2.17.14 Problems 1301 to 1400

Problem 1301

ODE	\[ \boxed {8 x^{2} y^{\prime \prime }-2 x \left (-x^{2}-4 x +3\right ) y^{\prime }+\left (x^{2}+6 x +3\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {3}{2}} \left (1-x +\frac {11 x^{2}}{26}-\frac {109 x^{3}}{1326}+\frac {5 x^{4}}{12376}+\frac {229 x^{5}}{71400}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{4}} \left (1+4 x -\frac {131 x^{2}}{24}+\frac {39 x^{3}}{14}-\frac {19865 x^{4}}{29568}+\frac {4421 x^{5}}{110880}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{4}} \left (1+4 x -\frac {131}{24} x^{2}+\frac {39}{14} x^{3}-\frac {19865}{29568} x^{4}+\frac {4421}{110880} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{\frac {3}{2}} \left (1-x +\frac {11}{26} x^{2}-\frac {109}{1326} x^{3}+\frac {5}{12376} x^{4}+\frac {229}{71400} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 1302

ODE	\[ \boxed {18 x^{2} \left (x +1\right ) y^{\prime \prime }+3 x \left (x^{2}+11 x +5\right ) y^{\prime }-\left (-5 x^{2}-2 x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1-\frac {x}{3}+\frac {2 x^{2}}{15}-\frac {5 x^{3}}{63}+\frac {23 x^{4}}{405}-\frac {458 x^{5}}{10395}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {x^{2}}{12}+\frac {x^{3}}{18}-\frac {11 x^{4}}{288}+\frac {31 x^{5}}{1080}+O\left (x^{6}\right )\right )}{x^{\frac {1}{6}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} \sqrt {x}\, \left (1-\frac {1}{3} x +\frac {2}{15} x^{2}-\frac {5}{63} x^{3}+\frac {23}{405} x^{4}-\frac {458}{10395} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1-\frac {1}{12} x^{2}+\frac {1}{18} x^{3}-\frac {11}{288} x^{4}+\frac {31}{1080} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {1}{6}}} \]

Problem 1303

ODE	\[ \boxed {x \left (x^{2}+x +3\right ) y^{\prime \prime }+\left (-x^{2}+x +4\right ) y^{\prime }+y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-\frac {x^{2}}{14}+\frac {x^{3}}{105}-\frac {x^{4}}{3640}-\frac {23 x^{5}}{54600}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {x}{18}-\frac {71 x^{2}}{405}+\frac {719 x^{3}}{34992}-\frac {1678 x^{4}}{1082565}-\frac {513547 x^{5}}{992023200}+O\left (x^{6}\right )\right )}{x^{\frac {1}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-\frac {1}{18} x -\frac {71}{405} x^{2}+\frac {719}{34992} x^{3}-\frac {1678}{1082565} x^{4}-\frac {513547}{992023200} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {1}{3}}}+c_{2} \left (1-\frac {1}{14} x^{2}+\frac {1}{105} x^{3}-\frac {1}{3640} x^{4}-\frac {23}{54600} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 1304

ODE	\[ \boxed {10 x^{2} \left (2 x^{2}+x +1\right ) y^{\prime \prime }+x \left (66 x^{2}+13 x +13\right ) y^{\prime }-\left (10 x^{2}+4 x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{5}} \left (1+\frac {3 x}{17}-\frac {7 x^{2}}{153}-\frac {547 x^{3}}{5661}+\frac {26942 x^{4}}{266067}+\frac {200432 x^{5}}{3991005}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+x +\frac {14 x^{2}}{13}-\frac {556 x^{3}}{897}-\frac {5314 x^{4}}{9867}+\frac {2092186 x^{5}}{2121405}+O\left (x^{6}\right )\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1+x +\frac {14}{13} x^{2}-\frac {556}{897} x^{3}-\frac {5314}{9867} x^{4}+\frac {2092186}{2121405} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+c_{2} x^{\frac {1}{5}} \left (1+\frac {3}{17} x -\frac {7}{153} x^{2}-\frac {547}{5661} x^{3}+\frac {26942}{266067} x^{4}+\frac {200432}{3991005} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 1305

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }+x \left (2 x +3\right ) y^{\prime }-\left (1-x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {2 x}{5}+\frac {4 x^{2}}{35}-\frac {8 x^{3}}{315}+\frac {16 x^{4}}{3465}-\frac {32 x^{5}}{45045}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-x +\frac {x^{2}}{2}-\frac {x^{3}}{6}+\frac {x^{4}}{24}-\frac {x^{5}}{120}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {3}{2}} \left (1-\frac {2}{5} x +\frac {4}{35} x^{2}-\frac {8}{315} x^{3}+\frac {16}{3465} x^{4}-\frac {32}{45045} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1-x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 1306

ODE	\[ \boxed {x^{2} \left (x +3\right ) y^{\prime \prime }+x \left (4 x +5\right ) y^{\prime }-\left (1-2 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1-\frac {4 x}{9}+\frac {14 x^{2}}{81}-\frac {140 x^{3}}{2187}+\frac {455 x^{4}}{19683}-\frac {1456 x^{5}}{177147}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {4}{3}} \left (1-\frac {4}{9} x +\frac {14}{81} x^{2}-\frac {140}{2187} x^{3}+\frac {455}{19683} x^{4}-\frac {1456}{177147} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 1307

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }+x \left (x +5\right ) y^{\prime }-\left (-3 x +2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {x}{2}+\frac {x^{2}}{8}-\frac {x^{3}}{48}+\frac {x^{4}}{384}-\frac {x^{5}}{3840}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+\frac {x}{3}+\frac {x^{2}}{3}-\frac {x^{3}}{3}+\frac {x^{4}}{9}-\frac {x^{5}}{45}+O\left (x^{6}\right )\right )}{x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {5}{2}} \left (1-\frac {1}{2} x +\frac {1}{8} x^{2}-\frac {1}{48} x^{3}+\frac {1}{384} x^{4}-\frac {1}{3840} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1+\frac {1}{3} x +\frac {1}{3} x^{2}-\frac {1}{3} x^{3}+\frac {1}{9} x^{4}-\frac {1}{45} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

Problem 1308

ODE	\[ \boxed {3 x^{2} y^{\prime \prime }+x \left (x +1\right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {x}{7}+\frac {x^{2}}{70}-\frac {x^{3}}{910}+\frac {x^{4}}{14560}-\frac {x^{5}}{276640}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {x}{3}+\frac {x^{2}}{18}-\frac {x^{3}}{162}+\frac {x^{4}}{1944}-\frac {x^{5}}{29160}+O\left (x^{6}\right )\right )}{x^{\frac {1}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-\frac {1}{3} x +\frac {1}{18} x^{2}-\frac {1}{162} x^{3}+\frac {1}{1944} x^{4}-\frac {1}{29160} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {1}{3}}}+c_{2} x \left (1-\frac {1}{7} x +\frac {1}{70} x^{2}-\frac {1}{910} x^{3}+\frac {1}{14560} x^{4}-\frac {1}{276640} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 1309

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }-y^{\prime } x +\left (1-2 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1+\frac {2 x}{3}+\frac {2 x^{2}}{15}+\frac {4 x^{3}}{315}+\frac {2 x^{4}}{2835}+\frac {4 x^{5}}{155925}+O\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1+2 x +\frac {2 x^{2}}{3}+\frac {4 x^{3}}{45}+\frac {2 x^{4}}{315}+\frac {4 x^{5}}{14175}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+2 x +\frac {2}{3} x^{2}+\frac {4}{45} x^{3}+\frac {2}{315} x^{4}+\frac {4}{14175} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x \left (1+\frac {2}{3} x +\frac {2}{15} x^{2}+\frac {4}{315} x^{3}+\frac {2}{2835} x^{4}+\frac {4}{155925} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 1310

ODE	\[ \boxed {9 x^{2} y^{\prime \prime }+9 y^{\prime } x -\left (3 x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1+\frac {x}{5}+\frac {x^{2}}{80}+\frac {x^{3}}{2640}+\frac {x^{4}}{147840}+\frac {x^{5}}{12566400}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+x +\frac {x^{2}}{8}+\frac {x^{3}}{168}+\frac {x^{4}}{6720}+\frac {x^{5}}{436800}+O\left (x^{6}\right )\right )}{x^{\frac {1}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {2}{3}} \left (1+\frac {1}{5} x +\frac {1}{80} x^{2}+\frac {1}{2640} x^{3}+\frac {1}{147840} x^{4}+\frac {1}{12566400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1+x +\frac {1}{8} x^{2}+\frac {1}{168} x^{3}+\frac {1}{6720} x^{4}+\frac {1}{436800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {1}{3}}} \]

Problem 1311

ODE	\[ \boxed {3 x^{2} y^{\prime \prime }+x \left (x +1\right ) y^{\prime }-\left (3 x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1+\frac {2 x}{7}+\frac {x^{2}}{70}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {10 x}{3}-\frac {35 x^{2}}{18}-\frac {14 x^{3}}{81}-\frac {7 x^{4}}{3888}+\frac {7 x^{5}}{320760}+O\left (x^{6}\right )\right )}{x^{\frac {1}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-\frac {10}{3} x -\frac {35}{18} x^{2}-\frac {14}{81} x^{3}-\frac {7}{3888} x^{4}+\frac {7}{320760} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {1}{3}}}+c_{2} x \left (1+\frac {2}{7} x +\frac {1}{70} x^{2}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 1312

ODE	\[ \boxed {2 x^{2} \left (x +3\right ) y^{\prime \prime }+x \left (1+5 x \right ) y^{\prime }+\left (x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {3 x}{7}+\frac {15 x^{2}}{91}-\frac {15 x^{3}}{247}+\frac {27 x^{4}}{1235}-\frac {297 x^{5}}{38285}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{3}} \left (1-\frac {4 x}{9}+\frac {14 x^{2}}{81}-\frac {140 x^{3}}{2187}+\frac {455 x^{4}}{19683}-\frac {1456 x^{5}}{177147}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{3}} \left (1-\frac {4}{9} x +\frac {14}{81} x^{2}-\frac {140}{2187} x^{3}+\frac {455}{19683} x^{4}-\frac {1456}{177147} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1-\frac {3}{7} x +\frac {15}{91} x^{2}-\frac {15}{247} x^{3}+\frac {27}{1235} x^{4}-\frac {297}{38285} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 1313

ODE	\[ \boxed {x^{2} \left (x +4\right ) y^{\prime \prime }-x \left (-3 x +1\right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {3 x}{7}+\frac {12 x^{2}}{77}-\frac {4 x^{3}}{77}+\frac {24 x^{4}}{1463}-\frac {24 x^{5}}{4807}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{4}} \left (1-\frac {9 x}{16}+\frac {117 x^{2}}{512}-\frac {663 x^{3}}{8192}+\frac {13923 x^{4}}{524288}-\frac {69615 x^{5}}{8388608}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{4}} \left (1-\frac {9}{16} x +\frac {117}{512} x^{2}-\frac {663}{8192} x^{3}+\frac {13923}{524288} x^{4}-\frac {69615}{8388608} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x \left (1-\frac {3}{7} x +\frac {12}{77} x^{2}-\frac {4}{77} x^{3}+\frac {24}{1463} x^{4}-\frac {24}{4807} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 1314

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

Problem 1315

ODE	\[ \boxed {x^{2} \left (4 x +3\right ) y^{\prime \prime }+x \left (5+18 x \right ) y^{\prime }-\left (1-12 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1-\frac {22 x}{9}+\frac {374 x^{2}}{81}-\frac {17204 x^{3}}{2187}+\frac {249458 x^{4}}{19683}-\frac {3492412 x^{5}}{177147}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+2 x -6 x^{2}+12 x^{3}-21 x^{4}+\frac {378 x^{5}}{11}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {4}{3}} \left (1-\frac {22}{9} x +\frac {374}{81} x^{2}-\frac {17204}{2187} x^{3}+\frac {249458}{19683} x^{4}-\frac {3492412}{177147} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1+2 x -6 x^{2}+12 x^{3}-21 x^{4}+\frac {378}{11} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 1316

ODE	\[ \boxed {6 x^{2} y^{\prime \prime }+x \left (10-x \right ) y^{\prime }-\left (2+x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1+\frac {2 x}{21}+\frac {x^{2}}{180}+\frac {x^{3}}{4212}+\frac {x^{4}}{124416}+\frac {x^{5}}{4432320}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {4}{3}} \left (1+\frac {2}{21} x +\frac {1}{180} x^{2}+\frac {1}{4212} x^{3}+\frac {1}{124416} x^{4}+\frac {1}{4432320} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 1317

ODE	\[ \boxed {x^{2} \left (x +8\right ) y^{\prime \prime }+x \left (3 x +2\right ) y^{\prime }+\left (x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {9 x}{40}+\frac {5 x^{2}}{128}-\frac {245 x^{3}}{39936}+\frac {6615 x^{4}}{7241728}-\frac {7623 x^{5}}{57933824}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{4}} \left (1-\frac {25 x}{96}+\frac {675 x^{2}}{14336}-\frac {38025 x^{3}}{5046272}+\frac {732615 x^{4}}{645922816}-\frac {9230949 x^{5}}{56103010304}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{4}} \left (1-\frac {25}{96} x +\frac {675}{14336} x^{2}-\frac {38025}{5046272} x^{3}+\frac {732615}{645922816} x^{4}-\frac {9230949}{56103010304} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1-\frac {9}{40} x +\frac {5}{128} x^{2}-\frac {245}{39936} x^{3}+\frac {6615}{7241728} x^{4}-\frac {7623}{57933824} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 1318

ODE	\[ \boxed {x^{2} \left (4 x +3\right ) y^{\prime \prime }+x \left (11+4 x \right ) y^{\prime }-\left (4 x +3\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1+\frac {32 x}{117}-\frac {28 x^{2}}{1053}+\frac {4480 x^{3}}{540189}-\frac {15680 x^{4}}{4113747}+\frac {401408 x^{5}}{185118615}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+\frac {32 x}{7}+\frac {48 x^{2}}{7}+O\left (x^{6}\right )\right )}{x^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1+\frac {32}{7} x +\frac {48}{7} x^{2}+\operatorname {O}\left (x^{6}\right )\right )}{x^{3}}+c_{2} x^{\frac {1}{3}} \left (1+\frac {32}{117} x -\frac {28}{1053} x^{2}+\frac {4480}{540189} x^{3}-\frac {15680}{4113747} x^{4}+\frac {401408}{185118615} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 1319

ODE	\[ \boxed {2 x^{2} \left (3 x +2\right ) y^{\prime \prime }+x \left (4+11 x \right ) y^{\prime }-\left (1-x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {5 x}{8}+\frac {55 x^{2}}{96}-\frac {935 x^{3}}{1536}+\frac {4301 x^{4}}{6144}-\frac {124729 x^{5}}{147456}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+\frac {x}{4}-\frac {5 x^{2}}{32}+\frac {55 x^{3}}{384}-\frac {935 x^{4}}{6144}+\frac {4301 x^{5}}{24576}+O\left (x^{6}\right )\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x \left (1-\frac {5}{8} x +\frac {55}{96} x^{2}-\frac {935}{1536} x^{3}+\frac {4301}{6144} x^{4}-\frac {124729}{147456} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-\frac {5}{4} x +\frac {25}{32} x^{2}-\frac {275}{384} x^{3}+\frac {4675}{6144} x^{4}-\frac {21505}{24576} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

Problem 1320

ODE	\[ \boxed {x^{2} \left (2+x \right ) y^{\prime \prime }+5 x \left (1-x \right ) y^{\prime }-\left (-8 x +2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {3 x}{4}+\frac {5 x^{2}}{96}+\frac {5 x^{3}}{4224}+\frac {5 x^{4}}{292864}-\frac {x^{5}}{3514368}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+8 x +60 x^{2}-160 x^{3}+40 x^{4}+O\left (x^{6}\right )\right )}{x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {5}{2}} \left (1-\frac {3}{4} x +\frac {5}{96} x^{2}+\frac {5}{4224} x^{3}+\frac {5}{292864} x^{4}-\frac {1}{3514368} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1+8 x +60 x^{2}-160 x^{3}+40 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

Problem 1321

ODE	\[ \boxed {x^{2} \left (x +6\right ) y^{\prime \prime }+x \left (11+4 x \right ) y^{\prime }+\left (1+2 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1-\frac {10 x}{63}+\frac {200 x^{2}}{7371}-\frac {17600 x^{3}}{3781323}+\frac {3872 x^{4}}{4861701}-\frac {921536 x^{5}}{6782072895}+O\left (x^{6}\right )\right )}{x^{\frac {1}{3}}}+\frac {c_{2} \left (1-\frac {3 x}{20}+\frac {9 x^{2}}{352}-\frac {105 x^{3}}{23936}+\frac {6615 x^{4}}{8808448}-\frac {11907 x^{5}}{92889088}+O\left (x^{6}\right )\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} \left (1-\frac {10}{63} x +\frac {200}{7371} x^{2}-\frac {17600}{3781323} x^{3}+\frac {3872}{4861701} x^{4}-\frac {921536}{6782072895} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) x^{\frac {1}{6}}+c_{1} \left (1-\frac {3}{20} x +\frac {9}{352} x^{2}-\frac {105}{23936} x^{3}+\frac {6615}{8808448} x^{4}-\frac {11907}{92889088} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

Problem 1322

ODE	\[ \boxed {8 x^{2} y^{\prime \prime }+x \left (x^{2}+2\right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {x^{2}}{72}+\frac {5 x^{4}}{19584}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{4}} \left (1-\frac {x^{2}}{112}+\frac {3 x^{4}}{17920}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{4}} \left (1-\frac {1}{112} x^{2}+\frac {3}{17920} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1-\frac {1}{72} x^{2}+\frac {5}{19584} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 1323

ODE	\[ \boxed {8 x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+2 x \left (-13 x^{2}+1\right ) y^{\prime }+\left (-9 x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1+\frac {5 x^{2}}{9}+\frac {65 x^{4}}{153}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{4}} \left (1+\frac {x^{2}}{2}+\frac {3 x^{4}}{8}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{4}} \left (1+\frac {1}{2} x^{2}+\frac {3}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1+\frac {5}{9} x^{2}+\frac {65}{153} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 1324

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

Problem 1325

ODE	\[ \boxed {x \left (x^{2}+3\right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }-8 y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1+\frac {11 x^{2}}{18}+\frac {55 x^{4}}{648}+O\left (x^{6}\right )\right )+c_{2} \left (1+\frac {4 x^{2}}{5}+\frac {8 x^{4}}{55}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{3}} \left (1+\frac {11}{18} x^{2}+\frac {55}{648} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+\frac {4}{5} x^{2}+\frac {8}{55} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 1326

ODE	\[ \boxed {4 x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+x \left (-19 x^{2}+7\right ) y^{\prime }-\left (14 x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{4}} \left (1+\frac {9 x^{2}}{13}+\frac {51 x^{4}}{91}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+\frac {x^{2}}{2}+\frac {3 x^{4}}{8}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {5}{4}} \left (1+\frac {9}{13} x^{2}+\frac {51}{91} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1+\frac {1}{2} x^{2}+\frac {3}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 1327

ODE	\[ \boxed {3 x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }+x \left (-11 x^{2}+1\right ) y^{\prime }+\left (-5 x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1+\frac {3 x^{2}}{8}+\frac {21 x^{4}}{128}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{3}} \left (1+\frac {4 x^{2}}{11}+\frac {40 x^{4}}{253}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{3}} \left (1+\frac {4}{11} x^{2}+\frac {40}{253} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1+\frac {3}{8} x^{2}+\frac {21}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 1328

ODE	\[ \boxed {2 x^{2} \left (x^{2}+2\right ) y^{\prime \prime }-x \left (-7 x^{2}+12\right ) y^{\prime }+\left (3 x^{2}+7\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {7}{2}} \left (1-\frac {9 x^{2}}{8}+\frac {117 x^{4}}{128}+O\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1+\frac {3 x^{2}}{4}-\frac {21 x^{4}}{16}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (\left (1-\frac {9}{8} x^{2}+\frac {117}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) x^{3} c_{1} +\left (12+9 x^{2}-\frac {63}{4} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]

Problem 1329

ODE	\[ \boxed {2 x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+x \left (7 x^{2}+4\right ) y^{\prime }-\left (-3 x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {x^{2}}{4}+\frac {7 x^{4}}{80}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {x^{2}}{8}+\frac {5 x^{4}}{128}+O\left (x^{6}\right )\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x \left (1-\frac {1}{4} x^{2}+\frac {7}{80} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-\frac {1}{8} x^{2}+\frac {5}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

Problem 1330

ODE	\[ \boxed {2 x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+5 x \left (6 x^{2}+1\right ) y^{\prime }-\left (-40 x^{2}+2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-3 x^{2}+\frac {15 x^{4}}{2}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+2 x^{2}-\frac {20 x^{4}}{3}+O\left (x^{6}\right )\right )}{x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {5}{2}} \left (1-3 x^{2}+\frac {15}{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1+2 x^{2}-\frac {20}{3} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

Problem 1331

ODE	\[ \boxed {3 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+5 x \left (x^{2}+1\right ) y^{\prime }-\left (-5 x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1-\frac {3 x^{2}}{10}+\frac {39 x^{4}}{320}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {3 x^{2}}{2}+\frac {15 x^{4}}{32}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {4}{3}} \left (1-\frac {3}{10} x^{2}+\frac {39}{320} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1-\frac {3}{2} x^{2}+\frac {15}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 1332

ODE	\[ \boxed {x \left (x^{2}+1\right ) y^{\prime \prime }+\left (7 x^{2}+4\right ) y^{\prime }+8 y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-\frac {4 x^{2}}{5}+\frac {24 x^{4}}{35}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {x^{2}}{2}+\frac {3 x^{4}}{8}+O\left (x^{6}\right )\right )}{x^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (1-\frac {4}{5} x^{2}+\frac {24}{35} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (12-6 x^{2}+\frac {9}{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{3}} \]

Problem 1333

ODE	\[ \boxed {x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+x \left (x^{2}+3\right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {x^{2}}{56}+\frac {25 x^{4}}{9856}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {x^{2}}{2}+\frac {x^{4}}{40}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {3}{2}} \left (1-\frac {1}{56} x^{2}+\frac {25}{9856} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1-\frac {1}{2} x^{2}+\frac {1}{40} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 1334

ODE	\[ \boxed {2 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (8 x^{2}+3\right ) y^{\prime }-\left (-4 x^{2}+3\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {2 x^{2}}{3}+\frac {20 x^{4}}{39}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {x^{2}}{4}+\frac {5 x^{4}}{32}+O\left (x^{6}\right )\right )}{x^{\frac {3}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-\frac {1}{4} x^{2}+\frac {5}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {3}{2}}}+c_{2} x \left (1-\frac {2}{3} x^{2}+\frac {20}{39} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 1335

ODE	\[ \boxed {9 x^{2} y^{\prime \prime }+3 x \left (x^{2}+3\right ) y^{\prime }-\left (-5 x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1-\frac {x^{2}}{8}+\frac {x^{4}}{112}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {x^{2}}{6}+\frac {x^{4}}{72}+O\left (x^{6}\right )\right )}{x^{\frac {1}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {2}{3}} \left (1-\frac {1}{8} x^{2}+\frac {1}{112} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1-\frac {1}{6} x^{2}+\frac {1}{72} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {1}{3}}} \]

Problem 1336

ODE	\[ \boxed {6 x^{2} y^{\prime \prime }+x \left (6 x^{2}+1\right ) y^{\prime }+\left (9 x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {6 x^{2}}{13}+\frac {36 x^{4}}{325}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{3}} \left (1-\frac {x^{2}}{2}+\frac {x^{4}}{8}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{3}} \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1-\frac {6}{13} x^{2}+\frac {36}{325} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 1337

ODE	\[ \boxed {x^{2} \left (x^{2}+8\right ) y^{\prime \prime }+7 x \left (x^{2}+2\right ) y^{\prime }-\left (-9 x^{2}+2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{4}} \left (1-\frac {13 x^{2}}{64}+\frac {273 x^{4}}{8192}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {x^{2}}{3}+\frac {2 x^{4}}{33}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {5}{4}} \left (1-\frac {13}{64} x^{2}+\frac {273}{8192} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1-\frac {1}{3} x^{2}+\frac {2}{33} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 1338

ODE	\[ \boxed {9 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+3 x \left (13 x^{2}+3\right ) y^{\prime }-\left (-25 x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1-\frac {3 x^{2}}{4}+\frac {9 x^{4}}{14}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {2 x^{2}}{3}+\frac {5 x^{4}}{9}+O\left (x^{6}\right )\right )}{x^{\frac {1}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {2}{3}} \left (1-\frac {3}{4} x^{2}+\frac {9}{14} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1-\frac {2}{3} x^{2}+\frac {5}{9} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {1}{3}}} \]

Problem 1339

ODE	\[ \boxed {4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+4 x \left (6 x^{2}+1\right ) y^{\prime }-\left (-25 x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {3 x^{2}}{2}+\frac {15 x^{4}}{8}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-2 x^{2}+\frac {8 x^{4}}{3}+O\left (x^{6}\right )\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x \left (1-\frac {3}{2} x^{2}+\frac {15}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-2 x^{2}+\frac {8}{3} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

Problem 1340

ODE	\[ \boxed {8 x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+2 x \left (34 x^{2}+5\right ) y^{\prime }-\left (-30 x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{4}} \left (1-x^{2}+\frac {3 x^{4}}{2}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {2 x^{2}}{5}+\frac {36 x^{4}}{65}+O\left (x^{6}\right )\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {3}{4}} \left (1-x^{2}+\frac {3}{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1-\frac {2}{5} x^{2}+\frac {36}{65} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

Problem 1341

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

Problem 1342

ODE	\[ \boxed {6 x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+x \left (50 x^{2}+1\right ) y^{\prime }+\left (30 x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-2 x^{2}+4 x^{4}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{3}} \left (1-2 x^{2}+4 x^{4}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (4 x^{4}-2 x^{2}+1\right ) x^{\frac {1}{3}} \left (c_{2} x^{\frac {1}{6}}+c_{1} \right )+O\left (x^{6}\right ) \]

Problem 1343

ODE	\[ \boxed {28 x^{2} \left (-3 x +1\right ) y^{\prime \prime }-7 x \left (5+9 x \right ) y^{\prime }+7 \left (2+9 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1+3 x +9 x^{2}+27 x^{3}+81 x^{4}+243 x^{5}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{4}} \left (1+3 x +9 x^{2}+27 x^{3}+81 x^{4}+243 x^{5}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (243 x^{5}+81 x^{4}+27 x^{3}+9 x^{2}+3 x +1\right ) \left (c_{1} x^{\frac {1}{4}}+c_{2} x^{2}\right )+O\left (x^{6}\right ) \]

Problem 1344

ODE	\[ \boxed {9 x^{2} \left (x +5\right ) y^{\prime \prime }+9 x \left (5+9 x \right ) y^{\prime }-\left (5-8 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1-\frac {11 x}{25}+\frac {11 x^{2}}{50}-\frac {x^{3}}{10}+\frac {29 x^{4}}{700}-\frac {4727 x^{5}}{297500}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+x -\frac {x^{2}}{2}+\frac {17 x^{3}}{70}-\frac {187 x^{4}}{1750}+\frac {24497 x^{5}}{568750}+O\left (x^{6}\right )\right )}{x^{\frac {1}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {2}{3}} \left (1-\frac {11}{25} x +\frac {11}{50} x^{2}-\frac {1}{10} x^{3}+\frac {29}{700} x^{4}-\frac {4727}{297500} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1+x -\frac {1}{2} x^{2}+\frac {17}{70} x^{3}-\frac {187}{1750} x^{4}+\frac {24497}{568750} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {1}{3}}} \]

Problem 1345

ODE	\[ \boxed {8 x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }+2 x \left (-21 x^{2}+10\right ) y^{\prime }-\left (35 x^{2}+2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{4}} \left (1+\frac {x^{2}}{2}+\frac {x^{4}}{4}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+\frac {x^{2}}{2}+\frac {x^{4}}{4}+O\left (x^{6}\right )\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (1+\frac {1}{2} x^{2}+\frac {1}{4} x^{4}\right ) \left (x^{\frac {3}{4}} c_{2} +c_{1} \right )}{\sqrt {x}}+O\left (x^{6}\right ) \]

Problem 1346

ODE	\[ \boxed {4 x^{2} \left (x^{2}+3 x +1\right ) y^{\prime \prime }-4 x \left (-3 x^{2}-3 x +1\right ) y^{\prime }+3 \left (x^{2}-x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {3}{2}} \left (1-3 x +8 x^{2}-21 x^{3}+55 x^{4}-144 x^{5}+O\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1-3 x +8 x^{2}-21 x^{3}+55 x^{4}-144 x^{5}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (x \left (1-3 x +8 x^{2}-21 x^{3}+55 x^{4}-144 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{1} +\left (1-6 x +17 x^{2}-45 x^{3}+118 x^{4}-309 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]

Problem 1347

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

Problem 1348

ODE	\[ \boxed {4 x^{2} \left (x^{2}+2 x +3\right ) y^{\prime \prime }-x \left (-15 x^{2}-14 x +3\right ) y^{\prime }+\left (7 x^{2}+3\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {2 x}{3}+\frac {x^{2}}{9}+\frac {4 x^{3}}{27}-\frac {11 x^{4}}{81}+\frac {10 x^{5}}{243}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{4}} \left (1-\frac {2 x}{3}+\frac {x^{2}}{9}+\frac {4 x^{3}}{27}-\frac {11 x^{4}}{81}+\frac {10 x^{5}}{243}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {2}{3} x +\frac {1}{9} x^{2}+\frac {4}{27} x^{3}-\frac {11}{81} x^{4}+\frac {10}{243} x^{5}\right ) \left (c_{1} x^{\frac {1}{4}}+c_{2} x \right )+O\left (x^{6}\right ) \]

Problem 1349

ODE	\[ \boxed {x^{2} \left (x^{2}-2 x +1\right ) y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+\left (x +4\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (17 x^{2}+5 x +1+\frac {143 x^{3}}{3}+\frac {355 x^{4}}{3}+\frac {4043 x^{5}}{15}+O\left (x^{6}\right )\right )+c_{2} \left (x^{2} \left (17 x^{2}+5 x +1+\frac {143 x^{3}}{3}+\frac {355 x^{4}}{3}+\frac {4043 x^{5}}{15}+O\left (x^{6}\right )\right ) \ln \left (x \right )+x^{2} \left (-3 x -\frac {29 x^{2}}{2}-\frac {859 x^{3}}{18}-\frac {4693 x^{4}}{36}-\frac {285181 x^{5}}{900}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+5 x +17 x^{2}+\frac {143}{3} x^{3}+\frac {355}{3} x^{4}+\frac {4043}{15} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-3\right ) x -\frac {29}{2} x^{2}-\frac {859}{18} x^{3}-\frac {4693}{36} x^{4}-\frac {285181}{900} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) x^{2} \]

Problem 1350

ODE	\[ \boxed {2 x^{2} \left (2+x \right ) y^{\prime \prime }+5 y^{\prime } x^{2}+\left (x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {3 x}{4}+\frac {15 x^{2}}{32}-\frac {35 x^{3}}{128}+\frac {315 x^{4}}{2048}-\frac {693 x^{5}}{8192}+O\left (x^{6}\right )\right )+c_{2} \left (\sqrt {x}\, \left (1-\frac {3 x}{4}+\frac {15 x^{2}}{32}-\frac {35 x^{3}}{128}+\frac {315 x^{4}}{2048}-\frac {693 x^{5}}{8192}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\sqrt {x}\, \left (\frac {x}{4}-\frac {13 x^{2}}{64}+\frac {101 x^{3}}{768}-\frac {641 x^{4}}{8192}+\frac {7303 x^{5}}{163840}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {3}{4} x +\frac {15}{32} x^{2}-\frac {35}{128} x^{3}+\frac {315}{2048} x^{4}-\frac {693}{8192} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{4} x -\frac {13}{64} x^{2}+\frac {101}{768} x^{3}-\frac {641}{8192} x^{4}+\frac {7303}{163840} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]

Problem 1351

ODE	\[ \boxed {x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }-2 x \left (2 x^{2}+1\right ) y^{\prime }+\left (-2 x^{2}+2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1+\frac {3 x^{2}}{4}+\frac {15 x^{4}}{32}+O\left (x^{6}\right )\right )+c_{2} \left (x \left (1+\frac {3 x^{2}}{4}+\frac {15 x^{4}}{32}+O\left (x^{6}\right )\right ) \ln \left (x \right )+x \left (-\frac {x^{2}}{8}-\frac {13 x^{4}}{128}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+\frac {3}{4} x^{2}+\frac {15}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {1}{8} x^{2}-\frac {13}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) x \]

Problem 1352

ODE	\[ \boxed {x^{2} y^{\prime \prime }-x \left (-x +5\right ) y^{\prime }+\left (9-4 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{3} \left (x +1+O\left (x^{6}\right )\right )+c_{2} \left (x^{3} \left (x +1+O\left (x^{6}\right )\right ) \ln \left (x \right )+x^{3} \left (-3 x -\frac {x^{2}}{4}+\frac {x^{3}}{36}-\frac {x^{4}}{288}+\frac {x^{5}}{2400}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+x +\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-3\right ) x -\frac {1}{4} x^{2}+\frac {1}{36} x^{3}-\frac {1}{288} x^{4}+\frac {1}{2400} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) x^{3} \]

Problem 1353

ODE	\[ \boxed {x^{2} y^{\prime \prime }-x \left (1-x \right ) y^{\prime }+\left (-x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-x +\frac {3 x^{2}}{4}-\frac {13 x^{3}}{36}+\frac {79 x^{4}}{576}-\frac {67 x^{5}}{1600}+\frac {5593 x^{6}}{518400}-\frac {60859 x^{7}}{25401600}+O\left (x^{8}\right )\right )+c_{2} \left (x \left (1-x +\frac {3 x^{2}}{4}-\frac {13 x^{3}}{36}+\frac {79 x^{4}}{576}-\frac {67 x^{5}}{1600}+\frac {5593 x^{6}}{518400}-\frac {60859 x^{7}}{25401600}+O\left (x^{8}\right )\right ) \ln \left (x \right )+x \left (-x^{2}+x +\frac {65 x^{3}}{108}-\frac {895 x^{4}}{3456}+\frac {12547 x^{5}}{144000}-\frac {41729 x^{6}}{1728000}+\frac {10121677 x^{7}}{1778112000}+O\left (x^{8}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-x +\frac {3}{4} x^{2}-\frac {13}{36} x^{3}+\frac {79}{576} x^{4}-\frac {67}{1600} x^{5}+\frac {5593}{518400} x^{6}-\frac {60859}{25401600} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (x -x^{2}+\frac {65}{108} x^{3}-\frac {895}{3456} x^{4}+\frac {12547}{144000} x^{5}-\frac {41729}{1728000} x^{6}+\frac {10121677}{1778112000} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \right ) x \]

Problem 1354

ODE	\[ \boxed {x^{2} \left (2 x^{2}+x +1\right ) y^{\prime \prime }+x \left (7 x^{2}+6 x +3\right ) y^{\prime }+\left (-3 x^{2}+6 x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1-2 x +\frac {9 x^{2}}{2}-\frac {20 x^{3}}{3}+\frac {173 x^{4}}{24}-\frac {93 x^{5}}{20}-\frac {419 x^{6}}{720}+\frac {6697 x^{7}}{1260}+O\left (x^{8}\right )\right )}{x}+c_{2} \left (\frac {\left (1-2 x +\frac {9 x^{2}}{2}-\frac {20 x^{3}}{3}+\frac {173 x^{4}}{24}-\frac {93 x^{5}}{20}-\frac {419 x^{6}}{720}+\frac {6697 x^{7}}{1260}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{x}+\frac {x -\frac {15 x^{2}}{4}+\frac {133 x^{3}}{18}-\frac {3077 x^{4}}{288}+\frac {4217 x^{5}}{400}-\frac {70949 x^{6}}{14400}-\frac {125221 x^{7}}{29400}+O\left (x^{8}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-2 x +\frac {9}{2} x^{2}-\frac {20}{3} x^{3}+\frac {173}{24} x^{4}-\frac {93}{20} x^{5}-\frac {419}{720} x^{6}+\frac {6697}{1260} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (x -\frac {15}{4} x^{2}+\frac {133}{18} x^{3}-\frac {3077}{288} x^{4}+\frac {4217}{400} x^{5}-\frac {70949}{14400} x^{6}-\frac {125221}{29400} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2}}{x} \]

Problem 1355

ODE	\[ \boxed {x^{2} \left (x^{2}+2 x +1\right ) y^{\prime \prime }+x \left (4 x^{2}+3 x +1\right ) y^{\prime }-x \left (1-2 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (-x^{2}+x +1+\frac {x^{3}}{3}+\frac {x^{4}}{3}-\frac {11 x^{5}}{15}+\frac {37 x^{6}}{45}-\frac {209 x^{7}}{315}+O\left (x^{8}\right )\right )+c_{2} \left (\left (-x^{2}+x +1+\frac {x^{3}}{3}+\frac {x^{4}}{3}-\frac {11 x^{5}}{15}+\frac {37 x^{6}}{45}-\frac {209 x^{7}}{315}+O\left (x^{8}\right )\right ) \ln \left (x \right )+\frac {x^{2}}{2}-3 x +\frac {31 x^{3}}{18}-\frac {91 x^{4}}{36}+\frac {1897 x^{5}}{900}-\frac {301 x^{6}}{300}-\frac {3901 x^{7}}{14700}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+x -x^{2}+\frac {1}{3} x^{3}+\frac {1}{3} x^{4}-\frac {11}{15} x^{5}+\frac {37}{45} x^{6}-\frac {209}{315} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (\left (-3\right ) x +\frac {1}{2} x^{2}+\frac {31}{18} x^{3}-\frac {91}{36} x^{4}+\frac {1897}{900} x^{5}-\frac {301}{300} x^{6}-\frac {3901}{14700} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \]

Problem 1356

ODE	\[ \boxed {4 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+12 x^{2} \left (x +1\right ) y^{\prime }+\left (3 x^{2}+3 x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-2 x +\frac {5 x^{2}}{2}-2 x^{3}+\frac {5 x^{4}}{8}+\frac {17 x^{5}}{20}-\frac {121 x^{6}}{80}+x^{7}+O\left (x^{8}\right )\right )+c_{2} \left (\sqrt {x}\, \left (1-2 x +\frac {5 x^{2}}{2}-2 x^{3}+\frac {5 x^{4}}{8}+\frac {17 x^{5}}{20}-\frac {121 x^{6}}{80}+x^{7}+O\left (x^{8}\right )\right ) \ln \left (x \right )+\sqrt {x}\, \left (x -\frac {9 x^{2}}{4}+\frac {17 x^{3}}{6}-\frac {205 x^{4}}{96}+\frac {481 x^{5}}{1200}+\frac {2109 x^{6}}{1600}-\frac {1063 x^{7}}{560}+O\left (x^{8}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-2 x +\frac {5}{2} x^{2}-2 x^{3}+\frac {5}{8} x^{4}+\frac {17}{20} x^{5}-\frac {121}{80} x^{6}+x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (x -\frac {9}{4} x^{2}+\frac {17}{6} x^{3}-\frac {205}{96} x^{4}+\frac {481}{1200} x^{5}+\frac {2109}{1600} x^{6}-\frac {1063}{560} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \right ) \]

Problem 1357

ODE	\[ \boxed {x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }-x \left (-2 x^{2}-4 x +1\right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (-4 x +1+\frac {19 x^{2}}{2}-\frac {49 x^{3}}{3}+\frac {515 x^{4}}{24}-\frac {319 x^{5}}{15}+\frac {10093 x^{6}}{720}-\frac {647 x^{7}}{360}+O\left (x^{8}\right )\right )+c_{2} \left (x \left (-4 x +1+\frac {19 x^{2}}{2}-\frac {49 x^{3}}{3}+\frac {515 x^{4}}{24}-\frac {319 x^{5}}{15}+\frac {10093 x^{6}}{720}-\frac {647 x^{7}}{360}+O\left (x^{8}\right )\right ) \ln \left (x \right )+x \left (3 x -\frac {43 x^{2}}{4}+\frac {208 x^{3}}{9}-\frac {10379 x^{4}}{288}+\frac {76321 x^{5}}{1800}-\frac {172499 x^{6}}{4800}+\frac {39091 x^{7}}{2400}+O\left (x^{8}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-4 x +\frac {19}{2} x^{2}-\frac {49}{3} x^{3}+\frac {515}{24} x^{4}-\frac {319}{15} x^{5}+\frac {10093}{720} x^{6}-\frac {647}{360} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (3 x -\frac {43}{4} x^{2}+\frac {208}{9} x^{3}-\frac {10379}{288} x^{4}+\frac {76321}{1800} x^{5}-\frac {172499}{4800} x^{6}+\frac {39091}{2400} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \right ) x \]

Problem 1358

ODE	\[ \boxed {9 x^{2} y^{\prime \prime }+3 x \left (-2 x^{2}+3 x +5\right ) y^{\prime }+\left (-14 x^{2}+12 x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1-x +\frac {5 x^{2}}{6}-\frac {x^{3}}{2}+\frac {19 x^{4}}{72}-\frac {43 x^{5}}{360}+\frac {319 x^{6}}{6480}-\frac {167 x^{7}}{9072}+O\left (x^{8}\right )\right )}{x^{\frac {1}{3}}}+c_{2} \left (\frac {\left (1-x +\frac {5 x^{2}}{6}-\frac {x^{3}}{2}+\frac {19 x^{4}}{72}-\frac {43 x^{5}}{360}+\frac {319 x^{6}}{6480}-\frac {167 x^{7}}{9072}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{x^{\frac {1}{3}}}+\frac {x -\frac {11 x^{2}}{12}+\frac {25 x^{3}}{36}-\frac {113 x^{4}}{288}+\frac {4211 x^{5}}{21600}-\frac {32773 x^{6}}{388800}+\frac {126647 x^{7}}{3810240}+O\left (x^{8}\right )}{x^{\frac {1}{3}}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-x +\frac {5}{6} x^{2}-\frac {1}{2} x^{3}+\frac {19}{72} x^{4}-\frac {43}{360} x^{5}+\frac {319}{6480} x^{6}-\frac {167}{9072} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (x -\frac {11}{12} x^{2}+\frac {25}{36} x^{3}-\frac {113}{288} x^{4}+\frac {4211}{21600} x^{5}-\frac {32773}{388800} x^{6}+\frac {126647}{3810240} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2}}{x^{\frac {1}{3}}} \]

Problem 1359

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x \left (x^{2}+x +1\right ) y^{\prime }+x \left (2-x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-2 x +\frac {7 x^{2}}{4}-\frac {7 x^{3}}{9}+\frac {77 x^{4}}{576}+\frac {217 x^{5}}{7200}-\frac {8813 x^{6}}{518400}+\frac {143 x^{7}}{453600}+O\left (x^{8}\right )\right )+c_{2} \left (\left (1-2 x +\frac {7 x^{2}}{4}-\frac {7 x^{3}}{9}+\frac {77 x^{4}}{576}+\frac {217 x^{5}}{7200}-\frac {8813 x^{6}}{518400}+\frac {143 x^{7}}{453600}+O\left (x^{8}\right )\right ) \ln \left (x \right )+3 x -\frac {15 x^{2}}{4}+\frac {239 x^{3}}{108}-\frac {2021 x^{4}}{3456}-\frac {1241 x^{5}}{54000}+\frac {93859 x^{6}}{1728000}-\frac {311177 x^{7}}{42336000}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-2 x +\frac {7}{4} x^{2}-\frac {7}{9} x^{3}+\frac {77}{576} x^{4}+\frac {217}{7200} x^{5}-\frac {8813}{518400} x^{6}+\frac {143}{453600} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (3 x -\frac {15}{4} x^{2}+\frac {239}{108} x^{3}-\frac {2021}{3456} x^{4}-\frac {1241}{54000} x^{5}+\frac {93859}{1728000} x^{6}-\frac {311177}{42336000} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \]

Problem 1360

ODE	\[ \boxed {x^{2} \left (1+2 x \right ) y^{\prime \prime }+x \left (3 x^{2}+14 x +5\right ) y^{\prime }+\left (12 x^{2}+18 x +4\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1-2 x +\frac {5 x^{2}}{2}-3 x^{3}+\frac {33 x^{4}}{8}-\frac {129 x^{5}}{20}+\frac {867 x^{6}}{80}-\frac {1059 x^{7}}{56}+O\left (x^{8}\right )\right )}{x^{2}}+c_{2} \left (\frac {\left (1-2 x +\frac {5 x^{2}}{2}-3 x^{3}+\frac {33 x^{4}}{8}-\frac {129 x^{5}}{20}+\frac {867 x^{6}}{80}-\frac {1059 x^{7}}{56}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{x^{2}}+\frac {\frac {3 x^{2}}{4}-\frac {13 x^{3}}{6}+\frac {407 x^{4}}{96}-\frac {9047 x^{5}}{1200}+\frac {63851 x^{6}}{4800}-\frac {559033 x^{7}}{23520}+O\left (x^{8}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-2 x +\frac {5}{2} x^{2}-3 x^{3}+\frac {33}{8} x^{4}-\frac {129}{20} x^{5}+\frac {867}{80} x^{6}-\frac {1059}{56} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (\frac {3}{4} x^{2}-\frac {13}{6} x^{3}+\frac {407}{96} x^{4}-\frac {9047}{1200} x^{5}+\frac {63851}{4800} x^{6}-\frac {559033}{23520} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2}}{x^{2}} \]

Problem 1361

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+2 x \left (x^{2}+x +4\right ) y^{\prime }+\left (3 x^{2}+5 x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1-x +\frac {x^{2}}{4}+\frac {x^{3}}{18}-\frac {37 x^{4}}{1152}-\frac {17 x^{5}}{28800}+\frac {593 x^{6}}{259200}-\frac {1913 x^{7}}{12700800}+O\left (x^{8}\right )\right )}{\sqrt {x}}+c_{2} \left (\frac {\left (1-x +\frac {x^{2}}{4}+\frac {x^{3}}{18}-\frac {37 x^{4}}{1152}-\frac {17 x^{5}}{28800}+\frac {593 x^{6}}{259200}-\frac {1913 x^{7}}{12700800}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{\sqrt {x}}+\frac {\frac {3 x}{2}-\frac {13 x^{2}}{16}+\frac {x^{3}}{54}+\frac {1103 x^{4}}{13824}-\frac {19507 x^{5}}{1728000}-\frac {98531 x^{6}}{20736000}+\frac {982189 x^{7}}{889056000}+O\left (x^{8}\right )}{\sqrt {x}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-x +\frac {1}{4} x^{2}+\frac {1}{18} x^{3}-\frac {37}{1152} x^{4}-\frac {17}{28800} x^{5}+\frac {593}{259200} x^{6}-\frac {1913}{12700800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (\frac {3}{2} x -\frac {13}{16} x^{2}+\frac {1}{54} x^{3}+\frac {1103}{13824} x^{4}-\frac {19507}{1728000} x^{5}-\frac {98531}{20736000} x^{6}+\frac {982189}{889056000} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2}}{\sqrt {x}} \]

Problem 1362

ODE	\[ \boxed {16 x^{2} y^{\prime \prime }+4 x \left (2 x^{2}+x +6\right ) y^{\prime }+\left (18 x^{2}+5 x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1-\frac {x}{4}-\frac {7 x^{2}}{32}+\frac {23 x^{3}}{384}+\frac {145 x^{4}}{6144}-\frac {881 x^{5}}{122880}-\frac {4919 x^{6}}{2949120}+\frac {47207 x^{7}}{82575360}+O\left (x^{8}\right )\right )}{x^{\frac {1}{4}}}+c_{2} \left (\frac {\left (1-\frac {x}{4}-\frac {7 x^{2}}{32}+\frac {23 x^{3}}{384}+\frac {145 x^{4}}{6144}-\frac {881 x^{5}}{122880}-\frac {4919 x^{6}}{2949120}+\frac {47207 x^{7}}{82575360}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{x^{\frac {1}{4}}}+\frac {\frac {x}{4}+\frac {5 x^{2}}{64}-\frac {157 x^{3}}{2304}-\frac {841 x^{4}}{73728}+\frac {65017 x^{5}}{7372800}+\frac {50791 x^{6}}{58982400}-\frac {953509 x^{7}}{1284505600}+O\left (x^{8}\right )}{x^{\frac {1}{4}}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {1}{4} x -\frac {7}{32} x^{2}+\frac {23}{384} x^{3}+\frac {145}{6144} x^{4}-\frac {881}{122880} x^{5}-\frac {4919}{2949120} x^{6}+\frac {47207}{82575360} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (\frac {1}{4} x +\frac {5}{64} x^{2}-\frac {157}{2304} x^{3}-\frac {841}{73728} x^{4}+\frac {65017}{7372800} x^{5}+\frac {50791}{58982400} x^{6}-\frac {953509}{1284505600} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2}}{x^{\frac {1}{4}}} \]

Problem 1363

ODE	\[ \boxed {9 x^{2} \left (x +1\right ) y^{\prime \prime }+3 x \left (-x^{2}+11 x +5\right ) y^{\prime }+\left (-7 x^{2}+16 x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1-x +\frac {7 x^{2}}{6}-\frac {23 x^{3}}{18}+\frac {11 x^{4}}{8}-\frac {1577 x^{5}}{1080}+\frac {3319 x^{6}}{2160}-\frac {72853 x^{7}}{45360}+O\left (x^{8}\right )\right )}{x^{\frac {1}{3}}}+c_{2} \left (\frac {\left (1-x +\frac {7 x^{2}}{6}-\frac {23 x^{3}}{18}+\frac {11 x^{4}}{8}-\frac {1577 x^{5}}{1080}+\frac {3319 x^{6}}{2160}-\frac {72853 x^{7}}{45360}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{x^{\frac {1}{3}}}+\frac {-\frac {x^{2}}{12}+\frac {13 x^{3}}{108}-\frac {131 x^{4}}{864}+\frac {11449 x^{5}}{64800}-\frac {76919 x^{6}}{388800}+\frac {4118557 x^{7}}{19051200}+O\left (x^{8}\right )}{x^{\frac {1}{3}}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-x +\frac {7}{6} x^{2}-\frac {23}{18} x^{3}+\frac {11}{8} x^{4}-\frac {1577}{1080} x^{5}+\frac {3319}{2160} x^{6}-\frac {72853}{45360} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-\frac {1}{12} x^{2}+\frac {13}{108} x^{3}-\frac {131}{864} x^{4}+\frac {11449}{64800} x^{5}-\frac {76919}{388800} x^{6}+\frac {4118557}{19051200} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2}}{x^{\frac {1}{3}}} \]

Problem 1364

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

Problem 1365

ODE	\[ \boxed {36 x^{2} \left (1-2 x \right ) y^{\prime \prime }+24 x \left (1-9 x \right ) y^{\prime }+\left (1-70 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{6}} \left (1+\frac {8 x}{3}+\frac {56 x^{2}}{9}+\frac {1120 x^{3}}{81}+\frac {7280 x^{4}}{243}+\frac {46592 x^{5}}{729}+O\left (x^{6}\right )\right )+c_{2} \left (x^{\frac {1}{6}} \left (1+\frac {8 x}{3}+\frac {56 x^{2}}{9}+\frac {1120 x^{3}}{81}+\frac {7280 x^{4}}{243}+\frac {46592 x^{5}}{729}+O\left (x^{6}\right )\right ) \ln \left (x \right )+x^{\frac {1}{6}} \left (-\frac {2 x}{3}-2 x^{2}-\frac {1192 x^{3}}{243}-\frac {8168 x^{4}}{729}-\frac {270112 x^{5}}{10935}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{\frac {1}{6}} \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+\frac {8}{3} x +\frac {56}{9} x^{2}+\frac {1120}{81} x^{3}+\frac {7280}{243} x^{4}+\frac {46592}{729} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {2}{3} x -2 x^{2}-\frac {1192}{243} x^{3}-\frac {8168}{729} x^{4}-\frac {270112}{10935} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]

Problem 1366

ODE	\[ \boxed {x^{2} \left (x +1\right ) y^{\prime \prime }-x \left (-x +3\right ) y^{\prime }+4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (-36 x^{5}+25 x^{4}-16 x^{3}+9 x^{2}-4 x +1+O\left (x^{6}\right )\right )+c_{2} \left (x^{2} \left (-36 x^{5}+25 x^{4}-16 x^{3}+9 x^{2}-4 x +1+O\left (x^{6}\right )\right ) \ln \left (x \right )+x^{2} \left (60 x^{5}-40 x^{4}+24 x^{3}-12 x^{2}+4 x +O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{2} \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-4 x +9 x^{2}-16 x^{3}+25 x^{4}-36 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (4 x -12 x^{2}+24 x^{3}-40 x^{4}+60 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]

Problem 1367

ODE	\[ \boxed {x^{2} \left (1-2 x \right ) y^{\prime \prime }-x \left (5-4 x \right ) y^{\prime }+\left (9-4 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{3} \left (192 x^{5}+80 x^{4}+32 x^{3}+12 x^{2}+4 x +1+O\left (x^{6}\right )\right )+c_{2} \left (x^{3} \left (192 x^{5}+80 x^{4}+32 x^{3}+12 x^{2}+4 x +1+O\left (x^{6}\right )\right ) \ln \left (x \right )+x^{3} \left (-160 x^{5}-64 x^{4}-24 x^{3}-8 x^{2}-2 x +O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{3} \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+4 x +12 x^{2}+32 x^{3}+80 x^{4}+192 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-2\right ) x -8 x^{2}-24 x^{3}-64 x^{4}-160 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]

Problem 1368

ODE	\[ \boxed {25 x^{2} y^{\prime \prime }+x \left (15+x \right ) y^{\prime }+\left (x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{5}} \left (1-\frac {6 x}{125}+\frac {33 x^{2}}{31250}-\frac {88 x^{3}}{5859375}+\frac {77 x^{4}}{488281250}-\frac {1001 x^{5}}{762939453125}+O\left (x^{6}\right )\right )+c_{2} \left (x^{\frac {1}{5}} \left (1-\frac {6 x}{125}+\frac {33 x^{2}}{31250}-\frac {88 x^{3}}{5859375}+\frac {77 x^{4}}{488281250}-\frac {1001 x^{5}}{762939453125}+O\left (x^{6}\right )\right ) \ln \left (x \right )+x^{\frac {1}{5}} \left (\frac {7 x}{125}-\frac {113 x^{2}}{62500}+\frac {1091 x^{3}}{35156250}-\frac {1721 x^{4}}{4687500000}+\frac {609221 x^{5}}{183105468750000}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{\frac {1}{5}} \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {6}{125} x +\frac {33}{31250} x^{2}-\frac {88}{5859375} x^{3}+\frac {77}{488281250} x^{4}-\frac {1001}{762939453125} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {7}{125} x -\frac {113}{62500} x^{2}+\frac {1091}{35156250} x^{3}-\frac {1721}{4687500000} x^{4}+\frac {609221}{183105468750000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]

Problem 1369

ODE	\[ \boxed {2 x^{2} \left (2+x \right ) y^{\prime \prime }+y^{\prime } x^{2}+\left (1-x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1+\frac {x}{4}-\frac {x^{2}}{32}+\frac {x^{3}}{128}-\frac {5 x^{4}}{2048}+\frac {7 x^{5}}{8192}+O\left (x^{6}\right )\right )+c_{2} \left (\sqrt {x}\, \left (1+\frac {x}{4}-\frac {x^{2}}{32}+\frac {x^{3}}{128}-\frac {5 x^{4}}{2048}+\frac {7 x^{5}}{8192}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\sqrt {x}\, \left (-\frac {3 x}{4}+\frac {3 x^{2}}{64}-\frac {7 x^{3}}{768}+\frac {61 x^{4}}{24576}-\frac {391 x^{5}}{491520}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+\frac {1}{4} x -\frac {1}{32} x^{2}+\frac {1}{128} x^{3}-\frac {5}{2048} x^{4}+\frac {7}{8192} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {3}{4} x +\frac {3}{64} x^{2}-\frac {7}{768} x^{3}+\frac {61}{24576} x^{4}-\frac {391}{491520} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]

Problem 1370

ODE	\[ \boxed {x^{2} \left (4 x +9\right ) y^{\prime \prime }+3 y^{\prime } x +\left (x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1-\frac {x}{81}+\frac {25 x^{2}}{26244}-\frac {3025 x^{3}}{19131876}+\frac {874225 x^{4}}{24794911296}-\frac {18498601 x^{5}}{2008387814976}+O\left (x^{6}\right )\right )+c_{2} \left (x^{\frac {1}{3}} \left (1-\frac {x}{81}+\frac {25 x^{2}}{26244}-\frac {3025 x^{3}}{19131876}+\frac {874225 x^{4}}{24794911296}-\frac {18498601 x^{5}}{2008387814976}+O\left (x^{6}\right )\right ) \ln \left (x \right )+x^{\frac {1}{3}} \left (\frac {14 x}{81}-\frac {35 x^{2}}{2916}+\frac {110495 x^{3}}{57395628}-\frac {62786185 x^{4}}{148769467776}+\frac {1315043653 x^{5}}{12050326889856}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{\frac {1}{3}} \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {1}{81} x +\frac {25}{26244} x^{2}-\frac {3025}{19131876} x^{3}+\frac {874225}{24794911296} x^{4}-\frac {18498601}{2008387814976} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {14}{81} x -\frac {35}{2916} x^{2}+\frac {110495}{57395628} x^{3}-\frac {62786185}{148769467776} x^{4}+\frac {1315043653}{12050326889856} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]

Problem 1371

ODE	\[ \boxed {x^{2} y^{\prime \prime }-x \left (3-2 x \right ) y^{\prime }+\left (3 x +4\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-7 x +\frac {63 x^{2}}{4}-\frac {77 x^{3}}{4}+\frac {1001 x^{4}}{64}-\frac {3003 x^{5}}{320}+O\left (x^{6}\right )\right )+c_{2} \left (x^{2} \left (1-7 x +\frac {63 x^{2}}{4}-\frac {77 x^{3}}{4}+\frac {1001 x^{4}}{64}-\frac {3003 x^{5}}{320}+O\left (x^{6}\right )\right ) \ln \left (x \right )+x^{2} \left (12 x -\frac {157 x^{2}}{4}+\frac {2063 x^{3}}{36}-\frac {59875 x^{4}}{1152}+\frac {323399 x^{5}}{9600}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-7 x +\frac {63}{4} x^{2}-\frac {77}{4} x^{3}+\frac {1001}{64} x^{4}-\frac {3003}{320} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (12 x -\frac {157}{4} x^{2}+\frac {2063}{36} x^{3}-\frac {59875}{1152} x^{4}+\frac {323399}{9600} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) x^{2} \]

Problem 1372

ODE	\[ \boxed {x^{2} \left (-4 x +1\right ) y^{\prime \prime }+3 x \left (1-6 x \right ) y^{\prime }+\left (1-12 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (252 x^{5}+70 x^{4}+20 x^{3}+6 x^{2}+2 x +1+O\left (x^{6}\right )\right )}{x}+c_{2} \left (\frac {\left (252 x^{5}+70 x^{4}+20 x^{3}+6 x^{2}+2 x +1+O\left (x^{6}\right )\right ) \ln \left (x \right )}{x}+\frac {7 x^{2}+2 x +\frac {74 x^{3}}{3}+\frac {533 x^{4}}{6}+\frac {1627 x^{5}}{5}+O\left (x^{6}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+2 x +6 x^{2}+20 x^{3}+70 x^{4}+252 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (2 x +7 x^{2}+\frac {74}{3} x^{3}+\frac {533}{6} x^{4}+\frac {1627}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2}}{x} \]

Problem 1373

ODE	\[ \boxed {x^{2} \left (1+2 x \right ) y^{\prime \prime }+x \left (3+5 x \right ) y^{\prime }+\left (1-2 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (3 x +1+\frac {3 x^{2}}{2}-\frac {x^{3}}{2}+\frac {3 x^{4}}{8}-\frac {3 x^{5}}{8}+O\left (x^{6}\right )\right )}{x}+c_{2} \left (\frac {\left (3 x +1+\frac {3 x^{2}}{2}-\frac {x^{3}}{2}+\frac {3 x^{4}}{8}-\frac {3 x^{5}}{8}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{x}+\frac {-5 x -\frac {25 x^{2}}{4}+\frac {5 x^{3}}{4}-\frac {25 x^{4}}{32}+\frac {113 x^{5}}{160}+O\left (x^{6}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+3 x +\frac {3}{2} x^{2}-\frac {1}{2} x^{3}+\frac {3}{8} x^{4}-\frac {3}{8} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-5\right ) x -\frac {25}{4} x^{2}+\frac {5}{4} x^{3}-\frac {25}{32} x^{4}+\frac {113}{160} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2}}{x} \]

Problem 1374

ODE	\[ \boxed {2 x^{2} \left (x +1\right ) y^{\prime \prime }-x \left (6-x \right ) y^{\prime }+\left (8-x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-\frac {5 x}{2}+\frac {35 x^{2}}{8}-\frac {105 x^{3}}{16}+\frac {1155 x^{4}}{128}-\frac {3003 x^{5}}{256}+O\left (x^{6}\right )\right )+c_{2} \left (x^{2} \left (1-\frac {5 x}{2}+\frac {35 x^{2}}{8}-\frac {105 x^{3}}{16}+\frac {1155 x^{4}}{128}-\frac {3003 x^{5}}{256}+O\left (x^{6}\right )\right ) \ln \left (x \right )+x^{2} \left (\frac {3 x}{2}-\frac {57 x^{2}}{16}+\frac {583 x^{3}}{96}-\frac {13771 x^{4}}{1536}+\frac {187339 x^{5}}{15360}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {5}{2} x +\frac {35}{8} x^{2}-\frac {105}{16} x^{3}+\frac {1155}{128} x^{4}-\frac {3003}{256} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {3}{2} x -\frac {57}{16} x^{2}+\frac {583}{96} x^{3}-\frac {13771}{1536} x^{4}+\frac {187339}{15360} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) x^{2} \]

Problem 1375

ODE	\[ \boxed {x^{2} \left (1+2 x \right ) y^{\prime \prime }+x \left (5+9 x \right ) y^{\prime }+\left (3 x +4\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (3 x +1+\frac {3 x^{2}}{2}-\frac {x^{3}}{2}+\frac {3 x^{4}}{8}-\frac {3 x^{5}}{8}+\frac {7 x^{6}}{16}-\frac {9 x^{7}}{16}+O\left (x^{8}\right )\right )}{x^{2}}+c_{2} \left (\frac {\left (3 x +1+\frac {3 x^{2}}{2}-\frac {x^{3}}{2}+\frac {3 x^{4}}{8}-\frac {3 x^{5}}{8}+\frac {7 x^{6}}{16}-\frac {9 x^{7}}{16}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{x^{2}}+\frac {-5 x -\frac {25 x^{2}}{4}+\frac {5 x^{3}}{4}-\frac {25 x^{4}}{32}+\frac {113 x^{5}}{160}-\frac {247 x^{6}}{320}+\frac {2123 x^{7}}{2240}+O\left (x^{8}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+3 x +\frac {3}{2} x^{2}-\frac {1}{2} x^{3}+\frac {3}{8} x^{4}-\frac {3}{8} x^{5}+\frac {7}{16} x^{6}-\frac {9}{16} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (\left (-5\right ) x -\frac {25}{4} x^{2}+\frac {5}{4} x^{3}-\frac {25}{32} x^{4}+\frac {113}{160} x^{5}-\frac {247}{320} x^{6}+\frac {2123}{2240} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2}}{x^{2}} \]

Problem 1376

ODE	\[ \boxed {x^{2} \left (1-2 x \right ) y^{\prime \prime }-x \left (4 x +5\right ) y^{\prime }+\left (4 x +9\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{3} \left (337920 x^{7}+94080 x^{6}+24192 x^{5}+5600 x^{4}+1120 x^{3}+180 x^{2}+20 x +1+O\left (x^{8}\right )\right )+c_{2} \left (x^{3} \left (337920 x^{7}+94080 x^{6}+24192 x^{5}+5600 x^{4}+1120 x^{3}+180 x^{2}+20 x +1+O\left (x^{8}\right )\right ) \ln \left (x \right )+x^{3} \left (-324 x^{2}-26 x -\frac {6968 x^{3}}{3}-\frac {37780 x^{4}}{3}-57360 x^{5}-\frac {694736 x^{6}}{3}-\frac {2566144 x^{7}}{3}+O\left (x^{8}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+20 x +180 x^{2}+1120 x^{3}+5600 x^{4}+24192 x^{5}+94080 x^{6}+337920 x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (\left (-26\right ) x -324 x^{2}-\frac {6968}{3} x^{3}-\frac {37780}{3} x^{4}-57360 x^{5}-\frac {694736}{3} x^{6}-\frac {2566144}{3} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \right ) x^{3} \]

Problem 1377

ODE	\[ \boxed {x^{2} \left (4 x +1\right ) y^{\prime \prime }-x \left (-4 x +1\right ) y^{\prime }+\left (x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-5 x +\frac {85 x^{2}}{4}-\frac {3145 x^{3}}{36}+\frac {204425 x^{4}}{576}-\frac {825877 x^{5}}{576}+\frac {119752165 x^{6}}{20736}-\frac {23591176505 x^{7}}{1016064}+O\left (x^{8}\right )\right )+c_{2} \left (x \left (1-5 x +\frac {85 x^{2}}{4}-\frac {3145 x^{3}}{36}+\frac {204425 x^{4}}{576}-\frac {825877 x^{5}}{576}+\frac {119752165 x^{6}}{20736}-\frac {23591176505 x^{7}}{1016064}+O\left (x^{8}\right )\right ) \ln \left (x \right )+x \left (2 x -\frac {39 x^{2}}{4}+\frac {4499 x^{3}}{108}-\frac {594305 x^{4}}{3456}+\frac {2420617 x^{5}}{3456}-\frac {117547073 x^{6}}{41472}+\frac {162576422327 x^{7}}{14224896}+O\left (x^{8}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-5 x +\frac {85}{4} x^{2}-\frac {3145}{36} x^{3}+\frac {204425}{576} x^{4}-\frac {825877}{576} x^{5}+\frac {119752165}{20736} x^{6}-\frac {23591176505}{1016064} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (2 x -\frac {39}{4} x^{2}+\frac {4499}{108} x^{3}-\frac {594305}{3456} x^{4}+\frac {2420617}{3456} x^{5}-\frac {117547073}{41472} x^{6}+\frac {162576422327}{14224896} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \right ) x \]

Problem 1378

ODE	\[ \boxed {x^{2} \left (x +1\right ) y^{\prime \prime }+x \left (1+2 x \right ) y^{\prime }+y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-x +\frac {3 x^{2}}{4}-\frac {7 x^{3}}{12}+\frac {91 x^{4}}{192}-\frac {637 x^{5}}{1600}+\frac {19747 x^{6}}{57600}-\frac {17329 x^{7}}{57600}+O\left (x^{8}\right )\right )+c_{2} \left (\left (1-x +\frac {3 x^{2}}{4}-\frac {7 x^{3}}{12}+\frac {91 x^{4}}{192}-\frac {637 x^{5}}{1600}+\frac {19747 x^{6}}{57600}-\frac {17329 x^{7}}{57600}+O\left (x^{8}\right )\right ) \ln \left (x \right )+x -\frac {3 x^{2}}{4}+\frac {5 x^{3}}{9}-\frac {499 x^{4}}{1152}+\frac {16919 x^{5}}{48000}-\frac {56861 x^{6}}{192000}+\frac {1027717 x^{7}}{4032000}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-x +\frac {3}{4} x^{2}-\frac {7}{12} x^{3}+\frac {91}{192} x^{4}-\frac {637}{1600} x^{5}+\frac {19747}{57600} x^{6}-\frac {17329}{57600} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (x -\frac {3}{4} x^{2}+\frac {5}{9} x^{3}-\frac {499}{1152} x^{4}+\frac {16919}{48000} x^{5}-\frac {56861}{192000} x^{6}+\frac {1027717}{4032000} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \]

Problem 1379

ODE	\[ \boxed {x^{2} \left (1-x \right ) y^{\prime \prime }+x \left (7+x \right ) y^{\prime }+\left (9-x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (x^{4}+16 x^{3}+36 x^{2}+16 x +1+O\left (x^{8}\right )\right )}{x^{3}}+c_{2} \left (\frac {\left (x^{4}+16 x^{3}+36 x^{2}+16 x +1+O\left (x^{8}\right )\right ) \ln \left (x \right )}{x^{3}}+\frac {-150 x^{2}-40 x -\frac {280 x^{3}}{3}-\frac {25 x^{4}}{3}+O\left (x^{8}\right )}{x^{3}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+16 x +36 x^{2}+16 x^{3}+x^{4}+\operatorname {O}\left (x^{8}\right )\right )+\left (\left (-40\right ) x -150 x^{2}-\frac {280}{3} x^{3}-\frac {25}{3} x^{4}+\operatorname {O}\left (x^{8}\right )\right ) c_{2}}{x^{3}} \]

Problem 1380

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

Problem 1381

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

Problem 1382

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

Problem 1383

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

Problem 1384

ODE	\[ \boxed {2 x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+7 y^{\prime } x^{3}+\left (3 x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {3 x^{2}}{8}+\frac {21 x^{4}}{128}+O\left (x^{6}\right )\right )+c_{2} \left (\sqrt {x}\, \left (1-\frac {3 x^{2}}{8}+\frac {21 x^{4}}{128}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\sqrt {x}\, \left (-\frac {x^{2}}{16}+\frac {17 x^{4}}{512}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {3}{8} x^{2}+\frac {21}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {1}{16} x^{2}+\frac {17}{512} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]

Problem 1385

ODE	\[ \boxed {x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-4 x^{2}+1\right ) y^{\prime }+y \left (2 x^{2}+1\right )=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {3 x^{2}}{2}+\frac {15 x^{4}}{8}+O\left (x^{6}\right )\right )+c_{2} \left (x \left (1-\frac {3 x^{2}}{2}+\frac {15 x^{4}}{8}+O\left (x^{6}\right )\right ) \ln \left (x \right )+x \left (\frac {x^{2}}{4}-\frac {13 x^{4}}{32}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {3}{2} x^{2}+\frac {15}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{4} x^{2}-\frac {13}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]

Problem 1386

ODE	\[ \boxed {4 x^{2} \left (x^{2}+4\right ) y^{\prime \prime }+3 x \left (3 x^{2}+8\right ) y^{\prime }+\left (-9 x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1+\frac {5 x^{2}}{32}-\frac {15 x^{4}}{2048}+O\left (x^{6}\right )\right )}{x^{\frac {1}{4}}}+c_{2} \left (\frac {\left (1+\frac {5 x^{2}}{32}-\frac {15 x^{4}}{2048}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{x^{\frac {1}{4}}}+\frac {-\frac {13 x^{2}}{64}+\frac {13 x^{4}}{8192}+O\left (x^{6}\right )}{x^{\frac {1}{4}}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+\frac {5}{32} x^{2}-\frac {15}{2048} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {13}{64} x^{2}+\frac {13}{8192} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2}}{x^{\frac {1}{4}}} \]

Problem 1387

ODE	\[ \boxed {3 x^{2} \left (x^{2}+3\right ) y^{\prime \prime }+x \left (11 x^{2}+3\right ) y^{\prime }+\left (5 x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1-\frac {2 x^{2}}{9}+\frac {5 x^{4}}{81}+O\left (x^{6}\right )\right )+c_{2} \left (x^{\frac {1}{3}} \left (1-\frac {2 x^{2}}{9}+\frac {5 x^{4}}{81}+O\left (x^{6}\right )\right ) \ln \left (x \right )+x^{\frac {1}{3}} \left (-\frac {x^{2}}{18}+\frac {x^{4}}{54}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{\frac {1}{3}} \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {2}{9} x^{2}+\frac {5}{81} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {1}{18} x^{2}+\frac {1}{54} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]

Problem 1388

ODE	\[ \boxed {4 x^{2} \left (4 x^{2}+1\right ) y^{\prime \prime }+32 y^{\prime } x^{3}+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {3 x^{2}}{4}+\frac {105 x^{4}}{64}+O\left (x^{6}\right )\right )+c_{2} \left (\sqrt {x}\, \left (1-\frac {3 x^{2}}{4}+\frac {105 x^{4}}{64}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\sqrt {x}\, \left (-\frac {5 x^{2}}{4}+\frac {389 x^{4}}{128}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {3}{4} x^{2}+\frac {105}{64} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {5}{4} x^{2}+\frac {389}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]

Problem 1389

ODE	\[ \boxed {9 x^{2} y^{\prime \prime }-3 x \left (-2 x^{2}+7\right ) y^{\prime }+\left (2 x^{2}+25\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {5}{3}} \left (1-\frac {x^{2}}{3}+\frac {x^{4}}{18}+O\left (x^{6}\right )\right )+c_{2} \left (x^{\frac {5}{3}} \left (1-\frac {x^{2}}{3}+\frac {x^{4}}{18}+O\left (x^{6}\right )\right ) \ln \left (x \right )+x^{\frac {5}{3}} \left (\frac {x^{2}}{6}-\frac {x^{4}}{24}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{\frac {5}{3}} \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {1}{3} x^{2}+\frac {1}{18} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{6} x^{2}-\frac {1}{24} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]

Problem 1390

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

Problem 1391

ODE	\[ \boxed {x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (8 x^{2}+3\right ) y^{\prime }+\left (12 x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1-\frac {3 x^{2}}{2}+\frac {15 x^{4}}{8}+O\left (x^{6}\right )\right )}{x}+c_{2} \left (\frac {\left (1-\frac {3 x^{2}}{2}+\frac {15 x^{4}}{8}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{x}+\frac {\frac {x^{2}}{4}-\frac {13 x^{4}}{32}+O\left (x^{6}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {3}{2} x^{2}+\frac {15}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{4} x^{2}-\frac {13}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2}}{x} \]

Problem 1392

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

Problem 1393

ODE	\[ \boxed {x^{2} \left (-2 x^{2}+1\right ) y^{\prime \prime }+x \left (-9 x^{2}+5\right ) y^{\prime }+\left (-3 x^{2}+4\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1-\frac {3 x^{2}}{4}-\frac {9 x^{4}}{64}+O\left (x^{6}\right )\right )}{x^{2}}+c_{2} \left (\frac {\left (1-\frac {3 x^{2}}{4}-\frac {9 x^{4}}{64}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{x^{2}}+\frac {\frac {x^{2}}{2}-\frac {21 x^{4}}{128}+O\left (x^{6}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {3}{4} x^{2}-\frac {9}{64} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{2} x^{2}-\frac {21}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2}}{x^{2}} \]

Problem 1394

ODE	\[ \boxed {x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+x \left (-x^{2}+14\right ) y^{\prime }+2 \left (x^{2}+9\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1-\frac {17 x^{2}}{8}+\frac {85 x^{4}}{256}+O\left (x^{6}\right )\right )}{x^{3}}+c_{2} \left (\frac {\left (1-\frac {17 x^{2}}{8}+\frac {85 x^{4}}{256}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{x^{3}}+\frac {\frac {25 x^{2}}{8}-\frac {471 x^{4}}{512}+O\left (x^{6}\right )}{x^{3}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {17}{8} x^{2}+\frac {85}{256} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {25}{8} x^{2}-\frac {471}{512} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2}}{x^{3}} \]

Problem 1395

ODE	\[ \boxed {x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (7 x^{2}+3\right ) y^{\prime }+\left (8 x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1-\frac {3 x^{2}}{4}+\frac {45 x^{4}}{64}+O\left (x^{6}\right )\right )}{x}+c_{2} \left (\frac {\left (1-\frac {3 x^{2}}{4}+\frac {45 x^{4}}{64}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{x}+\frac {-\frac {x^{2}}{4}+\frac {33 x^{4}}{128}+O\left (x^{6}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {3}{4} x^{2}+\frac {45}{64} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {1}{4} x^{2}+\frac {33}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2}}{x} \]

Problem 1396

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

Problem 1397

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

Problem 1398

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

Problem 1399

ODE	\[ \boxed {4 x^{2} \left (x +1\right ) y^{\prime \prime }+4 y^{\prime } x^{2}+\left (1-5 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (x +1-\frac {x^{2}}{4}+\frac {5 x^{3}}{36}-\frac {55 x^{4}}{576}+\frac {209 x^{5}}{2880}+O\left (x^{6}\right )\right )+c_{2} \left (\sqrt {x}\, \left (x +1-\frac {x^{2}}{4}+\frac {5 x^{3}}{36}-\frac {55 x^{4}}{576}+\frac {209 x^{5}}{2880}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\sqrt {x}\, \left (-3 x +\frac {x^{2}}{4}-\frac {5 x^{3}}{54}+\frac {175 x^{4}}{3456}-\frac {2863 x^{5}}{86400}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+x -\frac {1}{4} x^{2}+\frac {5}{36} x^{3}-\frac {55}{576} x^{4}+\frac {209}{2880} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-3\right ) x +\frac {1}{4} x^{2}-\frac {5}{54} x^{3}+\frac {175}{3456} x^{4}-\frac {2863}{86400} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]

Problem 1400

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