2.17.55 Problems 5401 to 5500

Problem 5401

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

Problem 5402

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

Problem 5403

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

Problem 5404

ODE	\[ \boxed {y^{\prime \prime }-y=x \,{\mathrm e}^{3 x}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+\frac {c_{2} {\mathrm e}^{x}}{2}+\frac {x \,{\mathrm e}^{3 x}}{8}-\frac {3 \,{\mathrm e}^{3 x}}{32} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{2} {\mathrm e}^{-x}+{\mathrm e}^{x} c_{1} +\frac {\left (4 x -3\right ) {\mathrm e}^{3 x}}{32} \]

Problem 5405

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

Problem 5406

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

Problem 5407

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

Problem 5408

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

Problem 5409

ODE	\[ \boxed {x^{3} y^{\prime \prime \prime }+y^{\prime } x -y=3 x^{4}} \]
program solution	\[ y = x \left (c_{1} +c_{2} \ln \left (x \right )+c_{3} \ln \left (x \right )^{2}\right )+\frac {x^{4}}{9} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x \left (9 c_{3} \ln \left (x \right )^{2}+x^{3}+9 c_{2} \ln \left (x \right )+9 c_{1} \right )}{9} \]

Problem 5410

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

Problem 5411

ODE	\[ \boxed {\left (2 x +1\right )^{2} y^{\prime \prime }-2 \left (2 x +1\right ) y^{\prime }-12 y=6 x} \]
program solution	\[ y = \frac {512 c_{2} x^{4}+1024 c_{2} x^{3}+\left (768 c_{2} -24\right ) x^{2}+\left (256 c_{2} -8\right ) x -4 c_{1} +32 c_{2} -1}{32+64 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1}}{2 x +1}+\left (2 x +1\right )^{3} c_{2} +\frac {-24 x^{2}-8 x -1}{64 x +32} \]

Problem 5412

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

Problem 5413

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

Problem 5414

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

Problem 5415

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

Problem 5416

ODE	\[ \boxed {y^{\prime \prime }-2 \tan \left (x \right ) y^{\prime }-10 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-3 x} \sec \left (x \right )+\frac {c_{2} {\mathrm e}^{3 x} \sec \left (x \right )}{6} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sec \left (x \right ) \left (c_{1} \sinh \left (3 x \right )+c_{2} \cosh \left (3 x \right )\right ) \]

Problem 5417

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

Problem 5418

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

Problem 5419

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

Problem 5420

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

Problem 5421

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

Problem 5422

ODE	\[ \boxed {x^{8} y^{\prime \prime }+4 y^{\prime } x^{7}+y=\frac {1}{x^{3}}} \]
program solution	\[ y = c_{1} {\mathrm e}^{\frac {i}{3 x^{3}}}-\frac {i c_{2} {\mathrm e}^{-\frac {i}{3 x^{3}}}}{2}+\left (i {\mathrm e}^{-\frac {i}{3 x^{3}}}+{\mathrm e}^{\frac {i}{3 x^{3}}}\right ) \operatorname {undefined} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sin \left (\frac {1}{3 x^{3}}\right ) c_{2} +\cos \left (\frac {1}{3 x^{3}}\right ) c_{1} +\frac {1}{x^{3}} \]

Problem 5423

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

Problem 5424

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

Problem 5425

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

Problem 5426

ODE	\[ \boxed {x^{2} y^{\prime \prime }-4 y^{\prime } x +\left (9 x^{2}+6\right ) y=0} \]
program solution	\[ y = c_{1} x^{2} \cos \left (3 x \right )+\frac {c_{2} x^{2} \sin \left (3 x \right )}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{2} \left (c_{1} \sin \left (3 x \right )+c_{2} \cos \left (3 x \right )\right ) \]

Problem 5427

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

Problem 5428

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +2 y=\frac {-x^{2}+1}{x}} \]
program solution	\[ y = \frac {c_{1} \left (x^{2}+1\right )^{2}}{\left (i x +1\right )^{2}}+\frac {c_{2} \left (x^{2}+1\right )^{2} x}{\left (-x +i\right )^{2} \left (x +i\right )^{2}}+\frac {2 \left (\left (-\frac {1}{2} x^{2}+\frac {1}{2}+i x \right ) \left (\int _{0}^{x}\frac {\left (\alpha ^{2}-1\right ) \left (\alpha +i\right )^{3} \left (i-\alpha \right )^{4}}{\left (-\alpha ^{3}+3 i \alpha ^{2}+3 \alpha -i\right ) \alpha \left (\alpha ^{2}+1\right )^{3}}d \alpha \right )+\frac {x^{2}}{2}+\frac {1}{2}\right ) \left (x^{2}+1\right )^{2} x}{\left (-x +i\right )^{4} \left (x +i\right )^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2}+\ln \left (x \right ) x +c_{2} x -c_{1} +x \]

Problem 5429

ODE	\[ \boxed {y^{\prime \prime }+{y^{\prime }}^{2}=-1} \]
program solution	\[ \int _{}^{y}\frac {1}{\sqrt {-1+\frac {{\mathrm e}^{-2 \textit {\_a}}}{c_{1}^{2}}}}d \textit {\_a} = x +c_{2} \] Verified OK. \[ -\frac {{\mathrm e}^{-y} \sqrt {-\frac {{\mathrm e}^{2 y} c_{1}^{2}-1}{c_{1}^{2}}}\, \arctan \left (\frac {{\mathrm e}^{y}}{\sqrt {-\frac {{\mathrm e}^{2 y} c_{1}^{2}-1}{c_{1}^{2}}}}\right )}{\sqrt {-\frac {\left ({\mathrm e}^{2 y} c_{1}^{2}-1\right ) {\mathrm e}^{-2 y}}{c_{1}^{2}}}} = x +c_{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \ln \left (\cos \left (x \right ) c_{2} -c_{1} \sin \left (x \right )\right ) \]

Problem 5430

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

Problem 5431

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

Problem 5432

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

Problem 5433

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

Problem 5434

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

Problem 5435

ODE	\[ \boxed {y y^{\prime \prime }-{y^{\prime }}^{2} \left (1-y^{\prime } \cos \left (y\right )+y y^{\prime } \sin \left (y\right )\right )=0} \]
program solution	\[ \int _{}^{y}\frac {\textit {\_a} \cos \left (\textit {\_a} \right )-c_{2}}{\textit {\_a}}d \textit {\_a} = x +c_{3} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= c_{1} \\ \sin \left (y \left (x \right )\right )+c_{1} \ln \left (y \left (x \right )\right )-x -c_{2} &= 0 \\ \end{align*}

Problem 5436

ODE	\[ \boxed {\left (2 x -3\right ) y^{\prime \prime \prime }-\left (6 x -7\right ) y^{\prime \prime }+4 y^{\prime } x -4 y=8} \]
program solution
Maple solution	\[ y \left (x \right ) = -2+c_{1} x +{\mathrm e}^{x} c_{2} +c_{3} {\mathrm e}^{2 x} \]

Problem 5437

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

Problem 5438

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

Problem 5439

ODE	\[ \boxed {\left (x +2 y\right ) y^{\prime \prime }+2 {y^{\prime }}^{2}+2 y^{\prime }=2} \]
program solution	\[ \frac {\ln \left (-5 x^{2}+\left (5 y-5 c_{1} \right ) x +5 y^{2}-c_{1}^{2}\right )}{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {x}{2}-\frac {\sqrt {-4 c_{1} x +5 x^{2}+4 c_{2}}}{2} \\ y \left (x \right ) &= -\frac {x}{2}+\frac {\sqrt {-4 c_{1} x +5 x^{2}+4 c_{2}}}{2} \\ \end{align*}

Problem 5440

ODE
program solution
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (224+36 x^{4}-432 c_{1} x^{2}-432 c_{1}^{2}-864 c_{2} x +864 c_{3} +12 \sqrt {9 x^{8}-216 c_{1} x^{6}+1080 c_{1}^{2} x^{4}-432 c_{2} x^{5}+2592 c_{1}^{3} x^{2}+5184 c_{1} c_{2} x^{3}+432 c_{3} x^{4}+1296 c_{1}^{4}+5184 c_{1}^{2} c_{2} x -5184 c_{1} c_{3} x^{2}+5184 c_{2}^{2} x^{2}+112 x^{4}-5184 c_{1}^{2} c_{3} -1344 c_{1} x^{2}-10368 c_{2} c_{3} x -1344 c_{1}^{2}-2688 c_{2} x +5184 c_{3}^{2}+2688 c_{3} +576}\right )^{\frac {1}{3}}}{12}-\frac {8}{3 \left (224+36 x^{4}-432 c_{1} x^{2}-432 c_{1}^{2}-864 c_{2} x +864 c_{3} +12 \sqrt {9 x^{8}-216 c_{1} x^{6}+1080 c_{1}^{2} x^{4}-432 c_{2} x^{5}+2592 c_{1}^{3} x^{2}+5184 c_{1} c_{2} x^{3}+432 c_{3} x^{4}+1296 c_{1}^{4}+5184 c_{1}^{2} c_{2} x -5184 c_{1} c_{3} x^{2}+5184 c_{2}^{2} x^{2}+112 x^{4}-5184 c_{1}^{2} c_{3} -1344 c_{1} x^{2}-10368 c_{2} c_{3} x -1344 c_{1}^{2}-2688 c_{2} x +5184 c_{3}^{2}+2688 c_{3} +576}\right )^{\frac {1}{3}}}-\frac {1}{3} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (224+36 x^{4}-432 c_{1} x^{2}-432 c_{1}^{2}-864 c_{2} x +864 c_{3} +12 \sqrt {9 x^{8}-216 c_{1} x^{6}-432 c_{2} x^{5}+\left (1080 c_{1}^{2}+432 c_{3} +112\right ) x^{4}+5184 c_{1} c_{2} x^{3}+\left (2592 c_{1}^{3}+\left (-5184 c_{3} -1344\right ) c_{1} +5184 c_{2}^{2}\right ) x^{2}+5184 c_{2} \left (c_{1}^{2}-2 c_{3} -\frac {14}{27}\right ) x +1296 c_{1}^{4}+\left (-5184 c_{3} -1344\right ) c_{1}^{2}+5184 c_{3}^{2}+2688 c_{3} +576}\right )^{\frac {2}{3}}+32 i \sqrt {3}+8 \left (224+36 x^{4}-432 c_{1} x^{2}-432 c_{1}^{2}-864 c_{2} x +864 c_{3} +12 \sqrt {9 x^{8}-216 c_{1} x^{6}-432 c_{2} x^{5}+\left (1080 c_{1}^{2}+432 c_{3} +112\right ) x^{4}+5184 c_{1} c_{2} x^{3}+\left (2592 c_{1}^{3}+\left (-5184 c_{3} -1344\right ) c_{1} +5184 c_{2}^{2}\right ) x^{2}+5184 c_{2} \left (c_{1}^{2}-2 c_{3} -\frac {14}{27}\right ) x +1296 c_{1}^{4}+\left (-5184 c_{3} -1344\right ) c_{1}^{2}+5184 c_{3}^{2}+2688 c_{3} +576}\right )^{\frac {1}{3}}-32}{24 \left (224+36 x^{4}-432 c_{1} x^{2}-432 c_{1}^{2}-864 c_{2} x +864 c_{3} +12 \sqrt {9 x^{8}-216 c_{1} x^{6}-432 c_{2} x^{5}+\left (1080 c_{1}^{2}+432 c_{3} +112\right ) x^{4}+5184 c_{1} c_{2} x^{3}+\left (2592 c_{1}^{3}+\left (-5184 c_{3} -1344\right ) c_{1} +5184 c_{2}^{2}\right ) x^{2}+5184 c_{2} \left (c_{1}^{2}-2 c_{3} -\frac {14}{27}\right ) x +1296 c_{1}^{4}+\left (-5184 c_{3} -1344\right ) c_{1}^{2}+5184 c_{3}^{2}+2688 c_{3} +576}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (224+36 x^{4}-432 c_{1} x^{2}-432 c_{1}^{2}-864 c_{2} x +864 c_{3} +12 \sqrt {9 x^{8}-216 c_{1} x^{6}-432 c_{2} x^{5}+\left (1080 c_{1}^{2}+432 c_{3} +112\right ) x^{4}+5184 c_{1} c_{2} x^{3}+\left (2592 c_{1}^{3}+\left (-5184 c_{3} -1344\right ) c_{1} +5184 c_{2}^{2}\right ) x^{2}+5184 c_{2} \left (c_{1}^{2}-2 c_{3} -\frac {14}{27}\right ) x +1296 c_{1}^{4}+\left (-5184 c_{3} -1344\right ) c_{1}^{2}+5184 c_{3}^{2}+2688 c_{3} +576}\right )^{\frac {2}{3}}+32 i \sqrt {3}-8 \left (224+36 x^{4}-432 c_{1} x^{2}-432 c_{1}^{2}-864 c_{2} x +864 c_{3} +12 \sqrt {9 x^{8}-216 c_{1} x^{6}-432 c_{2} x^{5}+\left (1080 c_{1}^{2}+432 c_{3} +112\right ) x^{4}+5184 c_{1} c_{2} x^{3}+\left (2592 c_{1}^{3}+\left (-5184 c_{3} -1344\right ) c_{1} +5184 c_{2}^{2}\right ) x^{2}+5184 c_{2} \left (c_{1}^{2}-2 c_{3} -\frac {14}{27}\right ) x +1296 c_{1}^{4}+\left (-5184 c_{3} -1344\right ) c_{1}^{2}+5184 c_{3}^{2}+2688 c_{3} +576}\right )^{\frac {1}{3}}+32}{24 \left (224+36 x^{4}-432 c_{1} x^{2}-432 c_{1}^{2}-864 c_{2} x +864 c_{3} +12 \sqrt {9 x^{8}-216 c_{1} x^{6}-432 c_{2} x^{5}+\left (1080 c_{1}^{2}+432 c_{3} +112\right ) x^{4}+5184 c_{1} c_{2} x^{3}+\left (2592 c_{1}^{3}+\left (-5184 c_{3} -1344\right ) c_{1} +5184 c_{2}^{2}\right ) x^{2}+5184 c_{2} \left (c_{1}^{2}-2 c_{3} -\frac {14}{27}\right ) x +1296 c_{1}^{4}+\left (-5184 c_{3} -1344\right ) c_{1}^{2}+5184 c_{3}^{2}+2688 c_{3} +576}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 5441

ODE
program solution
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (8 \ln \left (x \right ) x -8 c_{3} x^{3}+\left (12 c_{1} -4\right ) x +8 c_{2} \right )^{\frac {1}{3}}}{2} \\ y \left (x \right ) &= -\frac {\left (8 \ln \left (x \right ) x -8 c_{3} x^{3}+\left (12 c_{1} -4\right ) x +8 c_{2} \right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{4} \\ y \left (x \right ) &= \frac {\left (8 \ln \left (x \right ) x -8 c_{3} x^{3}+\left (12 c_{1} -4\right ) x +8 c_{2} \right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{4} \\ \end{align*}

Problem 5442

ODE
program solution
Maple solution	\begin{align*} y \left (x \right ) &= \sqrt {-2 c_{3} x \,{\mathrm e}^{x}+{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} c_{2} -2 c_{1}} \\ y \left (x \right ) &= -\sqrt {{\mathrm e}^{2 x}+\left (-2 c_{3} x +2 c_{2} \right ) {\mathrm e}^{x}-2 c_{1}} \\ \end{align*}

Problem 5443

ODE	\[ \boxed {2 \left (y+1\right ) y^{\prime \prime }+2 {y^{\prime }}^{2}+y^{2}+2 y=0} \]
program solution	\[ -i \ln \left (i y^{2}+2 i y+\sqrt {-y^{4}-4 y^{3}-4 y^{2}+8 c_{1}}\right ) = x +c_{2} \] Verified OK. \[ i \ln \left (i y^{2}+2 i y+\sqrt {-y^{4}-4 y^{3}-4 y^{2}+8 c_{1}}\right ) = x +c_{3} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -1-\sqrt {1+2 \cos \left (x \right ) c_{2} -2 c_{1} \sin \left (x \right )} \\ y \left (x \right ) &= -1+\sqrt {1+2 \cos \left (x \right ) c_{2} -2 c_{1} \sin \left (x \right )} \\ \end{align*}

Problem 5444

ODE	\begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right )+{\mathrm e}^{2 t}-{\mathrm e}^{t}\\ y^{\prime }\left (t \right )&=-x \left (t \right )+y \left (t \right )+{\mathrm e}^{2 t} \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= -\frac {{\mathrm e}^{t}}{2}+\frac {{\mathrm e}^{2 t}}{3}+c_{2} {\mathrm e}^{-t} \\ y \left (t \right ) &= \frac {c_{2} {\mathrm e}^{-t}}{2}+\frac {2 \,{\mathrm e}^{2 t}}{3}+c_{1} {\mathrm e}^{t}+\frac {{\mathrm e}^{t} t}{2} \\ \end{align*}

Problem 5445

ODE	\begin {align*} x^{\prime }\left (t \right )&=3 x \left (t \right )-t^{2}+2 y \left (t \right )+t\\ y^{\prime }\left (t \right )&=t^{2}-5 x \left (t \right )-3 y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{2} \sin \left (t \right )+c_{1} \cos \left (t \right )-t^{2}+t +3 \\ y \left (t \right ) &= 2 t^{2}-\frac {3 c_{2} \sin \left (t \right )}{2}-\frac {3 c_{1} \cos \left (t \right )}{2}-3 t -4+\frac {c_{2} \cos \left (t \right )}{2}-\frac {c_{1} \sin \left (t \right )}{2} \\ \end{align*}

Problem 5446

ODE	\begin {align*} x^{\prime }\left (t \right )&=4-5 x \left (t \right )-y \left (t \right )-{\mathrm e}^{t}\\ y^{\prime }\left (t \right )&=2 x \left (t \right )-3 y \left (t \right )+{\mathrm e}^{t}-1 \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{-4 t} \sin \left (t \right ) c_{2} +{\mathrm e}^{-4 t} \cos \left (t \right ) c_{1} +\frac {13}{17}-\frac {5 \,{\mathrm e}^{t}}{26} \\ y \left (t \right ) &= -{\mathrm e}^{-4 t} \sin \left (t \right ) c_{2} -{\mathrm e}^{-4 t} \cos \left (t \right ) c_{2} -{\mathrm e}^{-4 t} \cos \left (t \right ) c_{1} +{\mathrm e}^{-4 t} \sin \left (t \right ) c_{1} +\frac {2 \,{\mathrm e}^{t}}{13}+\frac {3}{17} \\ \end{align*}

Problem 5447

ODE	\begin {align*} x^{\prime }\left (t \right )+y^{\prime }\left (t \right )&=x \left (t \right )-3 y \left (t \right )-1+{\mathrm e}^{-t}\\ x^{\prime }\left (t \right )+y^{\prime }\left (t \right )&=-2 x \left (t \right )-3 y \left (t \right )+{\mathrm e}^{2 t}+1 \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{2 t}}{3}+\frac {2}{3}-\frac {{\mathrm e}^{-t}}{3} \\ y \left (t \right ) &= -\frac {1}{9}-\frac {{\mathrm e}^{2 t}}{15}+\frac {{\mathrm e}^{-t}}{6}+c_{1} {\mathrm e}^{-3 t} \\ \end{align*}

Problem 5448

ODE	\begin {align*} x^{\prime }\left (t \right )&=1+x \left (t \right )+\frac {{\mathrm e}^{t}}{2}\\ y^{\prime }\left (t \right )&=-2 y \left (t \right )+\frac {{\mathrm e}^{t}}{2}\\ z^{\prime }\left (t \right )&=2-z \left (t \right )+\frac {{\mathrm e}^{t}}{2} \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= -1+\frac {{\mathrm e}^{t} \left (2 c_{3} +t \right )}{2} \\ y \left (t \right ) &= \frac {{\mathrm e}^{t}}{6}+c_{2} {\mathrm e}^{-2 t} \\ z \left (t \right ) &= 2+\frac {{\mathrm e}^{t}}{4}+{\mathrm e}^{-t} c_{1} \\ \end{align*}

Problem 5449

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

Problem 5450

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

Problem 5451

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

Problem 5452

ODE	\[ \boxed {y^{\prime }-3 y=2 x^{2}} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+3 x +\frac {9}{2} x^{2}+\frac {9}{2} x^{3}+\frac {27}{8} x^{4}+\frac {81}{40} x^{5}\right ) y \left (0\right )+\frac {2 x^{3}}{3}+\frac {x^{4}}{2}+\frac {3 x^{5}}{10}+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+3 x +\frac {9}{2} x^{2}+\frac {9}{2} x^{3}+\frac {27}{8} x^{4}+\frac {81}{40} x^{5}\right ) c_{1} +\frac {2 x^{3}}{3}+\frac {x^{4}}{2}+\frac {3 x^{5}}{10}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+3 x +\frac {9}{2} x^{2}+\frac {9}{2} x^{3}+\frac {27}{8} x^{4}+\frac {81}{40} x^{5}\right ) y \left (0\right )+\frac {2 x^{3}}{3}+\frac {x^{4}}{2}+\frac {3 x^{5}}{10}+O\left (x^{6}\right ) \]

Problem 5453

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

Problem 5454

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 5455

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

Problem 5456

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime } x +y x^{2}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{12} x^{4}-\frac {1}{90} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}-\frac {1}{40} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {x^{4}}{12}\right ) c_{1} +\left (x +\frac {1}{6} x^{3}-\frac {1}{40} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {x^{4}}{12}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}-\frac {1}{40} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5457

ODE	\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +p \left (p +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2} p^{2}-\frac {1}{2} x^{2} p +\frac {1}{24} p^{4} x^{4}+\frac {1}{12} p^{3} x^{4}-\frac {5}{24} p^{2} x^{4}-\frac {1}{4} p \,x^{4}-\frac {1}{720} x^{6} p^{6}-\frac {1}{240} x^{6} p^{5}+\frac {23}{720} x^{6} p^{4}+\frac {17}{240} x^{6} p^{3}-\frac {47}{360} x^{6} p^{2}-\frac {1}{6} x^{6} p \right ) y \left (0\right )+\left (x -\frac {1}{6} p^{2} x^{3}-\frac {1}{6} p \,x^{3}+\frac {1}{3} x^{3}+\frac {1}{120} x^{5} p^{4}+\frac {1}{60} x^{5} p^{3}-\frac {13}{120} x^{5} p^{2}-\frac {7}{60} x^{5} p +\frac {1}{5} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\left (-\frac {1}{2} p^{2}-\frac {1}{2} p \right ) x^{2}+\left (-\frac {5}{24} p^{2}-\frac {1}{4} p +\frac {1}{24} p^{4}+\frac {1}{12} p^{3}\right ) x^{4}\right ) c_{1} +\left (x +\left (-\frac {1}{6} p^{2}-\frac {1}{6} p +\frac {1}{3}\right ) x^{3}+\left (-\frac {13}{120} p^{2}-\frac {7}{60} p +\frac {1}{5}+\frac {1}{120} p^{4}+\frac {1}{60} p^{3}\right ) x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {p \left (p +1\right ) x^{2}}{2}+\frac {p \left (p^{3}+2 p^{2}-5 p -6\right ) x^{4}}{24}\right ) y \left (0\right )+\left (x -\frac {\left (p^{2}+p -2\right ) x^{3}}{6}+\frac {\left (p^{4}+2 p^{3}-13 p^{2}-14 p +24\right ) x^{5}}{120}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5458

ODE	\[ \boxed {y^{\prime \prime }+y x^{2}=x^{2}+x +1} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {x^{4}}{12}\right ) y \left (0\right )+\left (x -\frac {1}{20} x^{5}\right ) y^{\prime }\left (0\right )+\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{12}-\frac {x^{6}}{60}+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {x^{4}}{12}\right ) c_{1} +\left (x -\frac {1}{20} x^{5}\right ) c_{2} +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{12}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {x^{4}}{12}\right ) y \left (0\right )+\left (x -\frac {1}{20} x^{5}\right ) D\left (y \right )\left (0\right )+\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{12}+O\left (x^{6}\right ) \]

Problem 5459

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

Problem 5460

ODE	\[ \boxed {4 x y^{\prime \prime }+2 \left (1-x \right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1+\frac {x}{3}+\frac {x^{2}}{15}+\frac {x^{3}}{105}+\frac {x^{4}}{945}+\frac {x^{5}}{10395}+O\left (x^{6}\right )\right )+c_{2} \left (1+\frac {x}{2}+\frac {x^{2}}{8}+\frac {x^{3}}{48}+\frac {x^{4}}{384}+\frac {x^{5}}{3840}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{3} x +\frac {1}{15} x^{2}+\frac {1}{105} x^{3}+\frac {1}{945} x^{4}+\frac {1}{10395} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+\frac {1}{2} x +\frac {1}{8} x^{2}+\frac {1}{48} x^{3}+\frac {1}{384} x^{4}+\frac {1}{3840} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 5461

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }-y^{\prime } x +\left (-x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1+\frac {x^{2}}{10}+\frac {x^{4}}{360}+O\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1+\frac {x^{2}}{6}+\frac {x^{4}}{168}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{6} x^{2}+\frac {1}{168} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x \left (1+\frac {1}{10} x^{2}+\frac {1}{360} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 5462

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

Problem 5463

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

Problem 5464

ODE	\[ \boxed {x y^{\prime \prime }-2 y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{3} \left (1-\frac {x}{4}+\frac {x^{2}}{40}-\frac {x^{3}}{720}+\frac {x^{4}}{20160}-\frac {x^{5}}{806400}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {x^{3} \left (1-\frac {x}{4}+\frac {x^{2}}{40}-\frac {x^{3}}{720}+\frac {x^{4}}{20160}-\frac {x^{5}}{806400}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{12}+1+\frac {x}{2}+\frac {x^{2}}{4}-\frac {5 x^{4}}{192}+\frac {13 x^{5}}{3200}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{3} \left (1-\frac {1}{4} x +\frac {1}{40} x^{2}-\frac {1}{720} x^{3}+\frac {1}{20160} x^{4}-\frac {1}{806400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-x^{3}+\frac {1}{4} x^{4}-\frac {1}{40} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (12+6 x +3 x^{2}-\frac {5}{16} x^{4}+\frac {39}{800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

Problem 5465

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

Problem 5466

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

Problem 5467

ODE	\[ \boxed {2 x y^{\prime \prime }+y^{\prime }-y=1+x} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = x +\frac {x^{2}}{3}+\frac {x^{3}}{45}+\frac {x^{4}}{1260}+\frac {x^{5}}{56700}+O\left (x^{6}\right )+c_{1} \sqrt {x}\, \left (1+\frac {x}{3}+\frac {x^{2}}{30}+\frac {x^{3}}{630}+\frac {x^{4}}{22680}+\frac {x^{5}}{1247400}+O\left (x^{6}\right )\right )+c_{2} \left (1+x +\frac {x^{2}}{6}+\frac {x^{3}}{90}+\frac {x^{4}}{2520}+\frac {x^{5}}{113400}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{3} x +\frac {1}{30} x^{2}+\frac {1}{630} x^{3}+\frac {1}{22680} x^{4}+\frac {1}{1247400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+x +\frac {1}{6} x^{2}+\frac {1}{90} x^{3}+\frac {1}{2520} x^{4}+\frac {1}{113400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+x \left (1+\frac {1}{3} x +\frac {1}{45} x^{2}+\frac {1}{1260} x^{3}+\frac {1}{56700} x^{4}+\operatorname {O}\left (x^{5}\right )\right ) \]

Problem 5468

ODE	\[ \boxed {2 x^{3} y^{\prime \prime }+x^{2} y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = \infty \).
program solution	\[ y = c_{1} \left (1-\frac {1}{3 x}+\frac {1}{30 x^{2}}-\frac {1}{630 x^{3}}+\frac {1}{22680 x^{4}}-\frac {1}{1247400 x^{5}}+O\left (\frac {1}{x^{6}}\right )\right )+\frac {c_{2} \left (1-\frac {1}{x}+\frac {1}{6 x^{2}}-\frac {1}{90 x^{3}}+\frac {1}{2520 x^{4}}-\frac {1}{113400 x^{5}}+O\left (\frac {1}{x^{6}}\right )\right )}{\sqrt {\frac {1}{x}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {\left (x -\operatorname {Infinity} \right )^{2}}{4 \operatorname {Infinity}^{3}}+\frac {7 \left (x -\operatorname {Infinity} \right )^{3}}{24 \operatorname {Infinity}^{4}}+\frac {\left (-59 \operatorname {Infinity} +2\right ) \left (x -\operatorname {Infinity} \right )^{4}}{192 \operatorname {Infinity}^{6}}+\frac {\left (605 \operatorname {Infinity} -52\right ) \left (x -\operatorname {Infinity} \right )^{5}}{1920 \operatorname {Infinity}^{7}}\right ) y \left (\operatorname {Infinity} \right )+\left (x -\operatorname {Infinity} -\frac {\left (x -\operatorname {Infinity} \right )^{2}}{4 \operatorname {Infinity}}+\frac {\left (3 \operatorname {Infinity}^{2}-2 \operatorname {Infinity} \right ) \left (x -\operatorname {Infinity} \right )^{3}}{24 \operatorname {Infinity}^{4}}-\frac {5 \left (\operatorname {Infinity} -\frac {28}{15}\right ) \left (x -\operatorname {Infinity} \right )^{4}}{64 \operatorname {Infinity}^{4}}+\frac {\left (105 \operatorname {Infinity}^{3}-370 \operatorname {Infinity}^{2}+4 \operatorname {Infinity} \right ) \left (x -\operatorname {Infinity} \right )^{5}}{1920 \operatorname {Infinity}^{7}}\right ) D\left (y \right )\left (\operatorname {Infinity} \right )+O\left (x^{6}\right ) \]

Problem 5469

ODE	\[ \boxed {x^{3} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = \infty \).
program solution	\[ y = c_{1} \left (\frac {1}{x}+1+\frac {1}{2 x^{2}}+\frac {1}{6 x^{3}}+\frac {1}{24 x^{4}}+\frac {1}{120 x^{5}}+O\left (\frac {1}{x^{6}}\right )\right )+c_{2} \left (\left (\frac {1}{x}+1+\frac {1}{2 x^{2}}+\frac {1}{6 x^{3}}+\frac {1}{24 x^{4}}+\frac {1}{120 x^{5}}+O\left (\frac {1}{x^{6}}\right )\right ) \ln \left (\frac {1}{x}\right )-\frac {1}{x}-\frac {3}{4 x^{2}}-\frac {11}{36 x^{3}}-\frac {25}{288 x^{4}}-\frac {137}{7200 x^{5}}+O\left (\frac {1}{x^{6}}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {\left (x -\operatorname {Infinity} \right )^{2}}{2 \operatorname {Infinity}^{3}}+\frac {\left (-4 \operatorname {Infinity} -1\right ) \left (x -\operatorname {Infinity} \right )^{3}}{6 \operatorname {Infinity}^{5}}+\frac {\left (18 \operatorname {Infinity}^{2}+10 \operatorname {Infinity} +1\right ) \left (x -\operatorname {Infinity} \right )^{4}}{24 \operatorname {Infinity}^{7}}+\frac {\left (-96 \operatorname {Infinity}^{3}-86 \operatorname {Infinity}^{2}-18 \operatorname {Infinity} -1\right ) \left (x -\operatorname {Infinity} \right )^{5}}{120 \operatorname {Infinity}^{9}}\right ) y \left (\operatorname {Infinity} \right )+\left (x -\operatorname {Infinity} +\frac {\left (-\operatorname {Infinity}^{2}-\operatorname {Infinity} \right ) \left (x -\operatorname {Infinity} \right )^{2}}{2 \operatorname {Infinity}^{3}}+\frac {\left (2 \operatorname {Infinity}^{3}+5 \operatorname {Infinity}^{2}+\operatorname {Infinity} \right ) \left (x -\operatorname {Infinity} \right )^{3}}{6 \operatorname {Infinity}^{5}}+\frac {\left (-6 \operatorname {Infinity}^{4}-26 \operatorname {Infinity}^{3}-11 \operatorname {Infinity}^{2}-\operatorname {Infinity} \right ) \left (x -\operatorname {Infinity} \right )^{4}}{24 \operatorname {Infinity}^{7}}+\frac {\left (24 \operatorname {Infinity}^{5}+154 \operatorname {Infinity}^{4}+102 \operatorname {Infinity}^{3}+19 \operatorname {Infinity}^{2}+\operatorname {Infinity} \right ) \left (x -\operatorname {Infinity} \right )^{5}}{120 \operatorname {Infinity}^{9}}\right ) D\left (y \right )\left (\operatorname {Infinity} \right )+O\left (x^{6}\right ) \]

Problem 5470

ODE	\[ \boxed {z^{\prime \prime }+t z^{\prime }+\left (t^{2}-\frac {1}{9}\right ) z=0} \] With the expansion point for the power series method at \(t = 0\).
program solution	\[ z = \left (1+\frac {1}{18} t^{2}-\frac {179}{1944} t^{4}+\frac {5293}{524880} t^{6}\right ) z \left (0\right )+\left (t -\frac {4}{27} t^{3}-\frac {139}{4860} t^{5}\right ) z^{\prime }\left (0\right )+O\left (t^{6}\right ) \] Verified OK. \[ z = \left (1+\frac {1}{18} t^{2}-\frac {179}{1944} t^{4}\right ) c_{1} +\left (t -\frac {4}{27} t^{3}-\frac {139}{4860} t^{5}\right ) c_{2} +O\left (t^{6}\right ) \] Verified OK.
Maple solution	\[ z \left (t \right ) = \left (1+\frac {1}{18} t^{2}-\frac {179}{1944} t^{4}\right ) z \left (0\right )+\left (t -\frac {4}{27} t^{3}-\frac {139}{4860} t^{5}\right ) D\left (z \right )\left (0\right )+O\left (t^{6}\right ) \]

Problem 5471

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

Problem 5472

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

Problem 5473

ODE	Solve \begin {gather*} \boxed {x^{3} \left (1+x \right ) y^{\prime \prime \prime }-\left (2+4 x \right ) x^{2} y^{\prime \prime }+\left (4+10 x \right ) x y^{\prime }-\left (4+12 x \right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
program solution	N/A
Maple solution	\[ y \left (x \right ) = x \left (\left (2 x +\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right )^{2} c_{3} +\ln \left (x \right ) \left (2+\operatorname {O}\left (x^{6}\right )\right ) c_{2} x +2 \left (\left (-4\right ) x +\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right ) c_{3} +\left (5+\operatorname {O}\left (x^{6}\right )\right ) c_{2} x +c_{1} x \left (1+\operatorname {O}\left (x^{6}\right )\right )+\left (2+4 x +2 x^{2}+\operatorname {O}\left (x^{6}\right )\right ) c_{3} \right ) \]

Problem 5474

ODE	Solve \begin {gather*} \boxed {x^{3} \left (x^{2}+1\right ) y^{\prime \prime \prime }-\left (4 x^{2}+2\right ) x^{2} y^{\prime \prime }+\left (10 x^{2}+4\right ) x y^{\prime }-\left (12 x^{2}+4\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
program solution	N/A
Maple solution	\[ y \left (x \right ) = \left (c_{3} \left (2+2 x^{2}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (1+\operatorname {O}\left (x^{6}\right )\right ) c_{1} +c_{2} \left (\ln \left (x \right ) \left (2+\operatorname {O}\left (x^{6}\right )\right )+\left (5+\operatorname {O}\left (x^{6}\right )\right )\right )\right ) x \right ) x \]

Problem 5475

ODE	\[ \boxed {2 \left (-x +2\right ) x^{2} y^{\prime \prime }-\left (4-x \right ) x y^{\prime }+\left (-x +3\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {3}{2}} \left (1+\frac {x}{8}+\frac {x^{2}}{32}+\frac {5 x^{3}}{512}+\frac {7 x^{4}}{2048}+\frac {21 x^{5}}{16384}+O\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1-\frac {x}{4}-\frac {x^{2}}{32}-\frac {x^{3}}{128}-\frac {5 x^{4}}{2048}-\frac {7 x^{5}}{8192}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (x \left (1+\frac {1}{8} x +\frac {1}{32} x^{2}+\frac {5}{512} x^{3}+\frac {7}{2048} x^{4}+\frac {21}{16384} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{1} +\left (1+\frac {1}{4} x +\frac {1}{32} x^{2}+\frac {1}{128} x^{3}+\frac {5}{2048} x^{4}+\frac {7}{8192} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]

Problem 5476

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

Problem 5477

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

Problem 5478

ODE	\[ \boxed {x^{2} y^{\prime \prime }+4 \left (a +x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{2}+\frac {\sqrt {1-16 a}}{2}} \left (1-\frac {4 x}{1+\sqrt {1-16 a}}+\frac {8 x^{2}}{\left (1+\sqrt {1-16 a}\right ) \left (2+\sqrt {1-16 a}\right )}-\frac {32 x^{3}}{3 \left (1+\sqrt {1-16 a}\right ) \left (2+\sqrt {1-16 a}\right ) \left (3+\sqrt {1-16 a}\right )}+\frac {32 x^{4}}{3 \left (1+\sqrt {1-16 a}\right ) \left (2+\sqrt {1-16 a}\right ) \left (3+\sqrt {1-16 a}\right ) \left (4+\sqrt {1-16 a}\right )}-\frac {128 x^{5}}{15 \left (1+\sqrt {1-16 a}\right ) \left (2+\sqrt {1-16 a}\right ) \left (3+\sqrt {1-16 a}\right ) \left (4+\sqrt {1-16 a}\right ) \left (5+\sqrt {1-16 a}\right )}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{2}-\frac {\sqrt {1-16 a}}{2}} \left (1+\frac {4 x}{-1+\sqrt {1-16 a}}+\frac {8 x^{2}}{\left (-1+\sqrt {1-16 a}\right ) \left (-2+\sqrt {1-16 a}\right )}+\frac {32 x^{3}}{3 \left (-1+\sqrt {1-16 a}\right ) \left (-2+\sqrt {1-16 a}\right ) \left (-3+\sqrt {1-16 a}\right )}+\frac {32 x^{4}}{3 \left (-1+\sqrt {1-16 a}\right ) \left (-2+\sqrt {1-16 a}\right ) \left (-3+\sqrt {1-16 a}\right ) \left (-4+\sqrt {1-16 a}\right )}+\frac {128 x^{5}}{15 \left (-1+\sqrt {1-16 a}\right ) \left (-2+\sqrt {1-16 a}\right ) \left (-3+\sqrt {1-16 a}\right ) \left (-4+\sqrt {1-16 a}\right ) \left (-5+\sqrt {1-16 a}\right )}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (c_{1} x^{-\frac {\sqrt {1-16 a}}{2}} \left (1+4 \frac {1}{-1+\sqrt {1-16 a}} x +8 \frac {1}{\left (-1+\sqrt {1-16 a}\right ) \left (-2+\sqrt {1-16 a}\right )} x^{2}+\frac {32}{3} \frac {1}{\left (-1+\sqrt {1-16 a}\right ) \left (-2+\sqrt {1-16 a}\right ) \left (-3+\sqrt {1-16 a}\right )} x^{3}+\frac {32}{3} \frac {1}{\left (-1+\sqrt {1-16 a}\right ) \left (-2+\sqrt {1-16 a}\right ) \left (-3+\sqrt {1-16 a}\right ) \left (-4+\sqrt {1-16 a}\right )} x^{4}+\frac {128}{15} \frac {1}{\left (-1+\sqrt {1-16 a}\right ) \left (-2+\sqrt {1-16 a}\right ) \left (-3+\sqrt {1-16 a}\right ) \left (-4+\sqrt {1-16 a}\right ) \left (-5+\sqrt {1-16 a}\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{\frac {\sqrt {1-16 a}}{2}} \left (1-4 \frac {1}{1+\sqrt {1-16 a}} x +8 \frac {1}{\left (1+\sqrt {1-16 a}\right ) \left (2+\sqrt {1-16 a}\right )} x^{2}-\frac {32}{3} \frac {1}{\left (1+\sqrt {1-16 a}\right ) \left (2+\sqrt {1-16 a}\right ) \left (3+\sqrt {1-16 a}\right )} x^{3}+\frac {32}{3} \frac {1}{\left (1+\sqrt {1-16 a}\right ) \left (2+\sqrt {1-16 a}\right ) \left (3+\sqrt {1-16 a}\right ) \left (4+\sqrt {1-16 a}\right )} x^{4}-\frac {128}{15} \frac {1}{\left (1+\sqrt {1-16 a}\right ) \left (2+\sqrt {1-16 a}\right ) \left (3+\sqrt {1-16 a}\right ) \left (4+\sqrt {1-16 a}\right ) \left (5+\sqrt {1-16 a}\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

Problem 5479

ODE	\[ \boxed {x y^{\prime \prime }+\left (x^{3}+1\right ) y^{\prime }+b x y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-\frac {b \,x^{2}}{4}+\frac {b^{2} x^{4}}{64}+\frac {b \,x^{5}}{50}+O\left (x^{6}\right )\right )+c_{2} \left (\left (1-\frac {b \,x^{2}}{4}+\frac {b^{2} x^{4}}{64}+\frac {b \,x^{5}}{50}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\frac {b \,x^{2}}{4}-\frac {x^{3}}{9}-\frac {3 b^{2} x^{4}}{128}-\frac {61 b \,x^{5}}{4500}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {1}{4} b \,x^{2}+\frac {1}{64} b^{2} x^{4}+\frac {1}{50} b \,x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {b}{4} x^{2}-\frac {1}{9} x^{3}-\frac {3}{128} b^{2} x^{4}-\frac {61}{4500} b \,x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 5480

ODE	\[ \boxed {\left (x -1\right ) \left (x -2\right ) y^{\prime \prime }+\left (4 x -6\right ) y^{\prime }+2 y=0} \] 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	\[ y = x +2+\frac {x^{2}}{2}+\frac {x^{3}}{4}+\frac {x^{4}}{8}+\frac {x^{5}}{16}+\frac {x^{6}}{32}+O\left (x^{6}\right ) \] Verified OK. \[ y = 2+\frac {x^{2}}{2}+\frac {x^{3}}{4}+\frac {x^{4}}{8}+\frac {x^{5}}{16}+x +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2+x +\frac {1}{2} x^{2}+\frac {1}{4} x^{3}+\frac {1}{8} x^{4}+\frac {1}{16} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 5481

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

Problem 5482

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

Problem 5483

ODE	\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +12 y=0} \] 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	\[ y = -5 x^{3}+3 x +O\left (x^{6}\right ) \] Verified OK. \[ y = -5 x^{3}+3 x +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -5 x^{3}+3 x \]

Problem 5484

ODE	\[ \boxed {y^{\prime \prime }-\left (x -1\right ) y=0} \] 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	\[ y = 1-\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}-\frac {x^{5}}{30}+\frac {x^{6}}{240}+O\left (x^{6}\right ) \] Verified OK. \[ y = 1-\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}-\frac {x^{5}}{30}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1-\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{30} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 5485

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

Problem 5486

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

Problem 5487

ODE	\[ \boxed {y^{\prime \prime }+\left ({\mathrm e}^{x}-1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{6} x^{3}-\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\frac {1}{240} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{12} x^{4}-\frac {1}{40} x^{5}-\frac {1}{180} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{6} x^{3}-\frac {1}{24} x^{4}-\frac {1}{120} x^{5}\right ) c_{1} +\left (x -\frac {1}{12} x^{4}-\frac {1}{40} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{6} x^{3}-\frac {1}{24} x^{4}-\frac {1}{120} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{12} x^{4}-\frac {1}{40} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5488

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

Problem 5489

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

Problem 5490

ODE	\[ \boxed {\sin \left (x \right ) y^{\prime \prime }-2 \cos \left (x \right ) y^{\prime }-\sin \left (x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{3} \left (1-\frac {x^{2}}{10}+\frac {x^{4}}{280}-\frac {x^{6}}{15120}+O\left (x^{6}\right )\right )+c_{2} \left (1-\frac {x^{2}}{2}+\frac {x^{4}}{24}-\frac {x^{6}}{720}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{3} \left (1-\frac {1}{10} x^{2}+\frac {1}{280} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (12-6 x^{2}+\frac {1}{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 5491

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

Problem 5492

ODE	\[ \boxed {x \left (x +2\right ) y^{\prime \prime }+\left (1+x \right ) y^{\prime }-4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1+\frac {5 x}{4}+\frac {7 x^{2}}{32}-\frac {3 x^{3}}{128}+\frac {11 x^{4}}{2048}-\frac {13 x^{5}}{8192}+O\left (x^{6}\right )\right )+c_{2} \left (1+4 x +2 x^{2}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {5}{4} x +\frac {7}{32} x^{2}-\frac {3}{128} x^{3}+\frac {11}{2048} x^{4}-\frac {13}{8192} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+4 x +2 x^{2}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 5493

ODE	\[ \boxed {x y^{\prime \prime }+\left (\frac {1}{2}-x \right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1+x +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+O\left (x^{6}\right )\right )+c_{2} \left (1+2 x +\frac {4 x^{2}}{3}+\frac {8 x^{3}}{15}+\frac {16 x^{4}}{105}+\frac {32 x^{5}}{945}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+2 x +\frac {4}{3} x^{2}+\frac {8}{15} x^{3}+\frac {16}{105} x^{4}+\frac {32}{945} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 5494

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (x^{2}+\frac {1}{4}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {i}{2}} \left (1+\left (-\frac {1}{5}+\frac {i}{10}\right ) x^{2}+\left (\frac {7}{680}-\frac {3 i}{340}\right ) x^{4}+O\left (x^{6}\right )\right )+c_{2} x^{-\frac {i}{2}} \left (1+\left (-\frac {1}{5}-\frac {i}{10}\right ) x^{2}+\left (\frac {7}{680}+\frac {3 i}{340}\right ) x^{4}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{-\frac {i}{2}} \left (1+\left (-\frac {1}{5}-\frac {i}{10}\right ) x^{2}+\left (\frac {7}{680}+\frac {3 i}{340}\right ) x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{\frac {i}{2}} \left (1+\left (-\frac {1}{5}+\frac {i}{10}\right ) x^{2}+\left (\frac {7}{680}-\frac {3 i}{340}\right ) x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 5495

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (x^{2}+\frac {9}{4}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {3 i}{2}} \left (1+\left (-\frac {1}{13}+\frac {3 i}{26}\right ) x^{2}+\left (-\frac {1}{2600}-\frac {9 i}{1300}\right ) x^{4}+O\left (x^{6}\right )\right )+c_{2} x^{-\frac {3 i}{2}} \left (1+\left (-\frac {1}{13}-\frac {3 i}{26}\right ) x^{2}+\left (-\frac {1}{2600}+\frac {9 i}{1300}\right ) x^{4}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{-\frac {3 i}{2}} \left (1+\left (-\frac {1}{13}-\frac {3 i}{26}\right ) x^{2}+\left (-\frac {1}{2600}+\frac {9 i}{1300}\right ) x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{\frac {3 i}{2}} \left (1+\left (-\frac {1}{13}+\frac {3 i}{26}\right ) x^{2}+\left (-\frac {1}{2600}-\frac {9 i}{1300}\right ) x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 5496

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (x^{2}+\frac {25}{4}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {5 i}{2}} \left (1+\left (-\frac {1}{29}+\frac {5 i}{58}\right ) x^{2}+\left (-\frac {17}{9512}-\frac {15 i}{4756}\right ) x^{4}+O\left (x^{6}\right )\right )+c_{2} x^{-\frac {5 i}{2}} \left (1+\left (-\frac {1}{29}-\frac {5 i}{58}\right ) x^{2}+\left (-\frac {17}{9512}+\frac {15 i}{4756}\right ) x^{4}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{-\frac {5 i}{2}} \left (1+\left (-\frac {1}{29}-\frac {5 i}{58}\right ) x^{2}+\left (-\frac {17}{9512}+\frac {15 i}{4756}\right ) x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{\frac {5 i}{2}} \left (1+\left (-\frac {1}{29}+\frac {5 i}{58}\right ) x^{2}+\left (-\frac {17}{9512}-\frac {15 i}{4756}\right ) x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 5497

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

Problem 5498

ODE	\[ \boxed {y^{\prime }+y x=\cos \left (x \right )} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}\right ) y \left (0\right )+x -\frac {x^{3}}{2}+\frac {13 x^{5}}{120}+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}\right ) c_{1} +x -\frac {x^{3}}{2}+\frac {13 x^{5}}{120}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}\right ) y \left (0\right )+x -\frac {x^{3}}{2}+\frac {13 x^{5}}{120}+O\left (x^{6}\right ) \]

Problem 5499

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

Problem 5500

ODE	\[ \boxed {x^{3} y^{\prime \prime }+y=\frac {1}{x^{4}}} \] With the expansion point for the power series method at \(x = 0\).
program solution	N/A
Maple solution	\[ \text {No solution found} \]