2.17.75 Problems 7401 to 7500

Problem 7401

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

Problem 7402

ODE	\[ \boxed {{y^{\prime \prime }}^{2}+y^{\prime }=0} \]
program solution	\[ y = -\frac {1}{12} c_{3}^{3}-\frac {1}{4} c_{3}^{2} x -\frac {1}{4} c_{3} x^{2}-\frac {1}{12} x^{3}-c_{1} \] Verified OK. \[ y = -\frac {1}{12} c_{4}^{3}-\frac {1}{4} x \,c_{4}^{2}-\frac {1}{4} c_{4} x^{2}-\frac {1}{12} x^{3}-c_{1} \] Verified OK. \[ y = -\frac {1}{12} c_{5}^{3}-\frac {1}{4} x \,c_{5}^{2}-\frac {1}{4} c_{5} x^{2}-\frac {1}{12} x^{3}-c_{1} \] Verified OK. \[ y = -\frac {1}{12} c_{6}^{3}-\frac {1}{4} c_{6}^{2} x -\frac {1}{4} c_{6} x^{2}-\frac {1}{12} x^{3}-c_{2} \] Verified OK. \[ y = -\frac {1}{12} c_{7}^{3}-\frac {1}{4} c_{7}^{2} x -\frac {1}{4} c_{7} x^{2}-\frac {1}{12} x^{3}-c_{2} \] Verified OK. \[ y = -\frac {1}{12} c_{8}^{3}-\frac {1}{4} c_{8}^{2} x -\frac {1}{4} c_{8} x^{2}-\frac {1}{12} x^{3}-c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= c_{1} \\ y \left (x \right ) &= -\frac {1}{12} x^{3}+\frac {1}{2} c_{1} x^{2}-x \,c_{1}^{2}+c_{2} \\ \end{align*}

Problem 7403

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

Problem 7404

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

Problem 7405

ODE	\[ \boxed {{y^{\prime \prime }}^{2}+y^{\prime }=1} \]
program solution	\[ \int _{}^{y}-\frac {4 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}}{\left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}}+4 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}+16}d \textit {\_a} = x +c_{3} \] Verified OK. \[ \int _{}^{y}\frac {8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}}{i \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}} \sqrt {3}+16+\left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}}-8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}-16 i \sqrt {3}}d \textit {\_a} = x +c_{4} \] Verified OK. \[ \int _{}^{y}-\frac {8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}}{i \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}} \sqrt {3}-16-\left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}}+8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}-16 i \sqrt {3}}d \textit {\_a} = x +c_{5} \] Verified OK. \[ \int _{}^{y}-\frac {4 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {2}{3}}}{\left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {4}{3}}+4 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {2}{3}}+16}d \textit {\_a} = x +c_{6} \] Verified OK. \[ \int _{}^{y}\frac {8 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {2}{3}}}{i \sqrt {3}\, \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {4}{3}}+\left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {4}{3}}-16 i \sqrt {3}-8 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {2}{3}}+16}d \textit {\_a} = x +c_{7} \] Verified OK. \[ \int _{}^{y}-\frac {8 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {2}{3}}}{i \sqrt {3}\, \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {4}{3}}-\left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {4}{3}}-16 i \sqrt {3}+8 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {2}{3}}-16}d \textit {\_a} = x +c_{8} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= x +c_{1} \\ y \left (x \right ) &= -\frac {1}{12} x^{3}+\frac {1}{2} c_{1} x^{2}-x \,c_{1}^{2}+x +c_{2} \\ \end{align*}

Problem 7406

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

Problem 7407

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

Problem 7408

ODE	\[ \boxed {{y^{\prime \prime }}^{2}+y^{\prime }=x} \]
program solution	\[ y = \int \operatorname {RootOf}\left (-\left (\int _{}^{x -\textit {\_Z}}-\frac {1}{\sqrt {\tau }-1}d \tau \right )+x +c_{1} \right )d x +c_{2} \] Verified OK. \[ y = \int \operatorname {RootOf}\left (-\left (\int _{}^{x -\textit {\_Z}}\frac {1}{\sqrt {\tau }+1}d \tau \right )+x +c_{1} \right )d x +c_{3} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \int \left (-{\mathrm e}^{2 \operatorname {RootOf}\left (\textit {\_Z} -x -2 \,{\mathrm e}^{\textit {\_Z}}+2+c_{1} -\ln \left ({\mathrm e}^{\textit {\_Z}} \left ({\mathrm e}^{\textit {\_Z}}-2\right )^{2}\right )\right )}+2 \,{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z} -x -2 \,{\mathrm e}^{\textit {\_Z}}+2+c_{1} -\ln \left ({\mathrm e}^{\textit {\_Z}} \left ({\mathrm e}^{\textit {\_Z}}-2\right )^{2}\right )\right )}+x \right )d x -x +c_{2} \\ y \left (x \right ) &= \frac {2 \operatorname {LambertW}\left (-c_{1} {\mathrm e}^{-\frac {x}{2}-1}\right )^{3}}{3}+3 \operatorname {LambertW}\left (-c_{1} {\mathrm e}^{-\frac {x}{2}-1}\right )^{2}+4 \operatorname {LambertW}\left (-c_{1} {\mathrm e}^{-\frac {x}{2}-1}\right )+\frac {x^{2}}{2}-x +c_{2} \\ \end{align*}

Problem 7409

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

Problem 7410

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

Problem 7411

ODE	\[ \boxed {{y^{\prime \prime }}^{2}+y^{\prime }+y=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 7412

ODE	\[ \boxed {y^{\prime \prime }+{y^{\prime }}^{2}+y=0} \]
program solution	\[ \int _{}^{y}\frac {2 \,{\mathrm e}^{2 \textit {\_a}}}{\sqrt {-2 \,{\mathrm e}^{2 \textit {\_a}} \left (2 \,{\mathrm e}^{2 \textit {\_a}} \textit {\_a} -{\mathrm e}^{2 \textit {\_a}}-4 c_{1} \right )}}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}-\frac {2 \,{\mathrm e}^{2 \textit {\_a}}}{\sqrt {-2 \,{\mathrm e}^{2 \textit {\_a}} \left (2 \,{\mathrm e}^{2 \textit {\_a}} \textit {\_a} -{\mathrm e}^{2 \textit {\_a}}-4 c_{1} \right )}}d \textit {\_a} = x +c_{3} \] Verified OK.
Maple solution	\begin{align*} -2 \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {2+4 \,{\mathrm e}^{-2 \textit {\_a}} c_{1} -4 \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ 2 \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {2+4 \,{\mathrm e}^{-2 \textit {\_a}} c_{1} -4 \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*}

Problem 7413

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

Problem 7414

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

Problem 7415

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

Problem 7416

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

Problem 7417

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

Problem 7418

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

Problem 7419

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

Problem 7420

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

Problem 7421

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

Problem 7422

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

Problem 7423

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

Problem 7424

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

Problem 7425

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

Problem 7426

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

Problem 7427

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

Problem 7428

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

Problem 7429

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

Problem 7430

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

Problem 7431

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

Problem 7432

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

Problem 7433

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

Problem 7434

ODE	\[ \boxed {y {y^{\prime \prime }}^{2}+y^{\prime }=0} \]
program solution	\[ \int _{}^{y}\frac {4}{\left (\frac {\left (-4+12 \textit {\_a} \right ) \left (\sqrt {-\frac {1}{\textit {\_a}}}\, \sqrt {\textit {\_a}}+i\right )}{\sqrt {\textit {\_a}}}\right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{3} \] Verified OK. \[ \int _{}^{y}\frac {16}{\left (\frac {\left (-4+12 \textit {\_a} \right ) \left (\sqrt {-\frac {1}{\textit {\_a}}}\, \sqrt {\textit {\_a}}+i\right )}{\sqrt {\textit {\_a}}}\right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )^{2}}d \textit {\_a} = x +c_{4} \] Verified OK. \[ \int _{}^{y}\frac {16}{\left (\frac {\left (-4+12 \textit {\_a} \right ) \left (\sqrt {-\frac {1}{\textit {\_a}}}\, \sqrt {\textit {\_a}}+i\right )}{\sqrt {\textit {\_a}}}\right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )^{2}}d \textit {\_a} = x +c_{5} \] Verified OK. \[ \frac {3 y^{\frac {4}{3}} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {4}{3}\right ], \left [\frac {7}{3}\right ], -\frac {y \left (12 \sqrt {-\frac {1}{y}}\, \sqrt {y}+12 i\right )}{4 \sqrt {-\frac {1}{y}}\, \sqrt {y}+4 i}\right )}{\left (4 \sqrt {-\frac {1}{y}}\, \sqrt {y}+4 i\right )^{\frac {2}{3}}} = x +c_{6} \] Verified OK. \[ \frac {12 y^{\frac {4}{3}} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {4}{3}\right ], \left [\frac {7}{3}\right ], -\frac {y \left (12 \sqrt {-\frac {1}{y}}\, \sqrt {y}+12 i\right )}{4 \sqrt {-\frac {1}{y}}\, \sqrt {y}+4 i}\right )}{\left (4 \sqrt {-\frac {1}{y}}\, \sqrt {y}+4 i\right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )^{2}} = x +c_{7} \] Verified OK. \[ \frac {12 y^{\frac {4}{3}} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {4}{3}\right ], \left [\frac {7}{3}\right ], -\frac {y \left (12 \sqrt {-\frac {1}{y}}\, \sqrt {y}+12 i\right )}{4 \sqrt {-\frac {1}{y}}\, \sqrt {y}+4 i}\right )}{\left (4 \sqrt {-\frac {1}{y}}\, \sqrt {y}+4 i\right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )^{2}} = x +c_{8} \] Verified OK. \[ \int _{}^{y}\frac {4}{\left (\frac {\left (-4-12 \textit {\_a} \right ) \left (-\sqrt {-\frac {1}{\textit {\_a}}}\, \sqrt {\textit {\_a}}+i\right )}{\sqrt {\textit {\_a}}}\right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{9} \] Verified OK. \[ \int _{}^{y}\frac {16}{\left (\frac {\left (-4-12 \textit {\_a} \right ) \left (-\sqrt {-\frac {1}{\textit {\_a}}}\, \sqrt {\textit {\_a}}+i\right )}{\sqrt {\textit {\_a}}}\right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )^{2}}d \textit {\_a} = x +c_{10} \] Verified OK. \[ \int _{}^{y}\frac {16}{\left (\frac {\left (-4-12 \textit {\_a} \right ) \left (-\sqrt {-\frac {1}{\textit {\_a}}}\, \sqrt {\textit {\_a}}+i\right )}{\sqrt {\textit {\_a}}}\right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )^{2}}d \textit {\_a} = x +c_{11} \] Verified OK. \[ \int _{}^{y}\frac {4}{\left (\frac {\left (4-12 \textit {\_a} \right ) \left (-\sqrt {-\frac {1}{\textit {\_a}}}\, \sqrt {\textit {\_a}}+i\right )}{\sqrt {\textit {\_a}}}\right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{12} \] Verified OK. \[ \int _{}^{y}\frac {16}{\left (\frac {\left (4-12 \textit {\_a} \right ) \left (-\sqrt {-\frac {1}{\textit {\_a}}}\, \sqrt {\textit {\_a}}+i\right )}{\sqrt {\textit {\_a}}}\right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )^{2}}d \textit {\_a} = x +c_{13} \] Verified OK. \[ \int _{}^{y}\frac {16}{\left (\frac {\left (4-12 \textit {\_a} \right ) \left (-\sqrt {-\frac {1}{\textit {\_a}}}\, \sqrt {\textit {\_a}}+i\right )}{\sqrt {\textit {\_a}}}\right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )^{2}}d \textit {\_a} = x +c_{14} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= c_{1} \\ y \left (x \right ) &= 0 \\ -\left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\left (\textit {\_a}^{\frac {3}{2}} \left (c_{1} -3 \sqrt {\textit {\_a}}\right )\right )^{\frac {2}{3}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\left (\textit {\_a}^{\frac {3}{2}} \left (c_{1} +3 \sqrt {\textit {\_a}}\right )\right )^{\frac {2}{3}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \frac {-4 \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\left (\textit {\_a}^{\frac {3}{2}} \left (c_{1} -3 \sqrt {\textit {\_a}}\right )\right )^{\frac {2}{3}}}d \textit {\_a} \right )+2 i \left (-x -c_{2} \right ) \sqrt {3}+2 x +2 c_{2}}{\left (-i \sqrt {3}-1\right )^{2}} &= 0 \\ \frac {-4 \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\left (\textit {\_a}^{\frac {3}{2}} \left (c_{1} -3 \sqrt {\textit {\_a}}\right )\right )^{\frac {2}{3}}}d \textit {\_a} \right )+2 i \left (x +c_{2} \right ) \sqrt {3}+2 x +2 c_{2}}{\left (1-i \sqrt {3}\right )^{2}} &= 0 \\ \frac {-4 \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\left (\textit {\_a}^{\frac {3}{2}} \left (c_{1} +3 \sqrt {\textit {\_a}}\right )\right )^{\frac {2}{3}}}d \textit {\_a} \right )+2 i \left (-x -c_{2} \right ) \sqrt {3}+2 x +2 c_{2}}{\left (-i \sqrt {3}-1\right )^{2}} &= 0 \\ \frac {-4 \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\left (\textit {\_a}^{\frac {3}{2}} \left (c_{1} +3 \sqrt {\textit {\_a}}\right )\right )^{\frac {2}{3}}}d \textit {\_a} \right )+2 i \left (x +c_{2} \right ) \sqrt {3}+2 x +2 c_{2}}{\left (1-i \sqrt {3}\right )^{2}} &= 0 \\ \end{align*}

Problem 7435

ODE	\[ \boxed {y {y^{\prime \prime }}^{2}+{y^{\prime }}^{3}=0} \]
program solution	\[ -\frac {2 c_{1}^{2}}{c_{1}^{2}+4 y}-\ln \left (c_{1}^{2}+4 y\right )+\frac {2 \sqrt {-\frac {1}{y}}\, \sqrt {y}\, \left (-\arctan \left (\frac {2 \sqrt {y}}{c_{1}}\right ) c_{1}^{2}+2 c_{1} \sqrt {y}-4 \arctan \left (\frac {2 \sqrt {y}}{c_{1}}\right ) y\right )}{c_{1}^{2}+4 y} = x +c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= c_{1} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {c_{2} {\left (\operatorname {LambertW}\left (c_{1} {\mathrm e}^{-1+\frac {x}{2}}\right )+1\right )}^{2}}{\operatorname {LambertW}\left (c_{1} {\mathrm e}^{-1+\frac {x}{2}}\right )^{2}} \\ y \left (x \right ) &= \frac {c_{2} {\left (\operatorname {LambertW}\left (-c_{1} {\mathrm e}^{-1+\frac {x}{2}}\right )+1\right )}^{2}}{\operatorname {LambertW}\left (-c_{1} {\mathrm e}^{-1+\frac {x}{2}}\right )^{2}} \\ y \left (x \right ) &= {\mathrm e}^{-\left (\int {\mathrm e}^{2 \operatorname {RootOf}\left ({\mathrm e}^{\textit {\_Z}} \ln \left (\left ({\mathrm e}^{\textit {\_Z}}+1\right )^{2}\right )+c_{1} {\mathrm e}^{\textit {\_Z}}-2 \,{\mathrm e}^{\textit {\_Z}} \textit {\_Z} +x \,{\mathrm e}^{\textit {\_Z}}+\ln \left (\left ({\mathrm e}^{\textit {\_Z}}+1\right )^{2}\right )+c_{1} -2 \textit {\_Z} +x -2\right )}d x \right )-2 \left (\int {\mathrm e}^{\operatorname {RootOf}\left ({\mathrm e}^{\textit {\_Z}} \ln \left (\left ({\mathrm e}^{\textit {\_Z}}+1\right )^{2}\right )+c_{1} {\mathrm e}^{\textit {\_Z}}-2 \,{\mathrm e}^{\textit {\_Z}} \textit {\_Z} +x \,{\mathrm e}^{\textit {\_Z}}+\ln \left (\left ({\mathrm e}^{\textit {\_Z}}+1\right )^{2}\right )+c_{1} -2 \textit {\_Z} +x -2\right )}d x \right )-x +c_{2}} \\ \end{align*}

Problem 7436

ODE	\[ \boxed {y^{2} {y^{\prime \prime }}^{2}+y^{\prime }=0} \]
program solution	\[ \int _{}^{y}-\frac {4}{\left (12 \ln \left (\textit {\_a} \right )-12 c_{1} \right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{3} \] Verified OK. \[ \int _{}^{y}-\frac {16}{\left (12 \ln \left (\textit {\_a} \right )-12 c_{1} \right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )^{2}}d \textit {\_a} = x +c_{4} \] Verified OK. \[ \int _{}^{y}-\frac {16}{\left (12 \ln \left (\textit {\_a} \right )-12 c_{1} \right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )^{2}}d \textit {\_a} = x +c_{5} \] Verified OK. \[ \int _{}^{y}-\frac {4}{\left (-12 \ln \left (\textit {\_a} \right )-12 c_{2} \right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{6} \] Verified OK. \[ \int _{}^{y}-\frac {16}{\left (-12 \ln \left (\textit {\_a} \right )-12 c_{2} \right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )^{2}}d \textit {\_a} = x +c_{7} \] Verified OK. \[ \int _{}^{y}-\frac {16}{\left (-12 \ln \left (\textit {\_a} \right )-12 c_{2} \right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )^{2}}d \textit {\_a} = x +c_{8} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= c_{1} \\ y \left (x \right ) &= 0 \\ -4 \left (\int _{}^{y \left (x \right )}\frac {1}{\left (-12 \ln \left (\textit {\_a} \right )+8 c_{1} \right )^{\frac {2}{3}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -4 \left (\int _{}^{y \left (x \right )}\frac {1}{\left (12 \ln \left (\textit {\_a} \right )-8 c_{1} \right )^{\frac {2}{3}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \frac {-16 \left (\int _{}^{y \left (x \right )}\frac {1}{\left (-12 \ln \left (\textit {\_a} \right )+8 c_{1} \right )^{\frac {2}{3}}}d \textit {\_a} \right )+2 i \left (-x -c_{2} \right ) \sqrt {3}+2 x +2 c_{2}}{\left (-i \sqrt {3}-1\right )^{2}} &= 0 \\ \frac {-16 \left (\int _{}^{y \left (x \right )}\frac {1}{\left (-12 \ln \left (\textit {\_a} \right )+8 c_{1} \right )^{\frac {2}{3}}}d \textit {\_a} \right )+2 i \left (x +c_{2} \right ) \sqrt {3}+2 x +2 c_{2}}{\left (1-i \sqrt {3}\right )^{2}} &= 0 \\ \frac {-16 \left (\int _{}^{y \left (x \right )}\frac {1}{\left (12 \ln \left (\textit {\_a} \right )-8 c_{1} \right )^{\frac {2}{3}}}d \textit {\_a} \right )+2 i \left (-x -c_{2} \right ) \sqrt {3}+2 x +2 c_{2}}{\left (-i \sqrt {3}-1\right )^{2}} &= 0 \\ \frac {-16 \left (\int _{}^{y \left (x \right )}\frac {1}{\left (12 \ln \left (\textit {\_a} \right )-8 c_{1} \right )^{\frac {2}{3}}}d \textit {\_a} \right )+2 i \left (x +c_{2} \right ) \sqrt {3}+2 x +2 c_{2}}{\left (1-i \sqrt {3}\right )^{2}} &= 0 \\ \end{align*}

Problem 7437

ODE	\[ \boxed {y {y^{\prime \prime }}^{4}+{y^{\prime }}^{2}=0} \]
program solution	\[ \int _{}^{y}\frac {\textit {\_a}^{\frac {3}{2}}}{\left (i \sqrt {2}\, \textit {\_a}^{3}+\sqrt {2}\, \textit {\_a}^{3}-\left ({\mathrm e}^{2 c_{1}} \textit {\_a}^{3}\right )^{\frac {3}{4}}\right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{5} \] Verified OK. \[ \int _{}^{y}\frac {4 \textit {\_a}^{\frac {3}{2}}}{\left (i \sqrt {2}\, \textit {\_a}^{3}+\sqrt {2}\, \textit {\_a}^{3}-\left ({\mathrm e}^{2 c_{1}} \textit {\_a}^{3}\right )^{\frac {3}{4}}\right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )^{2}}d \textit {\_a} = x +c_{6} \] Verified OK. \[ \int _{}^{y}\frac {4 \textit {\_a}^{\frac {3}{2}}}{\left (i \sqrt {2}\, \textit {\_a}^{3}+\sqrt {2}\, \textit {\_a}^{3}-\left ({\mathrm e}^{2 c_{1}} \textit {\_a}^{3}\right )^{\frac {3}{4}}\right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )^{2}}d \textit {\_a} = x +c_{7} \] Verified OK. \[ \int _{}^{y}\frac {\textit {\_a}^{\frac {3}{2}}}{\left (i \sqrt {2}\, \textit {\_a}^{3}+\sqrt {2}\, \textit {\_a}^{3}+i \left ({\mathrm e}^{2 c_{1}} \textit {\_a}^{3}\right )^{\frac {3}{4}}\right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{8} \] Verified OK. \[ \int _{}^{y}\frac {4 \textit {\_a}^{\frac {3}{2}}}{\left (i \sqrt {2}\, \textit {\_a}^{3}+\sqrt {2}\, \textit {\_a}^{3}+i \left ({\mathrm e}^{2 c_{1}} \textit {\_a}^{3}\right )^{\frac {3}{4}}\right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )^{2}}d \textit {\_a} = x +c_{9} \] Verified OK. \[ \int _{}^{y}\frac {4 \textit {\_a}^{\frac {3}{2}}}{\left (i \sqrt {2}\, \textit {\_a}^{3}+\sqrt {2}\, \textit {\_a}^{3}+i \left ({\mathrm e}^{2 c_{1}} \textit {\_a}^{3}\right )^{\frac {3}{4}}\right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )^{2}}d \textit {\_a} = x +c_{10} \] Verified OK. \[ \int _{}^{y}\frac {\textit {\_a}^{\frac {3}{2}}}{\left (i \sqrt {2}\, \textit {\_a}^{3}+\sqrt {2}\, \textit {\_a}^{3}+\left ({\mathrm e}^{2 c_{1}} \textit {\_a}^{3}\right )^{\frac {3}{4}}\right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{11} \] Verified OK. \[ \int _{}^{y}\frac {4 \textit {\_a}^{\frac {3}{2}}}{\left (i \sqrt {2}\, \textit {\_a}^{3}+\sqrt {2}\, \textit {\_a}^{3}+\left ({\mathrm e}^{2 c_{1}} \textit {\_a}^{3}\right )^{\frac {3}{4}}\right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )^{2}}d \textit {\_a} = x +c_{12} \] Verified OK. \[ \int _{}^{y}\frac {4 \textit {\_a}^{\frac {3}{2}}}{\left (i \sqrt {2}\, \textit {\_a}^{3}+\sqrt {2}\, \textit {\_a}^{3}+\left ({\mathrm e}^{2 c_{1}} \textit {\_a}^{3}\right )^{\frac {3}{4}}\right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )^{2}}d \textit {\_a} = x +c_{13} \] Verified OK. \[ \int _{}^{y}\frac {\textit {\_a}^{\frac {3}{2}}}{\left (i \sqrt {2}\, \textit {\_a}^{3}+\sqrt {2}\, \textit {\_a}^{3}-i \left ({\mathrm e}^{2 c_{1}} \textit {\_a}^{3}\right )^{\frac {3}{4}}\right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{14} \] Verified OK. \[ \int _{}^{y}\frac {4 \textit {\_a}^{\frac {3}{2}}}{\left (i \sqrt {2}\, \textit {\_a}^{3}+\sqrt {2}\, \textit {\_a}^{3}-i \left ({\mathrm e}^{2 c_{1}} \textit {\_a}^{3}\right )^{\frac {3}{4}}\right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )^{2}}d \textit {\_a} = x +c_{15} \] Verified OK. \[ \int _{}^{y}\frac {4 \textit {\_a}^{\frac {3}{2}}}{\left (i \sqrt {2}\, \textit {\_a}^{3}+\sqrt {2}\, \textit {\_a}^{3}-i \left ({\mathrm e}^{2 c_{1}} \textit {\_a}^{3}\right )^{\frac {3}{4}}\right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )^{2}}d \textit {\_a} = x +c_{16} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= c_{1} \\ y \left (x \right ) &= 0 \\ \int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (2 \textit {\_a} -\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \left (-2 \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}}}}d \textit {\_a} -x -c_{2} &= 0 \\ \int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {-\textit {\_a}^{3} \left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}-2 \textit {\_a} \right ) {\left (\left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}-2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}}}}d \textit {\_a} -x -c_{2} &= 0 \\ \int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \left (-2 \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}}}}d \textit {\_a} -x -c_{2} &= 0 \\ \int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right ) {\left (-\left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}}}}d \textit {\_a} -x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \left (1+i \sqrt {3}\right ) \textit {\_a}^{3} \left (-2 \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\left (i-\sqrt {3}\right ) \textit {\_a}^{3} {\left (\left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}-2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \left (\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 i \textit {\_a} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {-2 \left (1+i \sqrt {3}\right ) \left (-2 \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \left (\textit {\_a} +\frac {\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}}{2}\right ) \textit {\_a}^{3}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {-\textit {\_a}^{3} \left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right ) {\left (-\left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (2 \textit {\_a} -\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \left (-2 \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {-\textit {\_a}^{3} \left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}-2 \textit {\_a} \right ) {\left (\left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}-2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \left (-2 \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right ) {\left (-\left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\left (1-i \sqrt {3}\right ) \textit {\_a}^{3} \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \left (-2 \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\left (1-i \sqrt {3}\right ) \textit {\_a}^{3} \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \left (-2 \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} {\left (\left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}-2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \left (\sqrt {3}+i\right ) \left (\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 i \textit {\_a} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} {\left (\left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}-2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \left (\sqrt {3}+i\right ) \left (\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 i \textit {\_a} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \left (-2 \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \left (-2 \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right ) {\left (-\left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{3} \left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right ) {\left (-\left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \left (1+i \sqrt {3}\right ) \textit {\_a}^{3} \left (-2 \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\left (i-\sqrt {3}\right ) \textit {\_a}^{3} {\left (\left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}-2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \left (\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 i \textit {\_a} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {-2 \left (1+i \sqrt {3}\right ) \left (-2 \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \left (\textit {\_a} +\frac {\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}}{2}\right ) \textit {\_a}^{3}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {-\textit {\_a}^{3} \left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right ) {\left (-\left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \int _{}^{y \left (x \right )}\frac {1}{\operatorname {RootOf}\left (-\ln \left (\textit {\_a} \right )-2 \left (\int _{}^{\textit {\_Z}}\frac {\textit {\_f}}{2 i \left (-\textit {\_f}^{2}\right )^{\frac {1}{4}}+\textit {\_f}^{2}}d \textit {\_f} \right )+c_{1} \right ) \sqrt {\textit {\_a}}}d \textit {\_a} -x -c_{2} &= 0 \\ \int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\textit {\_a} \left (-2 i \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \left (\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 i \textit {\_a} \right )}}d \textit {\_a} -x -c_{2} &= 0 \\ \int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\textit {\_a} \left (-2 i \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \left (2 i \textit {\_a} -\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )}}d \textit {\_a} -x -c_{2} &= 0 \\ \int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {-i \textit {\_a} {\left (i \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )}}d \textit {\_a} -x -c_{2} &= 0 \\ \int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {i \textit {\_a} {\left (-i \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )}}d \textit {\_a} -x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\left (i-\sqrt {3}\right ) \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) {\left (i \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\left (\sqrt {3}+i\right ) \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) {\left (i \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}-2 \textit {\_a} \right ) \left (-2 i \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \textit {\_a} \left (\sqrt {3}+i\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\left (-2 i \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \textit {\_a} \left (-i+\sqrt {3}\right ) \left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\textit {\_a} \left (-2 i \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \left (\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 i \textit {\_a} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\textit {\_a} \left (-2 i \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \left (2 i \textit {\_a} -\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {-i \textit {\_a} {\left (i \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {i \textit {\_a} {\left (-i \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\left (-i+\sqrt {3}\right ) \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) {\left (-i \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\left (-i+\sqrt {3}\right ) \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) {\left (-i \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {-\left (\sqrt {3}+i\right ) \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) {\left (-i \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {-\left (\sqrt {3}+i\right ) \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) {\left (-i \left (2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\left (i-\sqrt {3}\right ) \left (-2 i \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}-2 \textit {\_a} \right ) \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\left (i-\sqrt {3}\right ) \left (-2 i \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}-2 \textit {\_a} \right ) \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {-\left (\sqrt {3}+i\right ) \left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right ) \left (-2 i \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {-\left (\sqrt {3}+i\right ) \left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right ) \left (-2 i \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\left (i-\sqrt {3}\right ) \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) {\left (i \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\left (\sqrt {3}+i\right ) \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) {\left (i \left (-2 \textit {\_a} +\left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right ) \textit {\_a}^{2}\right )}^{\frac {1}{3}} \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}-2 \textit {\_a} \right ) \left (-2 i \textit {\_a}^{3}-\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \textit {\_a} \left (\sqrt {3}+i\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\left (-2 i \textit {\_a}^{3}+\textit {\_a}^{2} \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \textit {\_a} \left (-i+\sqrt {3}\right ) \left (i \left (c_{1} \textit {\_a} \right )^{\frac {1}{4}}+2 \textit {\_a} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \int _{}^{y \left (x \right )}\frac {1}{\operatorname {RootOf}\left (-\ln \left (\textit {\_a} \right )-2 \left (\int _{}^{\textit {\_Z}}\frac {\textit {\_f}}{\textit {\_f}^{2}+2 \left (-\textit {\_f}^{2}\right )^{\frac {1}{4}}}d \textit {\_f} \right )+c_{1} \right ) \sqrt {\textit {\_a}}}d \textit {\_a} -x -c_{2} &= 0 \\ \end{align*}

Problem 7438

ODE	\[ \boxed {y^{3} {y^{\prime \prime }}^{2}+y^{\prime } y=0} \]
program solution	\[ \int _{}^{y}-\frac {4}{\left (12 \ln \left (\textit {\_a} \right )-12 c_{1} \right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{3} \] Verified OK. \[ \int _{}^{y}-\frac {16}{\left (12 \ln \left (\textit {\_a} \right )-12 c_{1} \right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )^{2}}d \textit {\_a} = x +c_{4} \] Verified OK. \[ \int _{}^{y}-\frac {16}{\left (12 \ln \left (\textit {\_a} \right )-12 c_{1} \right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )^{2}}d \textit {\_a} = x +c_{5} \] Verified OK. \[ \int _{}^{y}-\frac {4}{\left (-12 \ln \left (\textit {\_a} \right )-12 c_{2} \right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{6} \] Verified OK. \[ \int _{}^{y}-\frac {16}{\left (-12 \ln \left (\textit {\_a} \right )-12 c_{2} \right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )^{2}}d \textit {\_a} = x +c_{7} \] Verified OK. \[ \int _{}^{y}-\frac {16}{\left (-12 \ln \left (\textit {\_a} \right )-12 c_{2} \right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )^{2}}d \textit {\_a} = x +c_{8} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= c_{1} \\ y \left (x \right ) &= 0 \\ -4 \left (\int _{}^{y \left (x \right )}\frac {1}{\left (-12 \ln \left (\textit {\_a} \right )+8 c_{1} \right )^{\frac {2}{3}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -4 \left (\int _{}^{y \left (x \right )}\frac {1}{\left (12 \ln \left (\textit {\_a} \right )-8 c_{1} \right )^{\frac {2}{3}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \frac {-16 \left (\int _{}^{y \left (x \right )}\frac {1}{\left (-12 \ln \left (\textit {\_a} \right )+8 c_{1} \right )^{\frac {2}{3}}}d \textit {\_a} \right )+2 i \left (-x -c_{2} \right ) \sqrt {3}+2 x +2 c_{2}}{\left (-i \sqrt {3}-1\right )^{2}} &= 0 \\ \frac {-16 \left (\int _{}^{y \left (x \right )}\frac {1}{\left (-12 \ln \left (\textit {\_a} \right )+8 c_{1} \right )^{\frac {2}{3}}}d \textit {\_a} \right )+2 i \left (x +c_{2} \right ) \sqrt {3}+2 x +2 c_{2}}{\left (1-i \sqrt {3}\right )^{2}} &= 0 \\ \frac {-16 \left (\int _{}^{y \left (x \right )}\frac {1}{\left (12 \ln \left (\textit {\_a} \right )-8 c_{1} \right )^{\frac {2}{3}}}d \textit {\_a} \right )+2 i \left (-x -c_{2} \right ) \sqrt {3}+2 x +2 c_{2}}{\left (-i \sqrt {3}-1\right )^{2}} &= 0 \\ \frac {-16 \left (\int _{}^{y \left (x \right )}\frac {1}{\left (12 \ln \left (\textit {\_a} \right )-8 c_{1} \right )^{\frac {2}{3}}}d \textit {\_a} \right )+2 i \left (x +c_{2} \right ) \sqrt {3}+2 x +2 c_{2}}{\left (1-i \sqrt {3}\right )^{2}} &= 0 \\ \end{align*}

Problem 7439

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

Problem 7440

ODE	\[ \boxed {y {y^{\prime \prime }}^{3}+y^{3} y^{\prime }=0} \]
program solution	\[ y = \frac {{\mathrm e}^{-x}}{c_{6}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= c_{1} \\ y \left (x \right ) &= {\mathrm e}^{\int \operatorname {RootOf}\left (x -\left (\int _{}^{\textit {\_Z}}-\frac {1}{\textit {\_f}^{2}-\left (-\textit {\_f} \right )^{\frac {1}{3}}}d \textit {\_f} \right )+c_{1} \right )d x +c_{2}} \\ y \left (x \right ) &= {\mathrm e}^{\int \operatorname {RootOf}\left (x +2 \left (\int _{}^{\textit {\_Z}}\frac {1}{i \sqrt {3}\, \left (-\textit {\_f} \right )^{\frac {1}{3}}+2 \textit {\_f}^{2}+\left (-\textit {\_f} \right )^{\frac {1}{3}}}d \textit {\_f} \right )+c_{1} \right )d x +c_{2}} \\ y \left (x \right ) &= {\mathrm e}^{\int \operatorname {RootOf}\left (x -2 \left (\int _{}^{\textit {\_Z}}\frac {1}{i \sqrt {3}\, \left (-\textit {\_f} \right )^{\frac {1}{3}}-2 \textit {\_f}^{2}-\left (-\textit {\_f} \right )^{\frac {1}{3}}}d \textit {\_f} \right )+c_{1} \right )d x +c_{2}} \\ \end{align*}

Problem 7441

ODE	\[ \boxed {y {y^{\prime \prime }}^{3}+y^{3} {y^{\prime }}^{5}=0} \]
program solution	\[ \int _{}^{y}\frac {1}{-\frac {\textit {\_a}^{5}}{125}+\frac {\textit {\_a}^{2} \left (-\textit {\_a}^{2}\right )^{\frac {2}{3}} c_{1}}{25}+\frac {\textit {\_a} \left (-\textit {\_a}^{2}\right )^{\frac {1}{3}} c_{1}^{2}}{15}+\frac {c_{1}^{3}}{27}}d \textit {\_a} = x +c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= c_{1} \\ \int _{}^{y \left (x \right )}\frac {1}{\operatorname {RootOf}\left (5 \left (\int _{\textit {\_g}}^{\textit {\_Z}}\frac {1}{\textit {\_a} \left (-\textit {\_f}^{2} \textit {\_a}^{2}\right )^{\frac {1}{3}}-5 \textit {\_f}}d \textit {\_f} \right )-\ln \left (\textit {\_a}^{5}+125\right )+5 c_{1} \right )}d \textit {\_a} -x -c_{2} &= 0 \\ \int _{}^{y \left (x \right )}\frac {1}{\operatorname {RootOf}\left (-i \ln \left (\textit {\_a}^{5}+125\right )+\sqrt {3}\, \ln \left (\textit {\_a}^{5}+125\right )+20 \left (\int _{\textit {\_g}}^{\textit {\_Z}}\frac {1}{2 i \textit {\_a} \left (-\textit {\_f}^{2} \textit {\_a}^{2}\right )^{\frac {1}{3}}+5 i \textit {\_f} +5 \sqrt {3}\, \textit {\_f}}d \textit {\_f} \right )-20 c_{1} \right )}d \textit {\_a} -x -c_{2} &= 0 \\ \int _{}^{y \left (x \right )}\frac {1}{\operatorname {RootOf}\left (20 \left (\int _{\textit {\_g}}^{\textit {\_Z}}\frac {1}{-2 i \textit {\_a} \left (-\textit {\_f}^{2} \textit {\_a}^{2}\right )^{\frac {1}{3}}-5 i \textit {\_f} +5 \sqrt {3}\, \textit {\_f}}d \textit {\_f} \right )+i \ln \left (\textit {\_a}^{5}+125\right )+\sqrt {3}\, \ln \left (\textit {\_a}^{5}+125\right )+20 c_{1} \right )}d \textit {\_a} -x -c_{2} &= 0 \\ \end{align*}

Problem 7442

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

Problem 7443

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

Problem 7444

ODE	\[ \boxed {y^{\prime \prime }+\left (1-x \right ) y^{\prime }+{y^{\prime }}^{2} y^{2}=0} \]
program solution
Maple solution	\[ c_{1} \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x -1\right )}{2}\right )-c_{2} +\frac {2 \,3^{\frac {5}{6}} y \left (x \right ) \pi }{9 \Gamma \left (\frac {2}{3}\right ) \left (-y \left (x \right )^{3}\right )^{\frac {1}{3}}}-\frac {y \left (x \right ) \Gamma \left (\frac {1}{3}, -\frac {y \left (x \right )^{3}}{3}\right ) 3^{\frac {1}{3}}}{3 \left (-y \left (x \right )^{3}\right )^{\frac {1}{3}}} = 0 \]

Problem 7445

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

Problem 7446

ODE	\[ \boxed {y^{\prime } y^{\prime \prime }+y^{2}=0} \]
program solution	\[ \int _{}^{y}\frac {1}{\left (-\textit {\_a}^{3}+3 c_{1} \right )^{\frac {1}{3}}}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}\frac {1}{-\frac {\left (-\textit {\_a}^{3}+3 c_{1} \right )^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, \left (-\textit {\_a}^{3}+3 c_{1} \right )^{\frac {1}{3}}}{2}}d \textit {\_a} = x +c_{3} \] Verified OK. \[ \int _{}^{y}\frac {1}{-\frac {\left (-\textit {\_a}^{3}+3 c_{1} \right )^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, \left (-\textit {\_a}^{3}+3 c_{1} \right )^{\frac {1}{3}}}{2}}d \textit {\_a} = x +c_{4} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= {\mathrm e}^{\frac {\sqrt {3}\, \left (\int \tan \left (\operatorname {RootOf}\left (-\sqrt {3}\, \ln \left (\cos \left (\textit {\_Z} \right )^{2}\right )-2 \sqrt {3}\, \ln \left (\tan \left (\textit {\_Z} \right )+\sqrt {3}\right )+6 \sqrt {3}\, c_{1} +6 \sqrt {3}\, x +6 \textit {\_Z} \right )\right )d x \right )}{2}+c_{2} +\frac {x}{2}} \\ \end{align*}

Problem 7447

ODE	\[ \boxed {y^{\prime } y^{\prime \prime }+y^{n}=0} \]
program solution	\[ \int _{}^{y}\frac {n +1}{{\left (\left (3 c_{1} n -3 \textit {\_a}^{n +1}+3 c_{1} \right ) \left (n +1\right )^{2}\right )}^{\frac {1}{3}}}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}-\frac {2 \left (n +1\right )}{{\left (\left (3 c_{1} n -3 \textit {\_a}^{n +1}+3 c_{1} \right ) \left (n +1\right )^{2}\right )}^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}d \textit {\_a} = x +c_{3} \] Verified OK. \[ \int _{}^{y}\frac {2 n +2}{{\left (\left (3 c_{1} n -3 \textit {\_a}^{n +1}+3 c_{1} \right ) \left (n +1\right )^{2}\right )}^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}d \textit {\_a} = x +c_{4} \] Verified OK.
Maple solution	\begin{align*} \frac {\left (-2-2 n \right ) \left (\int _{}^{y \left (x \right )}\frac {1}{{\left (-\left (3 \textit {\_a}^{1+n}-c_{1} \right ) \left (1+n \right )^{2}\right )}^{\frac {1}{3}}}d \textit {\_a} \right )-\left (1+i \sqrt {3}\right ) \left (x +c_{2} \right )}{1+i \sqrt {3}} &= 0 \\ -\frac {2 i \left (1+n \right ) \left (\int _{}^{y \left (x \right )}\frac {1}{{\left (-\left (3 \textit {\_a}^{1+n}-c_{1} \right ) \left (1+n \right )^{2}\right )}^{\frac {1}{3}}}d \textit {\_a} \right )+\left (x +c_{2} \right ) \left (\sqrt {3}+i\right )}{\sqrt {3}+i} &= 0 \\ \left (\int _{}^{y \left (x \right )}\frac {1}{{\left (-\left (3 \textit {\_a}^{1+n}-c_{1} \right ) \left (1+n \right )^{2}\right )}^{\frac {1}{3}}}d \textit {\_a} \right ) n +\int _{}^{y \left (x \right )}\frac {1}{{\left (-\left (3 \textit {\_a}^{1+n}-c_{1} \right ) \left (1+n \right )^{2}\right )}^{\frac {1}{3}}}d \textit {\_a} -c_{2} -x &= 0 \\ \end{align*}

Problem 7448

ODE	\[ \boxed {y^{\prime }-\left (x +y\right )^{4}=0} \]
program solution	\[ x = \frac {\sqrt {2}\, \left (\ln \left (\frac {\left (x +y\right )^{2}+\sqrt {2}\, \left (x +y\right )+1}{\left (x +y\right )^{2}-\sqrt {2}\, \left (x +y\right )+1}\right )+2 \arctan \left (\sqrt {2}\, \left (x +y\right )+1\right )+2 \arctan \left (\sqrt {2}\, \left (x +y\right )-1\right )\right )}{8}+c_{1} \] Verified OK.
Maple solution	\[ \text {Expression too large to display} \]

Problem 7449

ODE	\[ \boxed {y^{\prime \prime }+\left (x +3\right ) y^{\prime }+\left (3+y^{2}\right ) {y^{\prime }}^{2}=0} \]
program solution
Maple solution	\[ c_{1} \operatorname {erf}\left (\frac {\sqrt {2}\, \left (x +3\right )}{2}\right )-c_{2} +\int _{}^{y \left (x \right )}{\mathrm e}^{\frac {\textit {\_a} \left (\textit {\_a}^{2}+9\right )}{3}}d \textit {\_a} = 0 \]

Problem 7450

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

Problem 7451

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

Problem 7452

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

Problem 7453

ODE	\[ \boxed {10 y^{\prime \prime }+y^{\prime } x^{2}+\frac {3 {y^{\prime }}^{2}}{y}=0} \]
program solution	\[ \frac {10 y^{\frac {13}{10}}}{13}-\frac {10^{\frac {1}{3}} 9^{\frac {2}{3}} c_{2} \left (\frac {3 \,243^{\frac {1}{6}} 10^{\frac {5}{6}} x \,{\mathrm e}^{-\frac {x^{3}}{60}} \operatorname {WhittakerM}\left (\frac {1}{6}, \frac {2}{3}, \frac {x^{3}}{30}\right )}{40 \left (x^{3}\right )^{\frac {1}{6}}}+\frac {3 \,30^{\frac {5}{6}} {\mathrm e}^{-\frac {x^{3}}{60}} \operatorname {WhittakerM}\left (\frac {7}{6}, \frac {2}{3}, \frac {x^{3}}{30}\right )}{x^{2} \left (x^{3}\right )^{\frac {1}{6}}}\right )}{9}-c_{3} = 0 \] Verified OK.
Maple solution	\[ -\frac {3 \left (c_{1} x \operatorname {WhittakerM}\left (\frac {1}{6}, \frac {2}{3}, \frac {x^{3}}{30}\right ) {\mathrm e}^{-\frac {x^{3}}{60}} 30^{\frac {1}{6}}+\frac {4 \left (x^{3}\right )^{\frac {1}{6}} \left (c_{1} x \,{\mathrm e}^{-\frac {x^{3}}{30}}+c_{2} -\frac {10 y \left (x \right )^{\frac {13}{10}}}{13}\right )}{3}\right )}{4 \left (x^{3}\right )^{\frac {1}{6}}} = 0 \]

Problem 7454

ODE	\[ \boxed {10 y^{\prime \prime }+\left ({\mathrm e}^{x}+3 x \right ) y^{\prime }+\frac {3 \,{\mathrm e}^{y} {y^{\prime }}^{2}}{\sin \left (y\right )}=0} \]
program solution
Maple solution	\[ \int _{}^{y \left (x \right )}{\mathrm e}^{\frac {3 \left (\int \csc \left (\textit {\_b} \right ) {\mathrm e}^{\textit {\_b}}d \textit {\_b} \right )}{10}}d \textit {\_b} -c_{1} \left (\int {\mathrm e}^{-\frac {3 x^{2}}{20}-\frac {{\mathrm e}^{x}}{10}}d x \right )-c_{2} = 0 \]

Problem 7455

ODE	\[ \boxed {y^{\prime \prime }-\frac {2 y}{x^{2}}=x \,{\mathrm e}^{-\sqrt {x}}} \]
program solution	\[ y = \frac {28 \left (x^{3}+20 x^{2}+120 x +120 \sqrt {x}+60 x^{\frac {3}{2}}+5 x^{\frac {5}{2}}+\frac {x^{\frac {7}{2}}}{7}\right ) {\mathrm e}^{-\sqrt {x}}-\frac {c_{1} \sqrt {x}}{3}+x^{\frac {7}{2}} c_{2}}{x^{\frac {3}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {4 \,{\mathrm e}^{-\sqrt {x}} \left (7 x^{\frac {5}{2}}+140 x^{\frac {3}{2}}+x^{3}+35 x^{2}+840 \sqrt {x}+420 x +840\right )+c_{1} x^{3}+c_{2}}{x} \]

Problem 7456

ODE	\[ \boxed {y^{\prime \prime }-\frac {y^{\prime }}{\sqrt {x}}+\frac {\left (x +\sqrt {x}-8\right ) y}{4 x^{2}}=x} \]
program solution	\[ y = \frac {{\mathrm e}^{\sqrt {x}} \left (x^{3} c_{2} +3 c_{1} \right )}{3 x}+\frac {28 x^{\frac {5}{2}}+560 x^{\frac {3}{2}}+4 x^{3}+140 x^{2}+3360 \sqrt {x}+1680 x +3360}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {560 x^{\frac {3}{2}}+28 x^{\frac {5}{2}}+\left (c_{1} x^{3}+c_{2} \right ) {\mathrm e}^{\sqrt {x}}+4 x^{3}+140 x^{2}+1680 x +3360 \sqrt {x}+3360}{x} \]

Problem 7457

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

Problem 7458

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

Problem 7459

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

Problem 7460

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

Problem 7461

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

Problem 7462

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+\left (1+x \right ) y^{\prime }+y=4 \cos \left (\ln \left (1+x \right )\right )} \]
program solution
Maple solution	\[ y \left (x \right ) = \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}+\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right ) c_{2} +\left (x +i\right )^{\frac {1}{2}-\frac {i}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i}{2}, \frac {1}{2}-\frac {3 i}{2}\right ], \left [\frac {3}{2}-\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right ) c_{1} +80 \left (\int \frac {\left (-i x +1\right ) \cos \left (\ln \left (x +1\right )\right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i}{2}, \frac {1}{2}-\frac {3 i}{2}\right ], \left [\frac {3}{2}-\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )}{\left (x^{2}+1\right ) \left (10 \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i}{2}, \frac {1}{2}-\frac {3 i}{2}\right ], \left [\frac {3}{2}-\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right ) \left (\left (-1-i+\left (-1+i\right ) x \right ) \operatorname {hypergeom}\left (\left [1-i, 1+i\right ], \left [\frac {3}{2}+\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )+\left (1+i\right ) \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}+\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )\right )+\operatorname {hypergeom}\left (\left [\frac {3}{2}+\frac {i}{2}, \frac {3}{2}-\frac {3 i}{2}\right ], \left [\frac {5}{2}-\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right ) \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}+\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right ) \left (1+7 i+\left (7-i\right ) x \right )\right )}d x \right ) \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}+\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )-80 \left (\int \frac {\cos \left (\ln \left (x +1\right )\right ) \left (x +i\right )^{\frac {1}{2}+\frac {i}{2}} \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}+\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )}{7 \left (\frac {10 \left (\left (1-i+\left (-1-i\right ) x \right ) \operatorname {hypergeom}\left (\left [1-i, 1+i\right ], \left [\frac {3}{2}+\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )+\left (-1+i\right ) \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}+\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )\right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i}{2}, \frac {1}{2}-\frac {3 i}{2}\right ], \left [\frac {3}{2}-\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )}{7}+\left (-1+\frac {i}{7}+\left (\frac {1}{7}+i\right ) x \right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}+\frac {i}{2}, \frac {3}{2}-\frac {3 i}{2}\right ], \left [\frac {5}{2}-\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right ) \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}+\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )\right ) \left (x^{2}+1\right )}d x \right ) \left (x +i\right )^{\frac {1}{2}-\frac {i}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i}{2}, \frac {1}{2}-\frac {3 i}{2}\right ], \left [\frac {3}{2}-\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right ) \]

Problem 7463

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

Problem 7464

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

Problem 7465

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

Problem 7466

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

Problem 7467

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

Problem 7468

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime } x^{2}+y x=x^{m +1}} \]
program solution	\[ y = c_{1} x +\frac {c_{2} \left (-3 \,{\mathrm e}^{\frac {x^{3}}{3}} \left (-x^{3}\right )^{\frac {2}{3}}+x^{3} 3^{\frac {2}{3}} \left (\Gamma \left (\frac {2}{3}\right )-\Gamma \left (\frac {2}{3}, -\frac {x^{3}}{3}\right )\right )\right )}{3 \left (-x^{3}\right )^{\frac {2}{3}}}-\frac {\left (-x^{3}\right )^{\frac {2}{3}} x \left (3+m \right ) \left (\int _{0}^{x}\frac {\left (-3 \left (-\alpha ^{3}\right )^{\frac {2}{3}}+{\mathrm e}^{-\frac {\alpha ^{3}}{3}} 3^{\frac {2}{3}} \alpha ^{3} \left (\Gamma \left (\frac {2}{3}\right )-\Gamma \left (\frac {2}{3}, -\frac {\alpha ^{3}}{3}\right )\right )\right ) \alpha ^{m +1}}{\left (-\alpha ^{3}\right )^{\frac {2}{3}}}d \alpha \right )+9 \operatorname {WhittakerM}\left (\frac {m}{6}, \frac {m}{6}+\frac {1}{2}, \frac {x^{3}}{3}\right ) \left (x^{3}\right )^{-\frac {m}{6}} \left (\left (-x^{3}\right )^{\frac {2}{3}} {\mathrm e}^{\frac {x^{3}}{6}} x^{m} 3^{\frac {m}{6}}+\frac {3^{\frac {m}{6}+\frac {5}{3}} {\mathrm e}^{-\frac {x^{3}}{6}} x^{3+m} \left (\Gamma \left (\frac {2}{3}, -\frac {x^{3}}{3}\right )-\Gamma \left (\frac {2}{3}\right )\right )}{9}\right )}{\left (-x^{3}\right )^{\frac {2}{3}} \left (3 m +9\right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-3 \,3^{\frac {m}{6}} {\mathrm e}^{\frac {x^{3}}{6}} \left (x^{3}\right )^{-\frac {m}{6}} \operatorname {WhittakerM}\left (\frac {m}{6}, \frac {m}{6}+\frac {1}{2}, \frac {x^{3}}{3}\right ) x^{m}+\left (m +3\right ) \left (3^{\frac {1}{3}} {\mathrm e}^{\frac {x^{3}}{3}} c_{1} -\frac {\left (\int \frac {\left (-3 \left (-x^{3}\right )^{\frac {2}{3}}+x^{3} 3^{\frac {2}{3}} {\mathrm e}^{-\frac {x^{3}}{3}} \left (\Gamma \left (\frac {2}{3}\right )-\Gamma \left (\frac {2}{3}, -\frac {x^{3}}{3}\right )\right )\right ) x^{m +1}}{\left (-x^{3}\right )^{\frac {2}{3}}}d x -3 c_{2} \right ) x}{3}\right )\right ) \left (-x^{3}\right )^{\frac {2}{3}}-\frac {\left (\left (x^{3}\right )^{-\frac {m}{6}} x^{m +3} 3^{\frac {5}{3}+\frac {m}{6}} {\mathrm e}^{-\frac {x^{3}}{6}} \operatorname {WhittakerM}\left (\frac {m}{6}, \frac {m}{6}+\frac {1}{2}, \frac {x^{3}}{3}\right )-3 c_{1} x^{3} \left (m +3\right )\right ) \left (\Gamma \left (\frac {2}{3}, -\frac {x^{3}}{3}\right )-\Gamma \left (\frac {2}{3}\right )\right )}{3}}{\left (-x^{3}\right )^{\frac {2}{3}} \left (m +3\right )} \]

Problem 7469

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

Problem 7470

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

Problem 7471

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

Problem 7472

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

Problem 7473

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

Problem 7474

ODE	\[ \boxed {y^{\prime \prime }-2 \tan \left (x \right ) y^{\prime }+5 y={\mathrm e}^{x^{2}} \sec \left (x \right )} \]
program solution	\[ y = c_{1} \cos \left (\sqrt {6}\, x \right ) \sec \left (x \right )+\frac {c_{2} \sqrt {6}\, \sin \left (\sqrt {6}\, x \right ) \sec \left (x \right )}{6}-\frac {\left (\left (i \sin \left (\sqrt {6}\, x \right )-\cos \left (\sqrt {6}\, x \right )\right ) \operatorname {erf}\left (i x -\frac {\sqrt {6}}{2}\right )+\left (i \sin \left (\sqrt {6}\, x \right )+\cos \left (\sqrt {6}\, x \right )\right ) \operatorname {erf}\left (i x +\frac {\sqrt {6}}{2}\right )-2 \cos \left (\sqrt {6}\, x \right ) \operatorname {erf}\left (\frac {\sqrt {6}}{2}\right )\right ) \sqrt {6}\, \sec \left (x \right ) \sqrt {\pi }\, {\mathrm e}^{\frac {3}{2}}}{24} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\left (\sqrt {6}\, {\mathrm e}^{\frac {3}{2}} \sqrt {\pi }\, \left (i \sin \left (\sqrt {6}\, x \right )-\cos \left (\sqrt {6}\, x \right )\right ) \operatorname {erf}\left (i x -\frac {\sqrt {6}}{2}\right )+\sqrt {6}\, {\mathrm e}^{\frac {3}{2}} \left (i \sin \left (\sqrt {6}\, x \right )+\cos \left (\sqrt {6}\, x \right )\right ) \sqrt {\pi }\, \operatorname {erf}\left (i x +\frac {\sqrt {6}}{2}\right )-24 \sin \left (\sqrt {6}\, x \right ) c_{2} -24 \cos \left (\sqrt {6}\, x \right ) c_{1} \right ) \sec \left (x \right )}{24} \]

Problem 7475

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

Problem 7476

ODE	\[ \boxed {4 y^{\prime \prime } x^{2}+4 x^{5} y^{\prime }+\left (x^{8}+6 x^{4}+4\right ) y=0} \]
program solution	\[ y = c_{1} x^{\frac {1}{2}-\frac {i \sqrt {3}}{2}} {\mathrm e}^{-\frac {x^{4}}{8}}-\frac {i c_{2} \sqrt {3}\, x^{\frac {1}{2}+\frac {i \sqrt {3}}{2}} {\mathrm e}^{-\frac {x^{4}}{8}}}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, {\mathrm e}^{-\frac {x^{4}}{8}} \left (c_{1} x^{\frac {i \sqrt {3}}{2}}+c_{2} x^{-\frac {i \sqrt {3}}{2}}\right ) \]

Problem 7477

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

Problem 7478

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

Problem 7479

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

Problem 7480

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

Problem 7481

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

Problem 7482

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

Problem 7483

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-y^{\prime \prime \prime }-3 y^{\prime \prime }+5 y^{\prime }-2 y=x \,{\mathrm e}^{x}+3 \,{\mathrm e}^{-2 x}} \]
program solution	\[ y = \left (\left (c_{4} x^{2}+c_{3} x +c_{2} \right ) {\mathrm e}^{3 x}+c_{1} \right ) {\mathrm e}^{-2 x}-\frac {{\mathrm e}^{-2 x} x}{9}-\frac {x^{3} {\mathrm e}^{x}}{54}+\frac {{\mathrm e}^{x} x^{4}}{72} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (\left (x^{4}-\frac {4 x^{3}}{3}+\left (72 c_{4} +\frac {4}{3}\right ) x^{2}+\left (72 c_{3} -\frac {8}{9}\right ) x +72 c_{1} +\frac {8}{27}\right ) {\mathrm e}^{3 x}-8 x +72 c_{2} -8\right ) {\mathrm e}^{-2 x}}{72} \]

Problem 7484

ODE	\[ \boxed {y^{\prime \prime } x^{2}-x \left (x +6\right ) y^{\prime }+10 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{5} \left (1+\frac {5 x}{4}+\frac {3 x^{2}}{4}+\frac {7 x^{3}}{24}+\frac {x^{4}}{12}+\frac {3 x^{5}}{160}+O\left (x^{6}\right )\right )+c_{2} \left (2 x^{5} \left (1+\frac {5 x}{4}+\frac {3 x^{2}}{4}+\frac {7 x^{3}}{24}+\frac {x^{4}}{12}+\frac {3 x^{5}}{160}+O\left (x^{6}\right )\right ) \ln \left (x \right )+x^{2} \left (1-x +\frac {3 x^{2}}{2}-\frac {21 x^{4}}{8}-\frac {19 x^{5}}{8}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{2} \left (c_{1} x^{3} \left (1+\frac {5}{4} x +\frac {3}{4} x^{2}+\frac {7}{24} x^{3}+\frac {1}{12} x^{4}+\frac {3}{160} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (24 x^{3}+30 x^{4}+18 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (12-12 x +18 x^{2}+26 x^{3}+x^{4}-9 x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )\right ) \]

Problem 7485

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

Problem 7486

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

Problem 7487

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

Problem 7488

ODE	\[ \boxed {y^{\prime \prime \prime }-y x=0} \]
program solution
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {1}{2}, \frac {3}{4}\right ], \frac {x^{4}}{64}\right )+c_{2} x \operatorname {hypergeom}\left (\left [\right ], \left [\frac {3}{4}, \frac {5}{4}\right ], \frac {x^{4}}{64}\right )+c_{3} x^{2} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {5}{4}, \frac {3}{2}\right ], \frac {x^{4}}{64}\right ) \]

Problem 7489

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

Problem 7490

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

Problem 7491

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

Problem 7492

ODE	\[ \boxed {\left (x^{2}-1\right ) y^{\prime \prime }-6 x y^{\prime }+12 y=0} \]
program solution	\[ y = c_{1} \left (1+x \right )^{4}-c_{2} x \left (x^{2}+1\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (x^{3}+x \right )+c_{2} \left (x^{4}+6 x^{2}+1\right ) \]

Problem 7493

ODE	\[ \boxed {\left (x^{2}+3\right ) y^{\prime \prime }-7 x y^{\prime }+16 y=0} \]
program solution	\[ y = c_{1} \left (x^{4}-9 x^{2}+\frac {27}{8}\right )+\frac {c_{2} \left (-256 \left (x^{4}-9 x^{2}+\frac {27}{8}\right ) \left (\left (-x^{3}-\frac {3}{2} x \right ) \sqrt {x^{2}+3}+x^{4}+3 x^{2}+\frac {9}{8}\right ) \ln \left (\sqrt {x^{2}+3}-x \right )+\left (-3840 x^{5}-720 x^{3}+4680 x \right ) \sqrt {x^{2}+3}+3840 x^{6}+6480 x^{4}-7920 x^{2}-2025\right )}{\left (-256 x^{3}-384 x \right ) \sqrt {x^{2}+3}+256 x^{4}+768 x^{2}+288} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (x^{4}-9 x^{2}+\frac {27}{8}\right )+c_{2} \left (\frac {\ln \left (\sqrt {x^{2}+3}-x \right ) x^{4}}{64}+\frac {25 \sqrt {x^{2}+3}\, x^{3}}{768}+\frac {25 x^{4}}{768}-\frac {9 \ln \left (\sqrt {x^{2}+3}-x \right ) x^{2}}{64}-\frac {55 \sqrt {x^{2}+3}\, x}{512}-\frac {75 x^{2}}{256}+\frac {27 \ln \left (\sqrt {x^{2}+3}-x \right )}{512}+\frac {225}{2048}\right ) \]

Problem 7494

ODE	\[ \boxed {\left (x^{2}-1\right ) y^{\prime \prime }+8 x y^{\prime }+12 y=0} \]
program solution	\[ y = \frac {c_{1}}{\left (x -1\right )^{3}}+\frac {c_{2} \left (-3 x^{2}-1\right )}{3 \left (x -1\right )^{3} \left (1+x \right )^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (3 x^{2}+1\right )}{\left (x -1\right )^{3} \left (x +1\right )^{3}}+\frac {c_{2} \left (x^{3}+3 x \right )}{\left (x -1\right )^{3} \left (x +1\right )^{3}} \]

Problem 7495

ODE	\[ \boxed {3 y^{\prime \prime }+x y^{\prime }-4 y=0} \]
program solution	\[ y = c_{1} \left (x^{4}+18 x^{2}+27\right )+c_{2} \left (x^{4}+18 x^{2}+27\right ) \left (\int \frac {{\mathrm e}^{-\frac {x^{2}}{6}}}{\left (x^{4}+18 x^{2}+27\right )^{2}}d x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (x^{4}+18 x^{2}+27\right )+c_{2} \left (x^{4}+18 x^{2}+27\right ) \left (\int \frac {{\mathrm e}^{-\frac {x^{2}}{6}}}{\left (x^{4}+18 x^{2}+27\right )^{2}}d x \right ) \]

Problem 7496

ODE	\[ \boxed {5 y^{\prime \prime }-2 x y^{\prime }+10 y=0} \]
program solution	\[ y = c_{1} \left (x^{5}-25 x^{3}+\frac {375}{4} x \right )+16 c_{2} x \left (x^{4}-25 x^{2}+\frac {375}{4}\right ) \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{5}}}{x^{2} \left (4 x^{4}-100 x^{2}+375\right )^{2}}d x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (x^{5}-25 x^{3}+\frac {375}{4} x \right )+c_{2} \left (x^{5}-25 x^{3}+\frac {375}{4} x \right ) \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{5}}}{\left (4 x^{4}-100 x^{2}+375\right )^{2} x^{2}}d x \right ) \]

Problem 7497

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime } x^{2}-3 y x=0} \]
program solution	\[ y = c_{1} x \,{\mathrm e}^{\frac {x^{3}}{3}}+c_{2} x \,{\mathrm e}^{\frac {x^{3}}{3}} \left (\int \frac {{\mathrm e}^{-\frac {x^{3}}{3}}}{x^{2}}d x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{\frac {x^{3}}{3}} x -\frac {9 c_{2} {\mathrm e}^{\frac {x^{3}}{3}} 3^{\frac {2}{3}} {\mathrm e}^{-\frac {x^{3}}{6}} \left (x^{6} \operatorname {WhittakerM}\left (\frac {1}{3}, \frac {5}{6}, \frac {x^{3}}{3}\right )+5 \operatorname {WhittakerM}\left (\frac {4}{3}, \frac {5}{6}, \frac {x^{3}}{3}\right ) x^{3}+10 \operatorname {WhittakerM}\left (\frac {4}{3}, \frac {5}{6}, \frac {x^{3}}{3}\right )\right )}{10 x^{3} \left (x^{3}\right )^{\frac {1}{3}}} \]

Problem 7498

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

Problem 7499

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

Problem 7500

ODE	\[ \boxed {\left (x^{2}-6 x +10\right ) y^{\prime \prime }-4 \left (x -3\right ) y^{\prime }+6 y=0} \]
program solution	\[ y = \frac {c_{1} \left (i x -3 i-1\right )^{2} \left (x^{2}-6 x +10\right )}{-x +3+i}+\frac {c_{2} \left (-x^{2}+6 x -10\right ) \left (x^{2}-6 x +\frac {26}{3}\right )}{\left (-x +3-i\right ) \left (-x +3+i\right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (\frac {26}{3}+x^{2}-6 x \right )+c_{2} \left (x^{3}-30 x +60\right ) \]