}-\frac {243 x^{5}}{1024}+O\left (x^{6}\right )\right )+c_{2} \left (x \left (1-\frac {3 x}{4}+\frac {9 x^{2}}{16}-\frac {27 x^{3}}{64}+\frac {81 x^{4}}{256}-\frac {243 x^{5}}{1024}+O\left (x^{6}\right )\right ) \ln \left (x \right )+x O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {3}{4} x +\frac {9}{16} x^{2}-\frac {27}{64} x^{3}+\frac {81}{256} x^{4}-\frac {243}{1024} x^{5}\right ) x \left (c_{2} \ln \left (x \right )+c_{1} \right )+O\left (x^{6}\right ) \]

Problem 1410

ODE	\[ \boxed {4 x^{2} \left (x^{2}+3 x +1\right ) y^{\prime \prime }+8 x^{2} \left (2 x +3\right ) y^{\prime }+\left (9 x^{2}+3 x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (-144 x^{5}+55 x^{4}-21 x^{3}+8 x^{2}-3 x +1+O\left (x^{6}\right )\right )+c_{2} \left (\sqrt {x}\, \left (-144 x^{5}+55 x^{4}-21 x^{3}+8 x^{2}-3 x +1+O\left (x^{6}\right )\right ) \ln \left (x \right )+\sqrt {x}\, O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (-144 x^{5}+55 x^{4}-21 x^{3}+8 x^{2}-3 x +1\right ) \left (c_{2} \ln \left (x \right )+c_{1} \right )+O\left (x^{6}\right ) \]

Problem 1411

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

Problem 1412

ODE	\[ \boxed {9 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+3 x \left (13 x^{2}+7 x +1\right ) y^{\prime }+\left (25 x^{2}+4 x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (-x^{4}+x^{3}-x +1+O\left (x^{6}\right )\right )+c_{2} \left (x^{\frac {1}{3}} \left (-x^{4}+x^{3}-x +1+O\left (x^{6}\right )\right ) \ln \left (x \right )+x^{\frac {1}{3}} O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{\frac {1}{3}} \left (-x^{4}+x^{3}-x +1\right ) \left (c_{2} \ln \left (x \right )+c_{1} \right )+O\left (x^{6}\right ) \]

Problem 1413

ODE	\[ \boxed {2 x^{2} \left (2+x \right ) y^{\prime \prime }-x \left (4-7 x \right ) y^{\prime }-\left (5-3 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {5}{2}} \left (1-\frac {7 x}{4}+\frac {63 x^{2}}{32}-\frac {231 x^{3}}{128}+\frac {3003 x^{4}}{2048}-\frac {9009 x^{5}}{8192}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {15 x^{\frac {5}{2}} \left (1-\frac {7 x}{4}+\frac {63 x^{2}}{32}-\frac {231 x^{3}}{128}+\frac {3003 x^{4}}{2048}-\frac {9009 x^{5}}{8192}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{128}+\frac {1+\frac {x}{8}+\frac {3 x^{2}}{32}-\frac {75 x^{4}}{512}+\frac {1875 x^{5}}{8192}+O\left (x^{6}\right )}{\sqrt {x}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{3} \left (1-\frac {7}{4} x +\frac {63}{32} x^{2}-\frac {231}{128} x^{3}+\frac {3003}{2048} x^{4}-\frac {9009}{8192} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-\frac {45}{32} x^{3}+\frac {315}{128} x^{4}-\frac {2835}{1024} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (12+\frac {3}{2} x +\frac {9}{8} x^{2}-\frac {981}{64} x^{3}+\frac {6417}{256} x^{4}-\frac {28089}{1024} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{\sqrt {x}} \]

Problem 1414

ODE	\[ \boxed {x^{2} \left (1-2 x \right ) y^{\prime \prime }+x \left (8-9 x \right ) y^{\prime }+\left (6-3 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1-\frac {x}{3}-\frac {x^{2}}{14}-\frac {x^{3}}{28}-\frac {25 x^{4}}{1008}-\frac {x^{5}}{48}+O\left (x^{6}\right )\right )}{x}+\frac {c_{2} \left (1-\frac {33 x}{4}+\frac {99 x^{2}}{4}-\frac {231 x^{3}}{8}+\frac {693 x^{5}}{32}+O\left (x^{6}\right )\right )}{x^{6}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-\frac {1}{3} x -\frac {1}{14} x^{2}-\frac {1}{28} x^{3}-\frac {25}{1008} x^{4}-\frac {1}{48} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x}+\frac {c_{2} \left (2880-23760 x +71280 x^{2}-83160 x^{3}+62370 x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{6}} \]

Problem 1415

ODE	\[ \boxed {x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (10 x^{2}+3\right ) y^{\prime }-\left (-14 x^{2}+15\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{3} \left (1-\frac {5 x^{2}}{2}+\frac {35 x^{4}}{8}-\frac {105 x^{6}}{16}+\frac {1155 x^{8}}{128}+O\left (x^{9}\right )\right )+c_{2} \left (-\frac {3 x^{3} \left (1-\frac {5 x^{2}}{2}+\frac {35 x^{4}}{8}-\frac {105 x^{6}}{16}+\frac {1155 x^{8}}{128}+O\left (x^{9}\right )\right ) \ln \left (x \right )}{16}+\frac {1-\frac {x^{2}}{2}+\frac {x^{4}}{8}+\frac {x^{6}}{16}+O\left (x^{9}\right )}{x^{5}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{3} \left (1-\frac {5}{2} x^{2}+\frac {35}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (-203212800+101606400 x^{2}-25401600 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{5}} \]

Problem 1416

ODE	\[ \boxed {x^{2} \left (-2 x^{2}+1\right ) y^{\prime \prime }+x \left (-13 x^{2}+7\right ) y^{\prime }-14 x^{2} y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1+\frac {7 x^{2}}{8}+\frac {77 x^{4}}{80}+\frac {77 x^{6}}{64}+O\left (x^{7}\right )\right )+\frac {c_{2} \left (1-\frac {5 x^{2}}{2}+\frac {5 x^{4}}{8}+\frac {5 x^{6}}{16}+O\left (x^{7}\right )\right )}{x^{6}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (1+\frac {7}{8} x^{2}+\frac {77}{80} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (-86400+216000 x^{2}-54000 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{6}} \]

Problem 1417

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

Problem 1418

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 1419

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

Problem 1420

ODE	\[ \boxed {x \left (x +1\right ) y^{\prime \prime }+y^{\prime } x +y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-x +\frac {5 x^{2}}{6}-\frac {25 x^{3}}{36}+\frac {85 x^{4}}{144}-\frac {221 x^{5}}{432}+O\left (x^{6}\right )\right )+c_{2} \left (-x \left (1-x +\frac {5 x^{2}}{6}-\frac {25 x^{3}}{36}+\frac {85 x^{4}}{144}-\frac {221 x^{5}}{432}+O\left (x^{6}\right )\right ) \ln \left (x \right )+1-\frac {x^{2}}{2}+\frac {4 x^{3}}{9}-\frac {155 x^{4}}{432}+\frac {253 x^{5}}{864}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1-x +\frac {5}{6} x^{2}-\frac {25}{36} x^{3}+\frac {85}{144} x^{4}-\frac {221}{432} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-x +x^{2}-\frac {5}{6} x^{3}+\frac {25}{36} x^{4}-\frac {85}{144} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right ) c_{2} +\left (1-x +\frac {1}{2} x^{2}-\frac {7}{18} x^{3}+\frac {145}{432} x^{4}-\frac {257}{864} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 1421

ODE	\[ \boxed {2 x^{2} \left (3 x +2\right ) y^{\prime \prime }+x \left (4+21 x \right ) y^{\prime }-\left (1-9 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {9 x}{4}+\frac {135 x^{2}}{32}-\frac {945 x^{3}}{128}+\frac {25515 x^{4}}{2048}-\frac {168399 x^{5}}{8192}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {3 \sqrt {x}\, \left (1-\frac {9 x}{4}+\frac {135 x^{2}}{32}-\frac {945 x^{3}}{128}+\frac {25515 x^{4}}{2048}-\frac {168399 x^{5}}{8192}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{4}+\frac {1-\frac {9 x^{2}}{16}+\frac {351 x^{3}}{256}-\frac {2727 x^{4}}{1024}+\frac {155763 x^{5}}{32768}+O\left (x^{6}\right )}{\sqrt {x}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x \left (1-\frac {9}{4} x +\frac {135}{32} x^{2}-\frac {945}{128} x^{3}+\frac {25515}{2048} x^{4}-\frac {168399}{8192} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-\frac {3}{4} x +\frac {27}{16} x^{2}-\frac {405}{128} x^{3}+\frac {2835}{512} x^{4}-\frac {76545}{8192} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-\frac {15}{4} x +\frac {63}{8} x^{2}-\frac {3699}{256} x^{3}+\frac {25623}{1024} x^{4}-\frac {1375137}{32768} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{\sqrt {x}} \]

Problem 1422

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x \left (2+x \right ) y^{\prime }-\left (-3 x +2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-x +\frac {x^{2}}{2}-\frac {x^{3}}{6}+\frac {x^{4}}{24}-\frac {x^{5}}{120}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {x \left (1-x +\frac {x^{2}}{2}-\frac {x^{3}}{6}+\frac {x^{4}}{24}-\frac {x^{5}}{120}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{2}+\frac {1+\frac {x}{2}+\frac {x^{2}}{2}-\frac {x^{4}}{2}+\frac {3 x^{5}}{8}+O\left (x^{6}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{3} \left (1-x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (\left (-6\right ) x^{3}+6 x^{4}-3 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (12+6 x +6 x^{2}-11 x^{3}+5 x^{4}-x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]

Problem 1423

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+4 y^{\prime } x -\left (9-x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {3}{2}} \left (1-\frac {x}{16}+\frac {x^{2}}{640}-\frac {x^{3}}{46080}+\frac {x^{4}}{5160960}-\frac {x^{5}}{825753600}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {x^{\frac {3}{2}} \left (1-\frac {x}{16}+\frac {x^{2}}{640}-\frac {x^{3}}{46080}+\frac {x^{4}}{5160960}-\frac {x^{5}}{825753600}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{768}+\frac {1+\frac {x}{8}+\frac {x^{2}}{64}-\frac {5 x^{4}}{49152}+\frac {13 x^{5}}{3276800}+O\left (x^{6}\right )}{x^{\frac {3}{2}}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{3} \left (1-\frac {1}{16} x +\frac {1}{640} x^{2}-\frac {1}{46080} x^{3}+\frac {1}{5160960} x^{4}-\frac {1}{825753600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-\frac {1}{64} x^{3}+\frac {1}{1024} x^{4}-\frac {1}{40960} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (12+\frac {3}{2} x +\frac {3}{16} x^{2}-\frac {5}{4096} x^{4}+\frac {39}{819200} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{\frac {3}{2}}} \]

Problem 1424

ODE	\[ \boxed {x^{2} y^{\prime \prime }+10 y^{\prime } x +\left (x +14\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1-\frac {x}{6}+\frac {x^{2}}{84}-\frac {x^{3}}{2016}+\frac {x^{4}}{72576}-\frac {x^{5}}{3628800}+O\left (x^{6}\right )\right )}{x^{2}}+c_{2} \left (-\frac {\left (1-\frac {x}{6}+\frac {x^{2}}{84}-\frac {x^{3}}{2016}+\frac {x^{4}}{72576}-\frac {x^{5}}{3628800}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{2880 x^{2}}+\frac {1+\frac {x}{4}+\frac {x^{2}}{24}+\frac {x^{3}}{144}+\frac {x^{4}}{576}+O\left (x^{6}\right )}{x^{7}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-\frac {1}{6} x +\frac {1}{84} x^{2}-\frac {1}{2016} x^{3}+\frac {1}{72576} x^{4}-\frac {1}{3628800} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) x^{5}+c_{2} \left (\ln \left (x \right ) \left (-x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (2880+720 x +120 x^{2}+20 x^{3}+5 x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{7}} \]

Problem 1425

ODE	\[ \boxed {4 x^{2} \left (x +1\right ) y^{\prime \prime }+4 x \left (3+8 x \right ) y^{\prime }-\left (5-49 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-4 x +10 x^{2}-20 x^{3}+35 x^{4}-56 x^{5}+O\left (x^{6}\right )\right )+c_{2} \left (-3 \sqrt {x}\, \left (1-4 x +10 x^{2}-20 x^{3}+35 x^{4}-56 x^{5}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\frac {1+\frac {x}{2}+x^{2}-9 x^{4}+\frac {63 x^{5}}{2}+O\left (x^{6}\right )}{x^{\frac {5}{2}}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{3} \left (1-4 x +10 x^{2}-20 x^{3}+35 x^{4}-56 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (\left (-36\right ) x^{3}+144 x^{4}-360 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (12+6 x +12 x^{2}-240 x^{3}+852 x^{4}-2022 x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{\frac {5}{2}}} \]

Problem 1426

ODE	\[ \boxed {x^{2} \left (x +1\right ) y^{\prime \prime }-x \left (3+10 x \right ) y^{\prime }+30 y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{4} \left (1-\frac {2 x}{5}+O\left (x^{6}\right )\right )+c_{2} \left (-300 x^{4} \left (1-\frac {2 x}{5}+O\left (x^{6}\right )\right ) \ln \left (x \right )+1+10 x +50 x^{2}+200 x^{3}-324 x^{5}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{4} \left (1-\frac {2}{5} x +\operatorname {O}\left (x^{6}\right )\right )+\ln \left (x \right ) \left (43200 x^{4}-17280 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} +\left (-144-1440 x -7200 x^{2}-28800 x^{3}-90720 x^{4}+82944 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 1427

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x \left (x +1\right ) y^{\prime }-3 y \left (x +3\right )=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{3} \left (1+O\left (x^{7}\right )\right )+c_{2} \left (-\frac {x^{3} \left (1+O\left (x^{7}\right )\right ) \ln \left (x \right )}{120}+\frac {1-\frac {6 x}{5}+\frac {3 x^{2}}{4}-\frac {x^{3}}{3}+\frac {x^{4}}{8}-\frac {x^{5}}{20}+O\left (x^{7}\right )}{x^{3}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{3} \left (1+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (-86400+103680 x -64800 x^{2}+28800 x^{3}-10800 x^{4}+4320 x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{3}} \]

Problem 1428

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x \left (1-2 x \right ) y^{\prime }-\left (x +4\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1+x +\frac {7 x^{2}}{12}+\frac {x^{3}}{4}+\frac {11 x^{4}}{128}+\frac {143 x^{5}}{5760}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {x^{2} \left (1+x +\frac {7 x^{2}}{12}+\frac {x^{3}}{4}+\frac {11 x^{4}}{128}+\frac {143 x^{5}}{5760}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{16}+\frac {1+x +\frac {x^{2}}{4}-\frac {x^{3}}{12}+\frac {x^{5}}{20}+O\left (x^{6}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{4} \left (1+x +\frac {7}{12} x^{2}+\frac {1}{4} x^{3}+\frac {11}{128} x^{4}+\frac {143}{5760} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (9 x^{4}+9 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-144 x -36 x^{2}+12 x^{3}-\frac {36}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]

Problem 1429

ODE	\[ \boxed {x \left (x +1\right ) y^{\prime \prime }-4 y^{\prime }-2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{5} \left (1-3 x +6 x^{2}-10 x^{3}+15 x^{4}-21 x^{5}+O\left (x^{6}\right )\right )+c_{2} \left (1-\frac {x}{2}+\frac {x^{2}}{6}-\frac {x^{5}}{6}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{5} \left (1-3 x +6 x^{2}-10 x^{3}+15 x^{4}-21 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (2880-1440 x +480 x^{2}-480 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 1430

ODE	\[ \boxed {x^{2} \left (1+2 x \right ) y^{\prime \prime }+x \left (9+13 x \right ) y^{\prime }+\left (7+5 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1+\frac {4 x}{7}-\frac {5 x^{2}}{28}+\frac {5 x^{3}}{42}-\frac {5 x^{4}}{48}+\frac {7 x^{5}}{66}-\frac {21 x^{6}}{176}+O\left (x^{7}\right )\right )}{x}+\frac {c_{2} \left (1+\frac {26 x}{5}+\frac {143 x^{2}}{20}-\frac {3003 x^{6}}{160}+O\left (x^{7}\right )\right )}{x^{7}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1+\frac {4}{7} x -\frac {5}{28} x^{2}+\frac {5}{42} x^{3}-\frac {5}{48} x^{4}+\frac {7}{66} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x}+\frac {c_{2} \left (-86400-449280 x -617760 x^{2}+\operatorname {O}\left (x^{6}\right )\right )}{x^{7}} \]

Problem 1431

ODE	\[ \boxed {4 x^{2} \left (1+2 x \right ) y^{\prime \prime }-2 x \left (-x +4\right ) y^{\prime }-\left (7+5 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {7}{2}} \left (1-\frac {18 x}{5}+\frac {39 x^{2}}{4}-\frac {663 x^{3}}{28}+\frac {13923 x^{4}}{256}-\frac {7735 x^{5}}{64}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {35 x^{4}}{128}+\frac {63 x^{5}}{64}+O\left (x^{6}\right )\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{4} \left (1-\frac {18}{5} x +\frac {39}{4} x^{2}-\frac {663}{28} x^{3}+\frac {13923}{256} x^{4}-\frac {7735}{64} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (-144-\frac {405}{8} x^{4}+\frac {729}{4} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

Problem 1432

ODE	\[ \boxed {3 x^{2} \left (x +3\right ) y^{\prime \prime }-x \left (15+x \right ) y^{\prime }-20 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {10}{3}} \left (1-\frac {4 x}{9}+\frac {13 x^{2}}{81}-\frac {832 x^{3}}{15309}+\frac {2470 x^{4}}{137781}-\frac {21736 x^{5}}{3720087}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+\frac {4 x}{27}-\frac {x^{2}}{243}+\frac {7 x^{4}}{59049}-\frac {28 x^{5}}{531441}+O\left (x^{6}\right )\right )}{x^{\frac {2}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{4} \left (1-\frac {4}{9} x +\frac {13}{81} x^{2}-\frac {832}{15309} x^{3}+\frac {2470}{137781} x^{4}-\frac {21736}{3720087} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (-144-\frac {64}{3} x +\frac {16}{27} x^{2}-\frac {112}{6561} x^{4}+\frac {448}{59049} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {2}{3}}} \]

Problem 1433

ODE	\[ \boxed {x^{2} \left (x +1\right ) y^{\prime \prime }+x \left (1-10 x \right ) y^{\prime }-\left (9-10 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{3} \left (1+2 x +\frac {9 x^{2}}{4}+\frac {5 x^{3}}{3}+\frac {5 x^{4}}{6}+\frac {3 x^{5}}{11}+\frac {7 x^{6}}{132}+O\left (x^{7}\right )\right )+\frac {c_{2} \left (1+\frac {52 x}{5}+\frac {234 x^{2}}{5}+\frac {572 x^{3}}{5}+143 x^{4}-\frac {1716 x^{6}}{5}+O\left (x^{7}\right )\right )}{x^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{3} \left (1+2 x +\frac {9}{4} x^{2}+\frac {5}{3} x^{3}+\frac {5}{6} x^{4}+\frac {3}{11} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (-86400-898560 x -4043520 x^{2}-9884160 x^{3}-12355200 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{3}} \]

Problem 1434

ODE	\[ \boxed {x^{2} \left (x +1\right ) y^{\prime \prime }+3 y^{\prime } x^{2}-\left (6-x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{3} \left (1-\frac {8 x}{3}+\frac {100 x^{2}}{21}-\frac {50 x^{3}}{7}+\frac {175 x^{4}}{18}-\frac {112 x^{5}}{9}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+\frac {x}{4}+O\left (x^{6}\right )\right )}{x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{3} \left (1-\frac {8}{3} x +\frac {100}{21} x^{2}-\frac {50}{7} x^{3}+\frac {175}{18} x^{4}-\frac {112}{9} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (2880+720 x +\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

Problem 1435

ODE	\[ \boxed {x^{2} \left (1+2 x \right ) y^{\prime \prime }-2 x \left (3+14 x \right ) y^{\prime }+\left (6+100 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{6} \left (1+\frac {4 x}{3}+\frac {8 x^{2}}{7}+\frac {4 x^{3}}{7}+\frac {8 x^{4}}{63}+O\left (x^{6}\right )\right )+c_{2} x \left (1+18 x +144 x^{2}+672 x^{3}+2016 x^{4}+4032 x^{5}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{6} \left (1+\frac {4}{3} x +\frac {8}{7} x^{2}+\frac {4}{7} x^{3}+\frac {8}{63} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x \left (2880+51840 x +414720 x^{2}+1935360 x^{3}+5806080 x^{4}[phone] x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 1436

ODE	\[ \boxed {x^{2} \left (x +1\right ) y^{\prime \prime }-x \left (6+11 x \right ) y^{\prime }+\left (6+32 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{6} \left (1+\frac {2 x}{3}+\frac {x^{2}}{7}+O\left (x^{6}\right )\right )+c_{2} x \left (1+\frac {21 x}{4}+\frac {21 x^{2}}{2}+\frac {35 x^{3}}{4}-\frac {21 x^{5}}{4}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{6} \left (1+\frac {2}{3} x +\frac {1}{7} x^{2}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x \left (2880+15120 x +30240 x^{2}+25200 x^{3}-15120 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 1437

ODE	\[ \boxed {4 x^{2} \left (x +1\right ) y^{\prime \prime }+4 x \left (4 x +1\right ) y^{\prime }-\left (49+27 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {7}{2}} \left (1-2 x +3 x^{2}-4 x^{3}+5 x^{4}-6 x^{5}+7 x^{6}-8 x^{7}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1-\frac {5 x}{6}+\frac {2 x^{2}}{3}-\frac {x^{3}}{2}+\frac {x^{4}}{3}-\frac {x^{5}}{6}+\frac {x^{7}}{6}+O\left (x^{8}\right )\right )}{x^{\frac {7}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{7} \left (1-2 x +3 x^{2}-4 x^{3}+5 x^{4}-6 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (3628800-3024000 x +2419200 x^{2}-1814400 x^{3}+1209600 x^{4}-604800 x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {7}{2}}} \]

Problem 1438

ODE	\[ \boxed {x^{2} \left (1+2 x \right ) y^{\prime \prime }-x \left (9+8 x \right ) y^{\prime }-12 y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{10} \left (1-8 x +40 x^{2}-160 x^{3}+560 x^{4}-1792 x^{5}+5376 x^{6}-15360 x^{7}+42240 x^{8}-112640 x^{9}+292864 x^{10}+O\left (x^{11}\right )\right )+c_{2} \left (1-\frac {4 x}{3}+\frac {5 x^{2}}{3}-\frac {40 x^{3}}{21}+\frac {40 x^{4}}{21}-\frac {32 x^{5}}{21}+\frac {16 x^{6}}{21}-\frac {256 x^{10}}{21}+O\left (x^{11}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{10} \left (1-8 x +40 x^{2}-160 x^{3}+560 x^{4}-1792 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (-1316818944000+1755758592000 x -2194698240000 x^{2}[phone] x^{3}-2508226560000 x^{4}[phone] x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 1439

ODE	\[ \boxed {x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-2 x^{2}+7\right ) y^{\prime }+12 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{6} \left (1-\frac {7 x^{2}}{2}+\frac {63 x^{4}}{8}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {15 x^{6} \left (1-\frac {7 x^{2}}{2}+\frac {63 x^{4}}{8}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{2}+x^{2} \left (1+\frac {3 x^{2}}{2}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} x^{4} \left (1-\frac {7}{2} x^{2}+\frac {63}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (1080 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-216 x^{2}+2106 x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right )\right ) x^{2} \]

Problem 1440

ODE	\[ \boxed {x^{2} y^{\prime \prime }-x \left (-x^{2}+7\right ) y^{\prime }+12 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{6} \left (1-\frac {x^{2}}{2}+\frac {x^{4}}{8}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {x^{6} \left (1-\frac {x^{2}}{2}+\frac {x^{4}}{8}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{2}+x^{2} \left (1+\frac {x^{2}}{2}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} x^{4} \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (72 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-72 x^{2}+54 x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right )\right ) x^{2} \]

Problem 1441

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 1442

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

Problem 1443

ODE	\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x -\left (-x^{2}+3\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{3} \left (1-\frac {x^{2}}{12}+\frac {x^{4}}{384}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {x^{3} \left (1-\frac {x^{2}}{12}+\frac {x^{4}}{384}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{16}+\frac {1+\frac {x^{2}}{4}+O\left (x^{6}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{4} \left (1-\frac {1}{12} x^{2}+\frac {1}{384} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (9 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-36 x^{2}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]

Problem 1444

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+2 x \left (x^{2}+8\right ) y^{\prime }+\left (3 x^{2}+5\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1-\frac {x^{2}}{16}+\frac {x^{4}}{256}+O\left (x^{6}\right )\right )}{\sqrt {x}}+c_{2} \left (\frac {\left (1-\frac {x^{2}}{16}+\frac {x^{4}}{256}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{4 \sqrt {x}}+\frac {1-\frac {x^{4}}{256}+O\left (x^{6}\right )}{x^{\frac {5}{2}}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-\frac {1}{16} x^{2}+\frac {1}{256} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) x^{2}+c_{2} \left (\ln \left (x \right ) \left (-\frac {1}{2} x^{2}+\frac {1}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+\frac {1}{2} x^{2}-\frac {3}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{\frac {5}{2}}} \]

Problem 1445

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

Problem 1446

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x \left (-2 x^{2}+1\right ) y^{\prime }-4 y \left (2 x^{2}+1\right )=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1+x^{2}+\frac {x^{4}}{2}+O\left (x^{6}\right )\right )+c_{2} \left (-2 x^{2} \left (1+x^{2}+\frac {x^{4}}{2}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\frac {1-x^{2}+O\left (x^{6}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{4} \left (1+x^{2}+\frac {1}{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (288 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144+144 x^{2}+216 x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]

Problem 1447

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+8 y^{\prime } x -\left (-x^{2}+35\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {5}{2}} \left (1-\frac {x^{2}}{64}+\frac {x^{4}}{10240}-\frac {x^{6}}{2949120}+O\left (x^{7}\right )\right )+c_{2} \left (-\frac {x^{\frac {5}{2}} \left (1-\frac {x^{2}}{64}+\frac {x^{4}}{10240}-\frac {x^{6}}{2949120}+O\left (x^{7}\right )\right ) \ln \left (x \right )}{24576}+\frac {1+\frac {x^{2}}{32}+\frac {x^{4}}{1024}+O\left (x^{7}\right )}{x^{\frac {7}{2}}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{6} \left (1-\frac {1}{64} x^{2}+\frac {1}{10240} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (-86400-2700 x^{2}-\frac {675}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {7}{2}}} \]

Problem 1448

ODE	\[ \boxed {9 x^{2} y^{\prime \prime }-3 x \left (2 x^{2}+11\right ) y^{\prime }+\left (10 x^{2}+13\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {13}{3}} \left (1+\frac {4 x^{2}}{27}+\frac {7 x^{4}}{486}+O\left (x^{6}\right )\right )+c_{2} \left (\frac {2 x^{\frac {13}{3}} \left (1+\frac {4 x^{2}}{27}+\frac {7 x^{4}}{486}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{81}+x^{\frac {1}{3}} \left (1+\frac {2 x^{2}}{9}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{\frac {1}{3}} \left (\left (1+\frac {4}{27} x^{2}+\frac {7}{486} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) x^{4} c_{1} +c_{2} \left (\ln \left (x \right ) \left (-\frac {32}{9} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-32 x^{2}-\frac {8}{3} x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right )\right ) \]

Problem 1449

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x \left (-2 x^{2}+1\right ) y^{\prime }-4 \left (1-x^{2}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1+O\left (x^{6}\right )\right )+c_{2} \left (-2 x^{2} \left (1+O\left (x^{6}\right )\right ) \ln \left (x \right )+\frac {1+2 x^{2}+O\left (x^{6}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \left (1+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (\ln \left (x \right ) \left (288 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-288 x^{2}-216 x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]

Problem 1450

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x \left (-3 x^{2}+1\right ) y^{\prime }-4 \left (-3 x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-\frac {x^{2}}{2}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {27 x^{2} \left (1-\frac {x^{2}}{2}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{2}+\frac {1+\frac {9 x^{2}}{2}+O\left (x^{6}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \left (1-\frac {1}{2} x^{2}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (\ln \left (x \right ) \left (1944 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-648 x^{2}-810 x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]

Problem 1451

ODE	\[ \boxed {x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (11 x^{2}+5\right ) y^{\prime }+24 x^{2} y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (3 x^{4}-2 x^{2}+1+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-x^{4}+O\left (x^{6}\right )\right )}{x^{4}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (1-2 x^{2}+3 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (-144+432 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{4}} \]

Problem 1452

ODE	\[ \boxed {4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+8 y^{\prime } x -\left (-x^{2}+35\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {5}{2}} \left (1-\frac {x^{2}}{4}+\frac {x^{4}}{10}-\frac {x^{6}}{20}+O\left (x^{7}\right )\right )+\frac {c_{2} \left (1+2 x^{2}+x^{4}+O\left (x^{7}\right )\right )}{x^{\frac {7}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{6} \left (1-\frac {1}{4} x^{2}+\frac {1}{10} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (-86400-172800 x^{2}-86400 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {7}{2}}} \]

Problem 1453

ODE	\[ \boxed {x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-x^{2}+5\right ) y^{\prime }-\left (25 x^{2}+7\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{7} \left (1-\frac {6 x^{2}}{5}+\frac {7 x^{4}}{5}-\frac {8 x^{6}}{5}+\frac {9 x^{8}}{5}+O\left (x^{9}\right )\right )+\frac {c_{2} \left (1-2 x^{2}+3 x^{4}-4 x^{6}+5 x^{8}+O\left (x^{9}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{7} \left (1-\frac {6}{5} x^{2}+\frac {7}{5} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (-203212800+406425600 x^{2}-609638400 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 1454

ODE	\[ \boxed {x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (2 x^{2}+5\right ) y^{\prime }-21 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{3} \left (1-\frac {x^{2}}{2}+\frac {15 x^{4}}{56}-\frac {5 x^{6}}{32}+\frac {25 x^{8}}{256}-\frac {33 x^{10}}{512}+O\left (x^{11}\right )\right )+\frac {c_{2} \left (1+\frac {21 x^{2}}{8}+\frac {35 x^{4}}{16}+\frac {35 x^{6}}{64}+\frac {7 x^{10}}{1024}+O\left (x^{11}\right )\right )}{x^{7}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{3} \left (1-\frac {1}{2} x^{2}+\frac {15}{56} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (-1316818944000-3456649728000 x^{2}-2880541440000 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{7}} \]

Problem 1455

ODE	\[ \boxed {x^{2} \left (1+2 x^{2}\right ) y^{\prime \prime }-x \left (x^{2}+3\right ) y^{\prime }-2 y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{4} \left (1+\frac {2 x}{5}-\frac {8 x^{2}}{5}-\frac {86 x^{3}}{105}+\frac {445 x^{4}}{168}+\frac {9571 x^{5}}{6300}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {x^{4} \left (1+\frac {2 x}{5}-\frac {8 x^{2}}{5}-\frac {86 x^{3}}{105}+\frac {445 x^{4}}{168}+\frac {9571 x^{5}}{6300}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{6}+1-\frac {2 x}{3}+\frac {x^{2}}{3}+\frac {2 x^{5}}{25}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{4} \left (1+\frac {2}{5} x -\frac {8}{5} x^{2}-\frac {86}{105} x^{3}+\frac {445}{168} x^{4}+\frac {9571}{6300} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (24 x^{4}+\frac {48}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144+96 x -48 x^{2}+210 x^{4}+\frac {1812}{25} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

Problem 1456

ODE	\[ \boxed {4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+4 x \left (x^{2}+2\right ) y^{\prime }-\left (x^{2}+15\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {3}{2}} \left (1-\frac {x^{2}}{6}+\frac {x^{4}}{16}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+\frac {3 x^{2}}{2}+\frac {3 x^{4}}{8}+O\left (x^{6}\right )\right )}{x^{\frac {5}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{4} \left (1-\frac {1}{6} x^{2}+\frac {1}{16} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (-144-216 x^{2}-54 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {5}{2}}} \]

Problem 1457

ODE	\[ \boxed {x^{3} y^{\prime \prime \prime }-x^{2} y^{\prime \prime }-2 y^{\prime } x +6 y=0} \] With initial conditions \begin {align*} [y \left (-1\right ) = -4, y^{\prime }\left (-1\right ) = -14, y^{\prime \prime }\left (-1\right ) = -20] \end {align*}
program solution	\[ y = \frac {9 x^{4}+22 x^{3}+25}{3 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 3 x^{3}+\frac {22 x^{2}}{3}+\frac {25}{3 x} \]

Problem 1458

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+y^{\prime \prime \prime }-7 y^{\prime \prime }-y^{\prime }+6 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 5, y^{\prime }\left (0\right ) = -6, y^{\prime \prime }\left (0\right ) = 10, y^{\prime \prime \prime }\left (0\right ) = -36] \end {align*}
program solution	\[ y = 3 \,{\mathrm e}^{-x}+2 \,{\mathrm e}^{x}+{\mathrm e}^{-3 x}-{\mathrm e}^{2 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (-{\mathrm e}^{5 x}+2 \,{\mathrm e}^{4 x}+3 \,{\mathrm e}^{2 x}+1\right ) {\mathrm e}^{-3 x} \]

Problem 1459

ODE	\[ \boxed {x^{3} y^{\prime \prime \prime }-x^{2} y^{\prime \prime }-2 y^{\prime } x +6 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1, y^{\prime }\left (1\right ) = 0, y^{\prime \prime }\left (1\right ) = 0] \end {align*}
program solution	\[ y = -\frac {x^{4}-2 x^{3}-1}{2 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {x^{3}}{2}+x^{2}+\frac {1}{2 x} \]

Problem 1460

ODE	\[ \boxed {x^{3} y^{\prime \prime \prime }-x^{2} y^{\prime \prime }-2 y^{\prime } x +6 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 0, y^{\prime }\left (1\right ) = 1, y^{\prime \prime }\left (1\right ) = 0] \end {align*}
program solution	\[ y = \frac {x^{3}-1}{3 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x^{3}-1}{3 x} \]

Problem 1461

ODE	\[ \boxed {x^{3} y^{\prime \prime \prime }-x^{2} y^{\prime \prime }-2 y^{\prime } x +6 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 0, y^{\prime }\left (1\right ) = 0, y^{\prime \prime }\left (1\right ) = 1] \end {align*}
program solution	\[ y = \frac {3 x^{4}-4 x^{3}+1}{12 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {3 x^{4}-4 x^{3}+1}{12 x} \]

Problem 1462

ODE	\[ \boxed {x^{3} y^{\prime \prime \prime }-x^{2} y^{\prime \prime }-2 y^{\prime } x +6 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = k_{0}, y^{\prime }\left (1\right ) = k_{1}, y^{\prime \prime }\left (1\right ) = k_{2}] \end {align*}
program solution	\[ y = \frac {3 \left (k_{2} -2 k_{0} \right ) x^{4}+4 \left (k_{1} +3 k_{0} -k_{2} \right ) x^{3}+6 k_{0} -4 k_{1} +k_{2}}{12 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {3 \left (-2 k_{0} +k_{2} \right ) x^{4}+4 \left (k_{1} +3 k_{0} -k_{2} \right ) x^{3}+6 k_{0} -4 k_{1} +k_{2}}{12 x} \]

Problem 1463

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

Problem 1464

ODE	\[ \boxed {y^{\prime \prime \prime }-3 y^{\prime \prime }+7 y^{\prime }-5 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{x}+{\mathrm e}^{\left (1+2 i\right ) x} c_{2} +{\mathrm e}^{\left (1-2 i\right ) x} c_{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{x} \left (c_{1} +\sin \left (2 x \right ) c_{2} +\cos \left (2 x \right ) c_{3} \right ) \]

Problem 1465

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

Problem 1466

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+8 y^{\prime \prime }-9 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+c_{2} {\mathrm e}^{x}+{\mathrm e}^{-3 i x} c_{3} +{\mathrm e}^{3 i x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-x} c_{1} +c_{2} {\mathrm e}^{x}+c_{3} \sin \left (3 x \right )+c_{4} \cos \left (3 x \right ) \]

Problem 1467

ODE	\[ \boxed {y^{\prime \prime \prime }-y^{\prime \prime }+16 y^{\prime }-16 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{x}+{\mathrm e}^{-4 i x} c_{2} +{\mathrm e}^{4 i x} c_{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{x} c_{1} +\sin \left (4 x \right ) c_{2} +c_{3} \cos \left (4 x \right ) \]

Problem 1468

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

Problem 1469

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

Problem 1470

ODE	\[ \boxed {4 y^{\prime \prime \prime }-8 y^{\prime \prime }+5 y^{\prime }-y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{x}+c_{2} {\mathrm e}^{\frac {x}{2}}+x \,{\mathrm e}^{\frac {x}{2}} c_{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{3} x +c_{2} \right ) {\mathrm e}^{\frac {x}{2}}+{\mathrm e}^{x} c_{1} \]

Problem 1471

ODE	\[ \boxed {27 y^{\prime \prime \prime }+27 y^{\prime \prime }+9 y^{\prime }+y=0} \]
program solution	\[ y = {\mathrm e}^{-\frac {x}{3}} c_{1} +x \,{\mathrm e}^{-\frac {x}{3}} c_{2} +x^{2} {\mathrm e}^{-\frac {x}{3}} c_{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\frac {x}{3}} \left (c_{3} x^{2}+c_{2} x +c_{1} \right ) \]

Problem 1472

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

Problem 1473

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-16 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+c_{2} {\mathrm e}^{2 x}+{\mathrm e}^{-2 i x} c_{3} +{\mathrm e}^{2 i x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{2 x}+{\mathrm e}^{-2 x} c_{2} +c_{3} \sin \left (2 x \right )+c_{4} \cos \left (2 x \right ) \]

Problem 1474

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+12 y^{\prime \prime }+36 y=0} \]
program solution	\[ y = {\mathrm e}^{i \sqrt {6}\, x} c_{1} +x \,{\mathrm e}^{i \sqrt {6}\, x} c_{2} +{\mathrm e}^{-i \sqrt {6}\, x} c_{3} +x \,{\mathrm e}^{-i \sqrt {6}\, x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{4} x +c_{2} \right ) \cos \left (\sqrt {6}\, x \right )+\sin \left (\sqrt {6}\, x \right ) \left (c_{3} x +c_{1} \right ) \]

Problem 1475

ODE	\[ \boxed {16 y^{\prime \prime \prime \prime }-72 y^{\prime \prime }+81 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {3 x}{2}}+x \,{\mathrm e}^{-\frac {3 x}{2}} c_{2} +{\mathrm e}^{\frac {3 x}{2}} c_{3} +x \,{\mathrm e}^{\frac {3 x}{2}} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\frac {3 x}{2}} \left (c_{2} x +c_{1} \right )+{\mathrm e}^{\frac {3 x}{2}} \left (c_{4} x +c_{3} \right ) \]

Problem 1476

ODE	\[ \boxed {6 y^{\prime \prime \prime \prime }+5 y^{\prime \prime \prime }+7 y^{\prime \prime }+5 y^{\prime }+y=0} \]
program solution	\[ y = {\mathrm e}^{-i x} c_{1} +{\mathrm e}^{i x} c_{2} +{\mathrm e}^{-\frac {x}{2}} c_{3} +{\mathrm e}^{-\frac {x}{3}} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {x}{2}}+c_{2} {\mathrm e}^{-\frac {x}{3}}+c_{3} \sin \left (x \right )+c_{4} \cos \left (x \right ) \]

Problem 1477

ODE	\[ \boxed {4 y^{\prime \prime \prime \prime }+12 y^{\prime \prime \prime }+3 y^{\prime \prime }-13 y^{\prime }-6 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+c_{2} {\mathrm e}^{x}+{\mathrm e}^{-\frac {3 x}{2}} c_{3} +{\mathrm e}^{-\frac {x}{2}} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{4} {\mathrm e}^{3 x}+c_{3} {\mathrm e}^{\frac {3 x}{2}}+c_{1} {\mathrm e}^{\frac {x}{2}}+c_{2} \right ) {\mathrm e}^{-2 x} \]

Problem 1478

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime }+7 y^{\prime \prime }-6 y^{\prime }+2 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{x}+{\mathrm e}^{x} c_{2} x +{\mathrm e}^{\left (1+i\right ) x} c_{3} +{\mathrm e}^{\left (1-i\right ) x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{x} \left (c_{1} +c_{2} x +c_{3} \sin \left (x \right )+c_{4} \cos \left (x \right )\right ) \]

Problem 1479

ODE	\[ \boxed {y^{\prime \prime \prime }-2 y^{\prime \prime }+4 y^{\prime }-8 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = -2, y^{\prime \prime }\left (0\right ) = 2] \end {align*}
program solution	\[ y = \frac {5 \,{\mathrm e}^{2 x}}{4}+\frac {3 \cos \left (2 x \right )}{4}-\frac {9 \sin \left (2 x \right )}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {5 \,{\mathrm e}^{2 x}}{4}-\frac {9 \sin \left (2 x \right )}{4}+\frac {3 \cos \left (2 x \right )}{4} \]

Problem 1480

ODE	\[ \boxed {y^{\prime \prime \prime }+3 y^{\prime \prime }-y^{\prime }-3 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 14, y^{\prime \prime }\left (0\right ) = -40] \end {align*}
program solution	\[ y = 2 \,{\mathrm e}^{x}-5 \,{\mathrm e}^{-3 x}+3 \,{\mathrm e}^{-x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (2 \,{\mathrm e}^{4 x}+3 \,{\mathrm e}^{2 x}-5\right ) {\mathrm e}^{-3 x} \]

Problem 1481

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

Problem 1482

ODE	\[ \boxed {y^{\prime \prime \prime }-2 y^{\prime }-4 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 6, y^{\prime }\left (0\right ) = 3, y^{\prime \prime }\left (0\right ) = 22] \end {align*}
program solution	\[ y = \left (1-\frac {3 i}{2}\right ) {\mathrm e}^{\left (-1-i\right ) x}+\left (1+\frac {3 i}{2}\right ) {\mathrm e}^{\left (-1+i\right ) x}+4 \,{\mathrm e}^{2 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 4 \,{\mathrm e}^{2 x}+\left (-3 \sin \left (x \right )+2 \cos \left (x \right )\right ) {\mathrm e}^{-x} \]

Problem 1483

ODE	\[ \boxed {3 y^{\prime \prime \prime }-y^{\prime \prime }-7 y^{\prime }+5 y=0} \] With initial conditions \begin {align*} \left [y \left (0\right ) = {\frac {14}{5}}, y^{\prime }\left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = 10\right ] \end {align*}
program solution	\[ y = 2 x \,{\mathrm e}^{x}+\frac {9 \,{\mathrm e}^{-\frac {5 x}{3}}}{5}+{\mathrm e}^{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\frac {5 x}{3}} \left (\frac {9}{5}+\left (2 x +1\right ) {\mathrm e}^{\frac {8 x}{3}}\right ) \]

Problem 1484

ODE	\[ \boxed {y^{\prime \prime \prime }-6 y^{\prime \prime }+12 y^{\prime }-8 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = -1, y^{\prime \prime }\left (0\right ) = -4] \end {align*}
program solution	\[ y = {\mathrm e}^{2 x} \left (2 x^{2}-3 x +1\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{2 x} \left (2 x^{2}-3 x +1\right ) \]

Problem 1485

ODE	\[ \boxed {2 y^{\prime \prime \prime }-11 y^{\prime \prime }+12 y^{\prime }+9 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 6, y^{\prime }\left (0\right ) = 3, y^{\prime \prime }\left (0\right ) = 13] \end {align*}
program solution	\[ y = 4 \,{\mathrm e}^{-\frac {x}{2}}+\left (2-x \right ) {\mathrm e}^{3 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 4 \,{\mathrm e}^{-\frac {x}{2}}+\left (2-x \right ) {\mathrm e}^{3 x} \]

Problem 1486

ODE	\[ \boxed {8 y^{\prime \prime \prime }-4 y^{\prime \prime }-2 y^{\prime }+y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 4, y^{\prime }\left (0\right ) = -3, y^{\prime \prime }\left (0\right ) = -1] \end {align*}
program solution	\[ y = -2 \,{\mathrm e}^{\frac {x}{2}} x +3 \,{\mathrm e}^{-\frac {x}{2}}+{\mathrm e}^{\frac {x}{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 3 \,{\mathrm e}^{-\frac {x}{2}}+{\mathrm e}^{\frac {x}{2}}-2 \,{\mathrm e}^{\frac {x}{2}} x \]

Problem 1487

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-16 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 2, y^{\prime \prime }\left (0\right ) = -2, y^{\prime \prime \prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {{\mathrm e}^{-2 x}}{8}+\frac {5 \,{\mathrm e}^{2 x}}{8}+\frac {5 \cos \left (2 x \right )}{4}+\frac {\sin \left (2 x \right )}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {5 \,{\mathrm e}^{2 x}}{8}+\frac {{\mathrm e}^{-2 x}}{8}+\frac {\sin \left (2 x \right )}{2}+\frac {5 \cos \left (2 x \right )}{4} \]

Problem 1488

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-6 y^{\prime \prime \prime }+7 y^{\prime \prime }+6 y^{\prime }-8 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -2, y^{\prime }\left (0\right ) = -8, y^{\prime \prime }\left (0\right ) = -14, y^{\prime \prime \prime }\left (0\right ) = -62] \end {align*}
program solution	\[ y = 2 \,{\mathrm e}^{-x}-4 \,{\mathrm e}^{x}+{\mathrm e}^{2 x}-{\mathrm e}^{4 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{2 x}-{\mathrm e}^{4 x}+2 \,{\mathrm e}^{-x}-4 \,{\mathrm e}^{x} \]

Problem 1489

ODE	\[ \boxed {4 y^{\prime \prime \prime \prime }-13 y^{\prime \prime }+9 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 3, y^{\prime \prime }\left (0\right ) = 1, y^{\prime \prime \prime }\left (0\right ) = 3] \end {align*}
program solution	\[ y = -{\mathrm e}^{-x}+2 \,{\mathrm e}^{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -{\mathrm e}^{-x}+2 \,{\mathrm e}^{x} \]

Problem 1490

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }-2 y^{\prime \prime }-8 y^{\prime }-8 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 5, y^{\prime }\left (0\right ) = -2, y^{\prime \prime }\left (0\right ) = 6, y^{\prime \prime \prime }\left (0\right ) = 8] \end {align*}
program solution	\[ y = \left (\frac {3}{2}+\frac {i}{2}\right ) {\mathrm e}^{\left (-1-i\right ) x}+\left (\frac {3}{2}-\frac {i}{2}\right ) {\mathrm e}^{\left (-1+i\right ) x}+{\mathrm e}^{-2 x}+{\mathrm e}^{2 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\left (\sin \left (x \right )+3 \cos \left (x \right )\right ) {\mathrm e}^{x}+{\mathrm e}^{4 x}\right ) {\mathrm e}^{-2 x} \]

Problem 1491

ODE	\[ \boxed {4 y^{\prime \prime \prime \prime }+8 y^{\prime \prime \prime }+19 y^{\prime \prime }+32 y^{\prime }+12 y=0} \] With initial conditions \begin {align*} \left [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = -3, y^{\prime \prime }\left (0\right ) = -{\frac {7}{2}}, y^{\prime \prime \prime }\left (0\right ) = {\frac {31}{4}}\right ] \end {align*}
program solution	\[ y = 2 \,{\mathrm e}^{-\frac {x}{2}}+\cos \left (2 x \right )-\sin \left (2 x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2 \,{\mathrm e}^{-\frac {x}{2}}-\sin \left (2 x \right )+\cos \left (2 x \right ) \]

Problem 1492

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

Problem 1493

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

Problem 1494

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+64 y=0} \]
program solution	\[ y = {\mathrm e}^{\left (-2+2 i\right ) x} c_{1} +{\mathrm e}^{\left (2+2 i\right ) x} c_{2} +{\mathrm e}^{\left (-2-2 i\right ) x} c_{3} +{\mathrm e}^{\left (2-2 i\right ) x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} {\mathrm e}^{2 x}+c_{4} {\mathrm e}^{-2 x}\right ) \cos \left (2 x \right )+\sin \left (2 x \right ) \left (c_{1} {\mathrm e}^{2 x}+c_{3} {\mathrm e}^{-2 x}\right ) \]

Problem 1495

ODE	\[ \boxed {y^{\left (6\right )}-y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+{\mathrm e}^{\frac {\sqrt {-2+2 i \sqrt {3}}\, x}{2}} c_{2} +{\mathrm e}^{x} c_{3} +{\mathrm e}^{-\frac {\sqrt {-2 i \sqrt {3}-2}\, x}{2}} c_{4} +{\mathrm e}^{\frac {\sqrt {-2 i \sqrt {3}-2}\, x}{2}} c_{5} +{\mathrm e}^{-\frac {\sqrt {-2+2 i \sqrt {3}}\, x}{2}} c_{6} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left ({\mathrm e}^{\frac {x}{2}} c_{6} +{\mathrm e}^{\frac {3 x}{2}} c_{4} \right ) \cos \left (\frac {\sqrt {3}\, x}{2}\right )+\left ({\mathrm e}^{\frac {x}{2}} c_{5} +c_{3} {\mathrm e}^{\frac {3 x}{2}}\right ) \sin \left (\frac {\sqrt {3}\, x}{2}\right )+c_{2} {\mathrm e}^{2 x}+c_{1} \right ) {\mathrm e}^{-x} \]

Problem 1496

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+64 y=0} \]
program solution	\[ y = {\mathrm e}^{\left (-2+2 i\right ) x} c_{1} +{\mathrm e}^{\left (2+2 i\right ) x} c_{2} +{\mathrm e}^{\left (-2-2 i\right ) x} c_{3} +{\mathrm e}^{\left (2-2 i\right ) x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} {\mathrm e}^{2 x}+c_{4} {\mathrm e}^{-2 x}\right ) \cos \left (2 x \right )+\sin \left (2 x \right ) \left (c_{1} {\mathrm e}^{2 x}+c_{3} {\mathrm e}^{-2 x}\right ) \]

Problem 1497

ODE	\[ \boxed {y^{\left (5\right )}+y^{\prime \prime \prime \prime }+y^{\prime \prime \prime }+y^{\prime \prime }+y^{\prime }+y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+{\mathrm e}^{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x} c_{2} +{\mathrm e}^{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x} c_{3} +{\mathrm e}^{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x} c_{4} +{\mathrm e}^{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x} c_{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-x} \left (\left ({\mathrm e}^{\frac {x}{2}} c_{5} +c_{3} {\mathrm e}^{\frac {3 x}{2}}\right ) \cos \left (\frac {\sqrt {3}\, x}{2}\right )+\left (c_{2} {\mathrm e}^{\frac {3 x}{2}}+c_{4} {\mathrm e}^{\frac {x}{2}}\right ) \sin \left (\frac {\sqrt {3}\, x}{2}\right )+c_{1} \right ) \]

Problem 1498

ODE	\[ \boxed {y^{\prime \prime \prime }-6 y^{\prime \prime }+11 y^{\prime }-6 y=-{\mathrm e}^{x} \left (-24 x^{2}+76 x +4\right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{x}+c_{2} {\mathrm e}^{2 x}+{\mathrm e}^{3 x} c_{3} -17 x \,{\mathrm e}^{x}-x^{2} {\mathrm e}^{x}+4 \,{\mathrm e}^{x} x^{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{x} \left (c_{3} {\mathrm e}^{2 x}+4 x^{3}+c_{2} {\mathrm e}^{x}-x^{2}+c_{1} -17 x \right ) \]

Problem 1499

ODE	\[ \boxed {y^{\prime \prime \prime }-2 y^{\prime \prime }-5 y^{\prime }+6 y={\mathrm e}^{-3 x} \left (6 x^{2}-23 x +32\right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+c_{2} {\mathrm e}^{x}+{\mathrm e}^{3 x} c_{3} +\frac {x \,{\mathrm e}^{-3 x}}{4}-\frac {x^{2} {\mathrm e}^{-3 x}}{4}-\frac {3 \,{\mathrm e}^{-3 x}}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (4 c_{3} {\mathrm e}^{6 x}+4 \,{\mathrm e}^{4 x} c_{1} +4 c_{2} {\mathrm e}^{x}-x^{2}+x -3\right ) {\mathrm e}^{-3 x}}{4} \]

Problem 1500

ODE	\[ \boxed {4 y^{\prime \prime \prime }+8 y^{\prime \prime }-y^{\prime }-2 y=-{\mathrm e}^{x} \left (6 x^{2}+45 x +4\right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+c_{2} {\mathrm e}^{-\frac {x}{2}}+{\mathrm e}^{\frac {x}{2}} c_{3} -x \,{\mathrm e}^{x}-\frac {2 x^{2} {\mathrm e}^{x}}{3}+\frac {149 \,{\mathrm e}^{x}}{27} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-18 x^{2} {\mathrm e}^{3 x}-27 x \,{\mathrm e}^{3 x}+149 \,{\mathrm e}^{3 x}+27 c_{3} {\mathrm e}^{\frac {5 x}{2}}+27 c_{2} {\mathrm e}^{\frac {3 x}{2}}+27 c_{1} \right ) {\mathrm e}^{-2 x}}{27} \]