2.17.148 Problems 14701 to 14800

Problem 14701

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+y^{\prime \prime }=\sec \left (t \right )^{2}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1, y^{\prime \prime }\left (0\right ) = 0, y^{\prime \prime \prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {i {\mathrm e}^{-i t} \ln \left ({\mathrm e}^{i t}-i\right )}{2}-\frac {i {\mathrm e}^{-i t} \ln \left ({\mathrm e}^{i t}+i\right )}{2}+\frac {\left (-\pi -2\right ) {\mathrm e}^{-i t}}{4}+{\mathrm e}^{i t} \arctan \left ({\mathrm e}^{i t}\right )+\frac {\left (-\pi -2\right ) {\mathrm e}^{i t}}{4}+t -\ln \left (\cos \left (t \right )\right )+1 \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (\int _{0}^{t}\left (\left (-{\mathrm e}^{i \textit {\_z1}}-{\mathrm e}^{-i \textit {\_z1}}\right ) \ln \left (\frac {i {\mathrm e}^{i \textit {\_z1}}-1}{-{\mathrm e}^{i \textit {\_z1}}+i}\right )-2 i \ln \left ({\mathrm e}^{i \textit {\_z1}}\right )-2 \textit {\_z1} +2 \sin \left (\textit {\_z1} \right )\right )d \textit {\_z1} \right )}{2}+t \]

Problem 14702

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

Problem 14703

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

Problem 14704

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

Problem 14705

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

Problem 14706

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

Problem 14707

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

Problem 14708

ODE	\[ \boxed {t y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }=\frac {45}{8 t^{\frac {7}{2}}}} \] With initial conditions \begin {align*} [y \left (1\right ) = 0, y^{\prime }\left (1\right ) = 0, y^{\prime \prime }\left (1\right ) = 1, y^{\prime \prime \prime }\left (1\right ) = 0] \end {align*}
program solution
Maple solution	\[ y \left (t \right ) = \frac {13 t^{2}}{8}+\frac {2}{\sqrt {t}}-\frac {15 t \ln \left (t \right )}{4}+\frac {3 t}{2}-\frac {41}{8} \]

Problem 14709

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }-8 y^{\prime } x +5 y=0} \]
program solution	\[ y = c_{1} \sqrt {x}+\frac {c_{2} x^{\frac {5}{2}}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (c_{1} x^{2}+c_{2} \right ) \]

Problem 14710

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

Problem 14711

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }-8 y^{\prime } x +8 y=0} \]
program solution	\[ y = c_{1} x +\frac {1}{3} c_{2} x^{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x \left (c_{1} x^{3}+c_{2} \right ) \]

Problem 14712

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }-7 y^{\prime } x +7 y=0} \]
program solution	\[ y = c_{1} x +\frac {2 c_{2} x^{\frac {7}{2}}}{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{\frac {7}{2}} c_{1} +c_{2} x \]

Problem 14713

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+17 y=0} \]
program solution	\[ y = c_{1} x^{\frac {1}{2}-2 i}-\frac {i c_{2} x^{\frac {1}{2}+2 i}}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (c_{1} \sin \left (2 \ln \left (x \right )\right )+c_{2} \cos \left (2 \ln \left (x \right )\right )\right ) \]

Problem 14714

ODE	\[ \boxed {9 x^{2} y^{\prime \prime }-9 y^{\prime } x +10 y=0} \]
program solution	\[ y = c_{1} x^{1-\frac {i}{3}}-\frac {3 i c_{2} x^{1+\frac {i}{3}}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x \left (c_{1} \sin \left (\frac {\ln \left (x \right )}{3}\right )+c_{2} \cos \left (\frac {\ln \left (x \right )}{3}\right )\right ) \]

Problem 14715

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }-2 y^{\prime } x +20 y=0} \]
program solution	\[ y = x^{1-3 i} c_{1} -\frac {i c_{2} x^{1+3 i}}{6} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x \left (c_{1} \sin \left (3 \ln \left (x \right )\right )+c_{2} \cos \left (3 \ln \left (x \right )\right )\right ) \]

Problem 14716

ODE	\[ \boxed {x^{2} y^{\prime \prime }-5 y^{\prime } x +10 y=0} \]
program solution	\[ y = c_{1} x^{3-i}-\frac {i c_{2} x^{3+i}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{3} \left (c_{1} \sin \left (\ln \left (x \right )\right )+c_{2} \cos \left (\ln \left (x \right )\right )\right ) \]

Problem 14717

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+8 y^{\prime } x +y=0} \]
program solution	\[ y = \frac {c_{1}}{\sqrt {x}}+\frac {c_{2} \ln \left (x \right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} +c_{2} \ln \left (x \right )}{\sqrt {x}} \]

Problem 14718

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

Problem 14719

ODE	\[ \boxed {x^{2} y^{\prime \prime }-5 y^{\prime } x +9 y=0} \]
program solution	\[ y = c_{1} x^{3}+c_{2} x^{3} \ln \left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{3} \left (c_{1} +c_{2} \ln \left (x \right )\right ) \]

Problem 14720

ODE	\[ \boxed {x^{2} y^{\prime \prime }+7 y^{\prime } x +9 y=0} \]
program solution	\[ y = \frac {c_{1}}{x^{3}}+\frac {c_{2} \ln \left (x \right )}{x^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} +c_{2} \ln \left (x \right )}{x^{3}} \]

Problem 14721

ODE	\[ \boxed {x^{3} y^{\prime \prime \prime }+22 x^{2} y^{\prime \prime }+124 y^{\prime } x +140 y=0} \]
program solution	\[ y = \frac {c_{1}}{x^{2}}+\frac {c_{2}}{x^{7}}+\frac {c_{3}}{x^{10}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{3} x^{8}+c_{1} x^{3}+c_{2}}{x^{10}} \]

Problem 14722

ODE	\[ \boxed {x^{3} y^{\prime \prime \prime }-4 x^{2} y^{\prime \prime }-46 y^{\prime } x +100 y=0} \]
program solution	\[ y = c_{1} x^{2}+\frac {c_{2}}{x^{5}}+c_{3} x^{10} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{15}+c_{3} x^{7}+c_{1}}{x^{5}} \]

Problem 14723

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

Problem 14724

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

Problem 14725

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

Problem 14726

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

Problem 14727

ODE	\[ \boxed {x^{3} y^{\prime \prime \prime }+6 x^{2} y^{\prime \prime }+7 y^{\prime } x +y=0} \]
program solution	\[ y = \frac {c_{1}}{x}+\frac {c_{2} \ln \left (x \right )}{x}+\frac {c_{3} \ln \left (x \right )^{2}}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} +c_{2} \ln \left (x \right )+c_{3} \ln \left (x \right )^{2}}{x} \]

Problem 14728

ODE	\[ \boxed {x^{3} y^{\prime \prime \prime \prime }+6 x^{2} y^{\prime \prime \prime }+7 x y^{\prime \prime }+y^{\prime }=0} \]
program solution	\[ y = \frac {\ln \left (x \right )^{3} c_{3}}{3}+\frac {\ln \left (x \right )^{2} c_{2}}{2}+c_{1} \ln \left (x \right )+c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{4} \ln \left (x \right )^{3}+c_{3} \ln \left (x \right )^{2}+c_{2} \ln \left (x \right )+c_{1} \]

Problem 14729

ODE	\[ \boxed {x^{2} y^{\prime \prime }+5 y^{\prime } x +4 y=\frac {1}{x^{5}}} \]
program solution	\[ y = \frac {c_{2} \ln \left (x \right )+c_{1}}{x^{2}}+\frac {1}{9 x^{5}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2}}{x^{2}}+\frac {\ln \left (x \right ) c_{1}}{x^{2}}+\frac {1}{9 x^{5}} \]

Problem 14730

ODE	\[ \boxed {x^{2} y^{\prime \prime }-5 y^{\prime } x +9 y=x^{3}} \]
program solution	\[ y = x^{3} \left (c_{2} \ln \left (x \right )+c_{1} \right )+\frac {\ln \left (x \right )^{2} x^{3}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{3} \left (c_{2} +c_{1} \ln \left (x \right )+\frac {\ln \left (x \right )^{2}}{2}\right ) \]

Problem 14731

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +y=\frac {1}{x^{2}}} \]
program solution	\[ y = x^{-i} c_{1} -\frac {i c_{2} x^{i}}{2}+\left (i x^{i}+x^{-i}\right ) \operatorname {undefined} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{2} \sin \left (\ln \left (x \right )\right )+\cos \left (\ln \left (x \right )\right ) c_{1} +\frac {1}{5 x^{2}} \]

Problem 14732

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

Problem 14733

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

Problem 14734

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x -16 y=\ln \left (x \right )} \]
program solution	\[ y = \frac {c_{1}}{x^{4}}+\frac {c_{2} x^{4}}{8}-\frac {\ln \left (x \right )}{16} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2}}{x^{4}}+x^{4} c_{1} -\frac {\ln \left (x \right )}{16} \]

Problem 14735

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

Problem 14736

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +36 y=x^{2}} \]
program solution	\[ y = x^{-6 i} c_{1} -\frac {i c_{2} x^{6 i}}{12}+\frac {x^{2}}{40} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sin \left (6 \ln \left (x \right )\right ) c_{2} +\cos \left (6 \ln \left (x \right )\right ) c_{1} +\frac {x^{2}}{40} \]

Problem 14737

ODE	\[ \boxed {x^{3} y^{\prime \prime \prime }+3 x^{2} y^{\prime \prime }-11 y^{\prime } x +16 y=\frac {1}{x^{3}}} \]
program solution	\[ y = c_{1} x^{2}+c_{2} \ln \left (x \right ) x^{2}+\frac {c_{3}}{x^{4}}+\frac {1}{25 x^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {25 c_{3} \ln \left (x \right ) x^{6}+25 c_{1} x^{6}+25 c_{2} +x}{25 x^{4}} \]

Problem 14738

ODE	\[ \boxed {x^{3} y^{\prime \prime \prime }+16 x^{2} y^{\prime \prime }+70 y^{\prime } x +80 y=\frac {1}{x^{13}}} \]
program solution	\[ y = \frac {c_{2} x \ln \left (x \right )+c_{1} x +c_{3}}{x^{5}}-\frac {1}{648 x^{13}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {1}{648 x^{13}}+\frac {c_{1}}{x^{4}}+\frac {c_{2}}{x^{5}}+\frac {c_{3} \ln \left (x \right )}{x^{4}} \]

Problem 14739

ODE	\[ \boxed {3 x^{2} y^{\prime \prime }-4 y^{\prime } x +2 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 2, y^{\prime }\left (1\right ) = 1] \end {align*}
program solution	\[ y = \frac {x^{2}}{5}+\frac {9 x^{\frac {1}{3}}}{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {9 x^{\frac {1}{3}}}{5}+\frac {x^{2}}{5} \]

Problem 14740

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }-7 y^{\prime } x +7 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = -1, y^{\prime }\left (1\right ) = 1] \end {align*}
program solution	\[ y = \frac {4 x^{\frac {7}{2}}}{5}-\frac {9 x}{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {4 x^{\frac {7}{2}}}{5}-\frac {9 x}{5} \]

Problem 14741

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

Problem 14742

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

Problem 14743

ODE	\[ \boxed {x^{3} y^{\prime \prime \prime }+10 x^{2} y^{\prime \prime }-20 y^{\prime } x +20 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 0, y^{\prime }\left (1\right ) = -1, y^{\prime \prime }\left (1\right ) = 1] \end {align*}
program solution	\[ y = -\frac {33 x^{12}-32 x^{11}-1}{44 x^{10}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {8 x}{11}+\frac {1}{44 x^{10}}-\frac {3 x^{2}}{4} \]

Problem 14744

ODE	\[ \boxed {x^{3} y^{\prime \prime \prime }+15 x^{2} y^{\prime \prime }+54 y^{\prime } x +42 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 5, y^{\prime }\left (1\right ) = 0, y^{\prime \prime }\left (1\right ) = 0] \end {align*}
program solution	\[ y = \frac {42 x^{5}-35 x^{4}+3}{2 x^{7}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {35}{2 x^{3}}+\frac {3}{2 x^{7}}+\frac {21}{x^{2}} \]

Problem 14745

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

Problem 14746

ODE	\[ \boxed {x^{3} y^{\prime \prime \prime }-6 x^{2} y^{\prime \prime }+17 y^{\prime } x -17 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = -2, y^{\prime }\left (1\right ) = 0, y^{\prime \prime }\left (1\right ) = 0] \end {align*}
program solution	\[ y = -\frac {11 x^{4} \sin \left (\ln \left (x \right )\right )}{5}+\frac {7 x^{4} \cos \left (\ln \left (x \right )\right )}{5}-\frac {17 x}{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-11 \sin \left (\ln \left (x \right )\right )+7 \cos \left (\ln \left (x \right )\right )\right ) x^{4}}{5}-\frac {17 x}{5} \]

Problem 14747

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }+3 y^{\prime } x -y=\frac {1}{x^{2}}} \] With initial conditions \begin {align*} [y \left (1\right ) = 0, y^{\prime }\left (1\right ) = 2] \end {align*}
program solution	\[ y = \frac {22 x^{\frac {5}{2}}-25 x +3}{15 x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {22 x^{\frac {5}{2}}-25 x +3}{15 x^{2}} \]

Problem 14748

ODE	\[ \boxed {x^{2} y^{\prime \prime }+4 y^{\prime } x +2 y=\ln \left (x \right )} \] With initial conditions \begin {align*} [y \left (1\right ) = 2, y^{\prime }\left (1\right ) = 0] \end {align*}
program solution	\[ y = \frac {2 \ln \left (x \right ) x^{2}-3 x^{2}+20 x -9}{4 x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {9}{4 x^{2}}+\frac {5}{x}+\frac {\ln \left (x \right )}{2}-\frac {3}{4} \]

Problem 14749

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+y=x^{3}} \] With initial conditions \begin {align*} [y \left (1\right ) = 1, y^{\prime }\left (1\right ) = -1] \end {align*}
program solution	\[ y = \frac {8 \left (3-5 \ln \left (x \right )\right ) \sqrt {x}}{25}+\frac {x^{3}}{25} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {8 \left (3-5 \ln \left (x \right )\right ) \sqrt {x}}{25}+\frac {x^{3}}{25} \]

Problem 14750

ODE	\[ \boxed {9 x^{2} y^{\prime \prime }+27 y^{\prime } x +10 y=\frac {1}{x}} \] With initial conditions \begin {align*} [y \left (1\right ) = 0, y^{\prime }\left (1\right ) = -1] \end {align*}
program solution	N/A
Maple solution	\[ y \left (x \right ) = \frac {-3 \sin \left (\frac {\ln \left (x \right )}{3}\right )-\cos \left (\frac {\ln \left (x \right )}{3}\right )+1}{x} \]

Problem 14751

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

Problem 14752

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

Problem 14753

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

Problem 14754

ODE	\[ \boxed {x^{3} y^{\prime \prime \prime }+3 x^{2} y^{\prime \prime }+37 y^{\prime } x=0} \]
program solution	\[ y = c_{1} +c_{2} \cos \left (6 \ln \left (x \right )\right )+c_{3} \sin \left (6 \ln \left (x \right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} +\sin \left (6 \ln \left (x \right )\right ) c_{2} +c_{3} \cos \left (6 \ln \left (x \right )\right ) \]

Problem 14755

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

Problem 14756

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

Problem 14757

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

Problem 14758

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

Problem 14759

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

Problem 14760

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

Problem 14761

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

Problem 14762

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

Problem 14763

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

Problem 14764

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

Problem 14765

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

Problem 14766

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +y=x^{2}} \]
program solution	\[ y = x^{-i} c_{1} -\frac {i c_{2} x^{i}}{2}+\frac {x^{2}}{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{2} \sin \left (\ln \left (x \right )\right )+\cos \left (\ln \left (x \right )\right ) c_{1} +\frac {x^{2}}{5} \]

Problem 14767

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

Problem 14768

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

Problem 14769

ODE	\[ \boxed {x^{3} y^{\prime \prime \prime }+16 x^{2} y^{\prime \prime }+79 y^{\prime } x +125 y=0} \]
program solution	\[ y = \frac {c_{1}}{x^{5}}+\frac {c_{2} \cos \left (3 \ln \left (x \right )\right )+c_{3} \sin \left (3 \ln \left (x \right )\right )}{x^{4}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} \sin \left (3 \ln \left (x \right )\right ) x +c_{3} \cos \left (3 \ln \left (x \right )\right ) x +c_{1}}{x^{5}} \]

Problem 14770

ODE	\[ \boxed {x^{4} y^{\prime \prime \prime \prime }+5 x^{3} y^{\prime \prime \prime }-12 x^{2} y^{\prime \prime }-12 y^{\prime } x +48 y=0} \]
program solution	\[ y = \frac {c_{1}}{x^{2}}+\frac {c_{2}}{x^{3}}+x^{2} c_{3} +c_{4} x^{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{7}+c_{3} x^{5}+c_{4} x +c_{1}}{x^{3}} \]

Problem 14771

ODE	\[ \boxed {x^{4} y^{\prime \prime \prime \prime }+14 x^{3} y^{\prime \prime \prime }+55 x^{2} y^{\prime \prime }+65 y^{\prime } x +15 y=0} \]
program solution	\[ y = \frac {c_{1}}{x}+\frac {c_{2}}{x^{3}}+\frac {c_{3} \cos \left (\ln \left (x \right )\right )+c_{4} \sin \left (\ln \left (x \right )\right )}{x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{3} \sin \left (\ln \left (x \right )\right ) x +c_{4} \cos \left (\ln \left (x \right )\right ) x +c_{2} x^{2}+c_{1}}{x^{3}} \]

Problem 14772

ODE	\[ \boxed {x^{4} y^{\prime \prime \prime \prime }+8 x^{3} y^{\prime \prime \prime }+27 x^{2} y^{\prime \prime }+35 y^{\prime } x +45 y=0} \]
program solution	\[ y = c_{1} \cos \left (3 \ln \left (x \right )\right )+c_{2} \sin \left (3 \ln \left (x \right )\right )+\frac {c_{3} \cos \left (2 \ln \left (x \right )\right )+c_{4} \sin \left (2 \ln \left (x \right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \sin \left (2 \ln \left (x \right )\right )}{x}+\frac {c_{2} \cos \left (2 \ln \left (x \right )\right )}{x}+c_{3} \sin \left (3 \ln \left (x \right )\right )+c_{4} \cos \left (3 \ln \left (x \right )\right ) \]

Problem 14773

ODE	\[ \boxed {x^{4} y^{\prime \prime \prime \prime }+10 x^{3} y^{\prime \prime \prime }+27 x^{2} y^{\prime \prime }+21 y^{\prime } x +4 y=0} \]
program solution	\[ y = \frac {c_{1} \cos \left (\ln \left (x \right )\right )+c_{2} \sin \left (\ln \left (x \right )\right )}{x}+\frac {\ln \left (x \right ) \left (c_{3} \cos \left (\ln \left (x \right )\right )+c_{4} \sin \left (\ln \left (x \right )\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (c_{4} \ln \left (x \right )+c_{2} \right ) \cos \left (\ln \left (x \right )\right )+\sin \left (\ln \left (x \right )\right ) \left (c_{3} \ln \left (x \right )+c_{1} \right )}{x} \]

Problem 14774

ODE	\[ \boxed {x^{3} y^{\prime \prime \prime }+9 x^{2} y^{\prime \prime }+44 y^{\prime } x +58 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 2, y^{\prime }\left (1\right ) = 10, y^{\prime \prime }\left (1\right ) = -2] \end {align*}
program solution	\[ y = \frac {70 \sin \left (5 \ln \left (x \right )\right )-56 \cos \left (5 \ln \left (x \right )\right )+106}{25 x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\frac {106}{25}+\frac {14 \sin \left (5 \ln \left (x \right )\right )}{5}-\frac {56 \cos \left (5 \ln \left (x \right )\right )}{25}}{x^{2}} \]

Problem 14775

ODE	\[ \boxed {6 x^{2} y^{\prime \prime }+5 y^{\prime } x -y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = a, y^{\prime }\left (1\right ) = b] \end {align*}
program solution	\[ y = \frac {\frac {2 \left (3 b +a \right ) x^{\frac {5}{6}}}{5}+\frac {3 a}{5}-\frac {6 b}{5}}{x^{\frac {1}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\frac {2 \left (a +3 b \right ) x^{\frac {5}{6}}}{5}+\frac {3 a}{5}-\frac {6 b}{5}}{x^{\frac {1}{3}}} \]

Problem 14776

ODE	\[ \boxed {x^{2} y^{\prime \prime }-2 y^{\prime } x +7 y=0} \] With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = \left (1-\frac {7 \left (x -1\right )^{2}}{2}+\frac {35 \left (x -1\right )^{4}}{24}-\frac {7 \left (x -1\right )^{5}}{6}+\frac {91 \left (x -1\right )^{6}}{144}\right ) y \left (1\right )+\left (x -1+\left (x -1\right )^{2}-\frac {5 \left (x -1\right )^{3}}{6}+\frac {7 \left (x -1\right )^{5}}{24}-\frac {7 \left (x -1\right )^{6}}{24}\right ) y^{\prime }\left (1\right )+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {7 \left (-1+x \right )^{2}}{2}+\frac {35 \left (-1+x \right )^{4}}{24}-\frac {7 \left (-1+x \right )^{5}}{6}\right ) y \left (1\right )+\left (-1+x +\left (-1+x \right )^{2}-\frac {5 \left (-1+x \right )^{3}}{6}+\frac {7 \left (-1+x \right )^{5}}{24}\right ) D\left (y \right )\left (1\right )+O\left (x^{6}\right ) \]

Problem 14777

ODE	\[ \boxed {\left (x -2\right ) y^{\prime \prime }+y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = -2\).
program solution	\[ y = \left (1-\frac {\left (x +2\right )^{2}}{8}-\frac {\left (x +2\right )^{3}}{48}-\frac {\left (x +2\right )^{4}}{768}+\frac {\left (x +2\right )^{6}}{92160}\right ) y \left (-2\right )+\left (x +2+\frac {\left (x +2\right )^{2}}{8}-\frac {\left (x +2\right )^{3}}{48}-\frac {5 \left (x +2\right )^{4}}{768}-\frac {\left (x +2\right )^{5}}{960}-\frac {\left (x +2\right )^{6}}{6144}\right ) y^{\prime }\left (-2\right )+O\left (\left (x +2\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {\left (x +2\right )^{2}}{8}-\frac {\left (x +2\right )^{3}}{48}-\frac {\left (x +2\right )^{4}}{768}\right ) y \left (-2\right )+\left (x +2+\frac {\left (x +2\right )^{2}}{8}-\frac {\left (x +2\right )^{3}}{48}-\frac {5 \left (x +2\right )^{4}}{768}-\frac {\left (x +2\right )^{5}}{960}\right ) D\left (y \right )\left (-2\right )+O\left (x^{6}\right ) \]

Problem 14778

ODE	\[ \boxed {\left (x^{2}-4\right ) y^{\prime \prime }+16 y^{\prime } \left (x +2\right )-y=0} \] With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = \left (1-\frac {\left (x -1\right )^{2}}{6}-\frac {25 \left (x -1\right )^{3}}{27}-\frac {2699 \left (x -1\right )^{4}}{648}-\frac {6404 \left (x -1\right )^{5}}{405}-\frac {3071417 \left (x -1\right )^{6}}{58320}\right ) y \left (1\right )+\left (x -1+8 \left (x -1\right )^{2}+\frac {815 \left (x -1\right )^{3}}{18}+\frac {10991 \left (x -1\right )^{4}}{54}+\frac {834547 \left (x -1\right )^{5}}{1080}+\frac {12507983 \left (x -1\right )^{6}}{4860}\right ) y^{\prime }\left (1\right )+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {\left (-1+x \right )^{2}}{6}-\frac {25 \left (-1+x \right )^{3}}{27}-\frac {2699 \left (-1+x \right )^{4}}{648}-\frac {6404 \left (-1+x \right )^{5}}{405}\right ) y \left (1\right )+\left (-1+x +8 \left (-1+x \right )^{2}+\frac {815 \left (-1+x \right )^{3}}{18}+\frac {10991 \left (-1+x \right )^{4}}{54}+\frac {834547 \left (-1+x \right )^{5}}{1080}\right ) D\left (y \right )\left (1\right )+O\left (x^{6}\right ) \]

Problem 14779

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }-18 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+9 x^{2}-9 x^{3}+\frac {81}{4} x^{4}-\frac {81}{4} x^{5}+\frac {891}{40} x^{6}\right ) y \left (0\right )+\left (x -\frac {3}{2} x^{2}+\frac {9}{2} x^{3}-\frac {45}{8} x^{4}+\frac {297}{40} x^{5}-\frac {567}{80} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+9 x^{2}-9 x^{3}+\frac {81}{4} x^{4}-\frac {81}{4} x^{5}\right ) c_{1} +\left (x -\frac {3}{2} x^{2}+\frac {9}{2} x^{3}-\frac {45}{8} x^{4}+\frac {297}{40} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+9 x^{2}-9 x^{3}+\frac {81}{4} x^{4}-\frac {81}{4} x^{5}\right ) y \left (0\right )+\left (x -\frac {3}{2} x^{2}+\frac {9}{2} x^{3}-\frac {45}{8} x^{4}+\frac {297}{40} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 14780

ODE	\[ \boxed {y^{\prime \prime }-11 y^{\prime }+30 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-15 x^{2}-55 x^{3}-\frac {455}{4} x^{4}-\frac {671}{4} x^{5}-\frac {4651}{24} x^{6}\right ) y \left (0\right )+\left (x +\frac {11}{2} x^{2}+\frac {91}{6} x^{3}+\frac {671}{24} x^{4}+\frac {4651}{120} x^{5}+\frac {31031}{720} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-15 x^{2}-55 x^{3}-\frac {455}{4} x^{4}-\frac {671}{4} x^{5}\right ) c_{1} +\left (x +\frac {11}{2} x^{2}+\frac {91}{6} x^{3}+\frac {671}{24} x^{4}+\frac {4651}{120} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-15 x^{2}-55 x^{3}-\frac {455}{4} x^{4}-\frac {671}{4} x^{5}\right ) y \left (0\right )+\left (x +\frac {11}{2} x^{2}+\frac {91}{6} x^{3}+\frac {671}{24} x^{4}+\frac {4651}{120} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 14781

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

Problem 14782

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }-2 y={\mathrm e}^{-x}} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+x^{2}+\frac {1}{3} x^{3}+\frac {1}{4} x^{4}+\frac {1}{12} x^{5}+\frac {11}{360} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{2} x^{3}+\frac {5}{24} x^{4}+\frac {11}{120} x^{5}+\frac {7}{240} x^{6}\right ) y^{\prime }\left (0\right )+\frac {x^{2}}{2}+\frac {x^{4}}{8}+\frac {x^{5}}{60}+\frac {x^{6}}{80}+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+x^{2}+\frac {1}{3} x^{3}+\frac {1}{4} x^{4}+\frac {1}{12} x^{5}\right ) c_{1} +\left (x +\frac {1}{2} x^{2}+\frac {1}{2} x^{3}+\frac {5}{24} x^{4}+\frac {11}{120} x^{5}\right ) c_{2} +\frac {x^{2}}{2}+\frac {x^{4}}{8}+\frac {x^{5}}{60}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+x^{2}+\frac {1}{3} x^{3}+\frac {1}{4} x^{4}+\frac {1}{12} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{2} x^{3}+\frac {5}{24} x^{4}+\frac {11}{120} x^{5}\right ) D\left (y \right )\left (0\right )+\frac {x^{2}}{2}+\frac {x^{4}}{8}+\frac {x^{5}}{60}+O\left (x^{6}\right ) \]

Problem 14783

ODE	\[ \boxed {\left (-2 x -2\right ) y^{\prime \prime }+2 y^{\prime }+4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+x^{2}+\frac {1}{6} x^{4}-\frac {1}{15} x^{5}+\frac {2}{45} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{3} x^{3}+\frac {1}{30} x^{5}-\frac {1}{60} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+x^{2}+\frac {1}{6} x^{4}-\frac {1}{15} x^{5}\right ) c_{1} +\left (x +\frac {1}{2} x^{2}+\frac {1}{3} x^{3}+\frac {1}{30} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+x^{2}+\frac {1}{6} x^{4}-\frac {1}{15} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{3} x^{3}+\frac {1}{30} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 14784

ODE	\[ \boxed {\left (2+3 x \right ) y^{\prime \prime }+3 y^{\prime } x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = y \left (0\right )+\left (x -\frac {1}{4} x^{3}+\frac {3}{16} x^{4}-\frac {9}{80} x^{5}+\frac {3}{40} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = c_{1} +\left (x -\frac {1}{4} x^{3}+\frac {3}{16} x^{4}-\frac {9}{80} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = y \left (0\right )+\left (x -\frac {1}{4} x^{3}+\frac {3}{16} x^{4}-\frac {9}{80} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 14785

ODE	\[ \boxed {\left (3 x +1\right ) y^{\prime \prime }-3 y^{\prime }-2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+x^{2}+\frac {1}{6} x^{4}-\frac {1}{5} x^{5}+\frac {14}{45} x^{6}\right ) y \left (0\right )+\left (x +\frac {3}{2} x^{2}+\frac {1}{3} x^{3}+\frac {1}{30} x^{5}-\frac {1}{20} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+x^{2}+\frac {1}{6} x^{4}-\frac {1}{5} x^{5}\right ) c_{1} +\left (x +\frac {3}{2} x^{2}+\frac {1}{3} x^{3}+\frac {1}{30} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+x^{2}+\frac {1}{6} x^{4}-\frac {1}{5} x^{5}\right ) y \left (0\right )+\left (x +\frac {3}{2} x^{2}+\frac {1}{3} x^{3}+\frac {1}{30} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 14786

ODE	\[ \boxed {\left (-x^{2}+2\right ) y^{\prime \prime }+2 y^{\prime } \left (x -1\right )+4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-x^{2}-\frac {1}{3} x^{3}+\frac {1}{6} x^{4}+\frac {1}{15} x^{5}+\frac {1}{90} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}-\frac {1}{3} x^{3}-\frac {5}{24} x^{4}-\frac {1}{120} x^{5}-\frac {1}{720} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-x^{2}-\frac {1}{3} x^{3}+\frac {1}{6} x^{4}+\frac {1}{15} x^{5}\right ) c_{1} +\left (x +\frac {1}{2} x^{2}-\frac {1}{3} x^{3}-\frac {5}{24} x^{4}-\frac {1}{120} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-x^{2}-\frac {1}{3} x^{3}+\frac {1}{6} x^{4}+\frac {1}{15} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}-\frac {1}{3} x^{3}-\frac {5}{24} x^{4}-\frac {1}{120} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 14787

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

Problem 14788

ODE	\[ \boxed {\left (2 x^{2}+2\right ) y^{\prime \prime }+2 y^{\prime } x -3 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {3}{4} x^{2}-\frac {5}{32} x^{4}+\frac {29}{384} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{12} x^{3}-\frac {1}{32} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {3}{4} x^{2}-\frac {5}{32} x^{4}\right ) c_{1} +\left (x +\frac {1}{12} x^{3}-\frac {1}{32} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {3}{4} x^{2}-\frac {5}{32} x^{4}\right ) y \left (0\right )+\left (x +\frac {1}{12} x^{3}-\frac {1}{32} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 14789

ODE	\[ \boxed {\left (3-2 x \right ) y^{\prime \prime }+2 y^{\prime }-2 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = -2] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 3-2 x +\frac {5 x^{2}}{3}-\frac {2 x^{3}}{9}+\frac {x^{4}}{18}+\frac {x^{5}}{135}+\frac {x^{6}}{270}+O\left (x^{6}\right ) \] Verified OK. \[ y = 3+\frac {5 x^{2}}{3}+\frac {x^{4}}{18}+\frac {x^{5}}{135}-2 x -\frac {2 x^{3}}{9}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 3-2 x +\frac {5}{3} x^{2}-\frac {2}{9} x^{3}+\frac {1}{18} x^{4}+\frac {1}{135} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 14790

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

Problem 14791

ODE	\[ \boxed {\left (2 x^{2}-1\right ) y^{\prime \prime }+2 y^{\prime } x -3 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -2, y^{\prime }\left (0\right ) = 2] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 3 x^{2}+2 x -2-\frac {x^{3}}{3}+\frac {5 x^{4}}{4}-\frac {x^{5}}{4}+\frac {29 x^{6}}{24}+O\left (x^{6}\right ) \] Verified OK. \[ y = -2+3 x^{2}+\frac {5 x^{4}}{4}+2 x -\frac {x^{3}}{3}-\frac {x^{5}}{4}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -2+2 x +3 x^{2}-\frac {1}{3} x^{3}+\frac {5}{4} x^{4}-\frac {1}{4} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 14792

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

Problem 14793

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }+y x=\cos \left (x \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = x -\frac {x^{4}}{8}+\frac {x^{5}}{40}-\frac {x^{6}}{360}+O\left (x^{6}\right ) \] Verified OK. \[ y = x -\frac {x^{4}}{8}+\frac {x^{5}}{40}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x -\frac {1}{8} x^{4}+\frac {1}{40} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 14794

ODE	\[ \boxed {y^{\prime \prime }+\left (-1+y^{2}\right ) y^{\prime }+y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {x^{2}}{2}+x -\frac {x^{4}}{8}-\frac {x^{5}}{8}-\frac {7 x^{6}}{120}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x +\frac {1}{2} x^{2}-\frac {1}{8} x^{4}-\frac {1}{8} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 14795

ODE	\[ \boxed {y^{\prime \prime }+\left (\frac {{y^{\prime }}^{2}}{3}-1\right ) y^{\prime }+y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = -\frac {x^{2}}{2}+1-\frac {x^{3}}{6}+\frac {x^{5}}{40}+\frac {x^{6}}{48}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1-\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{40} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 14796

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

Problem 14797

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

Problem 14798

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

Problem 14799

ODE	\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x +9 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {9}{2} x^{2}+\frac {15}{8} x^{4}+\frac {7}{16} x^{6}\right ) y \left (0\right )+\left (x -\frac {4}{3} x^{3}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {9}{2} x^{2}+\frac {15}{8} x^{4}\right ) c_{1} +\left (x -\frac {4}{3} x^{3}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {9}{2} x^{2}+\frac {15}{8} x^{4}\right ) y \left (0\right )+\left (x -\frac {4}{3} x^{3}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 14800

ODE	\[ \boxed {y^{\prime \prime }-\cos \left (x \right ) y=\sin \left (x \right )} \] 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	\[ y = \frac {x^{2}}{2}+1+\frac {x^{3}}{6}-\frac {x^{6}}{144}+O\left (x^{6}\right ) \] Verified OK. \[ y = \frac {x^{2}}{2}+1+\frac {x^{3}}{6}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1+\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\operatorname {O}\left (x^{6}\right ) \]