2.17.66 Problems 6501 to 6600

Problem 6501

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

Problem 6502

ODE	\[ \boxed {L i^{\prime }+R i=E_{0} \operatorname {Heaviside}\left (t \right )} \] With initial conditions \begin {align*} [i \left (0\right ) = 0] \end {align*}
program solution	\[ i = \frac {E_{0} \left (1-{\mathrm e}^{-\frac {R t}{L}}\right )}{R} \] Verified OK.
Maple solution	\[ i \left (t \right ) = -\frac {E_{0} \left ({\mathrm e}^{-\frac {R t}{L}}-1\right )}{R} \]

Problem 6503

ODE	\[ \boxed {L i^{\prime }+R i=E_{0} \delta \left (t \right )} \] With initial conditions \begin {align*} [i \left (0\right ) = 0] \end {align*}
program solution	\[ i = \frac {E_{0} {\mathrm e}^{-\frac {R t}{L}}}{L} \] Verified OK.
Maple solution	\[ i \left (t \right ) = \frac {{\mathrm e}^{-\frac {R t}{L}} E_{0}}{L} \]

Problem 6504

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

Problem 6505

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }-5 y=1} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \frac {13 \sqrt {29}\, \sinh \left (\frac {t \sqrt {29}}{2}\right ) {\mathrm e}^{-\frac {3 t}{2}}}{145}+\frac {\cosh \left (\frac {t \sqrt {29}}{2}\right ) {\mathrm e}^{-\frac {3 t}{2}}}{5}-\frac {1}{5} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {13 \,{\mathrm e}^{-\frac {3 t}{2}} \sqrt {29}\, \sinh \left (\frac {t \sqrt {29}}{2}\right )}{145}+\frac {{\mathrm e}^{-\frac {3 t}{2}} \cosh \left (\frac {t \sqrt {29}}{2}\right )}{5}-\frac {1}{5} \]

Problem 6506

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }-2 y=-6 \,{\mathrm e}^{\pi -t}} \] With initial conditions \begin {align*} [y \left (\pi \right ) = 1, y^{\prime }\left (\pi \right ) = 4] \end {align*}
program solution	\[ y = -\frac {19 \left (-\frac {51 \,{\mathrm e}^{-\frac {\pi }{2}-t}}{19}+{\mathrm e}^{-\frac {3 t}{2}} \left (\left (-\sqrt {17}\, \sinh \left (\frac {t \sqrt {17}}{2}\right )+\frac {17 \cosh \left (\frac {t \sqrt {17}}{2}\right )}{19}\right ) \cosh \left (\frac {\pi \sqrt {17}}{2}\right )+\sinh \left (\frac {\pi \sqrt {17}}{2}\right ) \left (\sqrt {17}\, \cosh \left (\frac {t \sqrt {17}}{2}\right )-\frac {17 \sinh \left (\frac {t \sqrt {17}}{2}\right )}{19}\right )\right )\right ) {\mathrm e}^{\frac {3 \pi }{2}}}{34} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {19 \sinh \left (\frac {\left (\pi -t \right ) \sqrt {17}}{2}\right ) \sqrt {17}\, {\mathrm e}^{-\frac {3 t}{2}+\frac {3 \pi }{2}}}{34}-\frac {\cosh \left (\frac {\left (\pi -t \right ) \sqrt {17}}{2}\right ) {\mathrm e}^{-\frac {3 t}{2}+\frac {3 \pi }{2}}}{2}+\frac {3 \,{\mathrm e}^{\pi -t}}{2} \]

Problem 6507

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

Problem 6508

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

Problem 6509

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

Problem 6510

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

Problem 6511

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }+2 y=t} \]
program solution	\[ y = -\frac {1}{4}+\frac {\sqrt {7}\, \left (c_{1} +2 c_{2} -\frac {3}{4}\right ) {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {7}\, t}{2}\right )}{7}+\frac {\left (1+4 c_{1} \right ) {\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {7}\, t}{2}\right )}{4}+\frac {t}{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {1}{4}+\frac {t}{2}+\frac {\left (7 \cos \left (\frac {\sqrt {7}\, t}{2}\right ) \left (1+4 y \left (0\right )\right )+\sin \left (\frac {\sqrt {7}\, t}{2}\right ) \sqrt {7}\, \left (8 D\left (y \right )\left (0\right )+4 y \left (0\right )-3\right )\right ) {\mathrm e}^{-\frac {t}{2}}}{28} \]

Problem 6512

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

Problem 6513

ODE	\[ \boxed {i^{\prime \prime }+2 i^{\prime }+3 i=\left \{\begin {array}{cc} 30 & 0
program solution	\[ i = -\sqrt {2}\, {\mathrm e}^{-t} \sin \left (\sqrt {2}\, t \right )-2 \,{\mathrm e}^{-t} \cos \left (\sqrt {2}\, t \right )+5 \left (\left \{\begin {array}{cc} 2 & t <2 \pi \\ -\frac {i \sqrt {2}\, {\mathrm e}^{\left (-1+i \sqrt {2}\right ) \left (t -2 \pi \right )}}{2}+\frac {i \sqrt {2}\, {\mathrm e}^{-\left (i \sqrt {2}+1\right ) \left (t -2 \pi \right )}}{2}+{\mathrm e}^{\left (-1+i \sqrt {2}\right ) \left (t -2 \pi \right )}+{\mathrm e}^{-\left (i \sqrt {2}+1\right ) \left (t -2 \pi \right )} & t <5 \pi \\ \frac {\left (-i \sqrt {2}-2\right ) {\mathrm e}^{-\left (i \sqrt {2}+1\right ) \left (t -5 \pi \right )}}{6}-\frac {i \sqrt {2}\, {\mathrm e}^{\left (-1+i \sqrt {2}\right ) \left (t -2 \pi \right )}}{2}+\frac {i \sqrt {2}\, {\mathrm e}^{\left (-1+i \sqrt {2}\right ) \left (t -5 \pi \right )}}{6}+\frac {i \sqrt {2}\, {\mathrm e}^{-\left (i \sqrt {2}+1\right ) \left (t -2 \pi \right )}}{2}+{\mathrm e}^{\left (-1+i \sqrt {2}\right ) \left (t -2 \pi \right )}-\frac {{\mathrm e}^{\left (-1+i \sqrt {2}\right ) \left (t -5 \pi \right )}}{3}+{\mathrm e}^{-\left (i \sqrt {2}+1\right ) \left (t -2 \pi \right )}+\frac {2}{3} & 5 \pi \le t \end {array}\right .\right ) \] Verified OK.
Maple solution	\[ \text {No solution found} \]

Problem 6514

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

Problem 6515

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

Problem 6516

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

Problem 6517

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

Problem 6518

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

Problem 6519

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

Problem 6520

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

Problem 6521

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

Problem 6522

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

Problem 6523

ODE	\begin {align*} x^{\prime }\left (t \right )&=4 x \left (t \right )-3 y \left (t \right )\\ y^{\prime }\left (t \right )&=8 x \left (t \right )-6 y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} +c_{2} {\mathrm e}^{-2 t} \\ y \left (t \right ) &= 2 c_{2} {\mathrm e}^{-2 t}+\frac {4 c_{1}}{3} \\ \end{align*}

Problem 6524

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

Problem 6525

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

Problem 6526

ODE	\begin {align*} x^{\prime }\left (t \right )&=7 x \left (t \right )+6 y \left (t \right )\\ y^{\prime }\left (t \right )&=2 x \left (t \right )+6 y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{10 t}+c_{2} {\mathrm e}^{3 t} \\ y \left (t \right ) &= \frac {c_{1} {\mathrm e}^{10 t}}{2}-\frac {2 c_{2} {\mathrm e}^{3 t}}{3} \\ \end{align*}

Problem 6527

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

Problem 6528

ODE	\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right )-5 t +2\\ y^{\prime }\left (t \right )&=4 x \left (t \right )-2 y \left (t \right )-8 t -8 \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{-3 t}+c_{1} {\mathrm e}^{2 t}+3 t +2 \\ y \left (t \right ) &= -4 c_{2} {\mathrm e}^{-3 t}+c_{1} {\mathrm e}^{2 t}-1+2 t \\ \end{align*}

Problem 6529

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

Problem 6530

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

Problem 6531

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

Problem 6532

ODE	\begin {align*} x^{\prime }\left (t \right )&=5 x \left (t \right )+3 y \left (t \right )\\ y^{\prime }\left (t \right )&=-6 x \left (t \right )-4 y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} c_{1} +c_{2} {\mathrm e}^{2 t} \\ y \left (t \right ) &= -2 \,{\mathrm e}^{-t} c_{1} -c_{2} {\mathrm e}^{2 t} \\ \end{align*}

Problem 6533

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

Problem 6534

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

Problem 6535

ODE	\begin {align*} x^{\prime }\left (t \right )&=3 x \left (t \right )-5 y \left (t \right )\\ y^{\prime }\left (t \right )&=-x \left (t \right )+2 y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{\frac {\left (5+\sqrt {21}\right ) t}{2}}+c_{2} {\mathrm e}^{-\frac {\left (-5+\sqrt {21}\right ) t}{2}} \\ y \left (t \right ) &= -\frac {c_{1} {\mathrm e}^{\frac {\left (5+\sqrt {21}\right ) t}{2}} \sqrt {21}}{10}+\frac {c_{2} {\mathrm e}^{-\frac {\left (-5+\sqrt {21}\right ) t}{2}} \sqrt {21}}{10}+\frac {c_{1} {\mathrm e}^{\frac {\left (5+\sqrt {21}\right ) t}{2}}}{10}+\frac {c_{2} {\mathrm e}^{-\frac {\left (-5+\sqrt {21}\right ) t}{2}}}{10} \\ \end{align*}

Problem 6536

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

Problem 6537

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

Problem 6538

ODE	\begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right )+y \left (t \right )-z \left (t \right )\\ y^{\prime }\left (t \right )&=2 x \left (t \right )-y \left (t \right )-4 z \left (t \right )\\ z^{\prime }\left (t \right )&=3 x \left (t \right )-y \left (t \right )+z \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{\frac {\left (13+\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+c_{3} {\mathrm e}^{\frac {\left (13+\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+c_{1} {\mathrm e}^{-\frac {\left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right ) t}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}} \\ \text {Expression too large to display} \\ \text {Expression too large to display} \\ \end{align*}

Problem 6539

ODE	\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )+2 y \left (t \right )-4 t +1\\ y^{\prime }\left (t \right )&=-x \left (t \right )+2 y \left (t \right )+3 t +4 \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{\frac {3 t}{2}} \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_{2} +{\mathrm e}^{\frac {3 t}{2}} \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_{1} +\frac {7 t}{2}+\frac {25}{8} \\ y \left (t \right ) &= \frac {{\mathrm e}^{\frac {3 t}{2}} \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_{2}}{4}+\frac {{\mathrm e}^{\frac {3 t}{2}} \sqrt {7}\, \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_{2}}{4}+\frac {{\mathrm e}^{\frac {3 t}{2}} \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_{1}}{4}-\frac {{\mathrm e}^{\frac {3 t}{2}} \sqrt {7}\, \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_{1}}{4}-\frac {5}{16}+\frac {t}{4} \\ \end{align*}

Problem 6540

ODE	\begin {align*} x^{\prime }\left (t \right )&=-2 x \left (t \right )+y \left (t \right )-t +3\\ y^{\prime }\left (t \right )&=x \left (t \right )+4 y \left (t \right )+t -2 \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 {5 t}{9}+\frac {145}{81} \\ y \left (t \right ) &= {\mathrm e}^{\left (1+\sqrt {10}\right ) t} c_{2} \sqrt {10}-{\mathrm e}^{-\left (-1+\sqrt {10}\right ) t} c_{1} \sqrt {10}+3 \,{\mathrm e}^{\left (1+\sqrt {10}\right ) t} c_{2} +3 \,{\mathrm e}^{-\left (-1+\sqrt {10}\right ) t} c_{1} -\frac {t}{9}+\frac {2}{81} \\ \end{align*}

Problem 6541

ODE	\begin {align*} x^{\prime }\left (t \right )&=-4 x \left (t \right )+y \left (t \right )-t +3\\ y^{\prime }\left (t \right )&=-x \left (t \right )-5 y \left (t \right )+t +1 \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{-\frac {9 t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} +{\mathrm e}^{-\frac {9 t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} -\frac {4 t}{21}+\frac {39}{49} \\ y \left (t \right ) &= -\frac {{\mathrm e}^{-\frac {9 t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2}}{2}+\frac {{\mathrm e}^{-\frac {9 t}{2}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2}}{2}-\frac {{\mathrm e}^{-\frac {9 t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1}}{2}-\frac {{\mathrm e}^{-\frac {9 t}{2}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1}}{2}-\frac {1}{147}+\frac {5 t}{21} \\ \end{align*}

Problem 6542

ODE	\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right ) y \left (t \right )+1\\ y^{\prime }\left (t \right )&=-x \left (t \right )+y \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 2, y \left (0\right ) = -1] \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 6543

ODE	\begin {align*} x^{\prime }\left (t \right )&=t y \left (t \right )+1\\ y^{\prime }\left (t \right )&=-x \left (t \right ) t +y \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 0, y \left (0\right ) = -1] \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 6544

ODE	\[ \boxed {y^{\prime }-y^{2}=-x} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {x^{2}}{2}+x +1+\frac {2 x^{3}}{3}+\frac {7 x^{4}}{12}+\frac {11 x^{5}}{20}+\frac {22 x^{6}}{45}+\frac {559 x^{7}}{1260}+O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1+x +\frac {1}{2} x^{2}+\frac {2}{3} x^{3}+\frac {7}{12} x^{4}+\frac {11}{20} x^{5}+\frac {22}{45} x^{6}+\frac {559}{1260} x^{7}+\operatorname {O}\left (x^{8}\right ) \]

Problem 6545

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

Problem 6546

ODE	\[ \boxed {-2 y+y^{\prime }=x^{2}} \] With initial conditions \begin {align*} [y \left (1\right ) = 1] \end {align*} With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = 3 \left (x -1\right )^{3}+4 \left (x -1\right )^{2}+3 x -2+\frac {3 \left (x -1\right )^{4}}{2}+\frac {3 \left (x -1\right )^{5}}{5}+\frac {\left (x -1\right )^{6}}{5}+\frac {2 \left (x -1\right )^{7}}{35}+O\left (\left (x -1\right )^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1+3 \left (x -1\right )+4 \left (x -1\right )^{2}+3 \left (x -1\right )^{3}+\frac {3}{2} \left (x -1\right )^{4}+\frac {3}{5} \left (x -1\right )^{5}+\frac {1}{5} \left (x -1\right )^{6}+\frac {2}{35} \left (x -1\right )^{7}+\operatorname {O}\left (\left (x -1\right )^{8}\right ) \]

Problem 6547

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

Problem 6548

ODE	\[ \boxed {y^{\prime }-y-{\mathrm e}^{y} x=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{6}+\frac {x^{5}}{15}+\frac {43 x^{6}}{720}+\frac {151 x^{7}}{5040}+O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{6} x^{4}+\frac {1}{15} x^{5}+\frac {43}{720} x^{6}+\frac {151}{5040} x^{7}+\operatorname {O}\left (x^{8}\right ) \]

Problem 6549

ODE	\[ \boxed {y^{\prime }-y-{\mathrm e}^{y} x=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution
Maple solution	\[ \text {No solution found} \]

Problem 6550

ODE	\[ \boxed {y^{\prime \prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (\frac {1}{40320} x^{8}-\frac {1}{720} x^{6}+\frac {1}{24} x^{4}-\frac {1}{2} x^{2}+1\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {1}{120} x^{5}-\frac {1}{5040} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}-\frac {1}{720} x^{6}\right ) c_{1} +\left (x -\frac {1}{6} x^{3}+\frac {1}{120} x^{5}-\frac {1}{5040} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}-\frac {1}{720} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {1}{120} x^{5}-\frac {1}{5040} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6551

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

Problem 6552

ODE	\[ \boxed {y^{\prime \prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\frac {1}{720} x^{6}+\frac {1}{40320} x^{8}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{120} x^{5}+\frac {1}{5040} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\frac {1}{720} x^{6}\right ) c_{1} +\left (x +\frac {1}{6} x^{3}+\frac {1}{120} x^{5}+\frac {1}{5040} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\frac {1}{720} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{120} x^{5}+\frac {1}{5040} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6553

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

Problem 6554

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}+\frac {1}{40320} x^{8}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = c_{1} +\left (x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6555

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

Problem 6556

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = y \left (0\right )+\left (x -x^{2}+\frac {2}{3} x^{3}-\frac {1}{3} x^{4}+\frac {2}{15} x^{5}-\frac {2}{45} x^{6}+\frac {4}{315} x^{7}-\frac {1}{315} x^{8}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = c_{1} +\left (x -x^{2}+\frac {2}{3} x^{3}-\frac {1}{3} x^{4}+\frac {2}{15} x^{5}-\frac {2}{45} x^{6}+\frac {4}{315} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = y \left (0\right )+\left (x -x^{2}+\frac {2}{3} x^{3}-\frac {1}{3} x^{4}+\frac {2}{15} x^{5}-\frac {2}{45} x^{6}+\frac {4}{315} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6557

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

Problem 6558

ODE	\[ \boxed {y^{\prime \prime }-y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{6} x^{3}+\frac {1}{180} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{12} x^{4}+\frac {1}{504} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{6} x^{3}+\frac {1}{180} x^{6}\right ) c_{1} +\left (x +\frac {1}{12} x^{4}+\frac {1}{504} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{6} x^{3}+\frac {1}{180} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{12} x^{4}+\frac {1}{504} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6559

ODE	\[ \boxed {y^{\prime \prime }+y x^{2}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{12} x^{4}+\frac {1}{672} x^{8}\right ) y \left (0\right )+\left (x -\frac {1}{20} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {x^{4}}{12}\right ) c_{1} +\left (x -\frac {1}{20} x^{5}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {x^{4}}{12}\right ) y \left (0\right )+\left (x -\frac {1}{20} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6560

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime } x +y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{8} x^{4}-\frac {7}{240} x^{6}-\frac {11}{1920} x^{8}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{24} x^{5}+\frac {1}{112} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{8} x^{4}-\frac {7}{240} x^{6}\right ) c_{1} +\left (x +\frac {1}{6} x^{3}+\frac {1}{24} x^{5}+\frac {1}{112} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}-\frac {1}{8} x^{4}-\frac {7}{240} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{24} x^{5}+\frac {1}{112} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6561

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime } x +2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (-x^{2}+1\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}-\frac {1}{120} x^{5}-\frac {1}{1680} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (-x^{2}+1\right ) c_{1} +\left (x -\frac {1}{6} x^{3}-\frac {1}{120} x^{5}-\frac {1}{1680} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (-x^{2}+1\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}-\frac {1}{120} x^{5}-\frac {1}{1680} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6562

ODE	\[ \boxed {y^{\prime \prime }+x^{2} y^{\prime }+y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{6} x^{3}+\frac {1}{45} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{4}+\frac {5}{252} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{6} x^{3}+\frac {1}{45} x^{6}\right ) c_{1} +\left (x -\frac {1}{6} x^{4}+\frac {5}{252} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{6} x^{3}+\frac {1}{45} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{4}+\frac {5}{252} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6563

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	\[ y = \left (1-x^{2}+\frac {1}{2} x^{4}-\frac {1}{6} x^{6}+\frac {1}{24} x^{8}\right ) y \left (0\right )+\left (x -\frac {2}{3} x^{3}+\frac {4}{15} x^{5}-\frac {8}{105} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-x^{2}+\frac {1}{2} x^{4}-\frac {1}{6} x^{6}\right ) c_{1} +\left (x -\frac {2}{3} x^{3}+\frac {4}{15} x^{5}-\frac {8}{105} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-x^{2}+\frac {1}{2} x^{4}-\frac {1}{6} x^{6}\right ) y \left (0\right )+\left (x -\frac {2}{3} x^{3}+\frac {4}{15} x^{5}-\frac {8}{105} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6564

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

Problem 6565

ODE	\[ \boxed {\left (x +2\right ) y^{\prime \prime }+y^{\prime } x -y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{4} x^{2}-\frac {1}{24} x^{3}+\frac {1}{480} x^{5}-\frac {1}{1440} x^{6}+\frac {1}{6720} x^{7}-\frac {1}{40320} x^{8}\right ) y \left (0\right )+y^{\prime }\left (0\right ) x +O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{4} x^{2}-\frac {1}{24} x^{3}+\frac {1}{480} x^{5}-\frac {1}{1440} x^{6}+\frac {1}{6720} x^{7}\right ) c_{1} +c_{2} x +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{4} x^{2}-\frac {1}{24} x^{3}+\frac {1}{480} x^{5}-\frac {1}{1440} x^{6}+\frac {1}{6720} x^{7}\right ) y \left (0\right )+D\left (y \right )\left (0\right ) x +O\left (x^{8}\right ) \]

Problem 6566

ODE	\[ \boxed {y^{\prime \prime }-\left (1+x \right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{6} x^{4}+\frac {1}{15} x^{5}+\frac {7}{180} x^{6}+\frac {19}{1260} x^{7}+\frac {17}{2520} x^{8}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{2} x^{3}+\frac {1}{4} x^{4}+\frac {3}{20} x^{5}+\frac {1}{15} x^{6}+\frac {13}{420} x^{7}+\frac {41}{3360} x^{8}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{6} x^{4}+\frac {1}{15} x^{5}+\frac {7}{180} x^{6}+\frac {19}{1260} x^{7}\right ) c_{1} +\left (x +\frac {1}{2} x^{2}+\frac {1}{2} x^{3}+\frac {1}{4} x^{4}+\frac {3}{20} x^{5}+\frac {1}{15} x^{6}+\frac {13}{420} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{6} x^{4}+\frac {1}{15} x^{5}+\frac {7}{180} x^{6}+\frac {19}{1260} x^{7}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{2} x^{3}+\frac {1}{4} x^{4}+\frac {3}{20} x^{5}+\frac {1}{15} x^{6}+\frac {13}{420} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6567

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

Problem 6568

ODE	\[ \boxed {\left (x^{2}+2\right ) y^{\prime \prime }+3 y^{\prime } x -y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{4} x^{2}-\frac {7}{96} x^{4}+\frac {161}{5760} x^{6}-\frac {1081}{92160} x^{8}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {7}{120} x^{5}-\frac {17}{720} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{4} x^{2}-\frac {7}{96} x^{4}+\frac {161}{5760} x^{6}\right ) c_{1} +\left (x -\frac {1}{6} x^{3}+\frac {7}{120} x^{5}-\frac {17}{720} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{4} x^{2}-\frac {7}{96} x^{4}+\frac {161}{5760} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {7}{120} x^{5}-\frac {17}{720} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6569

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	\[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{8} x^{4}-\frac {1}{16} x^{6}-\frac {5}{128} x^{8}\right ) y \left (0\right )+y^{\prime }\left (0\right ) x +O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{8} x^{4}-\frac {1}{16} x^{6}\right ) c_{1} +c_{2} x +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}-\frac {1}{8} x^{4}-\frac {1}{16} x^{6}\right ) y \left (0\right )+D\left (y \right )\left (0\right ) x +O\left (x^{8}\right ) \]

Problem 6570

ODE	\[ \boxed {\left (x -1\right ) y^{\prime \prime }-y^{\prime } x +y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -2, y^{\prime }\left (0\right ) = 6] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = -x^{2}+6 x -2-\frac {x^{3}}{3}-\frac {x^{4}}{12}-\frac {x^{5}}{60}-\frac {x^{6}}{360}-\frac {x^{7}}{2520}-\frac {x^{8}}{20160}+O\left (x^{8}\right ) \] Verified OK. \[ y = -2-x^{2}-\frac {x^{3}}{3}-\frac {x^{4}}{12}-\frac {x^{5}}{60}-\frac {x^{6}}{360}-\frac {x^{7}}{2520}+6 x +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -2+6 x -x^{2}-\frac {1}{3} x^{3}-\frac {1}{12} x^{4}-\frac {1}{60} x^{5}-\frac {1}{360} x^{6}-\frac {1}{2520} x^{7}+\operatorname {O}\left (x^{8}\right ) \]

Problem 6571

ODE	\[ \boxed {\left (1+x \right ) y^{\prime \prime }-\left (-x +2\right ) y^{\prime }+y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = -1] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = -2 x^{2}-x +2-\frac {x^{3}}{3}+\frac {x^{4}}{2}-\frac {x^{5}}{30}-\frac {13 x^{6}}{180}+\frac {x^{7}}{28}-\frac {89 x^{8}}{10080}+O\left (x^{8}\right ) \] Verified OK. \[ y = 2-2 x^{2}-\frac {x^{3}}{3}+\frac {x^{4}}{2}-\frac {x^{5}}{30}-\frac {13 x^{6}}{180}+\frac {x^{7}}{28}-x +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2-x -2 x^{2}-\frac {1}{3} x^{3}+\frac {1}{2} x^{4}-\frac {1}{30} x^{5}-\frac {13}{180} x^{6}+\frac {1}{28} x^{7}+\operatorname {O}\left (x^{8}\right ) \]

Problem 6572

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

Problem 6573

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+2 y^{\prime } x=0} \] 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	\[ y = -\frac {x^{3}}{3}+x +\frac {x^{5}}{5}-\frac {x^{7}}{7}+O\left (x^{8}\right ) \] Verified OK. \[ y = -\frac {x^{3}}{3}+x +\frac {x^{5}}{5}-\frac {x^{7}}{7}+O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x -\frac {1}{3} x^{3}+\frac {1}{5} x^{5}-\frac {1}{7} x^{7}+\operatorname {O}\left (x^{8}\right ) \]

Problem 6574

ODE	\[ \boxed {y^{\prime \prime }+\sin \left (x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{6} x^{3}+\frac {1}{120} x^{5}+\frac {1}{180} x^{6}-\frac {1}{5040} x^{7}-\frac {13}{20160} x^{8}\right ) y \left (0\right )+\left (x -\frac {1}{12} x^{4}+\frac {1}{180} x^{6}+\frac {1}{504} x^{7}-\frac {1}{6720} x^{8}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{6} x^{3}+\frac {1}{120} x^{5}+\frac {1}{180} x^{6}-\frac {1}{5040} x^{7}\right ) c_{1} +\left (x -\frac {1}{12} x^{4}+\frac {1}{180} x^{6}+\frac {1}{504} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{6} x^{3}+\frac {1}{120} x^{5}+\frac {1}{180} x^{6}-\frac {1}{5040} x^{7}\right ) y \left (0\right )+\left (x -\frac {1}{12} x^{4}+\frac {1}{180} x^{6}+\frac {1}{504} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6575

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime } {\mathrm e}^{x}-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{2} x^{2}-\frac {1}{6} x^{3}-\frac {1}{120} x^{5}+\frac {1}{240} x^{6}+\frac {1}{840} x^{7}+\frac {1}{13440} x^{8}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{24} x^{4}+\frac {1}{120} x^{5}-\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}-\frac {1}{40320} x^{8}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{2} x^{2}-\frac {1}{6} x^{3}-\frac {1}{120} x^{5}+\frac {1}{240} x^{6}+\frac {1}{840} x^{7}\right ) c_{1} +\left (x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{24} x^{4}+\frac {1}{120} x^{5}-\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{2} x^{2}-\frac {1}{6} x^{3}-\frac {1}{120} x^{5}+\frac {1}{240} x^{6}+\frac {1}{840} x^{7}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{24} x^{4}+\frac {1}{120} x^{5}-\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6576

ODE	\[ \boxed {\cos \left (x \right ) y^{\prime \prime }+y^{\prime }+5 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {5}{2} x^{2}+\frac {5}{6} x^{3}+\frac {5}{8} x^{4}-\frac {5}{24} x^{5}+\frac {1}{16} x^{6}-\frac {13}{336} x^{7}+\frac {83}{8064} x^{8}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}-\frac {2}{3} x^{3}+\frac {1}{3} x^{4}+\frac {1}{80} x^{6}+\frac {11}{5040} x^{7}-\frac {1}{960} x^{8}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {5}{2} x^{2}+\frac {5}{6} x^{3}+\frac {5}{8} x^{4}-\frac {5}{24} x^{5}+\frac {1}{16} x^{6}-\frac {13}{336} x^{7}\right ) c_{1} +\left (x -\frac {1}{2} x^{2}-\frac {2}{3} x^{3}+\frac {1}{3} x^{4}+\frac {1}{80} x^{6}+\frac {11}{5040} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {5}{2} x^{2}+\frac {5}{6} x^{3}+\frac {5}{8} x^{4}-\frac {5}{24} x^{5}+\frac {1}{16} x^{6}-\frac {13}{336} x^{7}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}-\frac {2}{3} x^{3}+\frac {1}{3} x^{4}+\frac {1}{80} x^{6}+\frac {11}{5040} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6577

ODE	\[ \boxed {\cos \left (x \right ) y^{\prime \prime }+y^{\prime }+5 y=0} \] With the expansion point for the power series method at \(x = 1\).
program solution	\[ \text {Expression too large to display} \] Verified OK.
Maple solution	\[ \text {Expression too large to display} \]

Problem 6578

ODE	\[ \boxed {y^{\prime \prime }-y x=1} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{6} x^{3}+\frac {1}{180} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{12} x^{4}+\frac {1}{504} x^{7}\right ) y^{\prime }\left (0\right )+\frac {x^{2}}{2}+\frac {x^{5}}{40}+\frac {x^{8}}{2240}+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{6} x^{3}+\frac {1}{180} x^{6}\right ) c_{1} +\left (x +\frac {1}{12} x^{4}+\frac {1}{504} x^{7}\right ) c_{2} +\frac {x^{2}}{2}+\frac {x^{5}}{40}+O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{6} x^{3}+\frac {1}{180} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{12} x^{4}+\frac {1}{504} x^{7}\right ) D\left (y \right )\left (0\right )+\frac {x^{2}}{2}+\frac {x^{5}}{40}+O\left (x^{8}\right ) \]

Problem 6579

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime } x -4 y={\mathrm e}^{x}} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+2 x^{2}+2 x^{4}+\frac {4}{3} x^{6}+\frac {2}{3} x^{8}\right ) y \left (0\right )+\left (x +\frac {4}{3} x^{3}+\frac {16}{15} x^{5}+\frac {64}{105} x^{7}\right ) y^{\prime }\left (0\right )+\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {13 x^{4}}{24}+\frac {17 x^{5}}{120}+\frac {29 x^{6}}{80}+\frac {409 x^{7}}{5040}+\frac {7309 x^{8}}{40320}+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1+2 x^{2}+2 x^{4}+\frac {4}{3} x^{6}\right ) c_{1} +\left (x +\frac {4}{3} x^{3}+\frac {16}{15} x^{5}+\frac {64}{105} x^{7}\right ) c_{2} +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {13 x^{4}}{24}+\frac {17 x^{5}}{120}+\frac {29 x^{6}}{80}+\frac {409 x^{7}}{5040}+O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+2 x^{2}+2 x^{4}+\frac {4}{3} x^{6}\right ) y \left (0\right )+\left (x +\frac {4}{3} x^{3}+\frac {16}{15} x^{5}+\frac {64}{105} x^{7}\right ) D\left (y \right )\left (0\right )+\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {13 x^{4}}{24}+\frac {17 x^{5}}{120}+\frac {29 x^{6}}{80}+\frac {409 x^{7}}{5040}+O\left (x^{8}\right ) \]

Problem 6580

ODE	\[ \boxed {x y^{\prime \prime }+\sin \left (x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{18} x^{4}-\frac {53}{10800} x^{6}+\frac {467}{1411200} x^{8}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {1}{60} x^{5}-\frac {19}{15120} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{18} x^{4}-\frac {53}{10800} x^{6}\right ) c_{1} +\left (x -\frac {1}{6} x^{3}+\frac {1}{60} x^{5}-\frac {19}{15120} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{18} x^{4}-\frac {53}{10800} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {1}{60} x^{5}-\frac {19}{15120} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6581

ODE	\[ \boxed {y^{\prime \prime }+5 y^{\prime } x +\sqrt {x}\, y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 6582

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime } x +y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}-\frac {1}{48} x^{6}+\frac {1}{384} x^{8}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{15} x^{5}-\frac {1}{105} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}-\frac {1}{48} x^{6}\right ) c_{1} +\left (x -\frac {1}{3} x^{3}+\frac {1}{15} x^{5}-\frac {1}{105} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}-\frac {1}{48} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{15} x^{5}-\frac {1}{105} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6583

ODE	\[ \boxed {y^{\prime \prime }+y \cos \left (x \right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 1] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 1+x -\frac {x^{2}}{2}-\frac {x^{3}}{6}+\frac {x^{4}}{12}+\frac {x^{5}}{30}-\frac {x^{6}}{80}-\frac {19 x^{7}}{5040}+\frac {11 x^{8}}{8064}+O\left (x^{8}\right ) \] Verified OK. \[ y = 1-\frac {x^{2}}{2}+\frac {x^{4}}{12}-\frac {x^{6}}{80}+x -\frac {x^{3}}{6}+\frac {x^{5}}{30}-\frac {19 x^{7}}{5040}+O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1+x -\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{30} x^{5}-\frac {1}{80} x^{6}-\frac {19}{5040} x^{7}+\operatorname {O}\left (x^{8}\right ) \]

Problem 6584

ODE	\[ \boxed {x^{3} y^{\prime \prime }+4 x^{2} y^{\prime }+3 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 6585

ODE	\[ \boxed {x \left (x +3\right )^{2} y^{\prime \prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1+\frac {x}{18}-\frac {11 x^{2}}{972}+\frac {277 x^{3}}{104976}-\frac {12539 x^{4}}{18895680}+\frac {893821 x^{5}}{5101833600}-\frac {13183337 x^{6}}{275499014400}+\frac {265861081 x^{7}}{19835929036800}+O\left (x^{8}\right )\right )+c_{2} \left (\frac {x \left (1+\frac {x}{18}-\frac {11 x^{2}}{972}+\frac {277 x^{3}}{104976}-\frac {12539 x^{4}}{18895680}+\frac {893821 x^{5}}{5101833600}-\frac {13183337 x^{6}}{275499014400}+\frac {265861081 x^{7}}{19835929036800}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{9}+1-\frac {5 x^{2}}{108}+\frac {167 x^{3}}{26244}-\frac {13583 x^{4}}{11337408}+\frac {1327279 x^{5}}{5101833600}-\frac {21146863 x^{6}}{344373768000}+\frac {379766273 x^{7}}{24794911296000}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1+\frac {1}{18} x -\frac {11}{972} x^{2}+\frac {277}{104976} x^{3}-\frac {12539}{18895680} x^{4}+\frac {893821}{5101833600} x^{5}-\frac {13183337}{275499014400} x^{6}+\frac {265861081}{19835929036800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (\frac {1}{9} x +\frac {1}{162} x^{2}-\frac {11}{8748} x^{3}+\frac {277}{944784} x^{4}-\frac {12539}{170061120} x^{5}+\frac {893821}{45916502400} x^{6}-\frac {13183337}{2479491129600} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (1-\frac {5}{108} x^{2}+\frac {167}{26244} x^{3}-\frac {13583}{11337408} x^{4}+\frac {1327279}{5101833600} x^{5}-\frac {21146863}{344373768000} x^{6}+\frac {379766273}{24794911296000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right ) \]

Problem 6586

ODE	\[ \boxed {\left (x^{2}-9\right )^{2} y^{\prime \prime }+\left (x +3\right ) y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{81} x^{2}+\frac {1}{6561} x^{3}-\frac {289}{708588} x^{4}+\frac {304}{23914845} x^{5}-\frac {194981}{7748409780} x^{6}+\frac {1732937}{1464449448420} x^{7}-\frac {118799749}{63264216171744} x^{8}\right ) y \left (0\right )+\left (x -\frac {1}{54} x^{2}-\frac {13}{2187} x^{3}-\frac {131}{236196} x^{4}-\frac {596}{1594323} x^{5}-\frac {78469}{2582803260} x^{6}-\frac {13738871}{488149816140} x^{7}-\frac {29608903}{15062908612320} x^{8}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{81} x^{2}+\frac {1}{6561} x^{3}-\frac {289}{708588} x^{4}+\frac {304}{23914845} x^{5}-\frac {194981}{7748409780} x^{6}+\frac {1732937}{1464449448420} x^{7}\right ) c_{1} +\left (x -\frac {1}{54} x^{2}-\frac {13}{2187} x^{3}-\frac {131}{236196} x^{4}-\frac {596}{1594323} x^{5}-\frac {78469}{2582803260} x^{6}-\frac {13738871}{488149816140} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{81} x^{2}+\frac {1}{6561} x^{3}-\frac {289}{708588} x^{4}+\frac {304}{23914845} x^{5}-\frac {194981}{7748409780} x^{6}+\frac {1732937}{1464449448420} x^{7}\right ) y \left (0\right )+\left (x -\frac {1}{54} x^{2}-\frac {13}{2187} x^{3}-\frac {131}{236196} x^{4}-\frac {596}{1594323} x^{5}-\frac {78469}{2582803260} x^{6}-\frac {13738871}{488149816140} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6587

ODE	\[ \boxed {y^{\prime \prime }-\frac {y^{\prime }}{x}+\frac {y}{\left (x -1\right )^{3}}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1+\frac {x^{2}}{8}+\frac {x^{3}}{5}+\frac {49 x^{4}}{192}+\frac {423 x^{5}}{1400}+\frac {15941 x^{6}}{46080}+\frac {30511 x^{7}}{78400}+O\left (x^{8}\right )\right )+c_{2} \left (\frac {x^{2} \left (1+\frac {x^{2}}{8}+\frac {x^{3}}{5}+\frac {49 x^{4}}{192}+\frac {423 x^{5}}{1400}+\frac {15941 x^{6}}{46080}+\frac {30511 x^{7}}{78400}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{2}+1+x^{3}+\frac {45 x^{4}}{64}+\frac {17 x^{5}}{25}+\frac {1673 x^{6}}{2304}+\frac {313337 x^{7}}{392000}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \left (1+\frac {1}{8} x^{2}+\frac {1}{5} x^{3}+\frac {49}{192} x^{4}+\frac {423}{1400} x^{5}+\frac {15941}{46080} x^{6}+\frac {30511}{78400} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-x^{2}-\frac {1}{8} x^{4}-\frac {1}{5} x^{5}-\frac {49}{192} x^{6}-\frac {423}{1400} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-2-2 x^{3}-\frac {45}{32} x^{4}-\frac {34}{25} x^{5}-\frac {1673}{1152} x^{6}-\frac {313337}{196000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right ) \]

Problem 6588

ODE	\[ \boxed {\left (x^{3}+4 x \right ) y^{\prime \prime }-2 y^{\prime } x +6 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {x}{2}+\frac {x^{2}}{24}+\frac {x^{3}}{48}-\frac {x^{4}}{384}-\frac {5 x^{5}}{2304}+\frac {5 x^{6}}{21504}+\frac {15 x^{7}}{50176}+O\left (x^{8}\right )\right )+c_{2} \left (-\frac {3 x \left (1-\frac {x}{2}+\frac {x^{2}}{24}+\frac {x^{3}}{48}-\frac {x^{4}}{384}-\frac {5 x^{5}}{2304}+\frac {5 x^{6}}{21504}+\frac {15 x^{7}}{50176}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{2}+1-\frac {3 x^{2}}{2}+\frac {29 x^{3}}{96}+\frac {x^{4}}{32}-\frac {21 x^{5}}{1024}-\frac {287 x^{6}}{92160}+\frac {40961 x^{7}}{18063360}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1-\frac {1}{2} x +\frac {1}{24} x^{2}+\frac {1}{48} x^{3}-\frac {1}{384} x^{4}-\frac {5}{2304} x^{5}+\frac {5}{21504} x^{6}+\frac {15}{50176} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-\frac {3}{2} x +\frac {3}{4} x^{2}-\frac {1}{16} x^{3}-\frac {1}{32} x^{4}+\frac {1}{256} x^{5}+\frac {5}{1536} x^{6}-\frac {5}{14336} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \ln \left (x \right ) c_{2} +\left (1+\frac {1}{2} x -\frac {7}{4} x^{2}+\frac {31}{96} x^{3}+\frac {1}{24} x^{4}-\frac {67}{3072} x^{5}-\frac {43}{10240} x^{6}+\frac {43061}{18063360} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \]

Problem 6589

ODE	\[ \boxed {x^{2} \left (x -5\right )^{2} y^{\prime \prime }+4 y^{\prime } x +\left (x^{2}-25\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {21}{50}+\frac {\sqrt {2941}}{50}} \left (1+\frac {\left (21+\sqrt {2941}\right ) \left (-29+\sqrt {2941}\right ) x}{6250+250 \sqrt {2941}}+\frac {9 \left (79709+879 \sqrt {2941}\right ) x^{2}}{15625 \left (25+\sqrt {2941}\right ) \left (50+\sqrt {2941}\right )}+\frac {12 \left (75561897+1274257 \sqrt {2941}\right ) x^{3}}{1953125 \left (25+\sqrt {2941}\right ) \left (50+\sqrt {2941}\right ) \left (75+\sqrt {2941}\right )}+\frac {12 \left (122814219551+2200649681 \sqrt {2941}\right ) x^{4}}{244140625 \left (25+\sqrt {2941}\right ) \left (50+\sqrt {2941}\right ) \left (75+\sqrt {2941}\right ) \left (100+\sqrt {2941}\right )}+\frac {1152 \left (8688311436917+157371578127 \sqrt {2941}\right ) x^{5}}{152587890625 \left (25+\sqrt {2941}\right ) \left (50+\sqrt {2941}\right ) \left (75+\sqrt {2941}\right ) \left (100+\sqrt {2941}\right ) \left (125+\sqrt {2941}\right )}+\frac {48 \left (-278565456347597849-5212232693970344 \sqrt {2941}\right ) x^{6}}{19073486328125 \left (25+\sqrt {2941}\right ) \left (50+\sqrt {2941}\right ) \left (75+\sqrt {2941}\right ) \left (100+\sqrt {2941}\right ) \left (125+\sqrt {2941}\right ) \left (150+\sqrt {2941}\right )}+\frac {96 \left (-20689947387639015669859-381820145596656632404 \sqrt {2941}\right ) x^{7}}{16689300537109375 \left (25+\sqrt {2941}\right ) \left (50+\sqrt {2941}\right ) \left (75+\sqrt {2941}\right ) \left (100+\sqrt {2941}\right ) \left (125+\sqrt {2941}\right ) \left (150+\sqrt {2941}\right ) \left (175+\sqrt {2941}\right )}+O\left (x^{8}\right )\right )+c_{2} x^{\frac {21}{50}-\frac {\sqrt {2941}}{50}} \left (1-\frac {\left (-21+\sqrt {2941}\right ) \left (29+\sqrt {2941}\right ) x}{-6250+250 \sqrt {2941}}+\frac {9 \left (79709-879 \sqrt {2941}\right ) x^{2}}{15625 \left (-25+\sqrt {2941}\right ) \left (-50+\sqrt {2941}\right )}+\frac {12 \left (-75561897+1274257 \sqrt {2941}\right ) x^{3}}{1953125 \left (-25+\sqrt {2941}\right ) \left (-50+\sqrt {2941}\right ) \left (-75+\sqrt {2941}\right )}+\frac {12 \left (122814219551-2200649681 \sqrt {2941}\right ) x^{4}}{244140625 \left (-25+\sqrt {2941}\right ) \left (-50+\sqrt {2941}\right ) \left (-75+\sqrt {2941}\right ) \left (-100+\sqrt {2941}\right )}+\frac {1152 \left (-8688311436917+157371578127 \sqrt {2941}\right ) x^{5}}{152587890625 \left (-25+\sqrt {2941}\right ) \left (-50+\sqrt {2941}\right ) \left (-75+\sqrt {2941}\right ) \left (-100+\sqrt {2941}\right ) \left (-125+\sqrt {2941}\right )}+\frac {48 \left (-278565456347597849+5212232693970344 \sqrt {2941}\right ) x^{6}}{19073486328125 \left (-25+\sqrt {2941}\right ) \left (-50+\sqrt {2941}\right ) \left (-75+\sqrt {2941}\right ) \left (-100+\sqrt {2941}\right ) \left (-125+\sqrt {2941}\right ) \left (-150+\sqrt {2941}\right )}+\frac {96 \left (20689947387639015669859-381820145596656632404 \sqrt {2941}\right ) x^{7}}{16689300537109375 \left (-25+\sqrt {2941}\right ) \left (-50+\sqrt {2941}\right ) \left (-75+\sqrt {2941}\right ) \left (-100+\sqrt {2941}\right ) \left (-125+\sqrt {2941}\right ) \left (-150+\sqrt {2941}\right ) \left (-175+\sqrt {2941}\right )}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{\frac {21}{50}} \left (c_{1} x^{-\frac {\sqrt {2941}}{50}} \left (1+\frac {-1166-4 \sqrt {2941}}{-3125+125 \sqrt {2941}} x -\frac {9}{15625} \frac {879 \sqrt {2941}-79709}{\left (-25+\sqrt {2941}\right ) \left (-50+\sqrt {2941}\right )} x^{2}+\frac {\frac {15291084 \sqrt {2941}}{1953125}-\frac {906742764}{1953125}}{\left (-25+\sqrt {2941}\right ) \left (-50+\sqrt {2941}\right ) \left (-75+\sqrt {2941}\right )} x^{3}-\frac {12}{244140625} \frac {-122814219551+2200649681 \sqrt {2941}}{\left (-25+\sqrt {2941}\right ) \left (-50+\sqrt {2941}\right ) \left (-75+\sqrt {2941}\right ) \left (-100+\sqrt {2941}\right )} x^{4}+\frac {-\frac {10008934775328384}{152587890625}+\frac {181292058002304 \sqrt {2941}}{152587890625}}{\left (-25+\sqrt {2941}\right ) \left (-50+\sqrt {2941}\right ) \left (-75+\sqrt {2941}\right ) \left (-100+\sqrt {2941}\right ) \left (-125+\sqrt {2941}\right )} x^{5}+\frac {-\frac {13371141904684696752}{19073486328125}+\frac {250187169310576512 \sqrt {2941}}{19073486328125}}{\left (-25+\sqrt {2941}\right ) \left (-50+\sqrt {2941}\right ) \left (-75+\sqrt {2941}\right ) \left (-100+\sqrt {2941}\right ) \left (-125+\sqrt {2941}\right ) \left (-150+\sqrt {2941}\right )} x^{6}-\frac {96}{16689300537109375} \frac {-20689947387639015669859+381820145596656632404 \sqrt {2941}}{\left (-25+\sqrt {2941}\right ) \left (-50+\sqrt {2941}\right ) \left (-75+\sqrt {2941}\right ) \left (-100+\sqrt {2941}\right ) \left (-125+\sqrt {2941}\right ) \left (-150+\sqrt {2941}\right ) \left (-175+\sqrt {2941}\right )} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x^{\frac {\sqrt {2941}}{50}} \left (1+\frac {1166-4 \sqrt {2941}}{125 \sqrt {2941}+3125} x +\frac {\frac {7911 \sqrt {2941}}{15625}+\frac {717381}{15625}}{\left (\sqrt {2941}+25\right ) \left (50+\sqrt {2941}\right )} x^{2}+\frac {\frac {15291084 \sqrt {2941}}{1953125}+\frac {906742764}{1953125}}{\left (\sqrt {2941}+25\right ) \left (50+\sqrt {2941}\right ) \left (\sqrt {2941}+75\right )} x^{3}+\frac {\frac {1473770634612}{244140625}+\frac {26407796172 \sqrt {2941}}{244140625}}{\left (\sqrt {2941}+25\right ) \left (50+\sqrt {2941}\right ) \left (\sqrt {2941}+75\right ) \left (100+\sqrt {2941}\right )} x^{4}+\frac {\frac {10008934775328384}{152587890625}+\frac {181292058002304 \sqrt {2941}}{152587890625}}{\left (\sqrt {2941}+25\right ) \left (50+\sqrt {2941}\right ) \left (\sqrt {2941}+75\right ) \left (100+\sqrt {2941}\right ) \left (125+\sqrt {2941}\right )} x^{5}-\frac {48}{19073486328125} \frac {278565456347597849+5212232693970344 \sqrt {2941}}{\left (\sqrt {2941}+25\right ) \left (50+\sqrt {2941}\right ) \left (\sqrt {2941}+75\right ) \left (100+\sqrt {2941}\right ) \left (125+\sqrt {2941}\right ) \left (150+\sqrt {2941}\right )} x^{6}-\frac {96}{16689300537109375} \frac {20689947387639015669859+381820145596656632404 \sqrt {2941}}{\left (\sqrt {2941}+25\right ) \left (50+\sqrt {2941}\right ) \left (\sqrt {2941}+75\right ) \left (100+\sqrt {2941}\right ) \left (125+\sqrt {2941}\right ) \left (150+\sqrt {2941}\right ) \left (175+\sqrt {2941}\right )} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right ) \]

Problem 6590

ODE	\[ \boxed {\left (x^{2}+x -6\right ) y^{\prime \prime }+\left (x +3\right ) y^{\prime }+y \left (x -2\right )=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{6} x^{2}-\frac {1}{108} x^{3}-\frac {17}{2592} x^{4}-\frac {7}{2160} x^{5}-\frac {139}{116640} x^{6}-\frac {5377}{9797760} x^{7}-\frac {5221}{22394880} x^{8}\right ) y \left (0\right )+\left (x +\frac {1}{4} x^{2}+\frac {1}{36} x^{3}+\frac {23}{864} x^{4}+\frac {13}{1440} x^{5}+\frac {619}{155520} x^{6}+\frac {689}{408240} x^{7}+\frac {19493}{26127360} x^{8}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{6} x^{2}-\frac {1}{108} x^{3}-\frac {17}{2592} x^{4}-\frac {7}{2160} x^{5}-\frac {139}{116640} x^{6}-\frac {5377}{9797760} x^{7}\right ) c_{1} +\left (x +\frac {1}{4} x^{2}+\frac {1}{36} x^{3}+\frac {23}{864} x^{4}+\frac {13}{1440} x^{5}+\frac {619}{155520} x^{6}+\frac {689}{408240} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{6} x^{2}-\frac {1}{108} x^{3}-\frac {17}{2592} x^{4}-\frac {7}{2160} x^{5}-\frac {139}{116640} x^{6}-\frac {5377}{9797760} x^{7}\right ) y \left (0\right )+\left (x +\frac {1}{4} x^{2}+\frac {1}{36} x^{3}+\frac {23}{864} x^{4}+\frac {13}{1440} x^{5}+\frac {619}{155520} x^{6}+\frac {689}{408240} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6591

ODE	\[ \boxed {x \left (x^{2}+1\right )^{2} y^{\prime \prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {x}{2}+\frac {x^{2}}{12}+\frac {23 x^{3}}{144}-\frac {167 x^{4}}{2880}-\frac {7993 x^{5}}{86400}+\frac {23599 x^{6}}{518400}+\frac {1860281 x^{7}}{29030400}+O\left (x^{8}\right )\right )+c_{2} \left (-x \left (1-\frac {x}{2}+\frac {x^{2}}{12}+\frac {23 x^{3}}{144}-\frac {167 x^{4}}{2880}-\frac {7993 x^{5}}{86400}+\frac {23599 x^{6}}{518400}+\frac {1860281 x^{7}}{29030400}+O\left (x^{8}\right )\right ) \ln \left (x \right )+1-\frac {3 x^{2}}{4}+\frac {19 x^{3}}{36}+\frac {85 x^{4}}{1728}-\frac {21907 x^{5}}{86400}+\frac {787 x^{6}}{81000}+\frac {5987917 x^{7}}{36288000}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1-\frac {1}{2} x +\frac {1}{12} x^{2}+\frac {23}{144} x^{3}-\frac {167}{2880} x^{4}-\frac {7993}{86400} x^{5}+\frac {23599}{518400} x^{6}+\frac {1860281}{29030400} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-x +\frac {1}{2} x^{2}-\frac {1}{12} x^{3}-\frac {23}{144} x^{4}+\frac {167}{2880} x^{5}+\frac {7993}{86400} x^{6}-\frac {23599}{518400} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (1-\frac {3}{4} x^{2}+\frac {19}{36} x^{3}+\frac {85}{1728} x^{4}-\frac {21907}{86400} x^{5}+\frac {787}{81000} x^{6}+\frac {5987917}{36288000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right ) \]

Problem 6592

ODE	\[ \boxed {x^{3} \left (x^{2}-25\right ) \left (x -2\right )^{2} y^{\prime \prime }+3 x \left (x -2\right ) y^{\prime }+7 \left (x +5\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 6593

ODE	\[ \boxed {\left (x^{3}-2 x^{2}+3 x \right )^{2} y^{\prime \prime }+x \left (x -3\right )^{2} y^{\prime }-y \left (1+x \right )=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1+\frac {x}{45}+\frac {149 x^{2}}{3240}+\frac {2701 x^{3}}{192456}+\frac {236933 x^{4}}{121247280}-\frac {67092967 x^{5}}{92754169200}-\frac {30839263691 x^{6}}{50087251368000}-\frac {14846109458423 x^{7}}{72576427232232000}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1+\frac {13 x}{9}-\frac {5 x^{2}}{162}+\frac {1591 x^{3}}{30618}+\frac {106583 x^{4}}{5511240}+\frac {7435523 x^{5}}{3224075400}-\frac {70024699 x^{6}}{43525017900}-\frac {2917066898 x^{7}}{2604972321315}+O\left (x^{8}\right )\right )}{x^{\frac {1}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {2}{3}} \left (1+\frac {1}{45} x +\frac {149}{3240} x^{2}+\frac {2701}{192456} x^{3}+\frac {236933}{121247280} x^{4}-\frac {67092967}{92754169200} x^{5}-\frac {30839263691}{50087251368000} x^{6}-\frac {14846109458423}{72576427232232000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{1} \left (1+\frac {13}{9} x -\frac {5}{162} x^{2}+\frac {1591}{30618} x^{3}+\frac {106583}{5511240} x^{4}+\frac {7435523}{3224075400} x^{5}-\frac {70024699}{43525017900} x^{6}-\frac {2917066898}{2604972321315} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x^{\frac {1}{3}}} \]

Problem 6594

ODE	\[ \boxed {\left (x^{2}-1\right ) y^{\prime \prime }+5 \left (1+x \right ) y^{\prime }+\left (x^{2}-x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{6} x^{3}-\frac {1}{8} x^{4}-\frac {3}{10} x^{5}-\frac {17}{45} x^{6}-\frac {199}{336} x^{7}-\frac {10373}{13440} x^{8}\right ) y \left (0\right )+\left (x +\frac {5}{2} x^{2}+5 x^{3}+\frac {26}{3} x^{4}+\frac {1661}{120} x^{5}+\frac {4967}{240} x^{6}+\frac {14881}{504} x^{7}+\frac {102149}{2520} x^{8}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{6} x^{3}-\frac {1}{8} x^{4}-\frac {3}{10} x^{5}-\frac {17}{45} x^{6}-\frac {199}{336} x^{7}\right ) c_{1} +\left (x +\frac {5}{2} x^{2}+5 x^{3}+\frac {26}{3} x^{4}+\frac {1661}{120} x^{5}+\frac {4967}{240} x^{6}+\frac {14881}{504} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{6} x^{3}-\frac {1}{8} x^{4}-\frac {3}{10} x^{5}-\frac {17}{45} x^{6}-\frac {199}{336} x^{7}\right ) y \left (0\right )+\left (x +\frac {5}{2} x^{2}+5 x^{3}+\frac {26}{3} x^{4}+\frac {1661}{120} x^{5}+\frac {4967}{240} x^{6}+\frac {14881}{504} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6595

ODE	\[ \boxed {x y^{\prime \prime }+\left (x +3\right ) y^{\prime }+7 y x^{2}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-\frac {7 x^{3}}{15}+\frac {7 x^{4}}{120}-\frac {x^{5}}{150}+\frac {11 x^{6}}{160}-\frac {197 x^{7}}{15120}+O\left (x^{8}\right )\right )+c_{2} \left (\left (-1+\frac {7 x^{3}}{15}-\frac {7 x^{4}}{120}+\frac {x^{5}}{150}-\frac {11 x^{6}}{160}+\frac {197 x^{7}}{15120}-O\left (x^{8}\right )\right ) \ln \left (x \right )+\frac {1-2 x -2 x^{3}+2 x^{4}-\frac {116 x^{5}}{225}+\frac {4699 x^{6}}{7200}-\frac {14969 x^{7}}{31500}+O\left (x^{8}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (1-\frac {7}{15} x^{3}+\frac {7}{120} x^{4}-\frac {1}{150} x^{5}+\frac {11}{160} x^{6}-\frac {197}{15120} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (\ln \left (x \right ) \left (2 x^{2}-\frac {14}{15} x^{5}+\frac {7}{60} x^{6}-\frac {1}{75} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-2+4 x -3 x^{2}+4 x^{3}-4 x^{4}+\frac {547}{225} x^{5}-\frac {5329}{3600} x^{6}+\frac {7642}{7875} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )}{x^{2}} \]

Problem 6596

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\left (\frac {5}{3} x +x^{2}\right ) y^{\prime }-\frac {y}{3}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1-\frac {x}{7}+\frac {x^{2}}{35}-\frac {x^{3}}{195}+\frac {x^{4}}{1248}-\frac {x^{5}}{9120}+\frac {x^{6}}{75240}-\frac {x^{7}}{693000}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1-3 x +O\left (x^{8}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {4}{3}} \left (1-\frac {1}{7} x +\frac {1}{35} x^{2}-\frac {1}{195} x^{3}+\frac {1}{1248} x^{4}-\frac {1}{9120} x^{5}+\frac {1}{75240} x^{6}-\frac {1}{693000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{1} \left (1-3 x +\operatorname {O}\left (x^{8}\right )\right )}{x} \]

Problem 6597

ODE	\[ \boxed {x y^{\prime \prime }+y^{\prime }+10 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (25 x^{2}-10 x +1-\frac {250 x^{3}}{9}+\frac {625 x^{4}}{36}-\frac {125 x^{5}}{18}+\frac {625 x^{6}}{324}-\frac {3125 x^{7}}{7938}+O\left (x^{8}\right )\right )+c_{2} \left (\left (25 x^{2}-10 x +1-\frac {250 x^{3}}{9}+\frac {625 x^{4}}{36}-\frac {125 x^{5}}{18}+\frac {625 x^{6}}{324}-\frac {3125 x^{7}}{7938}+O\left (x^{8}\right )\right ) \ln \left (x \right )-75 x^{2}+20 x +\frac {2750 x^{3}}{27}-\frac {15625 x^{4}}{216}+\frac {3425 x^{5}}{108}-\frac {6125 x^{6}}{648}+\frac {75625 x^{7}}{37044}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-10 x +25 x^{2}-\frac {250}{9} x^{3}+\frac {625}{36} x^{4}-\frac {125}{18} x^{5}+\frac {625}{324} x^{6}-\frac {3125}{7938} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (20 x -75 x^{2}+\frac {2750}{27} x^{3}-\frac {15625}{216} x^{4}+\frac {3425}{108} x^{5}-\frac {6125}{648} x^{6}+\frac {75625}{37044} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \]

Problem 6598

ODE	\[ \boxed {2 x y^{\prime \prime }-y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {3}{2}} \left (1-\frac {2 x}{5}+\frac {2 x^{2}}{35}-\frac {4 x^{3}}{945}+\frac {2 x^{4}}{10395}-\frac {4 x^{5}}{675675}+\frac {4 x^{6}}{30405375}-\frac {8 x^{7}}{3618239625}+O\left (x^{8}\right )\right )+c_{2} \left (1+2 x -2 x^{2}+\frac {4 x^{3}}{9}-\frac {2 x^{4}}{45}+\frac {4 x^{5}}{1575}-\frac {4 x^{6}}{42525}+\frac {8 x^{7}}{3274425}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {3}{2}} \left (1-\frac {2}{5} x +\frac {2}{35} x^{2}-\frac {4}{945} x^{3}+\frac {2}{10395} x^{4}-\frac {4}{675675} x^{5}+\frac {4}{30405375} x^{6}-\frac {8}{3618239625} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1+2 x -2 x^{2}+\frac {4}{9} x^{3}-\frac {2}{45} x^{4}+\frac {4}{1575} x^{5}-\frac {4}{42525} x^{6}+\frac {8}{3274425} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6599

ODE	\[ \boxed {2 x y^{\prime \prime }+5 y^{\prime }+y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-\frac {x^{2}}{14}+\frac {x^{4}}{616}-\frac {x^{6}}{55440}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1-\frac {x^{2}}{2}+\frac {x^{4}}{40}-\frac {x^{6}}{2160}+O\left (x^{8}\right )\right )}{x^{\frac {3}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-\frac {1}{2} x^{2}+\frac {1}{40} x^{4}-\frac {1}{2160} x^{6}+\operatorname {O}\left (x^{8}\right )\right )}{x^{\frac {3}{2}}}+c_{2} \left (1-\frac {1}{14} x^{2}+\frac {1}{616} x^{4}-\frac {1}{55440} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6600

ODE	\[ \boxed {4 x y^{\prime \prime }+\frac {y^{\prime }}{2}+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {7}{8}} \left (1-\frac {2 x}{15}+\frac {2 x^{2}}{345}-\frac {4 x^{3}}{32085}+\frac {2 x^{4}}{1251315}-\frac {4 x^{5}}{294059025}+\frac {4 x^{6}}{48519739125}-\frac {8 x^{7}}{21397204954125}+O\left (x^{8}\right )\right )+c_{2} \left (1-2 x +\frac {2 x^{2}}{9}-\frac {4 x^{3}}{459}+\frac {2 x^{4}}{11475}-\frac {4 x^{5}}{1893375}+\frac {4 x^{6}}{232885125}-\frac {8 x^{7}}{79879597875}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {7}{8}} \left (1-\frac {2}{15} x +\frac {2}{345} x^{2}-\frac {4}{32085} x^{3}+\frac {2}{1251315} x^{4}-\frac {4}{294059025} x^{5}+\frac {4}{48519739125} x^{6}-\frac {8}{21397204954125} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1-2 x +\frac {2}{9} x^{2}-\frac {4}{459} x^{3}+\frac {2}{11475} x^{4}-\frac {4}{1893375} x^{5}+\frac {4}{232885125} x^{6}-\frac {8}{79879597875} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]