2.17.13 Problems 1201 to 1300

Problem 1201

ODE	\[ \boxed {x^{2} \left (x +1\right ) y^{\prime \prime }-x \left (-x^{2}-6 x +1\right ) y^{\prime }+\left (x^{2}+6 x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-12 x +\frac {119 x^{2}}{2}-\frac {583 x^{3}}{3}+\frac {1981 x^{4}}{4}-\frac {80287 x^{5}}{75}+O\left (x^{6}\right )\right )+c_{2} \left (x \left (1-12 x +\frac {119 x^{2}}{2}-\frac {583 x^{3}}{3}+\frac {1981 x^{4}}{4}-\frac {80287 x^{5}}{75}+O\left (x^{6}\right )\right ) \ln \left (x \right )+x \left (17 x -\frac {471 x^{2}}{4}+445 x^{3}-\frac {118285 x^{4}}{96}+\frac {702451 x^{5}}{250}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-12 x +\frac {119}{2} x^{2}-\frac {583}{3} x^{3}+\frac {1981}{4} x^{4}-\frac {80287}{75} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (17 x -\frac {471}{4} x^{2}+445 x^{3}-\frac {118285}{96} x^{4}+\frac {702451}{250} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) x \]

Problem 1202

ODE	\[ \boxed {x^{2} \left (3 x +1\right ) y^{\prime \prime }+x \left (x^{2}+12 x +2\right ) y^{\prime }+2 x \left (x +3\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-3 x +\frac {26 x^{2}}{3}-\frac {101 x^{3}}{4}+\frac {4441 x^{4}}{60}-\frac {26141 x^{5}}{120}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-3 x +\frac {17 x^{2}}{2}-\frac {49 x^{3}}{2}+\frac {571 x^{4}}{8}-\frac {8369 x^{5}}{40}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (1-3 x +\frac {26}{3} x^{2}-\frac {101}{4} x^{3}+\frac {4441}{60} x^{4}-\frac {26141}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (1-6 x +\frac {35}{2} x^{2}-\frac {101}{2} x^{3}+\frac {1177}{8} x^{4}-\frac {17251}{40} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 1203

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

Problem 1204

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

Problem 1205

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

Problem 1206

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

Problem 1207

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }-8 y^{\prime } x +20 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (5 x^{4}-10 x^{2}+1\right ) y \left (0\right )+\left (x -2 x^{3}+\frac {1}{5} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (5 x^{4}-10 x^{2}+1\right ) c_{1} +\left (x -2 x^{3}+\frac {1}{5} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (5 x^{4}-10 x^{2}+1\right ) y \left (0\right )+\left (x -2 x^{3}+\frac {1}{5} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 1208

ODE	\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-8 y^{\prime } x -12 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (28 x^{6}+15 x^{4}+6 x^{2}+1\right ) y \left (0\right )+\left (x +\frac {10}{3} x^{3}+7 x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (15 x^{4}+6 x^{2}+1\right ) c_{1} +\left (x +\frac {10}{3} x^{3}+7 x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (15 x^{4}+6 x^{2}+1\right ) y \left (0\right )+\left (x +\frac {10}{3} x^{3}+7 x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 1209

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

Problem 1210

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+2 y^{\prime } x +\frac {y}{4}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{8} x^{2}+\frac {25}{384} x^{4}-\frac {45}{1024} x^{6}\right ) y \left (0\right )+\left (x -\frac {3}{8} x^{3}+\frac {147}{640} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{8} x^{2}+\frac {25}{384} x^{4}\right ) c_{1} +\left (x -\frac {3}{8} x^{3}+\frac {147}{640} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{8} x^{2}+\frac {25}{384} x^{4}\right ) y \left (0\right )+\left (x -\frac {3}{8} x^{3}+\frac {147}{640} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 1211

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

Problem 1212

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }-10 y^{\prime } x +28 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-14 x^{2}+\frac {35}{3} x^{4}\right ) y \left (0\right )+\left (x -3 x^{3}+\frac {3}{5} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-14 x^{2}+\frac {35}{3} x^{4}\right ) c_{1} +\left (x -3 x^{3}+\frac {3}{5} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {35}{3} x^{4}-14 x^{2}\right ) y \left (0\right )+\left (x -3 x^{3}+\frac {3}{5} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 1213

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

Problem 1214

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime } x +3 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {3}{2} x^{2}+\frac {7}{8} x^{4}-\frac {77}{240} x^{6}\right ) y \left (0\right )+\left (x -\frac {5}{6} x^{3}+\frac {3}{8} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {3}{2} x^{2}+\frac {7}{8} x^{4}\right ) c_{1} +\left (x -\frac {5}{6} x^{3}+\frac {3}{8} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {3}{2} x^{2}+\frac {7}{8} x^{4}\right ) y \left (0\right )+\left (x -\frac {5}{6} x^{3}+\frac {3}{8} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 1215

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+y^{\prime } x +y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = -1] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = -x^{2}-x +2+\frac {x^{3}}{3}+\frac {5 x^{4}}{12}-\frac {x^{5}}{6}-\frac {17 x^{6}}{72}+O\left (x^{6}\right ) \] Verified OK. \[ y = 2-x^{2}+\frac {5 x^{4}}{12}-x +\frac {x^{3}}{3}-\frac {x^{5}}{6}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2-x -x^{2}+\frac {1}{3} x^{3}+\frac {5}{12} x^{4}-\frac {1}{6} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 1216

ODE	\[ \boxed {\left (2 x^{2}+1\right ) y^{\prime \prime }-9 y^{\prime } x -6 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = -1] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 3 x^{2}-x +1-\frac {5 x^{3}}{2}+5 x^{4}-\frac {21 x^{5}}{8}+3 x^{6}+O\left (x^{6}\right ) \] Verified OK. \[ y = 5 x^{4}+3 x^{2}+1-x -\frac {5 x^{3}}{2}-\frac {21 x^{5}}{8}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1-x +3 x^{2}-\frac {5}{2} x^{3}+5 x^{4}-\frac {21}{8} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 1217

ODE	\[ \boxed {\left (8 x^{2}+1\right ) y^{\prime \prime }+2 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = -1] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = -2 x^{2}-x +2+\frac {x^{3}}{3}+3 x^{4}-\frac {5 x^{5}}{6}-\frac {49 x^{6}}{5}+O\left (x^{6}\right ) \] Verified OK. \[ y = 3 x^{4}-2 x^{2}+2-x +\frac {x^{3}}{3}-\frac {5 x^{5}}{6}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2-x -2 x^{2}+\frac {1}{3} x^{3}+3 x^{4}-\frac {5}{6} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 1218

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

Problem 1219

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

Problem 1220

ODE	\[ \boxed {\left (2 x^{2}-4 x +1\right ) y^{\prime \prime }+10 \left (x -1\right ) y^{\prime }+6 y=0} \] With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = \left (1+3 \left (x -1\right )^{2}+\frac {15 \left (x -1\right )^{4}}{2}+\frac {35 \left (x -1\right )^{6}}{2}\right ) y \left (1\right )+\left (x -1+\frac {8 \left (x -1\right )^{3}}{3}+\frac {32 \left (x -1\right )^{5}}{5}\right ) y^{\prime }\left (1\right )+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+3 \left (x -1\right )^{2}+\frac {15 \left (x -1\right )^{4}}{2}\right ) y \left (1\right )+\left (x -1+\frac {8 \left (x -1\right )^{3}}{3}+\frac {32 \left (x -1\right )^{5}}{5}\right ) D\left (y \right )\left (1\right )+O\left (x^{6}\right ) \]

Problem 1221

ODE	\[ \boxed {\left (2 x^{2}-8 x +11\right ) y^{\prime \prime }-16 \left (-2+x \right ) y^{\prime }+36 y=0} \] With the expansion point for the power series method at \(x = 2\).
program solution	\[ y = \left (1-6 \left (-2+x \right )^{2}+\frac {4 \left (-2+x \right )^{4}}{3}+\frac {8 \left (-2+x \right )^{6}}{135}\right ) y \left (2\right )+\left (-2+x -\frac {10 \left (-2+x \right )^{3}}{9}\right ) y^{\prime }\left (2\right )+O\left (\left (-2+x \right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-6 \left (-2+x \right )^{2}+\frac {4 \left (-2+x \right )^{4}}{3}\right ) y \left (2\right )+\left (-2+x -\frac {10 \left (-2+x \right )^{3}}{9}\right ) D\left (y \right )\left (2\right )+O\left (x^{6}\right ) \]

Problem 1222

ODE	\[ \boxed {\left (3 x^{2}+6 x +5\right ) y^{\prime \prime }+9 \left (x +1\right ) y^{\prime }+3 y=0} \] With the expansion point for the power series method at \(x = -1\).
program solution	\[ y = \left (1-\frac {3 \left (x +1\right )^{2}}{4}+\frac {27 \left (x +1\right )^{4}}{32}-\frac {135 \left (x +1\right )^{6}}{128}\right ) y \left (-1\right )+\left (x +1-\left (x +1\right )^{3}+\frac {6 \left (x +1\right )^{5}}{5}\right ) y^{\prime }\left (-1\right )+O\left (\left (x +1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {3 \left (x +1\right )^{2}}{4}+\frac {27 \left (x +1\right )^{4}}{32}\right ) y \left (-1\right )+\left (x +1-\left (x +1\right )^{3}+\frac {6 \left (x +1\right )^{5}}{5}\right ) D\left (y \right )\left (-1\right )+O\left (x^{6}\right ) \]

Problem 1223

ODE	\[ \boxed {\left (x^{2}-4\right ) y^{\prime \prime }-y^{\prime } x -3 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = 2] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 2 x -1+\frac {3 x^{2}}{8}-\frac {x^{3}}{3}-\frac {3 x^{4}}{128}-\frac {x^{6}}{1024}+O\left (x^{6}\right ) \] Verified OK. \[ y = -1+\frac {3 x^{2}}{8}-\frac {3 x^{4}}{128}+2 x -\frac {x^{3}}{3}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -1+2 x +\frac {3}{8} x^{2}-\frac {1}{3} x^{3}-\frac {3}{128} x^{4}+\operatorname {O}\left (x^{6}\right ) \]

Problem 1224

ODE	\[ \boxed {y^{\prime \prime }+\left (x -3\right ) y^{\prime }+3 y=0} \] With initial conditions \begin {align*} [y \left (3\right ) = -2, y^{\prime }\left (3\right ) = 3] \end {align*} With the expansion point for the power series method at \(x = 3\).
program solution	\[ y = -2 \left (x -3\right )^{3}+3 \left (x -3\right )^{2}+3 x -11-\frac {5 \left (x -3\right )^{4}}{4}+\frac {3 \left (x -3\right )^{5}}{5}+\frac {7 \left (x -3\right )^{6}}{24}+O\left (\left (x -3\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -2+3 \left (x -3\right )+3 \left (x -3\right )^{2}-2 \left (x -3\right )^{3}-\frac {5}{4} \left (x -3\right )^{4}+\frac {3}{5} \left (x -3\right )^{5}+\operatorname {O}\left (\left (x -3\right )^{6}\right ) \]

Problem 1225

ODE	\[ \boxed {\left (3 x^{2}-6 x +5\right ) y^{\prime \prime }+\left (x -1\right ) y^{\prime }+12 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = -1, y^{\prime }\left (1\right ) = 1] \end {align*} With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = 3 \left (x -1\right )^{2}+x -2-\frac {13 \left (x -1\right )^{3}}{12}-\frac {5 \left (x -1\right )^{4}}{2}+\frac {143 \left (x -1\right )^{5}}{160}+\frac {13 \left (x -1\right )^{6}}{6}+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -1+\left (x -1\right )+3 \left (x -1\right )^{2}-\frac {13}{12} \left (x -1\right )^{3}-\frac {5}{2} \left (x -1\right )^{4}+\frac {143}{160} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \]

Problem 1226

ODE	\[ \boxed {\left (4 x^{2}-24 x +37\right ) y^{\prime \prime }+y=0} \] With initial conditions \begin {align*} [y \left (3\right ) = 4, y^{\prime }\left (3\right ) = -6] \end {align*} With the expansion point for the power series method at \(x = 3\).
program solution	\[ y = \left (x -3\right )^{3}-2 \left (x -3\right )^{2}-6 x +22+\frac {3 \left (x -3\right )^{4}}{2}-\frac {5 \left (x -3\right )^{5}}{4}-\frac {49 \left (x -3\right )^{6}}{20}+O\left (\left (x -3\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 4-6 \left (x -3\right )-2 \left (x -3\right )^{2}+\left (x -3\right )^{3}+\frac {3}{2} \left (x -3\right )^{4}-\frac {5}{4} \left (x -3\right )^{5}+\operatorname {O}\left (\left (x -3\right )^{6}\right ) \]

Problem 1227

ODE	\[ \boxed {\left (x^{2}-8 x +14\right ) y^{\prime \prime }-8 \left (x -4\right ) y^{\prime }+20 y=0} \] With initial conditions \begin {align*} [y \left (4\right ) = 3, y^{\prime }\left (4\right ) = -4] \end {align*} With the expansion point for the power series method at \(x = 4\).
program solution	\[ y = -4 \left (x -4\right )^{3}+15 \left (x -4\right )^{2}-4 x +19+\frac {15 \left (x -4\right )^{4}}{4}-\frac {\left (x -4\right )^{5}}{5}+O\left (\left (x -4\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 3-4 \left (x -4\right )+15 \left (x -4\right )^{2}-4 \left (x -4\right )^{3}+\frac {15}{4} \left (x -4\right )^{4}-\frac {1}{5} \left (x -4\right )^{5}+\operatorname {O}\left (\left (x -4\right )^{6}\right ) \]

Problem 1228

ODE	\[ \boxed {\left (2 x^{2}+4 x +5\right ) y^{\prime \prime }-20 \left (x +1\right ) y^{\prime }+60 y=0} \] With initial conditions \begin {align*} [y \left (-1\right ) = 3, y^{\prime }\left (-1\right ) = -3] \end {align*} With the expansion point for the power series method at \(x = -1\).
program solution	\[ y = -30 \left (x +1\right )^{2}-3 x +\frac {20 \left (x +1\right )^{3}}{3}+20 \left (x +1\right )^{4}-\frac {4 \left (x +1\right )^{5}}{3}-\frac {8 \left (x +1\right )^{6}}{9}+O\left (\left (x +1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 3-3 \left (x +1\right )-30 \left (x +1\right )^{2}+\frac {20}{3} \left (x +1\right )^{3}+20 \left (x +1\right )^{4}-\frac {4}{3} \left (x +1\right )^{5}+\operatorname {O}\left (\left (x +1\right )^{6}\right ) \]

Problem 1229

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

Problem 1230

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime } x +2 \alpha y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\alpha \,x^{2}+\frac {1}{6} x^{4} \alpha ^{2}-\frac {1}{3} x^{4} \alpha -\frac {1}{90} x^{6} \alpha ^{3}+\frac {1}{15} x^{6} \alpha ^{2}-\frac {4}{45} x^{6} \alpha \right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3} \alpha +\frac {1}{3} x^{3}+\frac {1}{30} x^{5} \alpha ^{2}-\frac {2}{15} x^{5} \alpha +\frac {1}{10} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\alpha \,x^{2}+\left (\frac {1}{6} \alpha ^{2}-\frac {1}{3} \alpha \right ) x^{4}\right ) c_{1} +\left (x +\left (-\frac {\alpha }{3}+\frac {1}{3}\right ) x^{3}+\left (\frac {1}{30} \alpha ^{2}-\frac {2}{15} \alpha +\frac {1}{10}\right ) x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\alpha \,x^{2}+\frac {\alpha \left (\alpha -2\right ) x^{4}}{6}\right ) y \left (0\right )+\left (x -\frac {\left (\alpha -1\right ) x^{3}}{3}+\frac {\left (\alpha ^{2}-4 \alpha +3\right ) x^{5}}{30}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 1231

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

Problem 1232

ODE	\[ \boxed {\left (-2 x^{3}+1\right ) y^{\prime \prime }-10 y^{\prime } x^{2}-8 y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {4}{3} x^{3}+\frac {20}{9} x^{6}\right ) y \left (0\right )+\left (x +\frac {3}{2} x^{4}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {4 x^{3}}{3}\right ) c_{1} +\left (x +\frac {3}{2} x^{4}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {4 x^{3}}{3}\right ) y \left (0\right )+\left (x +\frac {3}{2} x^{4}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 1233

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

Problem 1234

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

Problem 1235

ODE	\[ \boxed {\left (-x^{3}+1\right ) y^{\prime \prime }+15 y^{\prime } x^{2}-36 y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+6 x^{3}-\frac {3}{5} x^{6}\right ) y \left (0\right )+\left (x +\frac {7}{4} x^{4}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (6 x^{3}+1\right ) c_{1} +\left (x +\frac {7}{4} x^{4}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (6 x^{3}+1\right ) y \left (0\right )+\left (x +\frac {7}{4} x^{4}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 1236

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

Problem 1237

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

Problem 1238

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

Problem 1239

ODE	\[ \boxed {\left (x^{8}+1\right ) y^{\prime \prime }-16 y^{\prime } x^{7}+72 y x^{6}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = y \left (0\right )+y^{\prime }\left (0\right ) x +O\left (x^{6}\right ) \] Verified OK. \[ y = c_{2} x +c_{1} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = y \left (0\right )+D\left (y \right )\left (0\right ) x \]

Problem 1240

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

Problem 1241

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

Problem 1242

ODE	\[ \boxed {\left (3 x +1\right ) y^{\prime \prime }+y^{\prime } x +2 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = -3] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = -2 x^{2}-3 x +2+\frac {7 x^{3}}{2}-\frac {55 x^{4}}{12}+\frac {59 x^{5}}{8}-\frac {83 x^{6}}{6}+O\left (x^{6}\right ) \] Verified OK. \[ y = 2-2 x^{2}+\frac {7 x^{3}}{2}-\frac {55 x^{4}}{12}+\frac {59 x^{5}}{8}-3 x +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2-3 x -2 x^{2}+\frac {7}{2} x^{3}-\frac {55}{12} x^{4}+\frac {59}{8} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 1243

ODE	\[ \boxed {\left (2 x^{2}+x +1\right ) y^{\prime \prime }+\left (2+8 x \right ) y^{\prime }+4 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = 2] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = -12 x^{6}+4 x^{5}+4 x^{4}-4 x^{3}+2 x -1+O\left (x^{6}\right ) \] Verified OK. \[ y = 4 x^{5}+4 x^{4}-4 x^{3}-1+2 x +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -1+2 x -4 x^{3}+4 x^{4}+4 x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 1244

ODE	\[ \boxed {\left (-2 x^{2}+1\right ) y^{\prime \prime }+\left (2-6 x \right ) y^{\prime }-2 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = x^{2}+1-\frac {2 x^{3}}{3}+\frac {11 x^{4}}{6}-\frac {9 x^{5}}{5}+\frac {329 x^{6}}{90}+O\left (x^{6}\right ) \] Verified OK. \[ y = x^{2}+1-\frac {2 x^{3}}{3}+\frac {11 x^{4}}{6}-\frac {9 x^{5}}{5}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1+x^{2}-\frac {2}{3} x^{3}+\frac {11}{6} x^{4}-\frac {9}{5} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 1245

ODE	\[ \boxed {\left (3 x^{2}+x +1\right ) y^{\prime \prime }+\left (2+15 x \right ) y^{\prime }+12 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = -x^{2}+x -\frac {7 x^{3}}{2}+\frac {15 x^{4}}{2}+\frac {45 x^{5}}{8}-\frac {261 x^{6}}{8}+O\left (x^{6}\right ) \] Verified OK. \[ y = -x^{2}+x -\frac {7 x^{3}}{2}+\frac {15 x^{4}}{2}+\frac {45 x^{5}}{8}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x -x^{2}-\frac {7}{2} x^{3}+\frac {15}{2} x^{4}+\frac {45}{8} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 1246

ODE	\[ \boxed {\left (2+x \right ) y^{\prime \prime }+\left (x +1\right ) y^{\prime }+3 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 4, y^{\prime }\left (0\right ) = 3] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 3 x +4-\frac {15 x^{2}}{4}+\frac {x^{3}}{4}+\frac {11 x^{4}}{16}-\frac {5 x^{5}}{16}+\frac {x^{6}}{20}+O\left (x^{6}\right ) \] Verified OK. \[ y = 4-\frac {15 x^{2}}{4}+\frac {x^{3}}{4}+\frac {11 x^{4}}{16}-\frac {5 x^{5}}{16}+3 x +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 4+3 x -\frac {15}{4} x^{2}+\frac {1}{4} x^{3}+\frac {11}{16} x^{4}-\frac {5}{16} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 1247

ODE	\[ \boxed {\left (x^{2}+3 x +3\right ) y^{\prime \prime }+\left (4 x +6\right ) y^{\prime }+2 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 7, y^{\prime }\left (0\right ) = 3] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 7+3 x -\frac {16 x^{2}}{3}+\frac {13 x^{3}}{3}-\frac {23 x^{4}}{9}+\frac {10 x^{5}}{9}-\frac {7 x^{6}}{27}+O\left (x^{6}\right ) \] Verified OK. \[ y = 7-\frac {16 x^{2}}{3}+\frac {13 x^{3}}{3}-\frac {23 x^{4}}{9}+\frac {10 x^{5}}{9}+3 x +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 7+3 x -\frac {16}{3} x^{2}+\frac {13}{3} x^{3}-\frac {23}{9} x^{4}+\frac {10}{9} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 1248

ODE	\[ \boxed {\left (x +4\right ) y^{\prime \prime }+\left (2+x \right ) y^{\prime }+2 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 5] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 5 x +2-\frac {7 x^{2}}{4}-\frac {3 x^{3}}{16}+\frac {37 x^{4}}{192}-\frac {7 x^{5}}{192}-\frac {x^{6}}{1920}+O\left (x^{6}\right ) \] Verified OK. \[ y = 2-\frac {7 x^{2}}{4}-\frac {3 x^{3}}{16}+\frac {37 x^{4}}{192}-\frac {7 x^{5}}{192}+5 x +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2+5 x -\frac {7}{4} x^{2}-\frac {3}{16} x^{3}+\frac {37}{192} x^{4}-\frac {7}{192} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 1249

ODE	\[ \boxed {\left (2 x^{2}-3 x +2\right ) y^{\prime \prime }-\left (4-6 x \right ) y^{\prime }+2 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1, y^{\prime }\left (1\right ) = -1] \end {align*} With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = 2-x +\frac {4 \left (x -1\right )^{3}}{3}-\frac {4 \left (x -1\right )^{4}}{3}-\frac {4 \left (x -1\right )^{5}}{5}+\frac {136 \left (x -1\right )^{6}}{45}+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1-\left (x -1\right )+\frac {4}{3} \left (x -1\right )^{3}-\frac {4}{3} \left (x -1\right )^{4}-\frac {4}{5} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \]

Problem 1250

ODE	\[ \boxed {\left (2 x^{2}+3 x \right ) y^{\prime \prime }+10 \left (x +1\right ) y^{\prime }+8 y=0} \] With initial conditions \begin {align*} [y \left (-1\right ) = 1, y^{\prime }\left (-1\right ) = -1] \end {align*} With the expansion point for the power series method at \(x = -1\).
program solution	\[ y = 4 \left (x +1\right )^{2}-x -\frac {13 \left (x +1\right )^{3}}{3}+\frac {77 \left (x +1\right )^{4}}{6}-\frac {278 \left (x +1\right )^{5}}{15}+\frac {1942 \left (x +1\right )^{6}}{45}+O\left (\left (x +1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1-\left (x +1\right )+4 \left (x +1\right )^{2}-\frac {13}{3} \left (x +1\right )^{3}+\frac {77}{6} \left (x +1\right )^{4}-\frac {278}{15} \left (x +1\right )^{5}+\operatorname {O}\left (\left (x +1\right )^{6}\right ) \]

Problem 1251

ODE	\[ \boxed {\left (x^{2}-x +1\right ) y^{\prime \prime }-\left (-4 x +1\right ) y^{\prime }+2 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 2, y^{\prime }\left (1\right ) = -1] \end {align*} With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = -x +3-\frac {\left (x -1\right )^{2}}{2}+\frac {5 \left (x -1\right )^{3}}{3}-\frac {19 \left (x -1\right )^{4}}{12}+\frac {7 \left (x -1\right )^{5}}{30}+\frac {59 \left (x -1\right )^{6}}{45}+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2-\left (x -1\right )-\frac {1}{2} \left (x -1\right )^{2}+\frac {5}{3} \left (x -1\right )^{3}-\frac {19}{12} \left (x -1\right )^{4}+\frac {7}{30} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \]

Problem 1252

ODE	\[ \boxed {\left (2+x \right ) y^{\prime \prime }+\left (2+x \right ) y^{\prime }+y=0} \] With initial conditions \begin {align*} [y \left (-1\right ) = -2, y^{\prime }\left (-1\right ) = 3] \end {align*} With the expansion point for the power series method at \(x = -1\).
program solution	\[ y = 1+3 x -\frac {\left (x +1\right )^{2}}{2}-\frac {2 \left (x +1\right )^{3}}{3}+\frac {5 \left (x +1\right )^{4}}{8}-\frac {11 \left (x +1\right )^{5}}{30}+\frac {29 \left (x +1\right )^{6}}{144}+O\left (\left (x +1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -2+3 \left (x +1\right )-\frac {1}{2} \left (x +1\right )^{2}-\frac {2}{3} \left (x +1\right )^{3}+\frac {5}{8} \left (x +1\right )^{4}-\frac {11}{30} \left (x +1\right )^{5}+\operatorname {O}\left (\left (x +1\right )^{6}\right ) \]

Problem 1253

ODE	\[ \boxed {x^{2} y^{\prime \prime }-\left (6-7 x \right ) y^{\prime }+8 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1, y^{\prime }\left (1\right ) = -2] \end {align*} With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = -4 \left (x -1\right )^{4}+8 \left (x -1\right )^{3}-3 \left (x -1\right )^{2}-2 x +3-\frac {42 \left (x -1\right )^{5}}{5}+19 \left (x -1\right )^{6}+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1-2 \left (x -1\right )-3 \left (x -1\right )^{2}+8 \left (x -1\right )^{3}-4 \left (x -1\right )^{4}-\frac {42}{5} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \]

Problem 1254

ODE	\[ \boxed {\left (2 x^{2}+x +1\right ) y^{\prime \prime }+\left (1+7 x \right ) y^{\prime }+2 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1, y^{\prime }\left (1\right ) = 0] \end {align*} With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = 1-\frac {\left (x -1\right )^{2}}{4}+\frac {13 \left (x -1\right )^{3}}{48}-\frac {77 \left (x -1\right )^{4}}{384}+\frac {287 \left (x -1\right )^{5}}{2560}-\frac {623 \left (x -1\right )^{6}}{15360}+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1-\frac {1}{4} \left (x -1\right )^{2}+\frac {13}{48} \left (x -1\right )^{3}-\frac {77}{384} \left (x -1\right )^{4}+\frac {287}{2560} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \]

Problem 1255

ODE	\[ \boxed {\left (x +3\right ) y^{\prime \prime }+\left (1+2 x \right ) y^{\prime }-\left (2-x \right ) y=0} \] With initial conditions \begin {align*} [y \left (-1\right ) = 1, y^{\prime }\left (-1\right ) = 0] \end {align*} With the expansion point for the power series method at \(x = -1\).
program solution	\[ y = 1+\frac {3 \left (x +1\right )^{2}}{4}-\frac {\left (x +1\right )^{3}}{12}-\frac {\left (x +1\right )^{4}}{48}-\frac {\left (x +1\right )^{5}}{120}+\frac {\left (x +1\right )^{6}}{192}+O\left (\left (x +1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1+\frac {3}{4} \left (x +1\right )^{2}-\frac {1}{12} \left (x +1\right )^{3}-\frac {1}{48} \left (x +1\right )^{4}-\frac {1}{120} \left (x +1\right )^{5}+\operatorname {O}\left (\left (x +1\right )^{6}\right ) \]

Problem 1256

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

Problem 1257

ODE	\[ \boxed {\left (4 x +2\right ) y^{\prime \prime }-4 y^{\prime }-\left (4 x +6\right ) y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = -7] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = -4 x^{2}-7 x +2-\frac {17 x^{3}}{6}-\frac {3 x^{4}}{4}-\frac {9 x^{5}}{40}-\frac {7 x^{6}}{180}+O\left (x^{6}\right ) \] Verified OK. \[ y = 2-4 x^{2}-\frac {17 x^{3}}{6}-\frac {3 x^{4}}{4}-\frac {9 x^{5}}{40}-7 x +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2-7 x -4 x^{2}-\frac {17}{6} x^{3}-\frac {3}{4} x^{4}-\frac {9}{40} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 1258

ODE	\[ \boxed {\left (1+2 x \right ) y^{\prime \prime }-\left (1-2 x \right ) y^{\prime }-\left (3-2 x \right ) y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1, y^{\prime }\left (1\right ) = -2] \end {align*} With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = 3+\frac {\left (x -1\right )^{2}}{2}-2 x -\frac {\left (x -1\right )^{3}}{6}+\frac {5 \left (x -1\right )^{4}}{36}-\frac {73 \left (x -1\right )^{5}}{1080}+\frac {173 \left (x -1\right )^{6}}{6480}+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1-2 \left (x -1\right )+\frac {1}{2} \left (x -1\right )^{2}-\frac {1}{6} \left (x -1\right )^{3}+\frac {5}{36} \left (x -1\right )^{4}-\frac {73}{1080} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \]

Problem 1259

ODE	\[ \boxed {\left (2 x +5\right ) y^{\prime \prime }-y^{\prime }+\left (x +5\right ) y=0} \] With initial conditions \begin {align*} [y \left (-2\right ) = 2, y^{\prime }\left (-2\right ) = -1] \end {align*} With the expansion point for the power series method at \(x = -2\).
program solution	\[ y = -x -\frac {7 \left (2+x \right )^{2}}{2}+\frac {4 \left (2+x \right )^{3}}{3}-\frac {\left (2+x \right )^{4}}{24}+\frac {\left (2+x \right )^{5}}{60}-\frac {43 \left (2+x \right )^{6}}{720}+O\left (\left (2+x \right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2-\left (2+x \right )-\frac {7}{2} \left (2+x \right )^{2}+\frac {4}{3} \left (2+x \right )^{3}-\frac {1}{24} \left (2+x \right )^{4}+\frac {1}{60} \left (2+x \right )^{5}+\operatorname {O}\left (\left (2+x \right )^{6}\right ) \]

Problem 1260

ODE	\[ \boxed {\left (x +4\right ) y^{\prime \prime }-\left (4+2 x \right ) y^{\prime }+\left (x +6\right ) y=0} \] With initial conditions \begin {align*} [y \left (-3\right ) = 2, y^{\prime }\left (-3\right ) = -2] \end {align*} With the expansion point for the power series method at \(x = -3\).
program solution	\[ y = \left (x +3\right )^{3}-\left (x +3\right )^{2}-2 x -4-\frac {11 \left (x +3\right )^{4}}{12}+\frac {67 \left (x +3\right )^{5}}{60}-\frac {469 \left (x +3\right )^{6}}{360}+O\left (\left (x +3\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2-2 \left (x +3\right )-\left (x +3\right )^{2}+\left (x +3\right )^{3}-\frac {11}{12} \left (x +3\right )^{4}+\frac {67}{60} \left (x +3\right )^{5}+\operatorname {O}\left (\left (x +3\right )^{6}\right ) \]

Problem 1261

ODE	\[ \boxed {\left (3 x +2\right ) y^{\prime \prime }-y^{\prime } x +2 y x=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = 2] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 2 x -1+\frac {x^{3}}{3}-\frac {5 x^{4}}{12}+\frac {2 x^{5}}{5}-\frac {79 x^{6}}{180}+O\left (x^{6}\right ) \] Verified OK. \[ y = -1+\frac {x^{3}}{3}-\frac {5 x^{4}}{12}+\frac {2 x^{5}}{5}+2 x +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -1+2 x +\frac {1}{3} x^{3}-\frac {5}{12} x^{4}+\frac {2}{5} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 1262

ODE	\[ \boxed {\left (2 x +3\right ) y^{\prime \prime }+3 y^{\prime }-y x=0} \] With initial conditions \begin {align*} [y \left (-1\right ) = 2, y^{\prime }\left (-1\right ) = -3] \end {align*} With the expansion point for the power series method at \(x = -1\).
program solution	\[ y = -1-3 x +\frac {7 \left (x +1\right )^{2}}{2}-5 \left (x +1\right )^{3}+\frac {197 \left (x +1\right )^{4}}{24}-\frac {287 \left (x +1\right )^{5}}{20}+\frac {3725 \left (x +1\right )^{6}}{144}+O\left (\left (x +1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2-3 \left (x +1\right )+\frac {7}{2} \left (x +1\right )^{2}-5 \left (x +1\right )^{3}+\frac {197}{24} \left (x +1\right )^{4}-\frac {287}{20} \left (x +1\right )^{5}+\operatorname {O}\left (\left (x +1\right )^{6}\right ) \]

Problem 1263

ODE	\[ \boxed {\left (2 x +3\right ) y^{\prime \prime }-3 y^{\prime }-\left (2+x \right ) y=0} \] With initial conditions \begin {align*} [y \left (-2\right ) = -2, y^{\prime }\left (-2\right ) = 3] \end {align*} With the expansion point for the power series method at \(x = -2\).
program solution	\[ y = 4+3 x -\frac {9 \left (2+x \right )^{2}}{2}+\frac {11 \left (2+x \right )^{3}}{6}+\frac {5 \left (2+x \right )^{4}}{24}+\frac {7 \left (2+x \right )^{5}}{20}+\frac {83 \left (2+x \right )^{6}}{360}+O\left (\left (2+x \right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -2+3 \left (2+x \right )-\frac {9}{2} \left (2+x \right )^{2}+\frac {11}{6} \left (2+x \right )^{3}+\frac {5}{24} \left (2+x \right )^{4}+\frac {7}{20} \left (2+x \right )^{5}+\operatorname {O}\left (\left (2+x \right )^{6}\right ) \]

Problem 1264

ODE	\[ \boxed {\left (10-2 x \right ) y^{\prime \prime }+\left (x +1\right ) y=0} \] With initial conditions \begin {align*} [y \left (2\right ) = 2, y^{\prime }\left (2\right ) = -4] \end {align*} With the expansion point for the power series method at \(x = 2\).
program solution	\[ y = 10-4 x -\frac {\left (-2+x \right )^{2}}{2}+\frac {2 \left (-2+x \right )^{3}}{9}+\frac {49 \left (-2+x \right )^{4}}{432}+\frac {23 \left (-2+x \right )^{5}}{1080}+\frac {25 \left (-2+x \right )^{6}}{15552}+O\left (\left (-2+x \right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2-4 \left (-2+x \right )-\frac {1}{2} \left (-2+x \right )^{2}+\frac {2}{9} \left (-2+x \right )^{3}+\frac {49}{432} \left (-2+x \right )^{4}+\frac {23}{1080} \left (-2+x \right )^{5}+\operatorname {O}\left (\left (-2+x \right )^{6}\right ) \]

Problem 1265

ODE	\[ \boxed {\left (7+x \right ) y^{\prime \prime }+\left (2 x +8\right ) y^{\prime }+\left (x +5\right ) y=0} \] With initial conditions \begin {align*} [y \left (-4\right ) = 1, y^{\prime }\left (-4\right ) = 2] \end {align*} With the expansion point for the power series method at \(x = -4\).
program solution	\[ y = 9+2 x -\frac {\left (x +4\right )^{2}}{6}-\frac {10 \left (x +4\right )^{3}}{27}+\frac {19 \left (x +4\right )^{4}}{648}+\frac {13 \left (x +4\right )^{5}}{324}-\frac {451 \left (x +4\right )^{6}}{58320}+O\left (\left (x +4\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1+2 \left (x +4\right )-\frac {1}{6} \left (x +4\right )^{2}-\frac {10}{27} \left (x +4\right )^{3}+\frac {19}{648} \left (x +4\right )^{4}+\frac {13}{324} \left (x +4\right )^{5}+\operatorname {O}\left (\left (x +4\right )^{6}\right ) \]

Problem 1266

ODE	\[ \boxed {\left (4 x +6\right ) y^{\prime \prime }+\left (1+2 x \right ) y=0} \] With initial conditions \begin {align*} [y \left (-1\right ) = -1, y^{\prime }\left (-1\right ) = 2] \end {align*} With the expansion point for the power series method at \(x = -1\).
program solution	\[ y = 1+2 x -\frac {\left (x +1\right )^{2}}{4}+\frac {\left (x +1\right )^{3}}{2}-\frac {65 \left (x +1\right )^{4}}{96}+\frac {67 \left (x +1\right )^{5}}{80}-\frac {6593 \left (x +1\right )^{6}}{5760}+O\left (\left (x +1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -1+2 \left (x +1\right )-\frac {1}{4} \left (x +1\right )^{2}+\frac {1}{2} \left (x +1\right )^{3}-\frac {65}{96} \left (x +1\right )^{4}+\frac {67}{80} \left (x +1\right )^{5}+\operatorname {O}\left (\left (x +1\right )^{6}\right ) \]

Problem 1267

ODE	\[ \boxed {\left (\beta \,x^{2}+\alpha x +1\right ) y^{\prime \prime }+\left (\delta x +\gamma \right ) y^{\prime }+\epsilon y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ \text {Expression too large to display} \] Verified OK. \[ y = \left (1-\frac {\epsilon \,x^{2}}{2}+\left (\frac {1}{6} \alpha \epsilon +\frac {1}{6} \epsilon \gamma \right ) x^{3}+\left (-\frac {1}{12} \epsilon \,\alpha ^{2}-\frac {1}{8} \epsilon \gamma \alpha +\frac {1}{12} \beta \epsilon +\frac {1}{12} \delta \epsilon +\frac {1}{24} \epsilon ^{2}-\frac {1}{24} \epsilon \,\gamma ^{2}\right ) x^{4}+\left (-\frac {1}{60} \epsilon ^{2} \gamma +\frac {1}{120} \epsilon \,\gamma ^{3}+\frac {1}{20} \epsilon \,\alpha ^{3}-\frac {1}{30} \alpha \,\epsilon ^{2}+\frac {11}{120} \epsilon \gamma \,\alpha ^{2}-\frac {1}{10} \beta \alpha \epsilon -\frac {3}{40} \alpha \delta \epsilon +\frac {1}{20} \epsilon \,\gamma ^{2} \alpha -\frac {1}{15} \beta \epsilon \gamma -\frac {1}{24} \delta \epsilon \gamma \right ) x^{5}\right ) c_{1} +\left (x -\frac {x^{2} \gamma }{2}+\left (\frac {1}{6} \alpha \gamma +\frac {1}{6} \gamma ^{2}-\frac {1}{6} \delta -\frac {1}{6} \epsilon \right ) x^{3}+\left (-\frac {1}{12} \alpha ^{2} \gamma -\frac {1}{8} \alpha \,\gamma ^{2}+\frac {1}{12} \alpha \delta +\frac {1}{12} \alpha \epsilon +\frac {1}{12} \beta \gamma +\frac {1}{8} \delta \gamma +\frac {1}{12} \epsilon \gamma -\frac {1}{24} \gamma ^{3}\right ) x^{4}+\left (\frac {1}{20} \beta \epsilon -\frac {1}{20} \delta \,\gamma ^{2}+\frac {1}{30} \delta \epsilon -\frac {1}{40} \epsilon \,\gamma ^{2}+\frac {1}{20} \alpha ^{3} \gamma +\frac {11}{120} \alpha ^{2} \gamma ^{2}-\frac {1}{20} \alpha ^{2} \delta -\frac {1}{20} \epsilon \,\alpha ^{2}+\frac {1}{20} \alpha \,\gamma ^{3}-\frac {1}{15} \beta \,\gamma ^{2}+\frac {1}{20} \beta \delta -\frac {1}{10} \alpha \beta \gamma -\frac {7}{60} \alpha \delta \gamma -\frac {3}{40} \epsilon \gamma \alpha +\frac {1}{40} \delta ^{2}+\frac {1}{120} \epsilon ^{2}+\frac {1}{120} \gamma ^{4}\right ) x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {\epsilon \,x^{2}}{2}+\frac {\epsilon \left (\alpha +\gamma \right ) x^{3}}{6}+\frac {\epsilon \left (-\alpha ^{2}-\frac {3}{2} \alpha \gamma -\frac {1}{2} \gamma ^{2}+\beta +\delta +\frac {1}{2} \epsilon \right ) x^{4}}{12}-\frac {\epsilon \left (\frac {\left (\alpha +\frac {\gamma }{2}\right ) \epsilon }{3}-\frac {\gamma ^{3}}{12}-\frac {\alpha \,\gamma ^{2}}{2}+\frac {\left (-\frac {11 \alpha ^{2}}{4}+2 \beta +\frac {5 \delta }{4}\right ) \gamma }{3}+\alpha \left (-\frac {\alpha ^{2}}{2}+\beta +\frac {3 \delta }{4}\right )\right ) x^{5}}{10}\right ) y \left (0\right )+\left (x -\frac {\gamma \,x^{2}}{2}+\frac {\left (\alpha \gamma +\gamma ^{2}-\delta -\epsilon \right ) x^{3}}{6}+\frac {\left (\left (2 \alpha +2 \gamma \right ) \epsilon -\gamma ^{3}-3 \alpha \,\gamma ^{2}+\left (-2 \alpha ^{2}+2 \beta +3 \delta \right ) \gamma +2 \delta \alpha \right ) x^{4}}{24}+\frac {\left (\epsilon ^{2}+\left (-6 \alpha ^{2}-9 \alpha \gamma -3 \gamma ^{2}+6 \beta +4 \delta \right ) \epsilon +\gamma ^{4}+6 \alpha \,\gamma ^{3}+\left (11 \alpha ^{2}-8 \beta -6 \delta \right ) \gamma ^{2}-12 \alpha \left (-\frac {\alpha ^{2}}{2}+\beta +\frac {7 \delta }{6}\right ) \gamma +6 \left (-\alpha ^{2}+\beta +\frac {\delta }{2}\right ) \delta \right ) x^{5}}{120}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 1268

ODE	\[ \boxed {\left (2 x^{2}+3 x +1\right ) y^{\prime \prime }+\left (6+8 x \right ) y^{\prime }+4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (-62 x^{6}+30 x^{5}-14 x^{4}+6 x^{3}-2 x^{2}+1\right ) y \left (0\right )+\left (-63 x^{6}+31 x^{5}-15 x^{4}+7 x^{3}-3 x^{2}+x \right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (30 x^{5}-14 x^{4}+6 x^{3}-2 x^{2}+1\right ) c_{1} +\left (31 x^{5}-15 x^{4}+7 x^{3}-3 x^{2}+x \right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (30 x^{5}-14 x^{4}+6 x^{3}-2 x^{2}+1\right ) y \left (0\right )+\left (31 x^{5}-15 x^{4}+7 x^{3}-3 x^{2}+x \right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 1269

ODE	\[ \boxed {\left (6 x^{2}-5 x +1\right ) y^{\prime \prime }-\left (10-24 x \right ) y^{\prime }+12 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (-1266 x^{6}-390 x^{5}-114 x^{4}-30 x^{3}-6 x^{2}+1\right ) y \left (0\right )+\left (665 x^{6}+211 x^{5}+65 x^{4}+19 x^{3}+5 x^{2}+x \right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (-390 x^{5}-114 x^{4}-30 x^{3}-6 x^{2}+1\right ) c_{1} +\left (211 x^{5}+65 x^{4}+19 x^{3}+5 x^{2}+x \right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (-390 x^{5}-114 x^{4}-30 x^{3}-6 x^{2}+1\right ) y \left (0\right )+\left (211 x^{5}+65 x^{4}+19 x^{3}+5 x^{2}+x \right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 1270

ODE	\[ \boxed {\left (4 x^{2}-4 x +1\right ) y^{\prime \prime }-\left (8-16 x \right ) y^{\prime }+8 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (-320 x^{6}-128 x^{5}-48 x^{4}-16 x^{3}-4 x^{2}+1\right ) y \left (0\right )+\left (192 x^{6}+80 x^{5}+32 x^{4}+12 x^{3}+4 x^{2}+x \right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (-128 x^{5}-48 x^{4}-16 x^{3}-4 x^{2}+1\right ) c_{1} +\left (80 x^{5}+32 x^{4}+12 x^{3}+4 x^{2}+x \right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (-128 x^{5}-48 x^{4}-16 x^{3}-4 x^{2}+1\right ) y \left (0\right )+\left (80 x^{5}+32 x^{4}+12 x^{3}+4 x^{2}+x \right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 1271

ODE	\[ \boxed {\left (x^{2}+4 x +4\right ) y^{\prime \prime }+\left (8+4 x \right ) y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{4} x^{2}+\frac {1}{4} x^{3}-\frac {3}{16} x^{4}+\frac {1}{8} x^{5}-\frac {5}{64} x^{6}\right ) y \left (0\right )+\left (x -x^{2}+\frac {3}{4} x^{3}-\frac {1}{2} x^{4}+\frac {5}{16} x^{5}-\frac {3}{16} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{4} x^{2}+\frac {1}{4} x^{3}-\frac {3}{16} x^{4}+\frac {1}{8} x^{5}\right ) c_{1} +\left (x -x^{2}+\frac {3}{4} x^{3}-\frac {1}{2} x^{4}+\frac {5}{16} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{4} x^{2}+\frac {1}{4} x^{3}-\frac {3}{16} x^{4}+\frac {1}{8} x^{5}\right ) y \left (0\right )+\left (x -x^{2}+\frac {3}{4} x^{3}-\frac {1}{2} x^{4}+\frac {5}{16} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 1272

ODE	\[ \boxed {\left (3 x^{2}+8 x +4\right ) y^{\prime \prime }+\left (16+12 x \right ) y^{\prime }+6 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {3}{4} x^{2}+\frac {3}{2} x^{3}-\frac {39}{16} x^{4}+\frac {15}{4} x^{5}-\frac {363}{64} x^{6}\right ) y \left (0\right )+\left (x -2 x^{2}+\frac {13}{4} x^{3}-5 x^{4}+\frac {121}{16} x^{5}-\frac {91}{8} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {3}{4} x^{2}+\frac {3}{2} x^{3}-\frac {39}{16} x^{4}+\frac {15}{4} x^{5}\right ) c_{1} +\left (x -2 x^{2}+\frac {13}{4} x^{3}-5 x^{4}+\frac {121}{16} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {3}{4} x^{2}+\frac {3}{2} x^{3}-\frac {39}{16} x^{4}+\frac {15}{4} x^{5}\right ) y \left (0\right )+\left (x -2 x^{2}+\frac {13}{4} x^{3}-5 x^{4}+\frac {121}{16} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 1273

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime } x +\left (2 x^{2}+3\right ) y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = -2] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 1-2 x -\frac {3 x^{2}}{2}+\frac {5 x^{3}}{3}+\frac {17 x^{4}}{24}-\frac {11 x^{5}}{20}-\frac {23 x^{6}}{144}+O\left (x^{6}\right ) \] Verified OK. \[ y = 1-\frac {3 x^{2}}{2}+\frac {17 x^{4}}{24}-2 x +\frac {5 x^{3}}{3}-\frac {11 x^{5}}{20}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1-2 x -\frac {3}{2} x^{2}+\frac {5}{3} x^{3}+\frac {17}{24} x^{4}-\frac {11}{20} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 1274

ODE	\[ \boxed {y^{\prime \prime }-3 y^{\prime } x +\left (2 x^{2}+5\right ) y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = -2] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 1-2 x -\frac {5 x^{2}}{2}+\frac {2 x^{3}}{3}-\frac {3 x^{4}}{8}+\frac {x^{5}}{3}+\frac {19 x^{6}}{240}+O\left (x^{6}\right ) \] Verified OK. \[ y = 1-\frac {5 x^{2}}{2}-\frac {3 x^{4}}{8}-2 x +\frac {2 x^{3}}{3}+\frac {x^{5}}{3}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1-2 x -\frac {5}{2} x^{2}+\frac {2}{3} x^{3}-\frac {3}{8} x^{4}+\frac {1}{3} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 1275

ODE	\[ \boxed {y^{\prime \prime }+5 y^{\prime } x -\left (-x^{2}+3\right ) y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 6, y^{\prime }\left (0\right ) = -2] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 9 x^{2}-2 x +6+\frac {2 x^{3}}{3}-\frac {23 x^{4}}{4}-\frac {3 x^{5}}{10}+\frac {71 x^{6}}{24}+O\left (x^{6}\right ) \] Verified OK. \[ y = 6+9 x^{2}-\frac {23 x^{4}}{4}-2 x +\frac {2 x^{3}}{3}-\frac {3 x^{5}}{10}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 6-2 x +9 x^{2}+\frac {2}{3} x^{3}-\frac {23}{4} x^{4}-\frac {3}{10} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 1276

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime } x -\left (3 x^{2}+2\right ) y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = -5] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 2 x^{2}-5 x +2-\frac {10 x^{3}}{3}+\frac {3 x^{4}}{2}-\frac {25 x^{5}}{12}+\frac {7 x^{6}}{10}+O\left (x^{6}\right ) \] Verified OK. \[ y = 2+2 x^{2}+\frac {3 x^{4}}{2}-5 x -\frac {10 x^{3}}{3}-\frac {25 x^{5}}{12}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2-5 x +2 x^{2}-\frac {10}{3} x^{3}+\frac {3}{2} x^{4}-\frac {25}{12} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 1277

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime } x +\left (4 x^{2}+2\right ) y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = 6] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = x^{4}-5 x^{3}-3 x^{2}+6 x +3+\frac {31 x^{5}}{20}-\frac {x^{6}}{15}+O\left (x^{6}\right ) \] Verified OK. \[ y = x^{4}-3 x^{2}+3+6 x -5 x^{3}+\frac {31 x^{5}}{20}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 3+6 x -3 x^{2}-5 x^{3}+x^{4}+\frac {31}{20} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 1278

ODE	\[ \boxed {2 y^{\prime \prime }+5 y^{\prime } x +\left (2 x^{2}+4\right ) y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = -2] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = -3 x^{2}-2 x +3+\frac {3 x^{3}}{2}+\frac {3 x^{4}}{2}-\frac {49 x^{5}}{80}-\frac {x^{6}}{2}+O\left (x^{6}\right ) \] Verified OK. \[ y = 3-3 x^{2}+\frac {3 x^{4}}{2}-2 x +\frac {3 x^{3}}{2}-\frac {49 x^{5}}{80}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 3-2 x -3 x^{2}+\frac {3}{2} x^{3}+\frac {3}{2} x^{4}-\frac {49}{80} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 1279

ODE	\[ \boxed {3 y^{\prime \prime }+2 y^{\prime } x +\left (-x^{2}+4\right ) y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -2, y^{\prime }\left (0\right ) = 3] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 3 x -2+\frac {4 x^{2}}{3}-x^{3}-\frac {19 x^{4}}{54}+\frac {13 x^{5}}{60}+\frac {5 x^{6}}{81}+O\left (x^{6}\right ) \] Verified OK. \[ y = -2+\frac {4 x^{2}}{3}-\frac {19 x^{4}}{54}+3 x -x^{3}+\frac {13 x^{5}}{60}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -2+3 x +\frac {4}{3} x^{2}-x^{3}-\frac {19}{54} x^{4}+\frac {13}{60} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 1280

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

Problem 1281

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

Problem 1282

ODE	\[ \boxed {\left (x +1\right ) y^{\prime \prime }+y^{\prime } x^{2}+\left (1+2 x \right ) y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -2, y^{\prime }\left (0\right ) = 3] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = x^{2}+3 x -2-\frac {x^{3}}{6}-\frac {3 x^{4}}{4}+\frac {31 x^{5}}{120}-\frac {43 x^{6}}{360}+O\left (x^{6}\right ) \] Verified OK. \[ y = -2+x^{2}-\frac {x^{3}}{6}-\frac {3 x^{4}}{4}+\frac {31 x^{5}}{120}+3 x +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -2+3 x +x^{2}-\frac {1}{6} x^{3}-\frac {3}{4} x^{4}+\frac {31}{120} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 1283

ODE	\[ \boxed {y^{\prime \prime }+\left (x^{2}+2 x +1\right ) y^{\prime }+2 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 3] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 3 x +2-\frac {7 x^{2}}{2}-\frac {5 x^{3}}{6}+\frac {41 x^{4}}{24}+\frac {41 x^{5}}{120}-\frac {391 x^{6}}{720}+O\left (x^{6}\right ) \] Verified OK. \[ y = 2-\frac {7 x^{2}}{2}-\frac {5 x^{3}}{6}+\frac {41 x^{4}}{24}+\frac {41 x^{5}}{120}+3 x +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2+3 x -\frac {7}{2} x^{2}-\frac {5}{6} x^{3}+\frac {41}{24} x^{4}+\frac {41}{120} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 1284

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+\left (x^{2}+2\right ) y^{\prime }+y x=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -3, y^{\prime }\left (0\right ) = 5] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = -5 x^{2}+5 x -3+\frac {23 x^{3}}{6}-\frac {23 x^{4}}{12}+\frac {11 x^{5}}{30}+\frac {2 x^{6}}{15}+O\left (x^{6}\right ) \] Verified OK. \[ y = -3+\frac {23 x^{3}}{6}-\frac {23 x^{4}}{12}+\frac {11 x^{5}}{30}+5 x -5 x^{2}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -3+5 x -5 x^{2}+\frac {23}{6} x^{3}-\frac {23}{12} x^{4}+\frac {11}{30} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 1285

ODE	\[ \boxed {\left (x +1\right ) y^{\prime \prime }+\left (2 x^{2}-3 x +1\right ) y^{\prime }-\left (x -4\right ) y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = -2, y^{\prime }\left (1\right ) = 3] \end {align*} With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = -5+3 x +\frac {3 \left (x -1\right )^{2}}{2}-\frac {17 \left (x -1\right )^{3}}{12}-\frac {\left (x -1\right )^{4}}{12}+\frac {\left (x -1\right )^{5}}{8}+\frac {31 \left (x -1\right )^{6}}{360}+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -2+3 \left (x -1\right )+\frac {3}{2} \left (x -1\right )^{2}-\frac {17}{12} \left (x -1\right )^{3}-\frac {1}{12} \left (x -1\right )^{4}+\frac {1}{8} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \]

Problem 1286

ODE	\[ \boxed {y^{\prime \prime }+\left (3 x^{2}+12 x +13\right ) y^{\prime }+\left (2 x +5\right ) y=0} \] With initial conditions \begin {align*} [y \left (-2\right ) = 2, y^{\prime }\left (-2\right ) = -3] \end {align*} With the expansion point for the power series method at \(x = -2\).
program solution	\[ y = -4-3 x +\frac {\left (2+x \right )^{2}}{2}-\frac {\left (2+x \right )^{3}}{3}+\frac {31 \left (2+x \right )^{4}}{24}-\frac {53 \left (2+x \right )^{5}}{120}+\frac {11 \left (2+x \right )^{6}}{72}+O\left (\left (2+x \right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2-3 \left (2+x \right )+\frac {1}{2} \left (2+x \right )^{2}-\frac {1}{3} \left (2+x \right )^{3}+\frac {31}{24} \left (2+x \right )^{4}-\frac {53}{120} \left (2+x \right )^{5}+\operatorname {O}\left (\left (2+x \right )^{6}\right ) \]

Problem 1287

ODE	\[ \boxed {\left (3 x^{2}+2 x +1\right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }+\left (x +1\right ) y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = -2] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 1-2 x +\frac {3 x^{2}}{2}-\frac {11 x^{3}}{6}+\frac {15 x^{4}}{8}-\frac {71 x^{5}}{60}-\frac {37 x^{6}}{80}+O\left (x^{6}\right ) \] Verified OK. \[ y = 1+\frac {3 x^{2}}{2}-\frac {11 x^{3}}{6}+\frac {15 x^{4}}{8}-\frac {71 x^{5}}{60}-2 x +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1-2 x +\frac {3}{2} x^{2}-\frac {11}{6} x^{3}+\frac {15}{8} x^{4}-\frac {71}{60} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 1288

ODE	\[ \boxed {\left (x^{2}+4 x +3\right ) y^{\prime \prime }-\left (-x^{2}+4 x +5\right ) y^{\prime }-\left (2+x \right ) y=0} \] With initial conditions \begin {align*} [y \left (-2\right ) = 2, y^{\prime }\left (-2\right ) = -1] \end {align*} With the expansion point for the power series method at \(x = -2\).
program solution	\[ y = -x -\frac {7 \left (2+x \right )^{2}}{2}-\frac {43 \left (2+x \right )^{3}}{6}-\frac {203 \left (2+x \right )^{4}}{24}-\frac {167 \left (2+x \right )^{5}}{30}-\frac {4 \left (2+x \right )^{6}}{3}+O\left (\left (2+x \right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2-\left (2+x \right )-\frac {7}{2} \left (2+x \right )^{2}-\frac {43}{6} \left (2+x \right )^{3}-\frac {203}{24} \left (2+x \right )^{4}-\frac {167}{30} \left (2+x \right )^{5}+\operatorname {O}\left (\left (2+x \right )^{6}\right ) \]

Problem 1289

ODE	\[ \boxed {\left (x^{2}+2 x +1\right ) y^{\prime \prime }+\left (1-x \right ) y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = -1] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = -x^{2}-x +2+\frac {7 x^{3}}{6}-x^{4}+\frac {89 x^{5}}{120}-\frac {31 x^{6}}{60}+O\left (x^{6}\right ) \] Verified OK. \[ y = 2-x^{2}+\frac {7 x^{3}}{6}-x^{4}+\frac {89 x^{5}}{120}-x +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2-x -x^{2}+\frac {7}{6} x^{3}-x^{4}+\frac {89}{120} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 1290

ODE	\[ \boxed {\left (-2 x^{2}+x \right ) y^{\prime \prime }+\left (-x^{2}+3 x +1\right ) y^{\prime }+\left (2+x \right ) y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1, y^{\prime }\left (1\right ) = 0] \end {align*} With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = 1+\frac {3 \left (x -1\right )^{2}}{2}+\frac {\left (x -1\right )^{3}}{6}-\frac {\left (x -1\right )^{5}}{8}+\frac {127 \left (x -1\right )^{6}}{720}+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1+\frac {3}{2} \left (x -1\right )^{2}+\frac {1}{6} \left (x -1\right )^{3}-\frac {1}{8} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \]

Problem 1291

ODE	\[ \boxed {\left (2 x^{2}-11 x +16\right ) y^{\prime \prime }+\left (x^{2}-6 x +10\right ) y^{\prime }-\left (2-x \right ) y=0} \] With initial conditions \begin {align*} [y \left (3\right ) = 1, y^{\prime }\left (3\right ) = -2] \end {align*} With the expansion point for the power series method at \(x = 3\).
program solution	\[ y = 7+\frac {\left (x -3\right )^{2}}{2}-2 x -\frac {\left (x -3\right )^{3}}{6}+\frac {\left (x -3\right )^{4}}{4}-\frac {\left (x -3\right )^{5}}{6}-\frac {17 \left (x -3\right )^{6}}{360}+O\left (\left (x -3\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1-2 \left (x -3\right )+\frac {1}{2} \left (x -3\right )^{2}-\frac {1}{6} \left (x -3\right )^{3}+\frac {1}{4} \left (x -3\right )^{4}-\frac {1}{6} \left (x -3\right )^{5}+\operatorname {O}\left (\left (x -3\right )^{6}\right ) \]

Problem 1292

ODE	\[ \boxed {2 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+x \left (11 x^{2}+11 x +9\right ) y^{\prime }+\left (7 x^{2}+10 x +6\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1-\frac {x}{3}+\frac {2 x^{2}}{5}-\frac {5 x^{3}}{21}+\frac {7 x^{4}}{135}+\frac {76 x^{5}}{1155}+O\left (x^{6}\right )\right )}{x^{\frac {3}{2}}}+\frac {c_{2} \left (1+\frac {x^{2}}{2}-\frac {x^{3}}{3}+\frac {x^{4}}{8}+\frac {x^{5}}{30}+O\left (x^{6}\right )\right )}{x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1+\frac {1}{2} x^{2}-\frac {1}{3} x^{3}+\frac {1}{8} x^{4}+\frac {1}{30} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}}+\frac {c_{2} \left (1-\frac {1}{3} x +\frac {2}{5} x^{2}-\frac {5}{21} x^{3}+\frac {7}{135} x^{4}+\frac {76}{1155} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {3}{2}}} \]

Problem 1293

ODE	\[ \boxed {x^{2} \left (x +3\right ) y^{\prime \prime }+5 x \left (x +1\right ) y^{\prime }-\left (-4 x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1-\frac {7 x}{9}+\frac {35 x^{2}}{81}-\frac {455 x^{3}}{2187}+\frac {1820 x^{4}}{19683}-\frac {6916 x^{5}}{177147}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+x -x^{2}+\frac {3 x^{3}}{5}-\frac {3 x^{4}}{10}+\frac {3 x^{5}}{22}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {4}{3}} \left (1-\frac {7}{9} x +\frac {35}{81} x^{2}-\frac {455}{2187} x^{3}+\frac {1820}{19683} x^{4}-\frac {6916}{177147} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1+x -x^{2}+\frac {3}{5} x^{3}-\frac {3}{10} x^{4}+\frac {3}{22} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 1294

ODE	\[ \boxed {x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }-x \left (4 x^{2}+3\right ) y^{\prime }+\left (2+2 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-\frac {2 x}{5}+\frac {27 x^{2}}{35}-\frac {34 x^{3}}{105}+\frac {584 x^{4}}{1155}-\frac {768 x^{5}}{3575}+O\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1+2 x -\frac {9 x^{2}}{8}+\frac {7 x^{3}}{4}-\frac {607 x^{4}}{640}+\frac {13347 x^{5}}{11200}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+2 x -\frac {9}{8} x^{2}+\frac {7}{4} x^{3}-\frac {607}{640} x^{4}+\frac {13347}{11200} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{2} \left (1-\frac {2}{5} x +\frac {27}{35} x^{2}-\frac {34}{105} x^{3}+\frac {584}{1155} x^{4}-\frac {768}{3575} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 1295

ODE	\[ \boxed {2 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+x \left (5 x^{2}+3 x +3\right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{-\frac {1}{4}+\frac {i \sqrt {7}}{4}} \left (1+\frac {x}{i \sqrt {7}+2}-\frac {5 \left (-1+i \sqrt {7}\right ) \left (i \sqrt {7}+\frac {9}{5}\right ) x^{2}}{16 \left (i \sqrt {7}+2\right ) \left (i \sqrt {7}+4\right )}+\frac {\left (49 i \sqrt {7}+89\right ) x^{3}}{432-444 i \sqrt {7}}+\frac {\left (1553 i+395 \sqrt {7}\right ) x^{4}}{26256 i+12480 \sqrt {7}}+\frac {\left (42423 i \sqrt {7}+45275\right ) x^{5}}{492720 i \sqrt {7}-1749600}+O\left (x^{6}\right )\right )+c_{2} x^{-\frac {i \sqrt {7}}{4}-\frac {1}{4}} \left (1+\frac {x}{-i \sqrt {7}+2}-\frac {5 \left (-i \sqrt {7}-1\right ) \left (-i \sqrt {7}+\frac {9}{5}\right ) x^{2}}{16 \left (-i \sqrt {7}+2\right ) \left (-i \sqrt {7}+4\right )}+\frac {\left (-49 i \sqrt {7}+89\right ) x^{3}}{432+444 i \sqrt {7}}+\frac {\left (-1553 i+395 \sqrt {7}\right ) x^{4}}{-26256 i+12480 \sqrt {7}}+\frac {\left (-42423 i \sqrt {7}+45275\right ) x^{5}}{-492720 i \sqrt {7}-1749600}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {i \sqrt {7}}{4}} \left (1+\frac {1}{2+i \sqrt {7}} x +\frac {1}{4} \frac {11-i \sqrt {7}}{\left (2+i \sqrt {7}\right ) \left (i \sqrt {7}+4\right )} x^{2}-\frac {1}{12} \frac {49 i \sqrt {7}+89}{\left (2+i \sqrt {7}\right ) \left (i \sqrt {7}+4\right ) \left (i \sqrt {7}+6\right )} x^{3}+\frac {1}{48} \frac {395 i \sqrt {7}-1553}{\left (2+i \sqrt {7}\right ) \left (i \sqrt {7}+4\right ) \left (i \sqrt {7}+6\right ) \left (i \sqrt {7}+8\right )} x^{4}+\frac {1}{240} \frac {42423 i \sqrt {7}+45275}{\left (2+i \sqrt {7}\right ) \left (i \sqrt {7}+4\right ) \left (i \sqrt {7}+6\right ) \left (i \sqrt {7}+8\right ) \left (i \sqrt {7}+10\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} x^{-\frac {i \sqrt {7}}{4}} \left (1+\frac {1}{2-i \sqrt {7}} x +\frac {-11-i \sqrt {7}}{-4+24 i \sqrt {7}} x^{2}+\frac {49 \sqrt {7}+89 i}{432 i-444 \sqrt {7}} x^{3}-\frac {1}{48} \frac {395 i \sqrt {7}+1553}{\left (2 i+\sqrt {7}\right ) \left (\sqrt {7}+4 i\right ) \left (\sqrt {7}+6 i\right ) \left (\sqrt {7}+8 i\right )} x^{4}+\frac {-42423 \sqrt {7}-45275 i}{1749600 i-492720 \sqrt {7}} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {1}{4}}} \]

Problem 1296

ODE	\[ \boxed {3 x^{2} y^{\prime \prime }+2 x \left (-2 x^{2}+x +1\right ) y^{\prime }+\left (-8 x^{2}+2 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1-\frac {2 x}{3}+\frac {8 x^{2}}{9}-\frac {40 x^{3}}{81}+\frac {92 x^{4}}{243}-\frac {664 x^{5}}{3645}+O\left (x^{6}\right )\right )+c_{2} \left (1-x +\frac {6 x^{2}}{5}-\frac {4 x^{3}}{5}+\frac {32 x^{4}}{55}-\frac {24 x^{5}}{77}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{3}} \left (1-\frac {2}{3} x +\frac {8}{9} x^{2}-\frac {40}{81} x^{3}+\frac {92}{243} x^{4}-\frac {664}{3645} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-x +\frac {6}{5} x^{2}-\frac {4}{5} x^{3}+\frac {32}{55} x^{4}-\frac {24}{77} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 1297

ODE	\[ \boxed {x^{2} \left (x^{2}+3 x +3\right ) y^{\prime \prime }+x \left (7 x^{2}+8 x +5\right ) y^{\prime }-\left (-9 x^{2}-2 x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1-\frac {4 x}{7}-\frac {7 x^{2}}{45}+\frac {970 x^{3}}{2457}-\frac {5707 x^{4}}{22680}+\frac {13568 x^{5}}{300105}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-x^{2}+\frac {2 x^{3}}{3}-\frac {10 x^{5}}{33}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {4}{3}} \left (1-\frac {4}{7} x -\frac {7}{45} x^{2}+\frac {970}{2457} x^{3}-\frac {5707}{22680} x^{4}+\frac {13568}{300105} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1-x^{2}+\frac {2}{3} x^{3}-\frac {10}{33} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 1298

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+x \left (4 x^{2}+2 x +7\right ) y^{\prime }-\left (-7 x^{2}-4 x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{4}} \left (1-\frac {x}{2}-\frac {19 x^{2}}{104}+\frac {1571 x^{3}}{10608}+\frac {3225 x^{4}}{198016}-\frac {752183 x^{5}}{29702400}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+2 x -\frac {11 x^{2}}{6}-\frac {x^{3}}{7}+\frac {895 x^{4}}{1848}-\frac {499 x^{5}}{13860}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {5}{4}} \left (1-\frac {1}{2} x -\frac {19}{104} x^{2}+\frac {1571}{10608} x^{3}+\frac {3225}{198016} x^{4}-\frac {752183}{29702400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1+2 x -\frac {11}{6} x^{2}-\frac {1}{7} x^{3}+\frac {895}{1848} x^{4}-\frac {499}{13860} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 1299

ODE	\[ \boxed {12 x^{2} \left (x +1\right ) y^{\prime \prime }+x \left (3 x^{2}+35 x +11\right ) y^{\prime }-\left (-5 x^{2}-10 x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1-x +\frac {28 x^{2}}{31}-\frac {1111 x^{3}}{1333}+\frac {57493 x^{4}}{73315}-\frac {3668716 x^{5}}{4912105}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-x +\frac {7 x^{2}}{8}-\frac {19 x^{3}}{24}+\frac {283 x^{4}}{384}-\frac {1339 x^{5}}{1920}+O\left (x^{6}\right )\right )}{x^{\frac {1}{4}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-x +\frac {7}{8} x^{2}-\frac {19}{24} x^{3}+\frac {283}{384} x^{4}-\frac {1339}{1920} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {1}{4}}}+c_{2} x^{\frac {1}{3}} \left (1-x +\frac {28}{31} x^{2}-\frac {1111}{1333} x^{3}+\frac {57493}{73315} x^{4}-\frac {3668716}{4912105} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 1300

ODE	\[ \boxed {x^{2} \left (10 x^{2}+x +5\right ) y^{\prime \prime }+x \left (48 x^{2}+3 x +4\right ) y^{\prime }+\left (36 x^{2}+x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{5}} \left (1-\frac {6 x}{25}-\frac {1217 x^{2}}{625}+\frac {41972 x^{3}}{46875}+\frac {1447799 x^{4}}{390625}-\frac {375253322 x^{5}}{146484375}+O\left (x^{6}\right )\right )+c_{2} \left (1-\frac {x}{4}-\frac {35 x^{2}}{18}+\frac {11 x^{3}}{12}+\frac {632 x^{4}}{171}-\frac {2671 x^{5}}{1026}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{5}} \left (1-\frac {6}{25} x -\frac {1217}{625} x^{2}+\frac {41972}{46875} x^{3}+\frac {1447799}{390625} x^{4}-\frac {375253322}{146484375} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-\frac {1}{4} x -\frac {35}{18} x^{2}+\frac {11}{12} x^{3}+\frac {632}{171} x^{4}-\frac {2671}{1026} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]