2.17.61 Problems 6001 to 6100

Problem 6001

ODE	\[ \boxed {y^{\prime \prime }+9 y=x^{2} {\mathrm e}^{3 x}} \]
program solution	\[ y = c_{1} \cos \left (3 x \right )+\frac {c_{2} \sin \left (3 x \right )}{3}-\frac {x \,{\mathrm e}^{3 x}}{27}+\frac {x^{2} {\mathrm e}^{3 x}}{18}+\frac {{\mathrm e}^{3 x}}{162} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (x -\frac {1}{3}\right )^{2} {\mathrm e}^{3 x}}{18}+\cos \left (3 x \right ) c_{1} +\sin \left (3 x \right ) c_{2} \]

Problem 6002

ODE	\[ \boxed {y^{\prime \prime }+y=x \,{\mathrm e}^{x} \cos \left (2 x \right )} \]
program solution	\[ y = c_{1} \cos \left (x \right )+c_{2} \sin \left (x \right )+\frac {11 \,{\mathrm e}^{x} \cos \left (2 x \right )}{50}-\frac {{\mathrm e}^{x} \sin \left (2 x \right )}{25}-\frac {x \,{\mathrm e}^{x} \cos \left (2 x \right )}{10}+\frac {x \,{\mathrm e}^{x} \sin \left (2 x \right )}{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (\left (-10 x +22\right ) \cos \left (x \right )^{2}+\left (20 x -4\right ) \sin \left (x \right ) \cos \left (x \right )+5 x -11\right ) {\mathrm e}^{x}}{50}+\cos \left (x \right ) c_{1} +\sin \left (x \right ) c_{2} \]

Problem 6003

ODE	\[ \boxed {y^{\prime \prime }+i y^{\prime }+2 y=2 \cosh \left (2 x \right )+{\mathrm e}^{-2 x}} \]
program solution	\[ y = \frac {{\mathrm e}^{-\frac {i x}{2}} \left (2 c_{2} \sin \left (\frac {3 x}{2}\right )+3 c_{1} \cos \left (\frac {3 x}{2}\right )\right )}{3}+\left (\frac {3}{10}+\frac {i}{10}\right ) {\mathrm e}^{-2 x}+\left (\frac {3}{20}-\frac {i}{20}\right ) {\mathrm e}^{2 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{2} {\mathrm e}^{i x}+{\mathrm e}^{-2 i x} c_{1} +\left (\frac {3}{10}+\frac {i}{10}\right ) {\mathrm e}^{-2 x}+\left (\frac {3}{20}-\frac {i}{20}\right ) {\mathrm e}^{2 x} \]

Problem 6004

ODE	\[ \boxed {y^{\prime \prime \prime }=x^{2}+{\mathrm e}^{-x} \sin \left (x \right )} \]
program solution	\[ y = c_{3} x^{2}+c_{2} x +c_{1} -\frac {{\mathrm e}^{-x} \cos \left (x \right )}{4}+\frac {{\mathrm e}^{-x} \sin \left (x \right )}{4}+\frac {x^{5}}{60} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{-x} \left (-\cos \left (x \right )+\sin \left (x \right )\right )}{4}+\frac {x^{5}}{60}+\frac {c_{1} x^{2}}{2}+c_{2} x +c_{3} \]

Problem 6005

ODE	\[ \boxed {y^{\prime \prime \prime }+3 y^{\prime \prime }+3 y^{\prime }+y=x^{2} {\mathrm e}^{-x}} \]
program solution	\[ y = {\mathrm e}^{-x} \left (c_{3} x^{2}+c_{2} x +c_{1} \right )+\frac {x^{5} {\mathrm e}^{-x}}{60} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-x} \left (\frac {1}{60} x^{5}+c_{1} +c_{2} x +c_{3} x^{2}\right ) \]

Problem 6006

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

Problem 6007

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

Problem 6008

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

Problem 6009

ODE	\[ \boxed {x^{2} y^{\prime \prime }-7 y^{\prime } x +15 y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= x^{3} \end {align*}
program solution	\[ y = c_{1} x^{3}+\frac {1}{2} c_{2} x^{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{3} \left (c_{1} x^{2}+c_{2} \right ) \]

Problem 6010

ODE	\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x +y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= x \end {align*}
program solution	\[ y = c_{1} x +c_{2} \ln \left (x \right ) x \] Verified OK.
Maple solution	\[ y \left (x \right ) = x \left (c_{2} \ln \left (x \right )+c_{1} \right ) \]

Problem 6011

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime } x +\left (4 x^{2}-2\right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{x^{2}} \end {align*}
program solution	\[ y = c_{1} {\mathrm e}^{x^{2}}+c_{2} x \,{\mathrm e}^{x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{x^{2}} \left (c_{2} x +c_{1} \right ) \]

Problem 6012

ODE	\[ \boxed {x y^{\prime \prime }-\left (1+x \right ) y^{\prime }+y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{x} \end {align*}
program solution	\[ y = c_{1} {\mathrm e}^{x}-c_{2} {\mathrm e}^{x} \left (1+x \right ) {\mathrm e}^{-x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{x} c_{2} +c_{1} x +c_{1} \]

Problem 6013

ODE	\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +2 y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= x \end {align*}
program solution	\[ y = c_{1} x +c_{2} x \left (-\frac {\ln \left (1+x \right )}{2}+\frac {\ln \left (x -1\right )}{2}+\frac {1}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {c_{2} \ln \left (x +1\right ) x}{2}+\frac {c_{2} \ln \left (x -1\right ) x}{2}+c_{1} x +c_{2} \]

Problem 6014

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime } x +2 y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= x \end {align*}
program solution	\[ y = c_{1} x +c_{2} x \left (-\frac {{\mathrm e}^{x^{2}}}{x}+\sqrt {\pi }\, \operatorname {erfi}\left (x \right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{x^{2}} c_{2} +x \left (-\sqrt {\pi }\, c_{2} \operatorname {erfi}\left (x \right )+c_{1} \right ) \]

Problem 6015

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

Problem 6016

ODE	\[ \boxed {x^{2} y^{\prime \prime }-2 y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= x^{2} \end {align*}
program solution	\[ y = c_{1} x^{2}-\frac {c_{2}}{3 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{3}+c_{2}}{x} \]

Problem 6017

ODE	\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x +y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= x \end {align*}
program solution	\[ y = c_{1} x +c_{2} \ln \left (x \right ) x \] Verified OK.
Maple solution	\[ y \left (x \right ) = x \left (c_{2} \ln \left (x \right )+c_{1} \right ) \]

Problem 6018

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

Problem 6019

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

Problem 6020

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

Problem 6021

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

Problem 6022

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

Problem 6023

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 6024

ODE	\[ \boxed {y^{\prime \prime }+\left (x -1\right )^{2} y^{\prime }-\left (x -1\right ) y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1, y^{\prime }\left (1\right ) = 0] \end {align*} With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = 1+\frac {\left (x -1\right )^{3}}{6}-\frac {\left (x -1\right )^{6}}{90}+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1+\frac {1}{6} \left (x -1\right )^{3}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \]

Problem 6025

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = x -\frac {x^{3}}{6}+\frac {7 x^{5}}{120}+O\left (x^{6}\right ) \] Verified OK. \[ y = x -\frac {x^{3}}{6}+\frac {7 x^{5}}{120}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x -\frac {1}{6} x^{3}+\frac {7}{120} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 6026

ODE	\[ \boxed {y^{\prime \prime }+{\mathrm e}^{x} y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 1-\frac {x^{2}}{2}-\frac {x^{3}}{6}+\frac {x^{5}}{40}+\frac {x^{6}}{80}+O\left (x^{6}\right ) \] Verified OK. \[ y = 1-\frac {x^{2}}{2}-\frac {x^{3}}{6}+\frac {x^{5}}{40}+O\left (x^{6}\right ) \] 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 6027

ODE	\[ \boxed {y^{\prime \prime \prime }-y x=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = 0] \end {align*}
program solution
Maple solution	\[ y \left (x \right ) = \operatorname {hypergeom}\left (\left [\right ], \left [\frac {1}{2}, \frac {3}{4}\right ], \frac {x^{4}}{64}\right ) \]

Problem 6028

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

Problem 6029

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

Problem 6030

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime } x +2 \alpha y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = x \left (\operatorname {KummerM}\left (\frac {1}{2}-\frac {\alpha }{2}, \frac {3}{2}, x^{2}\right ) c_{1} +\operatorname {KummerU}\left (\frac {1}{2}-\frac {\alpha }{2}, \frac {3}{2}, x^{2}\right ) c_{2} \right ) \]

Problem 6031

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

Problem 6032

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

Problem 6033

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

Problem 6034

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

Problem 6035

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

Problem 6036

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +4 y=1} \]
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	\[ y \left (x \right ) = \sin \left (2 \ln \left (x \right )\right ) c_{2} +\cos \left (2 \ln \left (x \right )\right ) c_{1} +\frac {1}{4} \]

Problem 6037

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

Problem 6038

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

Problem 6039

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

Problem 6040

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {x}{3}+\frac {x^{2}}{12}-\frac {x^{3}}{60}+\frac {x^{4}}{360}-\frac {x^{5}}{2520}+\frac {x^{6}}{20160}-\frac {x^{7}}{181440}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1-x +\frac {x^{2}}{2}-\frac {x^{3}}{6}+\frac {x^{4}}{24}-\frac {x^{5}}{120}+\frac {x^{6}}{720}-\frac {x^{7}}{5040}+O\left (x^{8}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1-\frac {1}{3} x +\frac {1}{12} x^{2}-\frac {1}{60} x^{3}+\frac {1}{360} x^{4}-\frac {1}{2520} x^{5}+\frac {1}{20160} x^{6}-\frac {1}{181440} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (-2+2 x -x^{2}+\frac {1}{3} x^{3}-\frac {1}{12} x^{4}+\frac {1}{60} x^{5}-\frac {1}{360} x^{6}+\frac {1}{2520} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x} \]

Problem 6041

ODE	\[ \boxed {3 x^{2} y^{\prime \prime }+y^{\prime } x^{6}+2 y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {x}{3}+\frac {x^{2}}{27}-\frac {x^{3}}{486}+\frac {x^{4}}{14580}-\frac {7291 x^{5}}{656100}+\frac {225991 x^{6}}{41334300}-\frac {2522341 x^{7}}{3472081200}+O\left (x^{8}\right )\right )+c_{2} \left (-\frac {2 x \left (1-\frac {x}{3}+\frac {x^{2}}{27}-\frac {x^{3}}{486}+\frac {x^{4}}{14580}-\frac {7291 x^{5}}{656100}+\frac {225991 x^{6}}{41334300}-\frac {2522341 x^{7}}{3472081200}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{3}+1-\frac {x^{2}}{3}+\frac {14 x^{3}}{243}-\frac {35 x^{4}}{8748}+\frac {101 x^{5}}{656100}+\frac {69199 x^{6}}{14762250}+\frac {19882543 x^{7}}{4340101500}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1-\frac {1}{3} x +\frac {1}{27} x^{2}-\frac {1}{486} x^{3}+\frac {1}{14580} x^{4}-\frac {7291}{656100} x^{5}+\frac {225991}{41334300} x^{6}-\frac {2522341}{3472081200} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-\frac {2}{3} x +\frac {2}{9} x^{2}-\frac {2}{81} x^{3}+\frac {1}{729} x^{4}-\frac {1}{21870} x^{5}+\frac {7291}{984150} x^{6}-\frac {225991}{62001450} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (1-\frac {1}{3} x^{2}+\frac {14}{243} x^{3}-\frac {35}{8748} x^{4}+\frac {101}{656100} x^{5}+\frac {69199}{14762250} x^{6}+\frac {19882543}{4340101500} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right ) \]

Problem 6042

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

Problem 6043

ODE	\[ \boxed {x y^{\prime \prime }+4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-2 x +\frac {4 x^{2}}{3}-\frac {4 x^{3}}{9}+\frac {4 x^{4}}{45}-\frac {8 x^{5}}{675}+\frac {16 x^{6}}{14175}-\frac {8 x^{7}}{99225}+O\left (x^{8}\right )\right )+c_{2} \left (-4 x \left (1-2 x +\frac {4 x^{2}}{3}-\frac {4 x^{3}}{9}+\frac {4 x^{4}}{45}-\frac {8 x^{5}}{675}+\frac {16 x^{6}}{14175}-\frac {8 x^{7}}{99225}+O\left (x^{8}\right )\right ) \ln \left (x \right )+1-12 x^{2}+\frac {112 x^{3}}{9}-\frac {140 x^{4}}{27}+\frac {808 x^{5}}{675}-\frac {1792 x^{6}}{10125}+\frac {9056 x^{7}}{496125}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1-2 x +\frac {4}{3} x^{2}-\frac {4}{9} x^{3}+\frac {4}{45} x^{4}-\frac {8}{675} x^{5}+\frac {16}{14175} x^{6}-\frac {8}{99225} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (\left (-4\right ) x +8 x^{2}-\frac {16}{3} x^{3}+\frac {16}{9} x^{4}-\frac {16}{45} x^{5}+\frac {32}{675} x^{6}-\frac {64}{14175} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (1-12 x^{2}+\frac {112}{9} x^{3}-\frac {140}{27} x^{4}+\frac {808}{675} x^{5}-\frac {1792}{10125} x^{6}+\frac {9056}{496125} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right ) \]

Problem 6044

ODE	\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +2 y=0} \] With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = c_{1} \left (x +O\left (\left (x -1\right )^{8}\right )\right )+c_{2} \left (\left (x +O\left (\left (x -1\right )^{8}\right )\right ) \ln \left (x -1\right )-\frac {5 x}{2}+\frac {5}{2}-\frac {3 \left (x -1\right )^{2}}{8}+\frac {\left (x -1\right )^{3}}{12}-\frac {5 \left (x -1\right )^{4}}{192}+\frac {3 \left (x -1\right )^{5}}{320}-\frac {7 \left (x -1\right )^{6}}{1920}+\frac {\left (x -1\right )^{7}}{672}+O\left (\left (x -1\right )^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (-\frac {5}{2} \left (x -1\right )-\frac {3}{8} \left (x -1\right )^{2}+\frac {1}{12} \left (x -1\right )^{3}-\frac {5}{192} \left (x -1\right )^{4}+\frac {3}{320} \left (x -1\right )^{5}-\frac {7}{1920} \left (x -1\right )^{6}+\frac {1}{672} \left (x -1\right )^{7}+\operatorname {O}\left (\left (x -1\right )^{8}\right )\right ) c_{2} +\left (1+\left (x -1\right )+\operatorname {O}\left (\left (x -1\right )^{8}\right )\right ) \left (c_{2} \ln \left (x -1\right )+c_{1} \right ) \]

Problem 6045

ODE	\[ \boxed {\left (x^{2}+x -2\right )^{2} y^{\prime \prime }+3 \left (x +2\right ) y^{\prime }+\left (x -1\right ) y=0} \] With the expansion point for the power series method at \(x = -2\).
program solution	\[ y = c_{1} \left (x +2\right ) \left (\frac {19}{21}-\frac {x}{21}-\frac {11 \left (x +2\right )^{2}}{1260}-\frac {53 \left (x +2\right )^{3}}{29484}-\frac {11093 \left (x +2\right )^{4}}{28304640}-\frac {709507 \left (x +2\right )^{5}}{8066822400}-\frac {5797423 \left (x +2\right )^{6}}{290405606400}-\frac {52991201 \left (x +2\right )^{7}}{11727918720000}+O\left (\left (x +2\right )^{8}\right )\right )+\frac {c_{2} \left (-\frac {1}{9}-\frac {5 x}{9}+\frac {23 \left (x +2\right )^{2}}{324}+\frac {271 \left (x +2\right )^{3}}{43740}+\frac {10517 \left (x +2\right )^{4}}{12597120}+\frac {778801 \left (x +2\right )^{5}}{6235574400}+\frac {16965493 \left (x +2\right )^{6}}{942818849280}+\frac {899971067 \left (x +2\right )^{7}}{458981357990400}+O\left (\left (x +2\right )^{8}\right )\right )}{\left (x +2\right )^{\frac {1}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-\frac {5}{9} \left (x +2\right )+\frac {23}{324} \left (x +2\right )^{2}+\frac {271}{43740} \left (x +2\right )^{3}+\frac {10517}{12597120} \left (x +2\right )^{4}+\frac {778801}{6235574400} \left (x +2\right )^{5}+\frac {16965493}{942818849280} \left (x +2\right )^{6}+\frac {899971067}{458981357990400} \left (x +2\right )^{7}+\operatorname {O}\left (\left (x +2\right )^{8}\right )\right )+c_{2} \left (x +2\right )^{\frac {4}{3}} \left (1-\frac {1}{21} \left (x +2\right )-\frac {11}{1260} \left (x +2\right )^{2}-\frac {53}{29484} \left (x +2\right )^{3}-\frac {11093}{28304640} \left (x +2\right )^{4}-\frac {709507}{8066822400} \left (x +2\right )^{5}-\frac {5797423}{290405606400} \left (x +2\right )^{6}-\frac {52991201}{11727918720000} \left (x +2\right )^{7}+\operatorname {O}\left (\left (x +2\right )^{8}\right )\right )}{\left (x +2\right )^{\frac {1}{3}}} \]

Problem 6046

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\sin \left (x \right ) y^{\prime }+y \cos \left (x \right )=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{i} \left (1+\left (\frac {1}{12}-\frac {i}{24}\right ) x^{2}+\left (\frac {29}{28800}-\frac {67 i}{28800}\right ) x^{4}+\left (-\frac {893}{14515200}+\frac {17 i}{4838400}\right ) x^{6}+O\left (x^{8}\right )\right )+c_{2} x^{-i} \left (1+\left (\frac {1}{12}+\frac {i}{24}\right ) x^{2}+\left (\frac {29}{28800}+\frac {67 i}{28800}\right ) x^{4}+\left (-\frac {893}{14515200}-\frac {17 i}{4838400}\right ) x^{6}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{-i} \left (1+\left (\frac {1}{12}+\frac {i}{24}\right ) x^{2}+\left (\frac {29}{28800}+\frac {67 i}{28800}\right ) x^{4}+\left (-\frac {893}{14515200}-\frac {17 i}{4838400}\right ) x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x^{i} \left (1+\left (\frac {1}{12}-\frac {i}{24}\right ) x^{2}+\left (\frac {29}{28800}-\frac {67 i}{28800}\right ) x^{4}+\left (-\frac {893}{14515200}+\frac {17 i}{4838400}\right ) x^{6}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6047

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

Problem 6048

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+\left (4 x^{4}-5 x \right ) y^{\prime }+y \left (x^{2}+2\right )=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-\frac {x^{2}}{30}-\frac {8 x^{3}}{57}+\frac {x^{4}}{2760}+\frac {64 x^{5}}{12825}+\frac {147181 x^{6}}{9753840}-\frac {4037 x^{7}}{72268875}+O\left (x^{8}\right )\right )+c_{2} x^{\frac {1}{4}} \left (1-\frac {x^{2}}{2}-\frac {x^{3}}{15}+\frac {x^{4}}{72}+\frac {137 x^{5}}{1950}+\frac {307 x^{6}}{36720}-\frac {7169 x^{7}}{3439800}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{4}} \left (1-\frac {1}{2} x^{2}-\frac {1}{15} x^{3}+\frac {1}{72} x^{4}+\frac {137}{1950} x^{5}+\frac {307}{36720} x^{6}-\frac {7169}{3439800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x^{2} \left (1-\frac {1}{30} x^{2}-\frac {8}{57} x^{3}+\frac {1}{2760} x^{4}+\frac {64}{12825} x^{5}+\frac {147181}{9753840} x^{6}-\frac {4037}{72268875} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6049

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\left (-3 x^{2}+x \right ) y^{\prime }+{\mathrm e}^{x} y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{i} \left (1+\left (1+i\right ) x +\left (\frac {7}{16}+\frac {13 i}{16}\right ) x^{2}+\left (\frac {7}{39}+\frac {395 i}{936}\right ) x^{3}+\left (\frac {2117}{29952}+\frac {5197 i}{29952}\right ) x^{4}+\left (\frac {5521}{217152}+\frac {642043 i}{10857600}\right ) x^{5}+\left (\frac {782461}{97718400}+\frac {8813057 i}{521164800}\right ) x^{6}+\left (\frac {1238071931}{580056422400}+\frac {3271304833 i}{812078991360}\right ) x^{7}+O\left (x^{8}\right )\right )+c_{2} x^{-i} \left (1+\left (1-i\right ) x +\left (\frac {7}{16}-\frac {13 i}{16}\right ) x^{2}+\left (\frac {7}{39}-\frac {395 i}{936}\right ) x^{3}+\left (\frac {2117}{29952}-\frac {5197 i}{29952}\right ) x^{4}+\left (\frac {5521}{217152}-\frac {642043 i}{10857600}\right ) x^{5}+\left (\frac {782461}{97718400}-\frac {8813057 i}{521164800}\right ) x^{6}+\left (\frac {1238071931}{580056422400}-\frac {3271304833 i}{812078991360}\right ) x^{7}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{-i} \left (1+\left (1-i\right ) x +\left (\frac {7}{16}-\frac {13 i}{16}\right ) x^{2}+\left (\frac {7}{39}-\frac {395 i}{936}\right ) x^{3}+\left (\frac {2117}{29952}-\frac {5197 i}{29952}\right ) x^{4}+\left (\frac {5521}{217152}-\frac {642043 i}{10857600}\right ) x^{5}+\left (\frac {782461}{97718400}-\frac {8813057 i}{521164800}\right ) x^{6}+\left (\frac {1238071931}{580056422400}-\frac {3271304833 i}{812078991360}\right ) x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x^{i} \left (1+\left (1+i\right ) x +\left (\frac {7}{16}+\frac {13 i}{16}\right ) x^{2}+\left (\frac {7}{39}+\frac {395 i}{936}\right ) x^{3}+\left (\frac {2117}{29952}+\frac {5197 i}{29952}\right ) x^{4}+\left (\frac {5521}{217152}+\frac {642043 i}{10857600}\right ) x^{5}+\left (\frac {782461}{97718400}+\frac {8813057 i}{521164800}\right ) x^{6}+\left (\frac {1238071931}{580056422400}+\frac {3271304833 i}{812078991360}\right ) x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6050

ODE	\[ \boxed {3 x^{2} y^{\prime \prime }+5 y^{\prime } x +3 y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-\frac {3 x}{5}+\frac {9 x^{2}}{80}-\frac {9 x^{3}}{880}+\frac {27 x^{4}}{49280}-\frac {81 x^{5}}{4188800}+\frac {81 x^{6}}{167552000}-\frac {243 x^{7}}{26975872000}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1-3 x +\frac {9 x^{2}}{8}-\frac {9 x^{3}}{56}+\frac {27 x^{4}}{2240}-\frac {81 x^{5}}{145600}+\frac {81 x^{6}}{4659200}-\frac {243 x^{7}}{619673600}+O\left (x^{8}\right )\right )}{x^{\frac {2}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-3 x +\frac {9}{8} x^{2}-\frac {9}{56} x^{3}+\frac {27}{2240} x^{4}-\frac {81}{145600} x^{5}+\frac {81}{4659200} x^{6}-\frac {243}{619673600} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x^{\frac {2}{3}}}+c_{2} \left (1-\frac {3}{5} x +\frac {9}{80} x^{2}-\frac {9}{880} x^{3}+\frac {27}{49280} x^{4}-\frac {81}{4188800} x^{5}+\frac {81}{167552000} x^{6}-\frac {243}{26975872000} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6051

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +y x^{2}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-\frac {x^{2}}{4}+\frac {x^{4}}{64}-\frac {x^{6}}{2304}+O\left (x^{8}\right )\right )+c_{2} \left (\left (1-\frac {x^{2}}{4}+\frac {x^{4}}{64}-\frac {x^{6}}{2304}+O\left (x^{8}\right )\right ) \ln \left (x \right )+\frac {x^{2}}{4}-\frac {3 x^{4}}{128}+\frac {11 x^{6}}{13824}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {1}{4} x^{2}+\frac {1}{64} x^{4}-\frac {1}{2304} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+\left (\frac {1}{4} x^{2}-\frac {3}{128} x^{4}+\frac {11}{13824} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \]

Problem 6052

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } {\mathrm e}^{x} x +y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{i} \left (1+\left (-\frac {2}{5}-\frac {i}{5}\right ) x +\left (\frac {3}{80}-\frac {i}{80}\right ) x^{2}+\left (\frac {67}{9360}+\frac {9 i}{1040}\right ) x^{3}+\left (-\frac {103}{149760}+\frac {229 i}{149760}\right ) x^{4}+\left (-\frac {2831}{7238400}-\frac {607 i}{4343040}\right ) x^{5}+\left (-\frac {59077}{1563494400}-\frac {26063 i}{260582400}\right ) x^{6}+\left (\frac {22952047}{2030197478400}-\frac {8634893 i}{580056422400}\right ) x^{7}+O\left (x^{8}\right )\right )+c_{2} x^{-i} \left (1+\left (-\frac {2}{5}+\frac {i}{5}\right ) x +\left (\frac {3}{80}+\frac {i}{80}\right ) x^{2}+\left (\frac {67}{9360}-\frac {9 i}{1040}\right ) x^{3}+\left (-\frac {103}{149760}-\frac {229 i}{149760}\right ) x^{4}+\left (-\frac {2831}{7238400}+\frac {607 i}{4343040}\right ) x^{5}+\left (-\frac {59077}{1563494400}+\frac {26063 i}{260582400}\right ) x^{6}+\left (\frac {22952047}{2030197478400}+\frac {8634893 i}{580056422400}\right ) x^{7}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{-i} \left (1+\left (-\frac {2}{5}+\frac {i}{5}\right ) x +\left (\frac {3}{80}+\frac {i}{80}\right ) x^{2}+\left (\frac {67}{9360}-\frac {9 i}{1040}\right ) x^{3}+\left (-\frac {103}{149760}-\frac {229 i}{149760}\right ) x^{4}+\left (-\frac {2831}{7238400}+\frac {607 i}{4343040}\right ) x^{5}+\left (-\frac {59077}{1563494400}+\frac {26063 i}{260582400}\right ) x^{6}+\left (\frac {22952047}{2030197478400}+\frac {8634893 i}{580056422400}\right ) x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x^{i} \left (1+\left (-\frac {2}{5}-\frac {i}{5}\right ) x +\left (\frac {3}{80}-\frac {i}{80}\right ) x^{2}+\left (\frac {67}{9360}+\frac {9 i}{1040}\right ) x^{3}+\left (-\frac {103}{149760}+\frac {229 i}{149760}\right ) x^{4}+\left (-\frac {2831}{7238400}-\frac {607 i}{4343040}\right ) x^{5}+\left (-\frac {59077}{1563494400}-\frac {26063 i}{260582400}\right ) x^{6}+\left (\frac {22952047}{2030197478400}-\frac {8634893 i}{580056422400}\right ) x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6053

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }+\left (x^{2}+5 x \right ) y^{\prime }+\left (x^{2}-2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {x}{14}-\frac {25 x^{2}}{504}+\frac {197 x^{3}}{33264}+\frac {1921 x^{4}}{3459456}-\frac {11653 x^{5}}{103783680}+\frac {12923 x^{6}}{21171870720}+\frac {917285 x^{7}}{1126343522304}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1-\frac {2 x}{3}+\frac {5 x^{2}}{6}+\frac {2 x^{3}}{9}-\frac {19 x^{4}}{216}-\frac {x^{5}}{540}+\frac {101 x^{6}}{45360}-\frac {4 x^{7}}{35721}+O\left (x^{8}\right )\right )}{x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {5}{2}} \left (1-\frac {1}{14} x -\frac {25}{504} x^{2}+\frac {197}{33264} x^{3}+\frac {1921}{3459456} x^{4}-\frac {11653}{103783680} x^{5}+\frac {12923}{21171870720} x^{6}+\frac {917285}{1126343522304} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{1} \left (1-\frac {2}{3} x +\frac {5}{6} x^{2}+\frac {2}{9} x^{3}-\frac {19}{216} x^{4}-\frac {1}{540} x^{5}+\frac {101}{45360} x^{6}-\frac {4}{35721} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x^{2}} \]

Problem 6054

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }-4 y^{\prime } {\mathrm e}^{x} x +3 y \cos \left (x \right )=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {3}{2}} \left (1+\frac {3 x}{4}+\frac {x^{2}}{2}+\frac {103 x^{3}}{384}+\frac {669 x^{4}}{5120}+\frac {54731 x^{5}}{921600}+\frac {123443 x^{6}}{4838400}+\frac {30273113 x^{7}}{2890137600}+O\left (x^{8}\right )\right )+c_{2} \left (\frac {x^{\frac {3}{2}} \left (1+\frac {3 x}{4}+\frac {x^{2}}{2}+\frac {103 x^{3}}{384}+\frac {669 x^{4}}{5120}+\frac {54731 x^{5}}{921600}+\frac {123443 x^{6}}{4838400}+\frac {30273113 x^{7}}{2890137600}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{2}+\sqrt {x}\, \left (1-\frac {13 x^{3}}{144}-\frac {1715 x^{4}}{27648}-\frac {1313 x^{5}}{36864}-\frac {2999423 x^{6}}{165888000}-\frac {204656267 x^{7}}{24385536000}+O\left (x^{8}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (x \left (1+\frac {3}{4} x +\frac {1}{2} x^{2}+\frac {103}{384} x^{3}+\frac {669}{5120} x^{4}+\frac {54731}{921600} x^{5}+\frac {123443}{4838400} x^{6}+\frac {30273113}{2890137600} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{1} +c_{2} \left (\ln \left (x \right ) \left (\frac {1}{2} x +\frac {3}{8} x^{2}+\frac {1}{4} x^{3}+\frac {103}{768} x^{4}+\frac {669}{10240} x^{5}+\frac {54731}{1843200} x^{6}+\frac {123443}{9676800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (1+x +\frac {3}{4} x^{2}+\frac {59}{144} x^{3}+\frac {5701}{27648} x^{4}+\frac {17519}{184320} x^{5}+\frac {6852157}{165888000} x^{6}+\frac {417496453}{24385536000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )\right ) \sqrt {x} \]

Problem 6055

ODE	\[ \boxed {\left (-x^{2}+1\right ) x^{2} y^{\prime \prime }+3 \left (x^{2}+x \right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (3 x +1+\frac {x^{2}}{2}-\frac {x^{3}}{6}+\frac {x^{4}}{16}-\frac {43 x^{5}}{1200}+\frac {161 x^{6}}{7200}-\frac {1837 x^{7}}{117600}+O\left (x^{8}\right )\right )}{x}+c_{2} \left (\frac {\left (3 x +1+\frac {x^{2}}{2}-\frac {x^{3}}{6}+\frac {x^{4}}{16}-\frac {43 x^{5}}{1200}+\frac {161 x^{6}}{7200}-\frac {1837 x^{7}}{117600}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{x}+\frac {-9 x -\frac {7 x^{2}}{2}+\frac {7 x^{3}}{9}-\frac {25 x^{4}}{96}+\frac {5141 x^{5}}{36000}-\frac {2083 x^{6}}{24000}+\frac {489941 x^{7}}{8232000}+O\left (x^{8}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+3 x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{16} x^{4}-\frac {43}{1200} x^{5}+\frac {161}{7200} x^{6}-\frac {1837}{117600} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (\left (-9\right ) x -\frac {7}{2} x^{2}+\frac {7}{9} x^{3}-\frac {25}{96} x^{4}+\frac {5141}{36000} x^{5}-\frac {2083}{24000} x^{6}+\frac {489941}{8232000} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2}}{x} \]

Problem 6056

ODE	\[ \boxed {x^{2} y^{\prime \prime }+3 y^{\prime } x +y \left (1+x \right )=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1-x +\frac {x^{2}}{4}-\frac {x^{3}}{36}+\frac {x^{4}}{576}-\frac {x^{5}}{14400}+\frac {x^{6}}{518400}-\frac {x^{7}}{25401600}+O\left (x^{8}\right )\right )}{x}+c_{2} \left (\frac {\left (1-x +\frac {x^{2}}{4}-\frac {x^{3}}{36}+\frac {x^{4}}{576}-\frac {x^{5}}{14400}+\frac {x^{6}}{518400}-\frac {x^{7}}{25401600}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{x}+\frac {2 x -\frac {3 x^{2}}{4}+\frac {11 x^{3}}{108}-\frac {25 x^{4}}{3456}+\frac {137 x^{5}}{432000}-\frac {49 x^{6}}{5184000}+\frac {121 x^{7}}{592704000}+O\left (x^{8}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-x +\frac {1}{4} x^{2}-\frac {1}{36} x^{3}+\frac {1}{576} x^{4}-\frac {1}{14400} x^{5}+\frac {1}{518400} x^{6}-\frac {1}{25401600} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (2 x -\frac {3}{4} x^{2}+\frac {11}{108} x^{3}-\frac {25}{3456} x^{4}+\frac {137}{432000} x^{5}-\frac {49}{5184000} x^{6}+\frac {121}{592704000} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2}}{x} \]

Problem 6057

ODE	\[ \boxed {x^{2} y^{\prime \prime }+2 x^{2} y^{\prime }-2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-x +\frac {3 x^{2}}{5}-\frac {4 x^{3}}{15}+\frac {2 x^{4}}{21}-\frac {x^{5}}{35}+\frac {x^{6}}{135}-\frac {8 x^{7}}{4725}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1-x +\frac {2 x^{3}}{3}-\frac {2 x^{4}}{3}+\frac {2 x^{5}}{5}-\frac {8 x^{6}}{45}+\frac {4 x^{7}}{63}+O\left (x^{8}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \left (1-x +\frac {3}{5} x^{2}-\frac {4}{15} x^{3}+\frac {2}{21} x^{4}-\frac {1}{35} x^{5}+\frac {1}{135} x^{6}-\frac {8}{4725} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (12-12 x +8 x^{3}-8 x^{4}+\frac {24}{5} x^{5}-\frac {32}{15} x^{6}+\frac {16}{21} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x} \]

Problem 6058

ODE	\[ \boxed {x^{2} y^{\prime \prime }+5 y^{\prime } x +\left (-x^{3}+3\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1+\frac {x^{3}}{15}+\frac {x^{6}}{720}+O\left (x^{8}\right )\right )}{x}+\frac {c_{2} \left (1+\frac {x^{3}}{3}+\frac {x^{6}}{72}+O\left (x^{8}\right )\right )}{x^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1+\frac {1}{15} x^{3}+\frac {1}{720} x^{6}+\operatorname {O}\left (x^{8}\right )\right )}{x}+\frac {c_{2} \left (-2-\frac {2}{3} x^{3}-\frac {1}{36} x^{6}+\operatorname {O}\left (x^{8}\right )\right )}{x^{3}} \]

Problem 6059

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

Problem 6060

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (x^{2}-1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {x^{2}}{8}+\frac {x^{4}}{192}-\frac {x^{6}}{9216}+O\left (x^{8}\right )\right )+c_{2} \left (-\frac {x \left (1-\frac {x^{2}}{8}+\frac {x^{4}}{192}-\frac {x^{6}}{9216}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{2}+\frac {1-\frac {3 x^{4}}{64}+\frac {7 x^{6}}{2304}+O\left (x^{8}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{2} \left (1-\frac {1}{8} x^{2}+\frac {1}{192} x^{4}-\frac {1}{9216} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (x^{2}-\frac {1}{8} x^{4}+\frac {1}{192} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+\left (-2+\frac {3}{32} x^{4}-\frac {7}{1152} x^{6}+\operatorname {O}\left (x^{8}\right )\right )\right )}{x} \]

Problem 6061

ODE	\[ \boxed {x^{2} y^{\prime \prime }-2 x^{2} y^{\prime }+\left (-2+4 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1+O\left (x^{8}\right )\right )+c_{2} \left (-4 x^{2} \left (1+O\left (x^{8}\right )\right ) \ln \left (x \right )+\frac {1+3 x +6 x^{2}-2 x^{4}-\frac {2 x^{5}}{5}-\frac {4 x^{6}}{45}-\frac {2 x^{7}}{105}+O\left (x^{8}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \left (1+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (\ln \left (x \right ) \left (\left (-48\right ) x^{3}+\operatorname {O}\left (x^{8}\right )\right )+\left (12+36 x +72 x^{2}+88 x^{3}-24 x^{4}-\frac {24}{5} x^{5}-\frac {16}{15} x^{6}-\frac {8}{35} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )}{x} \]

Problem 6062

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

Problem 6063

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

Problem 6064

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

Problem 6065

ODE	\[ \boxed {y^{\prime }-\frac {x^{2}+x}{y-y^{2}}=0} \]
program solution	\[ -\frac {x^{3}}{3}-\frac {y^{3}}{3}-\frac {x^{2}}{2}+\frac {y^{2}}{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (1-4 x^{3}-6 x^{2}-12 c_{1} +2 \sqrt {4 x^{6}+12 x^{5}+24 c_{1} x^{3}+9 x^{4}+36 c_{1} x^{2}-2 x^{3}+36 c_{1}^{2}-3 x^{2}-6 c_{1}}\right )^{\frac {1}{3}}}{2}+\frac {1}{2 \left (1-4 x^{3}-6 x^{2}-12 c_{1} +2 \sqrt {4 x^{6}+12 x^{5}+24 c_{1} x^{3}+9 x^{4}+36 c_{1} x^{2}-2 x^{3}+36 c_{1}^{2}-3 x^{2}-6 c_{1}}\right )^{\frac {1}{3}}}+\frac {1}{2} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (-4 x^{3}-6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}-12 c_{1} +1\right )^{\frac {2}{3}}-i \sqrt {3}-2 \left (-4 x^{3}-6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}-12 c_{1} +1\right )^{\frac {1}{3}}+1}{4 \left (-4 x^{3}-6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}-12 c_{1} +1\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (-4 x^{3}-6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}-12 c_{1} +1\right )^{\frac {2}{3}}-i \sqrt {3}+2 \left (-4 x^{3}-6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}-12 c_{1} +1\right )^{\frac {1}{3}}-1}{4 \left (-4 x^{3}-6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}-12 c_{1} +1\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 6066

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

Problem 6067

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

Problem 6068

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

Problem 6069

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

Problem 6070

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

Problem 6071

ODE	\[ \boxed {y^{\prime }-\frac {x +y}{x -y}=0} \]
program solution	\[ \frac {\ln \left (\frac {y^{2}}{x^{2}}+1\right )}{2}-\arctan \left (\frac {y}{x}\right )+\ln \left (x \right )-c_{2} = 0 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \tan \left (\operatorname {RootOf}\left (-2 \textit {\_Z} +\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+2 \ln \left (x \right )+2 c_{1} \right )\right ) x \]

Problem 6072

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

Problem 6073

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

Problem 6074

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

Problem 6075

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

Problem 6076

ODE	\[ \boxed {y^{\prime }-\frac {2 x +3 y+1}{-1+x -2 y}=0} \]
program solution	\[ \ln \left (\frac {\left (y+\frac {3}{7}\right )^{2}}{\left (x -\frac {1}{7}\right )^{2}}+\frac {y+\frac {3}{7}}{x -\frac {1}{7}}+1\right )-\frac {4 \sqrt {3}\, \arctan \left (\frac {\left (2 y+\frac {5}{7}+x \right ) \sqrt {3}}{3 x -\frac {3}{7}}\right )}{3}+2 \ln \left (x -\frac {1}{7}\right )-c_{3} = 0 \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {5}{14}-\frac {x}{2}+\frac {\sqrt {3}\, \left (7 x -1\right ) \tan \left (\operatorname {RootOf}\left (-2 \sqrt {3}\, \ln \left (2\right )+\sqrt {3}\, \ln \left (\sec \left (\textit {\_Z} \right )^{2} \left (7 x -1\right )^{2}\right )+\sqrt {3}\, \ln \left (3\right )+2 \sqrt {3}\, c_{1} -4 \textit {\_Z} \right )\right )}{14} \]

Problem 6077

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

Problem 6078

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

Problem 6079

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

Problem 6080

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

Problem 6081

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

Problem 6082

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

Problem 6083

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

Problem 6084

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

Problem 6085

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

Problem 6086

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

Problem 6087

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

Problem 6088

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

Problem 6089

ODE	\[ \boxed {5 x^{3} y^{2}+2 y+\left (3 y x^{4}+2 x \right ) y^{\prime }=0} \]
program solution	\[ y^{2} x^{2} \left (y x^{3}+1\right ) = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\frac {12^{\frac {2}{3}} \left (12^{\frac {1}{3}} c_{1}^{2}+{\left (\left (9 x^{2}+\sqrt {-12 c_{1}^{4}+81 x^{4}}\right ) c_{1} \right )}^{\frac {2}{3}}\right )^{2}}{36 c_{1}^{2} {\left (\left (9 x^{2}+\sqrt {-12 c_{1}^{4}+81 x^{4}}\right ) c_{1} \right )}^{\frac {2}{3}}}-1}{x^{3}} \\ y \left (x \right ) &= \frac {-\frac {c_{1} {\left (\left (9 x^{2}+\sqrt {-12 c_{1}^{4}+81 x^{4}}\right ) c_{1} \right )}^{\frac {2}{3}}}{3}+\frac {3 \,2^{\frac {1}{3}} \left (x^{2}+\frac {\sqrt {-12 c_{1}^{4}+81 x^{4}}}{9}\right ) \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) {\left (\left (9 x^{2}+\sqrt {-12 c_{1}^{4}+81 x^{4}}\right ) c_{1} \right )}^{\frac {1}{3}}}{4}-\frac {\left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} c_{1}^{3}}{6}}{c_{1} {\left (\left (9 x^{2}+\sqrt {-12 c_{1}^{4}+81 x^{4}}\right ) c_{1} \right )}^{\frac {2}{3}} x^{3}} \\ y \left (x \right ) &= -\frac {3 \left (\frac {4 c_{1} {\left (\left (9 x^{2}+\sqrt {-12 c_{1}^{4}+81 x^{4}}\right ) c_{1} \right )}^{\frac {2}{3}}}{9}+2^{\frac {1}{3}} \left (x^{2}+\frac {\sqrt {-12 c_{1}^{4}+81 x^{4}}}{9}\right ) \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right ) {\left (\left (9 x^{2}+\sqrt {-12 c_{1}^{4}+81 x^{4}}\right ) c_{1} \right )}^{\frac {1}{3}}-\frac {2 \left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} c_{1}^{3}}{9}\right )}{4 {\left (\left (9 x^{2}+\sqrt {-12 c_{1}^{4}+81 x^{4}}\right ) c_{1} \right )}^{\frac {2}{3}} x^{3} c_{1}} \\ \end{align*}

Problem 6090

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

Problem 6091

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

Problem 6092

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

Problem 6093

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

Problem 6094

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

Problem 6095

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

Problem 6096

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

Problem 6097

ODE	\[ \boxed {y^{\prime \prime }-{y^{\prime }}^{2}=1} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = 0 \] Warning, solution could not be verified
Maple solution	\[ y \left (x \right ) = \ln \left (\sec \left (x \right )\right ) \]

Problem 6098

ODE	\[ \boxed {y^{\prime \prime }+\frac {1}{2 {y^{\prime }}^{2}}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = -1] \end {align*}
program solution	\[ y = \frac {3}{2}+\frac {3 \left (x +\frac {2}{3}\right ) \left (i \sqrt {3}-1\right ) \left (-12 x -8\right )^{\frac {1}{3}}}{16} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {3 \left (x +\frac {2}{3}\right ) \left (-12 x -8\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{16}+\frac {3}{2} \]

Problem 6099

ODE	\[ \boxed {y^{\prime \prime }+\sin \left (y\right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = \beta ] \end {align*}
program solution	\[ \frac {2 \sqrt {\frac {\beta ^{2}+2 \cos \left (y\right )-2}{\beta ^{2}}}\, \operatorname {InverseJacobiAM}\left (\frac {y}{2}, \frac {2}{\beta }\right )}{\sqrt {\beta ^{2}+2 \cos \left (y\right )-2}} = x \] Verified OK. \[ -\frac {2 \sqrt {\frac {\beta ^{2}+2 \cos \left (y\right )-2}{\beta ^{2}}}\, \operatorname {InverseJacobiAM}\left (\frac {y}{2}, \frac {2}{\beta }\right )}{\sqrt {\beta ^{2}+2 \cos \left (y\right )-2}} = x \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \operatorname {RootOf}\left (-\left (\int _{0}^{\textit {\_Z}}\frac {1}{\sqrt {2 \cos \left (\textit {\_a} \right )+\beta ^{2}-2}}d \textit {\_a} \right )+x \right ) \\ y \left (x \right ) &= \operatorname {RootOf}\left (\int _{0}^{\textit {\_Z}}\frac {1}{\sqrt {2 \cos \left (\textit {\_a} \right )+\beta ^{2}-2}}d \textit {\_a} +x \right ) \\ \end{align*}

Problem 6100

ODE	\[ \boxed {y^{\prime \prime }+\sin \left (y\right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 2] \end {align*}
program solution	\[ \operatorname {InverseJacobiAM}\left (\frac {y}{2}, 1\right ) = x \] Verified OK. \[ -\operatorname {InverseJacobiAM}\left (\frac {y}{2}, 1\right ) = x \] Verified OK.
Maple solution	\[ y \left (x \right ) = \operatorname {RootOf}\left (-\left (\int _{0}^{\textit {\_Z}}\sec \left (\frac {\textit {\_a}}{2}\right ) \operatorname {csgn}\left (\cos \left (\frac {\textit {\_a}}{2}\right )\right )d \textit {\_a} \right )+2 x \right ) \]