2.17.5 Problems 401 to 500

Problem 401

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

Problem 402

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

Problem 403

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

Problem 404

ODE	\[ \boxed {y^{\prime \prime }+9 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {9}{2} x^{2}+\frac {27}{8} x^{4}-\frac {81}{80} x^{6}\right ) y \left (0\right )+\left (x -\frac {3}{2} x^{3}+\frac {27}{40} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {9}{2} x^{2}+\frac {27}{8} x^{4}\right ) c_{1} +\left (x -\frac {3}{2} x^{3}+\frac {27}{40} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {9}{2} x^{2}+\frac {27}{8} x^{4}\right ) y \left (0\right )+\left (x -\frac {3}{2} x^{3}+\frac {27}{40} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 405

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

Problem 406

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

Problem 407

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

Problem 408

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

Problem 409

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

Problem 410

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

Problem 411

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

Problem 412

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }+y=0} \] 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	\[ y = x^{2}+x +\frac {x^{3}}{2}+\frac {x^{4}}{6}+\frac {x^{5}}{24}+\frac {x^{6}}{120}+O\left (x^{6}\right ) \] Verified OK. \[ y = x^{2}+x +\frac {x^{3}}{2}+\frac {x^{4}}{6}+\frac {x^{5}}{24}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x +x^{2}+\frac {1}{2} x^{3}+\frac {1}{6} x^{4}+\frac {1}{24} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 413

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }-2 y=0} \] 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	\[ y = 2 x^{2}-2 x +1-\frac {4 x^{3}}{3}+\frac {2 x^{4}}{3}-\frac {4 x^{5}}{15}+\frac {4 x^{6}}{45}+O\left (x^{6}\right ) \] Verified OK. \[ y = 1+2 x^{2}-\frac {4 x^{3}}{3}+\frac {2 x^{4}}{3}-\frac {4 x^{5}}{15}-2 x +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1-2 x +2 x^{2}-\frac {4}{3} x^{3}+\frac {2}{3} x^{4}-\frac {4}{15} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 414

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x^{2}+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{2}+\frac {i \sqrt {3}}{2}} \left (1-\frac {x}{2}+\frac {\left (i \sqrt {3}+3\right ) x^{2}}{16+8 i \sqrt {3}}+\frac {\left (-i \sqrt {3}-5\right ) x^{3}}{48 i \sqrt {3}+96}+\frac {\left (i \sqrt {3}+5\right ) \left (i \sqrt {3}+7\right ) x^{4}}{384 \left (i \sqrt {3}+4\right ) \left (2+i \sqrt {3}\right )}-\frac {\left (i \sqrt {3}+7\right ) \left (i \sqrt {3}+9\right ) x^{5}}{3840 \left (i \sqrt {3}+4\right ) \left (2+i \sqrt {3}\right )}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{2}-\frac {i \sqrt {3}}{2}} \left (1-\frac {x}{2}+\frac {\left (-i \sqrt {3}+3\right ) x^{2}}{16-8 i \sqrt {3}}+\frac {\left (i \sqrt {3}-5\right ) x^{3}}{-48 i \sqrt {3}+96}+\frac {\left (-i \sqrt {3}+5\right ) \left (-i \sqrt {3}+7\right ) x^{4}}{384 \left (-i \sqrt {3}+4\right ) \left (2-i \sqrt {3}\right )}-\frac {\left (-i \sqrt {3}+7\right ) \left (-i \sqrt {3}+9\right ) x^{5}}{3840 \left (-i \sqrt {3}+4\right ) \left (2-i \sqrt {3}\right )}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (c_{2} x^{\frac {i \sqrt {3}}{2}} \left (1-\frac {1}{2} x +\frac {i \sqrt {3}+3}{8 i \sqrt {3}+16} x^{2}+\frac {-i \sqrt {3}-5}{48 i \sqrt {3}+96} x^{3}+\frac {1}{384} \frac {\left (i \sqrt {3}+5\right ) \left (i \sqrt {3}+7\right )}{\left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+2\right )} x^{4}-\frac {1}{3840} \frac {\left (i \sqrt {3}+7\right ) \left (i \sqrt {3}+9\right )}{\left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+2\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} x^{-\frac {i \sqrt {3}}{2}} \left (1-\frac {1}{2} x +\frac {\sqrt {3}+3 i}{8 \sqrt {3}+16 i} x^{2}+\frac {-\sqrt {3}-5 i}{48 \sqrt {3}+96 i} x^{3}+\frac {3 i \sqrt {3}-8}{576 i \sqrt {3}-480} x^{4}-\frac {1}{3840} \frac {\left (\sqrt {3}+7 i\right ) \left (\sqrt {3}+9 i\right )}{\left (\sqrt {3}+4 i\right ) \left (\sqrt {3}+2 i\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

Problem 415

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

Problem 416

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 417

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

Problem 418

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

Problem 419

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

Problem 420

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

Problem 421

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

Problem 422

ODE	\[ \boxed {\left (x^{2}+3\right ) y^{\prime \prime }-7 y^{\prime } x +16 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {8}{3} x^{2}+\frac {8}{27} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{3}+\frac {1}{120} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {8}{3} x^{2}+\frac {8}{27} x^{4}\right ) c_{1} +\left (x -\frac {1}{2} x^{3}+\frac {1}{120} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {8}{3} x^{2}+\frac {8}{27} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{3}+\frac {1}{120} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 423

ODE	\[ \boxed {\left (-x^{2}+2\right ) y^{\prime \prime }-y^{\prime } x +16 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (2 x^{4}-4 x^{2}+1\right ) y \left (0\right )+\left (x -\frac {5}{4} x^{3}+\frac {7}{32} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (2 x^{4}-4 x^{2}+1\right ) c_{1} +\left (x -\frac {5}{4} x^{3}+\frac {7}{32} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (2 x^{4}-4 x^{2}+1\right ) y \left (0\right )+\left (x -\frac {5}{4} x^{3}+\frac {7}{32} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 424

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 425

ODE	\[ \boxed {3 y^{\prime \prime }+x y^{\prime }-4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {2}{3} x^{2}+\frac {1}{27} x^{4}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{360} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {2}{3} x^{2}+\frac {1}{27} x^{4}\right ) c_{1} +\left (x +\frac {1}{6} x^{3}+\frac {1}{360} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {2}{3} x^{2}+\frac {1}{27} x^{4}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{360} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 426

ODE	\[ \boxed {5 y^{\prime \prime }-2 y^{\prime } x +10 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-x^{2}+\frac {1}{10} x^{4}-\frac {1}{750} x^{6}\right ) y \left (0\right )+\left (x -\frac {4}{15} x^{3}+\frac {4}{375} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-x^{2}+\frac {1}{10} x^{4}\right ) c_{1} +\left (x -\frac {4}{15} x^{3}+\frac {4}{375} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-x^{2}+\frac {1}{10} x^{4}\right ) y \left (0\right )+\left (\frac {4}{375} x^{5}-\frac {4}{15} x^{3}+x \right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 427

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

Problem 428

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

Problem 429

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 430

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 431

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+2 y^{\prime } x -2 y=0} \] 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	\[ y = x +O\left (x^{6}\right ) \] Verified OK. \[ y = x +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x \]

Problem 432

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime } x -2 y=0} \] 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	\[ y = x^{2}+1+O\left (x^{6}\right ) \] Verified OK. \[ y = x^{2}+1+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{2}+1 \]

Problem 433

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

Problem 434

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

Problem 435

ODE	\[ \boxed {\left (x^{2}-6 x +10\right ) y^{\prime \prime }-4 \left (x -3\right ) y^{\prime }+6 y=0} \] With initial conditions \begin {align*} [y \left (3\right ) = 2, y^{\prime }\left (3\right ) = 0] \end {align*} With the expansion point for the power series method at \(x = 3\).
program solution	\[ y = -6 \left (x -3\right )^{2}+2+O\left (\left (x -3\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -6 x^{2}+36 x -52 \]

Problem 436

ODE	\[ \boxed {\left (4 x^{2}+16 x +17\right ) y^{\prime \prime }-8 y=0} \] With initial conditions \begin {align*} [y \left (-2\right ) = 1, y^{\prime }\left (-2\right ) = 0] \end {align*} With the expansion point for the power series method at \(x = -2\).
program solution	\[ y = 4 \left (2+x \right )^{2}+1+O\left (\left (2+x \right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 4 x^{2}+16 x +17 \]

Problem 437

ODE	\[ \boxed {\left (x^{2}+6 x \right ) y^{\prime \prime }+\left (3 x +9\right ) y^{\prime }-3 y=0} \] With initial conditions \begin {align*} [y \left (-3\right ) = 1, y^{\prime }\left (-3\right ) = 0] \end {align*} With the expansion point for the power series method at \(x = -3\).
program solution	\[ y = 1-\frac {\left (x +3\right )^{2}}{6}-\frac {5 \left (x +3\right )^{4}}{648}-\frac {7 \left (x +3\right )^{6}}{11664}+O\left (\left (x +3\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1-\frac {1}{6} \left (x +3\right )^{2}-\frac {5}{648} \left (x +3\right )^{4}+\operatorname {O}\left (\left (x +3\right )^{6}\right ) \]

Problem 438

ODE	\[ \boxed {y^{\prime \prime }+\left (x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{30} x^{5}+\frac {1}{240} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}-\frac {1}{12} x^{4}+\frac {1}{120} x^{5}+\frac {1}{120} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{30} x^{5}\right ) c_{1} +\left (x -\frac {1}{6} x^{3}-\frac {1}{12} x^{4}+\frac {1}{120} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{30} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}-\frac {1}{12} x^{4}+\frac {1}{120} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 439

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

Problem 440

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

Problem 441

ODE	\[ \boxed {\left (x^{3}+1\right ) y^{\prime \prime }+y x^{4}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {x^{6}}{30}\right ) y \left (0\right )+x y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] 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 442

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime } x +y \left (2 x^{2}+1\right )=0} \] 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	\[ y = 1-x -\frac {x^{2}}{2}+\frac {x^{3}}{3}-\frac {x^{4}}{24}+\frac {x^{5}}{30}+\frac {29 x^{6}}{720}+O\left (x^{6}\right ) \] Verified OK. \[ y = 1-\frac {x^{2}}{2}-\frac {x^{4}}{24}-x +\frac {x^{3}}{3}+\frac {x^{5}}{30}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1-x -\frac {1}{2} x^{2}+\frac {1}{3} x^{3}-\frac {1}{24} x^{4}+\frac {1}{30} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 443

ODE	\[ \boxed {y^{\prime \prime }+{\mathrm e}^{-x} y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{40} x^{5}+\frac {1}{80} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {1}{12} x^{4}-\frac {1}{60} x^{5}-\frac {1}{360} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{40} x^{5}\right ) c_{1} +\left (x -\frac {1}{6} x^{3}+\frac {1}{12} x^{4}-\frac {1}{60} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{40} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {1}{12} x^{4}-\frac {1}{60} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 444

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

Problem 445

ODE	\[ \boxed {x y^{\prime \prime }+\sin \left (x \right ) y^{\prime }+y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{60} x^{5}+\frac {1}{180} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{18} x^{4}-\frac {7}{360} x^{5}+\frac {1}{900} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{60} x^{5}\right ) c_{1} +\left (x -\frac {1}{2} x^{2}+\frac {1}{18} x^{4}-\frac {7}{360} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{60} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{18} x^{4}-\frac {7}{360} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 446

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 447

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 448

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

Problem 449

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

Problem 450

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

Problem 451

ODE	\[ \boxed {\frac {y}{t}+y^{\prime }=3 \cos \left (2 t \right )} \]
program solution	\[ y = \frac {6 \sin \left (2 t \right ) t +3 \cos \left (2 t \right )+4 c_{1}}{4 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {4 c_{1} +6 \sin \left (2 t \right ) t +3 \cos \left (2 t \right )}{4 t} \]

Problem 452

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

Problem 453

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

Problem 454

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

Problem 455

ODE	\[ \boxed {4 y t +\left (t^{2}+1\right ) y^{\prime }=\frac {1}{\left (t^{2}+1\right )^{2}}} \]
program solution	\[ y t^{4}+2 y t^{2}-\arctan \left (t \right )+y = c_{1} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\arctan \left (t \right )+c_{1}}{\left (t^{2}+1\right )^{2}} \]

Problem 456

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

Problem 457

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

Problem 458

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

Problem 459

ODE	\[ \boxed {y+2 y^{\prime }=3 t^{2}} \]
program solution	\[ y = \frac {\left (6 t^{2} {\mathrm e}^{\frac {t}{2}}-24 t \,{\mathrm e}^{\frac {t}{2}}+48 \,{\mathrm e}^{\frac {t}{2}}+c_{1} \right ) {\mathrm e}^{-\frac {t}{2}}}{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = 3 t^{2}-12 t +24+{\mathrm e}^{-\frac {t}{2}} c_{1} \]

Problem 460

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

Problem 461

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

Problem 462

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

Problem 463

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

Problem 464

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

Problem 465

ODE	\[ \boxed {2 y+t y^{\prime }=\sin \left (t \right )} \] With initial conditions \begin {align*} \left [y \left (\frac {\pi }{2}\right ) = 1\right ] \end {align*}
program solution	\[ y = \frac {-4 \cos \left (t \right ) t +\pi ^{2}+4 \sin \left (t \right )-4}{4 t^{2}} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\sin \left (t \right )-\cos \left (t \right ) t +\frac {\pi ^{2}}{4}-1}{t^{2}} \]

Problem 466

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

Problem 467

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

Problem 468

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

Problem 469

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

Problem 470

ODE	\[ \boxed {-2 y+3 y^{\prime }={\mathrm e}^{-\frac {\pi t}{2}}} \] With initial conditions \begin {align*} [y \left (0\right ) = a] \end {align*}
program solution	\[ y = \frac {\left (\frac {9 \pi \left (3 \pi a +4 a +2\right )}{3 \pi +4}-6 \,{\mathrm e}^{-\frac {t \left (3 \pi +4\right )}{6}}+\frac {36 \pi a +48 a +24}{3 \pi +4}\right ) {\mathrm e}^{\frac {2 t}{3}}}{9 \pi +12} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (3 \pi a -2 \,{\mathrm e}^{t \left (-\frac {\pi }{2}-\frac {2}{3}\right )}+4 a +2\right ) {\mathrm e}^{\frac {2 t}{3}}}{3 \pi +4} \]

Problem 471

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

Problem 472

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

Problem 473

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

Problem 474

ODE	\[ \boxed {y^{\prime }+\frac {y}{2}=2 \cos \left (t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = -1] \end {align*}
program solution	\[ y = \frac {8 \sin \left (t \right )}{5}+\frac {4 \cos \left (t \right )}{5}-\frac {9 \,{\mathrm e}^{-\frac {t}{2}}}{5} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {4 \cos \left (t \right )}{5}+\frac {8 \sin \left (t \right )}{5}-\frac {9 \,{\mathrm e}^{-\frac {t}{2}}}{5} \]

Problem 475

ODE	\[ \boxed {\frac {2 y}{3}+y^{\prime }=-\frac {t}{2}+1} \]
program solution	\[ y = -\frac {\left (6 \,{\mathrm e}^{\frac {2 t}{3}} t -21 \,{\mathrm e}^{\frac {2 t}{3}}-8 c_{1} \right ) {\mathrm e}^{-\frac {2 t}{3}}}{8} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {3 t}{4}+\frac {21}{8}+{\mathrm e}^{-\frac {2 t}{3}} c_{1} \]

Problem 476

ODE	\[ \boxed {\frac {y}{4}+y^{\prime }=3+2 \cos \left (2 t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {64 \sin \left (2 t \right )}{65}+\frac {8 \cos \left (2 t \right )}{65}+12-\frac {788 \,{\mathrm e}^{-\frac {t}{4}}}{65} \] Verified OK.
Maple solution	\[ y \left (t \right ) = 12+\frac {8 \cos \left (2 t \right )}{65}+\frac {64 \sin \left (2 t \right )}{65}-\frac {788 \,{\mathrm e}^{-\frac {t}{4}}}{65} \]

Problem 477

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

Problem 478

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

Problem 479

ODE	\[ \boxed {y^{\prime }-\frac {x^{2}}{y}=0} \]
program solution	\[ -\frac {x^{3}}{3}+\frac {y^{2}}{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\sqrt {6 x^{3}+9 c_{1}}}{3} \\ y \left (x \right ) &= \frac {\sqrt {6 x^{3}+9 c_{1}}}{3} \\ \end{align*}

Problem 480

ODE	\[ \boxed {y^{\prime }-\frac {x^{2}}{\left (x^{3}+1\right ) y}=0} \]
program solution	\[ -\frac {\ln \left (x^{3}+1\right )}{3}+\frac {y^{2}}{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\sqrt {6 \ln \left (x^{3}+1\right )+9 c_{1}}}{3} \\ y \left (x \right ) &= \frac {\sqrt {6 \ln \left (x^{3}+1\right )+9 c_{1}}}{3} \\ \end{align*}

Problem 481

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

Problem 482

ODE	\[ \boxed {y^{\prime }-\frac {3 x^{2}-1}{3+2 y}=0} \]
program solution	\[ -x^{3}+y^{2}+3 y+x = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {3}{2}-\frac {\sqrt {4 x^{3}+4 c_{1} -4 x +9}}{2} \\ y \left (x \right ) &= -\frac {3}{2}+\frac {\sqrt {4 x^{3}+4 c_{1} -4 x +9}}{2} \\ \end{align*}

Problem 483

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

Problem 484

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

Problem 485

ODE	\[ \boxed {y^{\prime }-\frac {-{\mathrm e}^{-x}+x}{{\mathrm e}^{y}+x}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 486

ODE	\[ \boxed {y^{\prime }-\frac {x^{2}}{1+y^{2}}=0} \]
program solution	\[ -\frac {x^{3}}{3}+\frac {y^{3}}{3}+y = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {2}{3}}-4}{2 \left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {2}{3}}+4 i \sqrt {3}-4}{4 \left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {i \left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {2}{3}} \sqrt {3}+4 i \sqrt {3}-\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {2}{3}}+4}{4 \left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 487

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

Problem 488

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

Problem 489

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

Problem 490

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

Problem 491

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

Problem 492

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

Problem 493

ODE	\[ \boxed {y^{\prime }-\frac {2 x}{1+2 y}=0} \] With initial conditions \begin {align*} [y \left (2\right ) = 0] \end {align*}
program solution	\[ -\frac {x^{2}}{2}+\frac {y}{2}+\frac {y^{2}}{2} = -2 \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {1}{2}+\frac {\sqrt {4 x^{2}-15}}{2} \]

Problem 494

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

Problem 495

ODE	\[ \boxed {y^{\prime }-\frac {-{\mathrm e}^{x}+3 x^{2}}{-5+2 y}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
program solution	\[ x^{3}-y^{2}-{\mathrm e}^{x}+5 y = 3 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {5}{2}-\frac {\sqrt {13+4 x^{3}-4 \,{\mathrm e}^{x}}}{2} \]

Problem 496

ODE	\[ \boxed {y^{\prime }-\frac {-{\mathrm e}^{x}+{\mathrm e}^{-x}}{3+4 y}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
program solution	\[ -2 y^{2}-{\mathrm e}^{x}-3 y-{\mathrm e}^{-x} = -7 \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {3}{4}+\frac {\sqrt {{\mathrm e}^{x} \left (-8 \,{\mathrm e}^{2 x}+65 \,{\mathrm e}^{x}-8\right )}\, {\mathrm e}^{-x}}{4} \]

Problem 497

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

Problem 498

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

Problem 499

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

Problem 500

ODE	\[ \boxed {y^{\prime }-\frac {3 x^{2}}{-4+3 y^{2}}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 0] \end {align*}
program solution	\[ -\frac {x^{3}}{3}-\frac {4 y}{3}+\frac {y^{3}}{3} = -{\frac {1}{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\left (1+i \sqrt {3}\right ) \left (-108+108 x^{3}+12 \sqrt {81 x^{6}-162 x^{3}-687}\right )^{\frac {2}{3}}-48 i \sqrt {3}+48}{12 \left (-108+108 x^{3}+12 \sqrt {81 x^{6}-162 x^{3}-687}\right )^{\frac {1}{3}}} \]