2.17.73 Problems 7201 to 7300

Problem 7201

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

Problem 7202

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

Problem 7203

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

Problem 7204

ODE	\[ \boxed {y^{\prime \prime \prime }+y^{\prime }+y=x} \] With initial conditions \begin {align*} [y^{\prime }\left (0\right ) = 0, y \left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \frac {\left (\left (-35238 \sqrt {31}-113274 \sqrt {3}+44922 i+4658 i \sqrt {93}\right ) \left (108+12 \sqrt {93}\right )^{\frac {1}{3}}+\left (-3670 i \sqrt {93}-35392 i+6076 \sqrt {3}+1890 \sqrt {31}\right ) \left (108+12 \sqrt {93}\right )^{\frac {2}{3}}+91872 i \sqrt {93}+885984 i+388368 \sqrt {3}+120816 \sqrt {31}\right ) {\mathrm e}^{\frac {\left (-i \sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-12 i \sqrt {3}+\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-12\right ) x}{12 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}}+\left (\left (-37572 \sqrt {3}-11688 \sqrt {31}-120624 i-12508 i \sqrt {93}\right ) \left (108+12 \sqrt {93}\right )^{\frac {1}{3}}+\left (2381 i \sqrt {93}+22961 i-6355 \sqrt {3}-1977 \sqrt {31}\right ) \left (108+12 \sqrt {93}\right )^{\frac {2}{3}}-27144 i \sqrt {93}-261768 i+388368 \sqrt {3}+120816 \sqrt {31}\right ) {\mathrm e}^{\frac {\left (i \sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}+12 i \sqrt {3}+\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-12\right ) x}{12 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}}+\left (\left (16722 \sqrt {31}+53754 \sqrt {3}+75702 i+7850 i \sqrt {93}\right ) \left (108+12 \sqrt {93}\right )^{\frac {1}{3}}+\left (1289 i \sqrt {93}+12431 i+5487 \sqrt {3}+1707 \sqrt {31}\right ) \left (108+12 \sqrt {93}\right )^{\frac {2}{3}}+32364 i \sqrt {93}+312108 i+97092 \sqrt {3}+30204 \sqrt {31}\right ) {\mathrm e}^{-\frac {\left (\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-12\right ) x}{6 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}}+97092 \left (x -1\right ) \left (\left (-\sqrt {3}-\frac {839 \sqrt {31}}{2697}\right ) \left (108+12 \sqrt {93}\right )^{\frac {1}{3}}+\left (\frac {14 \sqrt {3}}{261}+\frac {15 \sqrt {31}}{899}\right ) \left (108+12 \sqrt {93}\right )^{\frac {2}{3}}+i \sqrt {93}+\frac {839 i}{87}+9 \sqrt {3}+\frac {2517 \sqrt {31}}{899}\right )}{\left (-97092 \sqrt {3}-30204 \sqrt {31}\right ) \left (108+12 \sqrt {93}\right )^{\frac {1}{3}}+\left (5208 \sqrt {3}+1620 \sqrt {31}\right ) \left (108+12 \sqrt {93}\right )^{\frac {2}{3}}+97092 i \sqrt {93}+936324 i+873828 \sqrt {3}+271836 \sqrt {31}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\frac {10 \,{\mathrm e}^{-\frac {x \left (108+12 \sqrt {93}\right )^{\frac {1}{3}} \left (-12+\left (\sqrt {93}-9\right ) \left (108+12 \sqrt {93}\right )^{\frac {1}{3}}\right )}{144}} \left (\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {31}+\frac {3 \sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}} \sqrt {31}}{5}-\frac {6 \sqrt {3}\, \sqrt {31}}{5}-\frac {39 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}{5}-\frac {31 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}}{5}+\frac {114}{5}\right ) \cos \left (\frac {\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}} \sqrt {3}\, \left (\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {31}-9 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}+12\right ) x}{144}\right )}{3}-26 \,{\mathrm e}^{-\frac {x \left (108+12 \sqrt {93}\right )^{\frac {1}{3}} \left (-12+\left (\sqrt {93}-9\right ) \left (108+12 \sqrt {93}\right )^{\frac {1}{3}}\right )}{144}} \left (\left (\sqrt {3}-\frac {5 \sqrt {31}}{13}\right ) \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}+\frac {38 \sqrt {3}}{13}-\frac {6 \sqrt {31}}{13}\right ) \sin \left (\frac {\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}} \sqrt {3}\, \left (\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {31}-9 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}+12\right ) x}{144}\right )+\left (-76-\frac {10 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {31}}{3}+\sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}} \sqrt {31}+4 \sqrt {3}\, \sqrt {31}+26 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}-\frac {31 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}}{3}\right ) {\mathrm e}^{\frac {x \left (108+12 \sqrt {93}\right )^{\frac {1}{3}} \left (-12+\left (\sqrt {93}-9\right ) \left (108+12 \sqrt {93}\right )^{\frac {1}{3}}\right )}{72}}+3 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}} \left (x -1\right ) \left (\sqrt {3}\, \sqrt {31}-\frac {31}{3}\right )}{\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}} \left (3 \sqrt {3}\, \sqrt {31}-31\right )} \]

Problem 7205

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

Problem 7206

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

Problem 7207

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

Problem 7208

ODE	\[ \boxed {x^{4} y^{\prime \prime \prime }+x^{3} y^{\prime \prime }+y^{\prime } x^{2}+y x=0} \]
program solution	\[ y = c_{1} x^{-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}}+x^{\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}} \left (c_{2} \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \ln \left (x \right )}{2}\right )-c_{3} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \ln \left (x \right )}{2}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{-\frac {\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}-4 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}-8}{6 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}}+c_{2} x^{\frac {-8+\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {2}{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}}\right )+c_{3} x^{\frac {-8+\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {2}{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}}\right ) \]

Problem 7209

ODE	\[ \boxed {x^{4} y^{\prime \prime \prime }+x^{3} y^{\prime \prime }+y^{\prime } x^{2}+y x=x} \]
program solution
Maple solution	\[ y \left (x \right ) = c_{2} x^{\frac {\left (47-3 \sqrt {249}\right ) \left (188+12 \sqrt {249}\right )^{\frac {2}{3}}}{192}+\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}+\frac {2}{3}} \cos \left (\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}} \sqrt {3}\, \left (3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {83}-47 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}+16\right ) \ln \left (x \right )}{192}\right )+c_{3} x^{\frac {\left (47-3 \sqrt {249}\right ) \left (188+12 \sqrt {249}\right )^{\frac {2}{3}}}{192}+\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}+\frac {2}{3}} \sin \left (\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}} \sqrt {3}\, \left (3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {83}-47 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}+16\right ) \ln \left (x \right )}{192}\right )+x^{\frac {\left (188+12 \sqrt {249}\right )^{\frac {2}{3}} \left (-47+3 \sqrt {249}\right )}{96}-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}+\frac {2}{3}} c_{1} +1 \]

Problem 7210

ODE	\[ \boxed {5 x^{5} y^{\prime \prime \prime \prime }+4 x^{4} y^{\prime \prime \prime }+y^{\prime } x^{2}+y x=0} \]
program solution	\[ \text {Expression too large to display} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \moverset {4}{\munderset {\textit {\_a} =1}{\sum }}x^{\operatorname {RootOf}\left (5 \textit {\_Z}^{4}-26 \textit {\_Z}^{3}+43 \textit {\_Z}^{2}-21 \textit {\_Z} +1, \operatorname {index} =\textit {\_a} \right )} \textit {\_C}_{\textit {\_a}} \]

Problem 7211

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

Problem 7212

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+{y^{\prime }}^{2}=x -1} \]
program solution	\[ \text {Expression too large to display} \] Warning, solution could not be verified
Maple solution	\[ y \left (x \right ) = -\left (\int \frac {-\left (\frac {1}{2}-\frac {i x}{2}\right )^{\frac {i \sqrt {-2+2 \sqrt {2}}}{2}} \left (x +i\right ) \left (\frac {1}{2}+\frac {i x}{2}\right )^{i \sqrt {-1+i}} \sqrt {-1+i}\, \operatorname {hypergeom}\left (\left [\frac {i \sqrt {-2+2 \sqrt {2}}}{2}, \frac {i \sqrt {-1+i}}{2}+\frac {\sqrt {1+i}}{2}+1\right ], \left [i \sqrt {-1+i}+1\right ], \frac {1}{2}+\frac {i x}{2}\right )-4 \left (\frac {1}{2}-\frac {i x}{2}\right )^{\frac {\sqrt {2+2 \sqrt {2}}}{2}} \left (-\frac {1}{2}+\frac {i x}{2}\right )^{i \sqrt {-1+i}} \left (x +i\right ) \sqrt {-1+i}\, c_{1} \operatorname {hypergeom}\left (\left [\frac {\sqrt {2+2 \sqrt {2}}}{2}, \frac {\sqrt {2+2 \sqrt {2}}}{2}+1\right ], \left [-i \sqrt {-1+i}+1\right ], \frac {1}{2}+\frac {i x}{2}\right )-8 \left (\operatorname {HeunCPrime}\left (0, -i \sqrt {-1+i}, -1, 0, \frac {1}{2}-\frac {i}{2}, \frac {x -i}{x +i}\right ) c_{1} \left (-\frac {1}{2}+\frac {i x}{2}\right )^{i \sqrt {-1+i}}-\frac {\operatorname {HeunCPrime}\left (0, i \sqrt {-1+i}, -1, 0, \frac {1}{2}-\frac {i}{2}, \frac {x -i}{x +i}\right ) \left (\frac {1}{2}+\frac {i x}{2}\right )^{i \sqrt {-1+i}}}{4}\right ) \left (i x +1\right )}{\left (4 \left (\frac {1}{2}-\frac {i x}{2}\right )^{\frac {\sqrt {2+2 \sqrt {2}}}{2}} c_{1} \operatorname {hypergeom}\left (\left [\frac {\sqrt {2+2 \sqrt {2}}}{2}, \frac {\sqrt {2+2 \sqrt {2}}}{2}+1\right ], \left [-i \sqrt {-1+i}+1\right ], \frac {1}{2}+\frac {i x}{2}\right ) \left (-\frac {1}{2}+\frac {i x}{2}\right )^{i \sqrt {-1+i}}-\left (\frac {1}{2}-\frac {i x}{2}\right )^{\frac {i \sqrt {-2+2 \sqrt {2}}}{2}} \operatorname {hypergeom}\left (\left [\frac {i \sqrt {-2+2 \sqrt {2}}}{2}, \frac {i \sqrt {-1+i}}{2}+\frac {\sqrt {1+i}}{2}+1\right ], \left [i \sqrt {-1+i}+1\right ], \frac {1}{2}+\frac {i x}{2}\right ) \left (\frac {1}{2}+\frac {i x}{2}\right )^{i \sqrt {-1+i}}\right ) \left (x +i\right )}d x \right )+c_{2} \]

Problem 7213

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

Problem 7214

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+y {y^{\prime }}^{2}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 7215

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+{y^{\prime }}^{2}=0} \]
program solution	\[ y = \frac {x +i}{\arctan \left (x \right )-c_{1}}+2 i \left (\int _{}^{\arctan \left (x \right )}\frac {1}{\left (-\textit {\_a} +c_{1} \right )^{2} \left ({\mathrm e}^{2 i \textit {\_a}}+1\right )}d \textit {\_a} \right )+c_{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \int \frac {1}{\arctan \left (x \right )+c_{1}}d x +c_{2} \]

Problem 7216

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

Problem 7217

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

Problem 7218

ODE	\[ \boxed {y^{\prime }-{\mathrm e}^{-\frac {y}{x}}=0} \]
program solution	\[ \ln \left (x \right ) = \int _{}^{\frac {y}{x}}\frac {1}{{\mathrm e}^{-\textit {\_a}}-\textit {\_a}}d \textit {\_a} +c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \operatorname {RootOf}\left (-\left (\int _{}^{\textit {\_Z}}-\frac {1}{-{\mathrm e}^{-\textit {\_a}}+\textit {\_a}}d \textit {\_a} \right )+\ln \left (x \right )+c_{1} \right ) x \]

Problem 7219

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

Problem 7220

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

Problem 7221

ODE	\[ \boxed {v v^{\prime }-\frac {2 v^{2}}{r^{3}}=\frac {\lambda r}{3}} \]
program solution	\[ -\frac {\lambda \,\operatorname {expIntegral}_{1}\left (-\frac {2}{r^{2}}\right )}{3}-\frac {{\mathrm e}^{\frac {2}{r^{2}}} \left (\lambda \,r^{2}-3 v^{2}\right )}{6} = c_{1} \] Verified OK.
Maple solution	\begin{align*} v \left (r \right ) &= -\frac {\sqrt {3}\, \sqrt {{\mathrm e}^{\frac {2}{r^{2}}} \left (\lambda \,{\mathrm e}^{\frac {2}{r^{2}}} r^{2}+2 \lambda \,\operatorname {expIntegral}_{1}\left (-\frac {2}{r^{2}}\right )+3 c_{1} \right )}\, {\mathrm e}^{-\frac {2}{r^{2}}}}{3} \\ v \left (r \right ) &= \frac {\sqrt {3}\, \sqrt {{\mathrm e}^{\frac {2}{r^{2}}} \left (\lambda \,{\mathrm e}^{\frac {2}{r^{2}}} r^{2}+2 \lambda \,\operatorname {expIntegral}_{1}\left (-\frac {2}{r^{2}}\right )+3 c_{1} \right )}\, {\mathrm e}^{-\frac {2}{r^{2}}}}{3} \\ \end{align*}

Problem 7222

ODE	\[ \boxed {2 y^{\prime \prime } x^{2}-x y^{\prime }+\left (1-x^{2}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1+\frac {x^{2}}{10}+\frac {x^{4}}{360}+O\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1+\frac {x^{2}}{6}+\frac {x^{4}}{168}+O\left (x^{6}\right )\right ) \] 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 7223

ODE	\[ \boxed {2 y^{\prime \prime } x^{2}-x y^{\prime }+\left (1-x^{2}\right ) y=1} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 1+\frac {x^{2}}{3}+\frac {x^{4}}{63}+O\left (x^{6}\right )+c_{1} x \left (1+\frac {x^{2}}{10}+\frac {x^{4}}{360}+O\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1+\frac {x^{2}}{6}+\frac {x^{4}}{168}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{6} x^{2}+\frac {1}{168} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x \left (1+\frac {1}{10} x^{2}+\frac {1}{360} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (1+\frac {1}{3} x^{2}+\frac {1}{63} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 7224

ODE	\[ \boxed {2 y^{\prime \prime } x^{2}-x y^{\prime }+\left (1-x^{2}\right ) y=1+x} \] With the expansion point for the power series method at \(x = 0\).
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 7225

ODE	\[ \boxed {2 y^{\prime \prime } x^{2}-x y^{\prime }+\left (1-x^{2}\right ) y=x} \] With the expansion point for the power series method at \(x = 0\).
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 7226

ODE	\[ \boxed {2 y^{\prime \prime } x^{2}-x y^{\prime }+\left (1-x^{2}\right ) y=x^{2}+x +1} \] With the expansion point for the power series method at \(x = 0\).
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 7227

ODE	\[ \boxed {2 y^{\prime \prime } x^{2}-x y^{\prime }+\left (1-x^{2}\right ) y=x^{2}} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {x^{2}}{3}+\frac {x^{4}}{63}+\frac {x^{6}}{3465}+O\left (x^{6}\right )+c_{1} x \left (1+\frac {x^{2}}{10}+\frac {x^{4}}{360}+O\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1+\frac {x^{2}}{6}+\frac {x^{4}}{168}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{6} x^{2}+\frac {1}{168} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x \left (1+\frac {1}{10} x^{2}+\frac {1}{360} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+x^{2} \left (\frac {1}{3}+\frac {1}{63} x^{2}+\operatorname {O}\left (x^{4}\right )\right ) \]

Problem 7228

ODE	\[ \boxed {2 y^{\prime \prime } x^{2}-x y^{\prime }+\left (1-x^{2}\right ) y=x^{2}+1} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 1+\frac {2 x^{2}}{3}+\frac {2 x^{4}}{63}+\frac {x^{6}}{3465}+O\left (x^{6}\right )+c_{1} x \left (1+\frac {x^{2}}{10}+\frac {x^{4}}{360}+O\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1+\frac {x^{2}}{6}+\frac {x^{4}}{168}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{6} x^{2}+\frac {1}{168} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x \left (1+\frac {1}{10} x^{2}+\frac {1}{360} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (1+\frac {2}{3} x^{2}+\frac {2}{63} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 7229

ODE	\[ \boxed {2 y^{\prime \prime } x^{2}-x y^{\prime }+\left (1-x^{2}\right ) y=x^{4}} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {x^{4}}{21}+O\left (x^{6}\right )+c_{1} x \left (1+\frac {x^{2}}{10}+\frac {x^{4}}{360}+O\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1+\frac {x^{2}}{6}+\frac {x^{4}}{168}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{6} x^{2}+\frac {1}{168} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x \left (1+\frac {1}{10} x^{2}+\frac {1}{360} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+x^{4} \left (\frac {1}{21}+\operatorname {O}\left (x^{2}\right )\right ) \]

Problem 7230

ODE	\[ \boxed {2 y^{\prime \prime } x^{2}-x y^{\prime }+\left (1-x^{2}\right ) y=\sin \left (x \right )} \] With the expansion point for the power series method at \(x = 0\).
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 7231

ODE	\[ \boxed {2 y^{\prime \prime } x^{2}-x y^{\prime }+\left (1-x^{2}\right ) y=1+\sin \left (x \right )} \] With the expansion point for the power series method at \(x = 0\).
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 7232

ODE	\[ \boxed {2 y^{\prime \prime } x^{2}-x y^{\prime }+\left (1-x^{2}\right ) y=x \sin \left (x \right )} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {x^{2}}{3}+\frac {x^{4}}{126}+O\left (x^{6}\right )+c_{1} x \left (1+\frac {x^{2}}{10}+\frac {x^{4}}{360}+O\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1+\frac {x^{2}}{6}+\frac {x^{4}}{168}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{6} x^{2}+\frac {1}{168} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x \left (1+\frac {1}{10} x^{2}+\frac {1}{360} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+x^{2} \left (\frac {1}{3}+\frac {1}{126} x^{2}+\operatorname {O}\left (x^{4}\right )\right ) \]

Problem 7233

ODE	\[ \boxed {2 y^{\prime \prime } x^{2}-x y^{\prime }+\left (1-x^{2}\right ) y=\cos \left (x \right )+\sin \left (x \right )} \] With the expansion point for the power series method at \(x = 0\).
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 7234

ODE	\[ \boxed {y^{\prime \prime } x^{2}+\left (\cos \left (x \right )-1\right ) y^{\prime }+{\mathrm e}^{x} y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{2}+\frac {i \sqrt {3}}{2}} \left (1+\frac {i \sqrt {3}\, x}{4}+\frac {\left (-i \sqrt {3}-11\right ) x^{2}}{32 i \sqrt {3}+64}+\frac {55 \left (\sqrt {3}+3 i\right ) x^{3}}{288 \left (i-\sqrt {3}\right ) \left (i \sqrt {3}+2\right ) \left (i \sqrt {3}+3\right )}+\frac {\left (112 i \sqrt {3}+199\right ) x^{4}}{384 \left (\sqrt {3}-2 i\right ) \left (i-\sqrt {3}\right ) \left (i \sqrt {3}+3\right ) \left (i \sqrt {3}+4\right )}+\frac {41 \left (451 \sqrt {3}+321 i\right ) x^{5}}{38400 \left (-i+\sqrt {3}\right ) \left (i \sqrt {3}+2\right ) \left (i \sqrt {3}+3\right ) \left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+5\right )}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{2}-\frac {i \sqrt {3}}{2}} \left (1-\frac {i \sqrt {3}\, x}{4}+\frac {\left (i \sqrt {3}-11\right ) x^{2}}{-32 i \sqrt {3}+64}+\frac {55 \left (-3 i+\sqrt {3}\right ) x^{3}}{288 \left (-i-\sqrt {3}\right ) \left (-i \sqrt {3}+2\right ) \left (-i \sqrt {3}+3\right )}+\frac {\left (-112 i \sqrt {3}+199\right ) x^{4}}{384 \left (\sqrt {3}+2 i\right ) \left (-i-\sqrt {3}\right ) \left (-i \sqrt {3}+3\right ) \left (-i \sqrt {3}+4\right )}+\frac {41 \left (451 \sqrt {3}-321 i\right ) x^{5}}{38400 \left (\sqrt {3}+i\right ) \left (-i \sqrt {3}+2\right ) \left (-i \sqrt {3}+3\right ) \left (-i \sqrt {3}+4\right ) \left (-i \sqrt {3}+5\right )}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (c_{2} x^{\frac {i \sqrt {3}}{2}} \left (1+\frac {1}{4} i \sqrt {3} x +\frac {-i \sqrt {3}-11}{32 i \sqrt {3}+64} x^{2}+\frac {\frac {55 \sqrt {3}}{288}+\frac {55 i}{96}}{\left (i-\sqrt {3}\right ) \left (i \sqrt {3}+2\right ) \left (i \sqrt {3}+3\right )} x^{3}+\frac {1}{384} \frac {112 i \sqrt {3}+199}{\left (-\sqrt {3}+2 i\right ) \left (-i+\sqrt {3}\right ) \left (i \sqrt {3}+3\right ) \left (i \sqrt {3}+4\right )} x^{4}+\frac {\frac {18491 \sqrt {3}}{38400}+\frac {4387 i}{12800}}{\left (-i+\sqrt {3}\right ) \left (i \sqrt {3}+2\right ) \left (i \sqrt {3}+3\right ) \left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+5\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} x^{-\frac {i \sqrt {3}}{2}} \left (1-\frac {1}{4} i \sqrt {3} x +\frac {-\sqrt {3}-11 i}{32 \sqrt {3}+64 i} x^{2}+\frac {55 \sqrt {3}-165 i}{3456 i-2304 \sqrt {3}} x^{3}+\frac {199 i+112 \sqrt {3}}{-27648 i+7680 \sqrt {3}} x^{4}+\frac {\frac {18491 \sqrt {3}}{38400}-\frac {4387 i}{12800}}{\left (\sqrt {3}+i\right ) \left (\sqrt {3}+2 i\right ) \left (\sqrt {3}+3 i\right ) \left (\sqrt {3}+4 i\right ) \left (\sqrt {3}+5 i\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

Problem 7235

ODE	\[ \boxed {\left (x -2\right ) y^{\prime \prime }+\frac {y^{\prime }}{x}+\left (1+x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {3}{2}} \left (1+\frac {3 x}{20}+\frac {25 x^{2}}{224}+\frac {1361 x^{3}}{17280}+\frac {80753 x^{4}}{2365440}+\frac {616517 x^{5}}{38707200}+O\left (x^{6}\right )\right )+c_{2} \left (1+\frac {x^{2}}{2}+\frac {2 x^{3}}{9}+\frac {11 x^{4}}{120}+\frac {82 x^{5}}{1575}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {3}{2}} \left (1+\frac {3}{20} x +\frac {25}{224} x^{2}+\frac {1361}{17280} x^{3}+\frac {80753}{2365440} x^{4}+\frac {616517}{38707200} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+\frac {1}{2} x^{2}+\frac {2}{9} x^{3}+\frac {11}{120} x^{4}+\frac {82}{1575} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 7236

ODE	\[ \boxed {\left (x -2\right ) y^{\prime \prime }+\frac {y^{\prime }}{x}+\left (1+x \right ) y=0} \] With the expansion point for the power series method at \(x = 2\).
program solution	\[ y = c_{1} \sqrt {x -2}\, \left (\frac {29}{6}-\frac {23 x}{12}+\frac {127 \left (x -2\right )^{2}}{160}+\frac {1621 \left (x -2\right )^{3}}{40320}-\frac {426599 \left (x -2\right )^{4}}{5806080}+\frac {4670443 \left (x -2\right )^{5}}{425779200}+O\left (\left (x -2\right )^{6}\right )\right )+c_{2} \left (13-6 x +\frac {31 \left (x -2\right )^{2}}{6}-\frac {37 \left (x -2\right )^{3}}{45}-\frac {299 \left (x -2\right )^{4}}{840}+\frac {6743 \left (x -2\right )^{5}}{56700}+O\left (\left (x -2\right )^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x -2}\, \left (1-\frac {23}{12} \left (x -2\right )+\frac {127}{160} \left (x -2\right )^{2}+\frac {1621}{40320} \left (x -2\right )^{3}-\frac {426599}{5806080} \left (x -2\right )^{4}+\frac {4670443}{425779200} \left (x -2\right )^{5}+\operatorname {O}\left (\left (x -2\right )^{6}\right )\right )+c_{2} \left (1-6 \left (x -2\right )+\frac {31}{6} \left (x -2\right )^{2}-\frac {37}{45} \left (x -2\right )^{3}-\frac {299}{840} \left (x -2\right )^{4}+\frac {6743}{56700} \left (x -2\right )^{5}+\operatorname {O}\left (\left (x -2\right )^{6}\right )\right ) \]

Problem 7237

ODE	\[ \boxed {\left (1+x \right ) \left (3 x -1\right ) y^{\prime \prime }+y^{\prime } \cos \left (x \right )-3 y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{3}-\frac {5}{8} x^{4}-\frac {53}{40} x^{5}-\frac {213}{80} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{2} x^{3}+\frac {7}{12} x^{4}+\frac {7}{6} x^{5}+\frac {1711}{720} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{3}-\frac {5}{8} x^{4}-\frac {53}{40} x^{5}\right ) c_{1} +\left (x +\frac {1}{2} x^{2}+\frac {1}{2} x^{3}+\frac {7}{12} x^{4}+\frac {7}{6} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{3}-\frac {5}{8} x^{4}-\frac {53}{40} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{2} x^{3}+\frac {7}{12} x^{4}+\frac {7}{6} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 7238

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

Problem 7239

ODE	\[ \boxed {2 y^{\prime \prime } x^{2}+3 x y^{\prime }-y x=x^{2}+2 x} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {2 x}{3}+\frac {x^{2}}{6}+\frac {x^{3}}{126}+\frac {x^{4}}{4536}+\frac {x^{5}}{249480}+O\left (x^{6}\right )+c_{1} \left (1+\frac {x}{3}+\frac {x^{2}}{30}+\frac {x^{3}}{630}+\frac {x^{4}}{22680}+\frac {x^{5}}{1247400}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+x +\frac {x^{2}}{6}+\frac {x^{3}}{90}+\frac {x^{4}}{2520}+\frac {x^{5}}{113400}+O\left (x^{6}\right )\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1+x +\frac {1}{6} x^{2}+\frac {1}{90} x^{3}+\frac {1}{2520} x^{4}+\frac {1}{113400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+c_{2} \left (1+\frac {1}{3} x +\frac {1}{30} x^{2}+\frac {1}{630} x^{3}+\frac {1}{22680} x^{4}+\frac {1}{1247400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+x \left (\frac {2}{3}+\frac {1}{6} x +\frac {1}{126} x^{2}+\frac {1}{4536} x^{3}+\frac {1}{249480} x^{4}+\operatorname {O}\left (x^{5}\right )\right ) \]

Problem 7240

ODE	\[ \boxed {2 y^{\prime \prime } x^{2}-x y^{\prime }+\left (1-x^{2}\right ) y=1} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 1+\frac {x^{2}}{3}+\frac {x^{4}}{63}+O\left (x^{6}\right )+c_{1} x \left (1+\frac {x^{2}}{10}+\frac {x^{4}}{360}+O\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1+\frac {x^{2}}{6}+\frac {x^{4}}{168}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{6} x^{2}+\frac {1}{168} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x \left (1+\frac {1}{10} x^{2}+\frac {1}{360} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (1+\frac {1}{3} x^{2}+\frac {1}{63} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 7241

ODE	\[ \boxed {2 y^{\prime \prime } x^{2}+2 x y^{\prime }-y x=1} \] With the expansion point for the power series method at \(x = 0\).
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 7242

ODE	\[ \boxed {y^{\prime \prime }+\left (-6+x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+3 x^{2}-\frac {1}{6} x^{3}+\frac {3}{2} x^{4}-\frac {1}{5} x^{5}+\frac {11}{36} x^{6}\right ) y \left (0\right )+\left (x +x^{3}-\frac {1}{12} x^{4}+\frac {3}{10} x^{5}-\frac {1}{20} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+3 x^{2}-\frac {1}{6} x^{3}+\frac {3}{2} x^{4}-\frac {1}{5} x^{5}\right ) c_{1} +\left (x +x^{3}-\frac {1}{12} x^{4}+\frac {3}{10} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+3 x^{2}-\frac {1}{6} x^{3}+\frac {3}{2} x^{4}-\frac {1}{5} x^{5}\right ) y \left (0\right )+\left (x +x^{3}-\frac {1}{12} x^{4}+\frac {3}{10} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 7243

ODE	\[ \boxed {y^{\prime \prime } x^{2}+\left (3 x^{2}+2 x \right ) y^{\prime }-2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {3 x}{4}+\frac {9 x^{2}}{20}-\frac {9 x^{3}}{40}+\frac {27 x^{4}}{280}-\frac {81 x^{5}}{2240}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-3 x +\frac {9 x^{2}}{2}-\frac {9 x^{3}}{2}+\frac {27 x^{4}}{8}-\frac {81 x^{5}}{40}+O\left (x^{6}\right )\right )}{x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1-\frac {3}{4} x +\frac {9}{20} x^{2}-\frac {9}{40} x^{3}+\frac {27}{280} x^{4}-\frac {81}{2240} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (12-36 x +54 x^{2}-54 x^{3}+\frac {81}{2} x^{4}-\frac {243}{10} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

Problem 7244

ODE	\[ \boxed {2 y^{\prime \prime } x^{2}-x y^{\prime }+\left (1-x^{2}\right ) y=x^{2}+\cos \left (x \right )} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 1+\frac {x^{2}}{2}+\frac {13 x^{4}}{504}+O\left (x^{6}\right )+c_{1} x \left (1+\frac {x^{2}}{10}+\frac {x^{4}}{360}+O\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1+\frac {x^{2}}{6}+\frac {x^{4}}{168}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{6} x^{2}+\frac {1}{168} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x \left (1+\frac {1}{10} x^{2}+\frac {1}{360} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (1+\frac {1}{2} x^{2}+\frac {13}{504} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 7245

ODE	\[ \boxed {2 y^{\prime \prime } x^{2}-x y^{\prime }+\left (1-x^{2}\right ) y=\cos \left (x \right )} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 1+\frac {x^{2}}{6}+\frac {5 x^{4}}{504}+O\left (x^{6}\right )+c_{1} x \left (1+\frac {x^{2}}{10}+\frac {x^{4}}{360}+O\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1+\frac {x^{2}}{6}+\frac {x^{4}}{168}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{6} x^{2}+\frac {1}{168} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x \left (1+\frac {1}{10} x^{2}+\frac {1}{360} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (1+\frac {1}{6} x^{2}+\frac {5}{504} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 7246

ODE	\[ \boxed {2 y^{\prime \prime } x^{2}-x y^{\prime }+\left (1-x^{2}\right ) y=x^{3}+\cos \left (x \right )} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 1+\frac {x^{2}}{6}+\frac {x^{3}}{10}+\frac {5 x^{4}}{504}+\frac {x^{5}}{360}+O\left (x^{6}\right )+c_{1} x \left (1+\frac {x^{2}}{10}+\frac {x^{4}}{360}+O\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1+\frac {x^{2}}{6}+\frac {x^{4}}{168}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{6} x^{2}+\frac {1}{168} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x \left (1+\frac {1}{10} x^{2}+\frac {1}{360} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (1+\frac {1}{6} x^{2}+\frac {1}{10} x^{3}+\frac {5}{504} x^{4}+\frac {1}{360} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 7247

ODE	\[ \boxed {2 y^{\prime \prime } x^{2}-x y^{\prime }+\left (1-x^{2}\right ) y=\cos \left (x \right ) x^{3}} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {x^{3}}{10}-\frac {x^{5}}{90}+O\left (x^{6}\right )+c_{1} x \left (1+\frac {x^{2}}{10}+\frac {x^{4}}{360}+O\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1+\frac {x^{2}}{6}+\frac {x^{4}}{168}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{6} x^{2}+\frac {1}{168} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x \left (1+\frac {1}{10} x^{2}+\frac {1}{360} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+x^{3} \left (\frac {1}{10}-\frac {1}{90} x^{2}+\operatorname {O}\left (x^{4}\right )\right ) \]

Problem 7248

ODE	\[ \boxed {2 y^{\prime \prime } x^{2}-x y^{\prime }+\left (1-x^{2}\right ) y=\cos \left (x \right ) x^{3}+\sin \left (x \right )^{2}} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {x^{2}}{3}+\frac {x^{3}}{10}-\frac {x^{5}}{90}+O\left (x^{6}\right )+c_{1} x \left (1+\frac {x^{2}}{10}+\frac {x^{4}}{360}+O\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1+\frac {x^{2}}{6}+\frac {x^{4}}{168}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{6} x^{2}+\frac {1}{168} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x \left (1+\frac {1}{10} x^{2}+\frac {1}{360} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+x^{2} \left (\frac {1}{3}+\frac {1}{10} x -\frac {1}{90} x^{3}+\operatorname {O}\left (x^{4}\right )\right ) \]

Problem 7249

ODE	\[ \boxed {2 y^{\prime \prime } x^{2}-x y^{\prime }+\left (1-x^{2}\right ) y=\ln \left (x \right )} \] With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = \left (1+\frac {\left (x -1\right )^{3}}{6}-\frac {5 \left (x -1\right )^{4}}{48}+\frac {37 \left (x -1\right )^{5}}{480}-\frac {323 \left (x -1\right )^{6}}{5760}\right ) y \left (1\right )+\left (x -1+\frac {\left (x -1\right )^{2}}{4}-\frac {\left (x -1\right )^{3}}{24}+\frac {19 \left (x -1\right )^{4}}{192}-\frac {119 \left (x -1\right )^{5}}{1920}+\frac {121 \left (x -1\right )^{6}}{2560}\right ) y^{\prime }\left (1\right )+\frac {\left (x -1\right )^{3}}{12}-\frac {3 \left (x -1\right )^{4}}{32}+\frac {89 \left (x -1\right )^{5}}{960}-\frac {991 \left (x -1\right )^{6}}{11520}+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {\left (x -1\right )^{3}}{6}-\frac {5 \left (x -1\right )^{4}}{48}+\frac {37 \left (x -1\right )^{5}}{480}\right ) y \left (1\right )+\left (x -1+\frac {\left (x -1\right )^{2}}{4}-\frac {\left (x -1\right )^{3}}{24}+\frac {19 \left (x -1\right )^{4}}{192}-\frac {119 \left (x -1\right )^{5}}{1920}\right ) D\left (y \right )\left (1\right )+\frac {\left (x -1\right )^{3}}{12}-\frac {3 \left (x -1\right )^{4}}{32}+\frac {89 \left (x -1\right )^{5}}{960}+O\left (x^{6}\right ) \]

Problem 7250

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

Problem 7251

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

Problem 7252

ODE	\[ \boxed {x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }-x \left (4 x^{2}+3\right ) y^{\prime }+\left (-2 x^{2}+2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1+\frac {6 x^{2}}{7}+\frac {45 x^{4}}{77}+O\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1+\frac {15 x^{2}}{8}+\frac {189 x^{4}}{128}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {15}{8} x^{2}+\frac {189}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{2} \left (1+\frac {6}{7} x^{2}+\frac {45}{77} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 7253

ODE	\[ \boxed {{y^{\prime }}^{2}+y^{2}=\sec \left (x \right )^{4}} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 7254

ODE	\[ \boxed {\left (y-2 x y^{\prime }\right )^{2}-{y^{\prime }}^{3}=0} \]
program solution	\[ y = 0 \] Verified OK. \[ x = \frac {\frac {432 \left (\left (16 \left (x^{3}-\frac {3 y}{16}\right ) \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}-128 x^{6}+160 x^{3} y-27 y^{2}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-2048 x \left (-\frac {\left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (x^{3}-\frac {y}{2}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}}{4}+x \left (-\frac {\left (x^{3}-\frac {15 y}{16}\right ) \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{4}+x^{6}-\frac {7 x^{3} y}{2}+\frac {135 y^{2}}{64}\right )\right )\right ) \sqrt {\frac {{\left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-4 x \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+16 x^{2}\right )}^{2}}{\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}}}}{5}-82944 \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}} c_{1}}{{\left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-4 x \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+16 x^{2}\right )}^{4}} \] Warning, solution could not be verified \[ x = \frac {\left (\left (82944 \left (x^{3}-\frac {3 y}{16}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {-32 x^{3} y+27 y^{2}}-221184 \left (i \sqrt {3}-1\right ) \left (x^{6}-\frac {5 x^{3} y}{4}+\frac {27 y^{2}}{128}\right )\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}+3538944 x \left (\frac {\left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (x^{3}-\frac {y}{2}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}}{2}+x \left (-\frac {3 \left (x^{3}-\frac {15 y}{16}\right ) \left (\frac {\sqrt {3}}{3}+i\right ) \sqrt {-32 x^{3} y+27 y^{2}}}{4}+\left (1+i \sqrt {3}\right ) \left (x^{6}-\frac {7 x^{3} y}{2}+\frac {135 y^{2}}{64}\right )\right )\right )\right ) \sqrt {\frac {{\left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right )}^{2} {\left (-i \left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}-4 x \right ) \sqrt {3}+\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right )}^{2}}{\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}}}-6635520 \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}} c_{1}}{5 \left (16 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}+16 x^{2}+8 x \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}\right )^{4}} \] Warning, solution could not be verified \[ x = \frac {\left (\left (-82944 \left (x^{3}-\frac {3 y}{16}\right ) \left (\frac {\sqrt {3}}{3}+i\right ) \sqrt {-32 x^{3} y+27 y^{2}}+221184 \left (x^{6}-\frac {5 x^{3} y}{4}+\frac {27 y^{2}}{128}\right ) \left (1+i \sqrt {3}\right )\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-3538944 x \left (-\frac {\left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (x^{3}-\frac {y}{2}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}}{2}+x \left (-\frac {3 \left (x^{3}-\frac {15 y}{16}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {-32 x^{3} y+27 y^{2}}}{4}+\left (i \sqrt {3}-1\right ) \left (x^{6}-\frac {7 x^{3} y}{2}+\frac {135 y^{2}}{64}\right )\right )\right )\right ) \sqrt {\frac {{\left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right )}^{2} {\left (i \left (\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}-4 x \right ) \sqrt {3}+\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right )}^{2}}{\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}}}-6635520 \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}} c_{1}}{5 \left (16 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-16 x^{2}-8 x \left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}-\left (108 y-64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}\right )^{4}} \] Warning, solution could not be verified \[ y = 0 \] Verified OK. \[ x = \frac {\frac {432 \left (\left (16 \left (x^{3}-\frac {3 y}{16}\right ) \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}+128 x^{6}-160 x^{3} y+27 y^{2}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}+2048 x \left (\frac {\left (x^{3}-\frac {y}{2}\right ) \left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}}{4}+x \left (\frac {\left (x^{3}-\frac {15 y}{16}\right ) \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{4}+x^{6}-\frac {7 x^{3} y}{2}+\frac {135 y^{2}}{64}\right )\right )\right ) \sqrt {\frac {{\left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}+4 x \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+16 x^{2}\right )}^{2}}{\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}}}}{5}+82944 \left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}} c_{2}}{{\left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}+4 x \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+16 x^{2}\right )}^{4}} \] Warning, solution could not be verified \[ x = \frac {\left (\left (82944 \left (x^{3}-\frac {3 y}{16}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {-32 x^{3} y+27 y^{2}}+221184 \left (i \sqrt {3}-1\right ) \left (x^{6}-\frac {5 x^{3} y}{4}+\frac {27 y^{2}}{128}\right )\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-3538944 x \left (-\frac {\left (x^{3}-\frac {y}{2}\right ) \left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}}{2}+x \left (\frac {3 \left (x^{3}-\frac {15 y}{16}\right ) \left (\frac {\sqrt {3}}{3}+i\right ) \sqrt {-32 x^{3} y+27 y^{2}}}{4}+\left (1+i \sqrt {3}\right ) \left (x^{6}-\frac {7 x^{3} y}{2}+\frac {135 y^{2}}{64}\right )\right )\right )\right ) \sqrt {\frac {{\left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}-4 x \right )}^{2} {\left (i \left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right ) \sqrt {3}-\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right )}^{2}}{\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}}}+6635520 \left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}} c_{2}}{5 \left (16 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}+16 x^{2}-8 x \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}\right )^{4}} \] Warning, solution could not be verified \[ x = \frac {\frac {27648 \sqrt {\frac {{\left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}-4 x \right )}^{2} {\left (i \left (\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+4 x \right ) \sqrt {3}+\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}-4 x \right )}^{2}}{\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}}}\, \left (-\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}} \left (3 \left (x^{3}-\frac {3 y}{16}\right ) \left (\frac {\sqrt {3}}{3}+i\right ) \sqrt {-32 x^{3} y+27 y^{2}}+8 \left (x^{6}-\frac {5 x^{3} y}{4}+\frac {27 y^{2}}{128}\right ) \left (1+i \sqrt {3}\right )\right )+128 x \left (\frac {\left (x^{3}-\frac {y}{2}\right ) \left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}}{2}+x \left (\frac {3 \left (x^{3}-\frac {15 y}{16}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {-32 x^{3} y+27 y^{2}}}{4}+\left (i \sqrt {3}-1\right ) \left (x^{6}-\frac {7 x^{3} y}{2}+\frac {135 y^{2}}{64}\right )\right )\right )\right )}{5}+1327104 \left (x^{3}+\frac {3 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}}{16}-\frac {27 y}{16}\right ) \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}} c_{2}}{{\left (i \sqrt {3}\, \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-16 i \sqrt {3}\, x^{2}+\left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {2}{3}}-8 x \left (-108 y+64 x^{3}+12 \sqrt {3}\, \sqrt {-32 x^{3} y+27 y^{2}}\right )^{\frac {1}{3}}+16 x^{2}\right )}^{4}} \] Warning, solution could not be verified
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ \left [x \left (\textit {\_T} \right ) &= \frac {3 \textit {\_T}^{\frac {5}{2}}+5 c_{1}}{5 \textit {\_T}^{2}}, y \left (\textit {\_T} \right ) &= \frac {\textit {\_T}^{\frac {5}{2}}+10 c_{1}}{5 \textit {\_T}}\right ] \\ \left [x \left (\textit {\_T} \right ) &= \frac {-3 \textit {\_T}^{\frac {5}{2}}+5 c_{1}}{5 \textit {\_T}^{2}}, y \left (\textit {\_T} \right ) &= \frac {-\textit {\_T}^{\frac {5}{2}}+10 c_{1}}{5 \textit {\_T}}\right ] \\ \end{align*}

Problem 7255

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

Problem 7256

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

Problem 7257

ODE	\[ \boxed {4 x y^{\prime \prime }+2 y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {x}{6}+\frac {x^{2}}{120}-\frac {x^{3}}{5040}+\frac {x^{4}}{362880}-\frac {x^{5}}{39916800}+O\left (x^{6}\right )\right )+c_{2} \left (1-\frac {x}{2}+\frac {x^{2}}{24}-\frac {x^{3}}{720}+\frac {x^{4}}{40320}-\frac {x^{5}}{3628800}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1-\frac {1}{6} x +\frac {1}{120} x^{2}-\frac {1}{5040} x^{3}+\frac {1}{362880} x^{4}-\frac {1}{39916800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-\frac {1}{2} x +\frac {1}{24} x^{2}-\frac {1}{720} x^{3}+\frac {1}{40320} x^{4}-\frac {1}{3628800} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 7258

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

Problem 7259

ODE	\[ \boxed {x y^{\prime \prime }+\left (1+x \right ) y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-2 x +\frac {3 x^{2}}{2}-\frac {2 x^{3}}{3}+\frac {5 x^{4}}{24}-\frac {x^{5}}{20}+O\left (x^{6}\right )\right )+c_{2} \left (\left (1-2 x +\frac {3 x^{2}}{2}-\frac {2 x^{3}}{3}+\frac {5 x^{4}}{24}-\frac {x^{5}}{20}+O\left (x^{6}\right )\right ) \ln \left (x \right )+3 x -\frac {13 x^{2}}{4}+\frac {31 x^{3}}{18}-\frac {173 x^{4}}{288}+\frac {187 x^{5}}{1200}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-2 x +\frac {3}{2} x^{2}-\frac {2}{3} x^{3}+\frac {5}{24} x^{4}-\frac {1}{20} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (3 x -\frac {13}{4} x^{2}+\frac {31}{18} x^{3}-\frac {173}{288} x^{4}+\frac {187}{1200} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 7260

ODE	\[ \boxed {x \left (x -1\right ) y^{\prime \prime }+3 x y^{\prime }+y=0} \] 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	\[ y \left (x \right ) = c_{1} x \left (1+2 x +3 x^{2}+4 x^{3}+5 x^{4}+6 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (x +2 x^{2}+3 x^{3}+4 x^{4}+5 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right ) c_{2} +\left (1+3 x +5 x^{2}+7 x^{3}+9 x^{4}+11 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 7261

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

Problem 7262

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

Problem 7263

ODE	\[ \boxed {2 y^{\prime \prime } x^{2}+x y^{\prime }+\left (x -5\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{4}+\frac {\sqrt {41}}{4}} \left (1+\frac {x}{-2-\sqrt {41}}+\frac {x^{2}}{2 \left (2+\sqrt {41}\right ) \left (4+\sqrt {41}\right )}-\frac {x^{3}}{3240+510 \sqrt {41}}+\frac {x^{4}}{187320+29280 \sqrt {41}}-\frac {x^{5}}{600 \left (1561+244 \sqrt {41}\right ) \left (10+\sqrt {41}\right )}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{4}-\frac {\sqrt {41}}{4}} \left (1+\frac {x}{-2+\sqrt {41}}+\frac {x^{2}}{2 \left (-2+\sqrt {41}\right ) \left (-4+\sqrt {41}\right )}+\frac {x^{3}}{-3240+510 \sqrt {41}}+\frac {x^{4}}{187320-29280 \sqrt {41}}-\frac {x^{5}}{600 \left (-1561+244 \sqrt {41}\right ) \left (-10+\sqrt {41}\right )}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{\frac {1}{4}} \left (c_{1} x^{-\frac {\sqrt {41}}{4}} \left (1+\frac {1}{-2+\sqrt {41}} x +\frac {1}{2} \frac {1}{\left (-2+\sqrt {41}\right ) \left (-4+\sqrt {41}\right )} x^{2}+\frac {1}{6} \frac {1}{\left (-2+\sqrt {41}\right ) \left (-4+\sqrt {41}\right ) \left (-6+\sqrt {41}\right )} x^{3}+\frac {1}{24} \frac {1}{\left (-2+\sqrt {41}\right ) \left (-4+\sqrt {41}\right ) \left (-6+\sqrt {41}\right ) \left (-8+\sqrt {41}\right )} x^{4}+\frac {1}{120} \frac {1}{\left (-2+\sqrt {41}\right ) \left (-4+\sqrt {41}\right ) \left (-6+\sqrt {41}\right ) \left (-8+\sqrt {41}\right ) \left (-10+\sqrt {41}\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{\frac {\sqrt {41}}{4}} \left (1+\frac {1}{-2-\sqrt {41}} x +\frac {1}{2} \frac {1}{\left (2+\sqrt {41}\right ) \left (4+\sqrt {41}\right )} x^{2}-\frac {1}{6} \frac {1}{\left (2+\sqrt {41}\right ) \left (4+\sqrt {41}\right ) \left (6+\sqrt {41}\right )} x^{3}+\frac {1}{24} \frac {1}{\left (2+\sqrt {41}\right ) \left (4+\sqrt {41}\right ) \left (6+\sqrt {41}\right ) \left (8+\sqrt {41}\right )} x^{4}-\frac {1}{120} \frac {1}{\left (2+\sqrt {41}\right ) \left (4+\sqrt {41}\right ) \left (6+\sqrt {41}\right ) \left (8+\sqrt {41}\right ) \left (10+\sqrt {41}\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

Problem 7264

ODE	\[ \boxed {2 y^{\prime \prime } x^{2}+2 x y^{\prime }-y x=\sin \left (x \right )} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {x}{2}+\frac {x^{2}}{16}-\frac {5 x^{3}}{864}-\frac {5 x^{4}}{27648}+\frac {1127 x^{5}}{6912000}+O\left (x^{6}\right )+c_{1} \left (\frac {x}{2}+1+\frac {x^{2}}{16}+\frac {x^{3}}{288}+\frac {x^{4}}{9216}+\frac {x^{5}}{460800}+O\left (x^{6}\right )\right )+c_{2} \left (\left (\frac {x}{2}+1+\frac {x^{2}}{16}+\frac {x^{3}}{288}+\frac {x^{4}}{9216}+\frac {x^{5}}{460800}+O\left (x^{6}\right )\right ) \ln \left (x \right )-x -\frac {3 x^{2}}{16}-\frac {11 x^{3}}{864}-\frac {25 x^{4}}{55296}-\frac {137 x^{5}}{13824000}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+\frac {1}{2} x +\frac {1}{16} x^{2}+\frac {1}{288} x^{3}+\frac {1}{9216} x^{4}+\frac {1}{460800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+x \left (\frac {1}{2}+\frac {1}{16} x -\frac {5}{864} x^{2}-\frac {5}{27648} x^{3}+\frac {1127}{6912000} x^{4}+\frac {1127}{497664000} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-x -\frac {3}{16} x^{2}-\frac {11}{864} x^{3}-\frac {25}{55296} x^{4}-\frac {137}{13824000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 7265

ODE	\[ \boxed {2 y^{\prime \prime } x^{2}+2 x y^{\prime }-y x=x \sin \left (x \right )} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {x^{2}}{8}+\frac {x^{3}}{144}-\frac {23 x^{4}}{4608}-\frac {23 x^{5}}{230400}+O\left (x^{6}\right )+c_{1} \left (\frac {x}{2}+1+\frac {x^{2}}{16}+\frac {x^{3}}{288}+\frac {x^{4}}{9216}+\frac {x^{5}}{460800}+O\left (x^{6}\right )\right )+c_{2} \left (\left (\frac {x}{2}+1+\frac {x^{2}}{16}+\frac {x^{3}}{288}+\frac {x^{4}}{9216}+\frac {x^{5}}{460800}+O\left (x^{6}\right )\right ) \ln \left (x \right )-x -\frac {3 x^{2}}{16}-\frac {11 x^{3}}{864}-\frac {25 x^{4}}{55296}-\frac {137 x^{5}}{13824000}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+\frac {1}{2} x +\frac {1}{16} x^{2}+\frac {1}{288} x^{3}+\frac {1}{9216} x^{4}+\frac {1}{460800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+x^{2} \left (\frac {1}{8}+\frac {1}{144} x -\frac {23}{4608} x^{2}-\frac {23}{230400} x^{3}+\operatorname {O}\left (x^{4}\right )\right )+\left (-x -\frac {3}{16} x^{2}-\frac {11}{864} x^{3}-\frac {25}{55296} x^{4}-\frac {137}{13824000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 7266

ODE	\[ \boxed {2 y^{\prime \prime } x^{2}+2 x y^{\prime }-y x=\cos \left (x \right ) \sin \left (x \right )} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {x}{2}+\frac {x^{2}}{16}-\frac {29 x^{3}}{864}-\frac {29 x^{4}}{27648}+\frac {18287 x^{5}}{6912000}+O\left (x^{6}\right )+c_{1} \left (\frac {x}{2}+1+\frac {x^{2}}{16}+\frac {x^{3}}{288}+\frac {x^{4}}{9216}+\frac {x^{5}}{460800}+O\left (x^{6}\right )\right )+c_{2} \left (\left (\frac {x}{2}+1+\frac {x^{2}}{16}+\frac {x^{3}}{288}+\frac {x^{4}}{9216}+\frac {x^{5}}{460800}+O\left (x^{6}\right )\right ) \ln \left (x \right )-x -\frac {3 x^{2}}{16}-\frac {11 x^{3}}{864}-\frac {25 x^{4}}{55296}-\frac {137 x^{5}}{13824000}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+\frac {1}{2} x +\frac {1}{16} x^{2}+\frac {1}{288} x^{3}+\frac {1}{9216} x^{4}+\frac {1}{460800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+x \left (\frac {1}{2}+\frac {1}{16} x -\frac {29}{864} x^{2}-\frac {29}{27648} x^{3}+\frac {18287}{6912000} x^{4}+\frac {18287}{497664000} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-x -\frac {3}{16} x^{2}-\frac {11}{864} x^{3}-\frac {25}{55296} x^{4}-\frac {137}{13824000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 7267

ODE	\[ \boxed {2 y^{\prime \prime } x^{2}+2 x y^{\prime }-y x=x^{3}+x \sin \left (x \right )} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {x^{2}}{8}+\frac {x^{3}}{16}-\frac {5 x^{4}}{1536}-\frac {x^{5}}{15360}+O\left (x^{6}\right )+c_{1} \left (\frac {x}{2}+1+\frac {x^{2}}{16}+\frac {x^{3}}{288}+\frac {x^{4}}{9216}+\frac {x^{5}}{460800}+O\left (x^{6}\right )\right )+c_{2} \left (\left (\frac {x}{2}+1+\frac {x^{2}}{16}+\frac {x^{3}}{288}+\frac {x^{4}}{9216}+\frac {x^{5}}{460800}+O\left (x^{6}\right )\right ) \ln \left (x \right )-x -\frac {3 x^{2}}{16}-\frac {11 x^{3}}{864}-\frac {25 x^{4}}{55296}-\frac {137 x^{5}}{13824000}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+\frac {1}{2} x +\frac {1}{16} x^{2}+\frac {1}{288} x^{3}+\frac {1}{9216} x^{4}+\frac {1}{460800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+x^{2} \left (\frac {1}{8}+\frac {1}{16} x -\frac {5}{1536} x^{2}-\frac {1}{15360} x^{3}+\operatorname {O}\left (x^{4}\right )\right )+\left (-x -\frac {3}{16} x^{2}-\frac {11}{864} x^{3}-\frac {25}{55296} x^{4}-\frac {137}{13824000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 7268

ODE	\[ \boxed {y^{\prime \prime } \cos \left (x \right )+2 x y^{\prime }-y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{6} x^{3}-\frac {1}{40} x^{5}+\frac {1}{180} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{12} x^{4}+\frac {1}{20} x^{5}-\frac {1}{60} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{6} x^{3}-\frac {1}{40} x^{5}\right ) c_{1} +\left (x -\frac {1}{3} x^{3}+\frac {1}{12} x^{4}+\frac {1}{20} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{6} x^{3}-\frac {1}{40} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{12} x^{4}+\frac {1}{20} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 7269

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

Problem 7270

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

Problem 7271

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

Problem 7272

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

Problem 7273

ODE	\[ \boxed {y^{\prime \prime } x^{2}+\left (x^{2}+6 x \right ) y^{\prime }+y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-\frac {x}{6}+\frac {x^{2}}{42}-\frac {x^{3}}{336}+\frac {x^{4}}{3024}-\frac {x^{5}}{30240}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-x +\frac {x^{2}}{2}-\frac {x^{3}}{6}+\frac {x^{4}}{24}-\frac {x^{5}}{120}+O\left (x^{6}\right )\right )}{x^{5}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (1-\frac {1}{6} x +\frac {1}{42} x^{2}-\frac {1}{336} x^{3}+\frac {1}{3024} x^{4}-\frac {1}{30240} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (2880-2880 x +1440 x^{2}-480 x^{3}+120 x^{4}-24 x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{5}} \]

Problem 7274

ODE	\[ \boxed {y^{\prime \prime } x^{2}-x y^{\prime }+\left (x^{2}-8\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{4} \left (1-\frac {x^{2}}{16}+\frac {x^{4}}{640}-\frac {x^{6}}{46080}+O\left (x^{7}\right )\right )+c_{2} \left (-\frac {x^{4} \left (1-\frac {x^{2}}{16}+\frac {x^{4}}{640}-\frac {x^{6}}{46080}+O\left (x^{7}\right )\right ) \ln \left (x \right )}{384}+\frac {1+\frac {x^{2}}{8}+\frac {x^{4}}{64}+O\left (x^{7}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{4} \left (1-\frac {1}{16} x^{2}+\frac {1}{640} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (-86400-10800 x^{2}-1350 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

Problem 7275

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

Problem 7276

ODE	\[ \boxed {y^{\prime \prime } x^{2}-x y^{\prime }-\left (x^{2}+\frac {5}{4}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {5}{2}} \left (1+\frac {x^{2}}{10}+\frac {x^{4}}{280}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {x^{2}}{2}-\frac {x^{4}}{8}+O\left (x^{6}\right )\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{3} \left (1+\frac {1}{10} x^{2}+\frac {1}{280} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (12-6 x^{2}-\frac {3}{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

Problem 7277

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

Problem 7278

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

Problem 7279

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

Problem 7280

ODE	\[ \boxed {2 y^{\prime \prime } x^{2}+5 x y^{\prime }+4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{-\frac {3}{4}+\frac {i \sqrt {23}}{4}} \left (1+O\left (x^{6}\right )\right )+c_{2} x^{-\frac {3}{4}-\frac {i \sqrt {23}}{4}} \left (1+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x^{-\frac {i \sqrt {23}}{4}} c_{1} +x^{\frac {i \sqrt {23}}{4}} c_{2}}{x^{\frac {3}{4}}}+O\left (x^{6}\right ) \]

Problem 7281

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

Problem 7282

ODE	\[ \boxed {y^{\prime \prime } x^{2}-y x=0} \] 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 7283

ODE	\[ \boxed {\left (1-x^{2}\right ) y^{\prime \prime }+y^{\prime }+y=x \,{\mathrm e}^{x}} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{12} x^{4}+\frac {7}{120} x^{5}-\frac {29}{720} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}-\frac {1}{24} x^{4}+\frac {1}{120} x^{5}-\frac {1}{60} x^{6}\right ) y^{\prime }\left (0\right )+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {7 x^{5}}{120}+\frac {x^{6}}{90}+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{12} x^{4}+\frac {7}{120} x^{5}\right ) c_{1} +\left (x -\frac {1}{2} x^{2}-\frac {1}{24} x^{4}+\frac {1}{120} x^{5}\right ) c_{2} +\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {7 x^{5}}{120}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{12} x^{4}+\frac {7}{120} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}-\frac {1}{24} x^{4}+\frac {1}{120} x^{5}\right ) D\left (y \right )\left (0\right )+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {7 x^{5}}{120}+O\left (x^{6}\right ) \]

Problem 7284

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

Problem 7285

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

Problem 7286

ODE	\[ \boxed {\frac {x y^{\prime \prime }}{1-x}+y x=0} \]
program solution	\[ y = -c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (1, 2 \sqrt {x}\right )-c_{2} \sqrt {x}\, \operatorname {BesselY}\left (1, 2 \sqrt {x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {AiryAi}\left (x -1\right )+c_{2} \operatorname {AiryBi}\left (x -1\right ) \]

Problem 7287

ODE	\[ \boxed {\frac {x y^{\prime \prime }}{1-x}+y=\cos \left (x \right )} \]
program solution	\[ y = -c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (1, 2 \sqrt {x}\right )-c_{2} \sqrt {x}\, \operatorname {BesselY}\left (1, 2 \sqrt {x}\right )+\pi \sqrt {x}\, \left (\left (\int _{0}^{x}\frac {\operatorname {BesselY}\left (1, 2 \sqrt {\alpha }\right ) \cos \left (\alpha \right ) \left (\alpha -1\right )}{\sqrt {\alpha }}d \alpha \right ) \operatorname {BesselJ}\left (1, 2 \sqrt {x}\right )-\left (\int _{0}^{x}\frac {\operatorname {BesselJ}\left (1, 2 \sqrt {\alpha }\right ) \cos \left (\alpha \right ) \left (\alpha -1\right )}{\sqrt {\alpha }}d \alpha \right ) \operatorname {BesselY}\left (1, 2 \sqrt {x}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\left (\left (\operatorname {BesselI}\left (0, -x \right )+\operatorname {BesselI}\left (1, -x \right )\right ) \left (\int -\frac {\cos \left (x \right ) \left (\operatorname {BesselK}\left (0, -x \right )-\operatorname {BesselK}\left (1, -x \right )\right ) \left (x -1\right )}{x \left (\operatorname {BesselI}\left (0, x\right ) \left (x +1\right ) \operatorname {BesselK}\left (1, -x \right )+1-\left (x +1\right ) \operatorname {BesselK}\left (0, -x \right ) \operatorname {BesselI}\left (1, x\right )\right )}d x \right )+\left (-\operatorname {BesselK}\left (0, -x \right )+\operatorname {BesselK}\left (1, -x \right )\right ) \left (\int -\frac {\cos \left (x \right ) \left (\operatorname {BesselI}\left (0, x\right )-\operatorname {BesselI}\left (1, x\right )\right ) \left (x -1\right )}{x \left (\operatorname {BesselI}\left (0, x\right ) \left (x +1\right ) \operatorname {BesselK}\left (1, -x \right )+1-\left (x +1\right ) \operatorname {BesselK}\left (0, -x \right ) \operatorname {BesselI}\left (1, x\right )\right )}d x \right )+\operatorname {BesselK}\left (1, -x \right ) c_{1} -\operatorname {BesselK}\left (0, -x \right ) c_{1} -\operatorname {BesselI}\left (0, -x \right ) c_{2} -\operatorname {BesselI}\left (1, -x \right ) c_{2} \right ) x \]

Problem 7288

ODE	\[ \boxed {\frac {x y^{\prime \prime }}{1-x^{2}}+y=0} \]
program solution	\[ y = -c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (1, 2 \sqrt {x}\right )-c_{2} \sqrt {x}\, \operatorname {BesselY}\left (1, 2 \sqrt {x}\right ) \] Verified OK.
Maple solution	\[ \text {No solution found} \]

Problem 7289

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

Problem 7290

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	\[ y = \left (1+\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{30} x^{5}+\frac {1}{144} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}-\frac {1}{12} x^{4}+\frac {1}{120} x^{5}-\frac {1}{120} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] 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 7291

ODE	\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )+2 y \left (t \right )+2 t +1\\ y^{\prime }\left (t \right )&=5 x \left (t \right )+y \left (t \right )+3 t -1 \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{\left (1+\sqrt {10}\right ) t} c_{2} +{\mathrm e}^{-\left (-1+\sqrt {10}\right ) t} c_{1} -\frac {4 t}{9}+\frac {17}{81} \\ y \left (t \right ) &= \frac {{\mathrm e}^{\left (1+\sqrt {10}\right ) t} c_{2} \sqrt {10}}{2}-\frac {{\mathrm e}^{-\left (-1+\sqrt {10}\right ) t} c_{1} \sqrt {10}}{2}-\frac {7 t}{9}-\frac {67}{81} \\ \end{align*}

Problem 7292

ODE	\[ \boxed {y^{\prime \prime }+20 y^{\prime }+500 y=100000 \cos \left (100 x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-10 x} \cos \left (20 x \right )+\frac {c_{2} {\mathrm e}^{-10 x} \sin \left (20 x \right )}{20}-\frac {3800 \cos \left (100 x \right )}{377}+\frac {800 \sin \left (100 x \right )}{377} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-10 x} \sin \left (20 x \right ) c_{2} +{\mathrm e}^{-10 x} \cos \left (20 x \right ) c_{1} -\frac {3800 \cos \left (100 x \right )}{377}+\frac {800 \sin \left (100 x \right )}{377} \]

Problem 7293

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

Problem 7294

ODE	\[ \boxed {y^{\prime \prime }-A y^{\frac {2}{3}}=0} \]
program solution	\[ \int _{}^{y}\frac {5}{\sqrt {30 A \,\textit {\_a}^{\frac {5}{3}}+50 c_{1}}}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}-\frac {5}{\sqrt {30 A \,\textit {\_a}^{\frac {5}{3}}+50 c_{1}}}d \textit {\_a} = x +c_{3} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ -5 \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {30 \textit {\_a}^{\frac {5}{3}} A -5 c_{1}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ 5 \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {30 \textit {\_a}^{\frac {5}{3}} A -5 c_{1}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*}

Problem 7295

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

Problem 7296

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

Problem 7297

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

Problem 7298

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

Problem 7299

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

Problem 7300

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