2.17.69 Problems 6801 to 6900

Problem 6801

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

Problem 6802

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

Problem 6803

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

Problem 6804

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

Problem 6805

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

Problem 6806

ODE	\[ \boxed {x^{6} {y^{\prime }}^{3}-3 y^{\prime } x -3 y=0} \]
program solution	\[ -\frac {3 \ln \left (x \right )}{2} = \int _{}^{y x^{\frac {3}{2}}}-\frac {3 \left (\sqrt {9 \textit {\_a}^{2}-4}+3 \textit {\_a} \right )^{\frac {1}{3}}}{2^{\frac {2}{3}} \left (\sqrt {9 \textit {\_a}^{2}-4}+3 \textit {\_a} \right )^{\frac {2}{3}}+3 \left (\sqrt {9 \textit {\_a}^{2}-4}+3 \textit {\_a} \right )^{\frac {1}{3}} \textit {\_a} +2 \,2^{\frac {1}{3}}}d \textit {\_a} +c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {2}{3 x^{\frac {3}{2}}} \\ y \left (x \right ) &= \frac {2}{3 x^{\frac {3}{2}}} \\ y \left (x \right ) &= \frac {c_{1}^{3}}{3}-\frac {c_{1}}{x} \\ \end{align*}

Problem 6807

ODE	\[ \boxed {y-x^{6} {y^{\prime }}^{3}+y^{\prime } x=0} \]
program solution	\[ -\frac {3 \ln \left (x \right )}{2} = \int _{}^{y x^{\frac {3}{2}}}-\frac {3 \left (3 \sqrt {3}\, \textit {\_a} +\sqrt {27 \textit {\_a}^{2}-4}\right )^{\frac {2}{3}} \sqrt {3}\, 2^{\frac {1}{3}} \textit {\_a} -\left (3 \sqrt {3}\, \textit {\_a} +\sqrt {27 \textit {\_a}^{2}-4}\right )^{\frac {2}{3}} 2^{\frac {1}{3}} \sqrt {27 \textit {\_a}^{2}-4}-2 \,2^{\frac {2}{3}} \left (3 \sqrt {3}\, \textit {\_a} +\sqrt {27 \textit {\_a}^{2}-4}\right )^{\frac {1}{3}}+2 \sqrt {27 \textit {\_a}^{2}-4}}{2 \textit {\_a} \sqrt {27 \textit {\_a}^{2}-4}}d \textit {\_a} +c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {2 \sqrt {3}}{9 x^{\frac {3}{2}}} \\ y \left (x \right ) &= \frac {2 \sqrt {3}}{9 x^{\frac {3}{2}}} \\ y \left (x \right ) &= c_{1}^{3}-\frac {c_{1}}{x} \\ \end{align*}

Problem 6808

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

Problem 6809

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

Problem 6810

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

Problem 6811

ODE	\[ \boxed {2 {y^{\prime }}^{3}+y^{\prime } x -2 y=0} \]
program solution	\[ y = 0 \] Verified OK. \[ x = \frac {\left (\left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}-6 x \right ) \left (\left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}+c_{1} \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {1}{3}}-6 x \right )}{6 \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}} \] Verified OK. \[ x = \frac {\left (i \sqrt {3}\, \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}+6 i x \sqrt {3}-\left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}+6 x \right ) \left (i \sqrt {3}\, \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}+6 i x \sqrt {3}-\left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}+2 c_{1} \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {1}{3}}+6 x \right )}{24 \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}} \] Warning, solution could not be verified \[ x = \frac {\left (i \sqrt {3}\, \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}+6 i x \sqrt {3}+\left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}-6 x \right ) \left (i \sqrt {3}\, \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}+6 i x \sqrt {3}+\left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}-2 c_{1} \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {1}{3}}-6 x \right )}{24 \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}} \] Warning, solution could not be verified
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (-c_{1}^{2}-24 x \right ) \sqrt {c_{1}^{2}+24 x}}{432}-\frac {c_{1}^{3}}{432}-\frac {c_{1} x}{12} \\ y \left (x \right ) &= \frac {\left (c_{1}^{2}+24 x \right )^{\frac {3}{2}}}{432}-\frac {c_{1}^{3}}{432}-\frac {c_{1} x}{12} \\ \end{align*}

Problem 6812

ODE	\[ \boxed {2 {y^{\prime }}^{2}+y^{\prime } x -2 y=0} \]
program solution	\[ y = 0 \] Verified OK. \[ x = \frac {\left (-x +\sqrt {x^{2}+16 y}\right ) \left (-8 \ln \left (2\right )+4 \ln \left (-x +\sqrt {x^{2}+16 y}\right )+c_{1} \right )}{4} \] Verified OK. \[ x = -\frac {\left (x +\sqrt {x^{2}+16 y}\right ) \left (-8 \ln \left (2\right )+4 \ln \left (-x -\sqrt {x^{2}+16 y}\right )+c_{1} \right )}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x^{2} \left (1+2 \operatorname {LambertW}\left (\frac {x \,{\mathrm e}^{\frac {c_{1}}{4}}}{4}\right )\right )}{16 \operatorname {LambertW}\left (\frac {x \,{\mathrm e}^{\frac {c_{1}}{4}}}{4}\right )^{2}} \]

Problem 6813

ODE	\[ \boxed {{y^{\prime }}^{3}+2 y^{\prime } x -y=0} \]
program solution	\[ y = 0 \] Verified OK. \[ x = -\frac {{\left (\left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-24 x \right )}^{2}}{48 \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}+\frac {36 c_{1} \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}{{\left (\left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-24 x \right )}^{2}} \] Verified OK. \[ x = \frac {3 {\left (\frac {\left (\sqrt {3}+i\right ) \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}{24}+x \left (-i+\sqrt {3}\right )\right )}^{2}}{\left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}+\frac {144 c_{1} \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}{{\left (i \sqrt {3}\, \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 i x \sqrt {3}-\left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 x \right )}^{2}} \] Verified OK. \[ x = \frac {3 {\left (\frac {\left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}} \left (-i+\sqrt {3}\right )}{24}+\left (\sqrt {3}+i\right ) x \right )}^{2}}{\left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}+\frac {144 c_{1} \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}{{\left (i \sqrt {3}\, \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 i x \sqrt {3}+\left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-24 x \right )}^{2}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {2 \left (-2 x +\sqrt {x^{2}+3 c_{1}}\right ) \sqrt {-6 \sqrt {x^{2}+3 c_{1}}-6 x}}{9} \\ y \left (x \right ) &= -\frac {2 \left (-2 x +\sqrt {x^{2}+3 c_{1}}\right ) \sqrt {-6 \sqrt {x^{2}+3 c_{1}}-6 x}}{9} \\ y \left (x \right ) &= -\frac {2 \left (2 x +\sqrt {x^{2}+3 c_{1}}\right ) \sqrt {6 \sqrt {x^{2}+3 c_{1}}-6 x}}{9} \\ y \left (x \right ) &= \frac {2 \left (2 x +\sqrt {x^{2}+3 c_{1}}\right ) \sqrt {6 \sqrt {x^{2}+3 c_{1}}-6 x}}{9} \\ \end{align*}

Problem 6814

ODE	\[ \boxed {4 x {y^{\prime }}^{2}-3 y^{\prime } y=-3} \]
program solution	\[ x = \frac {64 x^{2} \left (64 c_{1} x^{2}+9 y \sqrt {9 y^{2}-48 x}+27 y^{2}-72 x \right )}{\left (3 y+\sqrt {9 y^{2}-48 x}\right )^{4}} \] Verified OK. \[ x = -\frac {64 x^{2} \left (-64 c_{1} x^{2}+9 y \sqrt {9 y^{2}-48 x}-27 y^{2}+72 x \right )}{\left (-3 y+\sqrt {9 y^{2}-48 x}\right )^{4}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {2 x \left (6+\sqrt {16 c_{1} x +9}\right )}{3 \sqrt {x \left (3+\sqrt {16 c_{1} x +9}\right )}} \\ y \left (x \right ) &= \frac {2 x \left (6+\sqrt {16 c_{1} x +9}\right )}{3 \sqrt {x \left (3+\sqrt {16 c_{1} x +9}\right )}} \\ y \left (x \right ) &= \frac {2 x \left (-6+\sqrt {16 c_{1} x +9}\right )}{3 \sqrt {-x \left (-3+\sqrt {16 c_{1} x +9}\right )}} \\ y \left (x \right ) &= -\frac {2 x \left (-6+\sqrt {16 c_{1} x +9}\right )}{3 \sqrt {-x \left (-3+\sqrt {16 c_{1} x +9}\right )}} \\ \end{align*}

Problem 6815

ODE	\[ \boxed {{y^{\prime }}^{3}-y^{\prime } x +2 y=0} \]
program solution	\[ y = 0 \] Verified OK. \[ x = -\frac {\left (\left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+3 x \right ) \left (\left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-c_{1} \left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {1}{3}}+3 x \right )}{3 \left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}} \] Verified OK. \[ x = -\frac {\left (-i \sqrt {3}\, \left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+3 i x \sqrt {3}+\left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+3 x \right ) \left (-i \sqrt {3}\, \left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+3 i x \sqrt {3}+2 c_{1} \left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {1}{3}}+\left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+3 x \right )}{12 \left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}} \] Warning, solution could not be verified \[ x = -\frac {\left (i \sqrt {3}\, \left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-3 i x \sqrt {3}+\left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+3 x \right ) \left (i \sqrt {3}\, \left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-3 i x \sqrt {3}+\left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+2 c_{1} \left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {1}{3}}+3 x \right )}{12 \left (-27 y+3 \sqrt {-3 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}} \] Warning, solution could not be verified
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (c_{1}^{2}-12 x \right )^{\frac {3}{2}}}{108}-\frac {c_{1}^{3}}{108}+\frac {c_{1} x}{6} \\ y \left (x \right ) &= \frac {\left (-c_{1}^{2}+12 x \right ) \sqrt {c_{1}^{2}-12 x}}{108}-\frac {c_{1}^{3}}{108}+\frac {c_{1} x}{6} \\ \end{align*}

Problem 6816

ODE	\[ \boxed {5 {y^{\prime }}^{2}+6 y^{\prime } x -2 y=0} \]
program solution	\[ y = 0 \] Verified OK. \[ x = \frac {3 x}{5}-\frac {\sqrt {9 x^{2}+10 y}}{5}+\frac {125 c_{1}}{\left (-15 x +5 \sqrt {9 x^{2}+10 y}\right )^{\frac {3}{2}}} \] Verified OK. \[ x = \frac {3 x}{5}+\frac {\sqrt {9 x^{2}+10 y}}{5}+\frac {125 c_{1}}{\left (-15 x -5 \sqrt {9 x^{2}+10 y}\right )^{\frac {3}{2}}} \] Verified OK.
Maple solution	\begin{align*} \frac {c_{1}}{\left (-15 x -5 \sqrt {9 x^{2}+10 y \left (x \right )}\right )^{\frac {3}{2}}}+\frac {2 x}{5}-\frac {\sqrt {9 x^{2}+10 y \left (x \right )}}{5} &= 0 \\ \frac {c_{1}}{\left (-15 x +5 \sqrt {9 x^{2}+10 y \left (x \right )}\right )^{\frac {3}{2}}}+\frac {2 x}{5}+\frac {\sqrt {9 x^{2}+10 y \left (x \right )}}{5} &= 0 \\ \end{align*}

Problem 6817

ODE	\[ \boxed {2 x {y^{\prime }}^{2}+\left (-y+2 x \right ) y^{\prime }-y=-1} \]
program solution	\[ y = 1 \] Verified OK. \[ x = \frac {32 \left (\left (x +\frac {y}{2}+\frac {\sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}}{2}\right ) \ln \left (\frac {y+2 x +\sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}}{x}\right )+\left (\frac {c_{1}}{2}-\ln \left (2\right )\right ) \sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}+\left (-y-2 x \right ) \ln \left (2\right )+\left (c_{1} +2\right ) x +\frac {c_{1} y}{2}\right ) x^{2}}{\left (y+2 x +\sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}\right ) \left (-y+2 x -\sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}\right )^{2}} \] Verified OK. \[ x = \frac {32 \left (\left (x +\frac {y}{2}-\frac {\sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}}{2}\right ) \ln \left (\frac {2 x +y-\sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}}{x}\right )+\left (-\frac {c_{1}}{2}+\ln \left (2\right )\right ) \sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}+\left (-y-2 x \right ) \ln \left (2\right )+\left (c_{1} +2\right ) x +\frac {c_{1} y}{2}\right ) x^{2}}{\left (2 x +y-\sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}\right ) \left (2 x -y+\sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}\right )^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -2 \left (x \,{\mathrm e}^{\operatorname {RootOf}\left (-{\mathrm e}^{3 \textit {\_Z}} x +2 x \,{\mathrm e}^{2 \textit {\_Z}}+c_{1} {\mathrm e}^{\textit {\_Z}}+{\mathrm e}^{\textit {\_Z}} \textit {\_Z} -x \,{\mathrm e}^{\textit {\_Z}}+1\right )}-{\mathrm e}^{2 \operatorname {RootOf}\left (-{\mathrm e}^{3 \textit {\_Z}} x +2 x \,{\mathrm e}^{2 \textit {\_Z}}+c_{1} {\mathrm e}^{\textit {\_Z}}+{\mathrm e}^{\textit {\_Z}} \textit {\_Z} -x \,{\mathrm e}^{\textit {\_Z}}+1\right )} x -\frac {1}{2}\right ) {\mathrm e}^{-\operatorname {RootOf}\left (-{\mathrm e}^{3 \textit {\_Z}} x +2 x \,{\mathrm e}^{2 \textit {\_Z}}+c_{1} {\mathrm e}^{\textit {\_Z}}+{\mathrm e}^{\textit {\_Z}} \textit {\_Z} -x \,{\mathrm e}^{\textit {\_Z}}+1\right )} \]

Problem 6818

ODE	\[ \boxed {5 {y^{\prime }}^{2}+3 y^{\prime } x -y=0} \]
program solution	\[ y = 0 \] Verified OK. \[ x = \frac {3 x}{5}-\frac {\sqrt {9 x^{2}+20 y}}{5}+\frac {1000 c_{1}}{\left (-30 x +10 \sqrt {9 x^{2}+20 y}\right )^{\frac {3}{2}}} \] Verified OK. \[ x = \frac {3 x}{5}+\frac {\sqrt {9 x^{2}+20 y}}{5}+\frac {1000 c_{1}}{\left (-30 x -10 \sqrt {9 x^{2}+20 y}\right )^{\frac {3}{2}}} \] Verified OK.
Maple solution	\begin{align*} \frac {c_{1}}{\left (-30 x -10 \sqrt {9 x^{2}+20 y \left (x \right )}\right )^{\frac {3}{2}}}+\frac {2 x}{5}-\frac {\sqrt {9 x^{2}+20 y \left (x \right )}}{5} &= 0 \\ \frac {c_{1}}{\left (-30 x +10 \sqrt {9 x^{2}+20 y \left (x \right )}\right )^{\frac {3}{2}}}+\frac {2 x}{5}+\frac {\sqrt {9 x^{2}+20 y \left (x \right )}}{5} &= 0 \\ \end{align*}

Problem 6819

ODE	\[ \boxed {{y^{\prime }}^{2}+3 y^{\prime } x -y=0} \]
program solution	\[ y = 0 \] Verified OK. \[ x = \frac {3 x}{5}-\frac {\sqrt {9 x^{2}+4 y}}{5}+\frac {8 c_{1}}{\left (-6 x +2 \sqrt {9 x^{2}+4 y}\right )^{\frac {3}{2}}} \] Verified OK. \[ x = \frac {3 x}{5}+\frac {\sqrt {9 x^{2}+4 y}}{5}+\frac {8 c_{1}}{\left (-6 x -2 \sqrt {9 x^{2}+4 y}\right )^{\frac {3}{2}}} \] Verified OK.
Maple solution	\begin{align*} \frac {c_{1}}{\left (-6 x -2 \sqrt {9 x^{2}+4 y \left (x \right )}\right )^{\frac {3}{2}}}+\frac {2 x}{5}-\frac {\sqrt {9 x^{2}+4 y \left (x \right )}}{5} &= 0 \\ \frac {c_{1}}{\left (-6 x +2 \sqrt {9 x^{2}+4 y \left (x \right )}\right )^{\frac {3}{2}}}+\frac {2 x}{5}+\frac {\sqrt {9 x^{2}+4 y \left (x \right )}}{5} &= 0 \\ \end{align*}

Problem 6820

ODE	\[ \boxed {y-y^{\prime } x -x^{3} {y^{\prime }}^{2}=0} \]
program solution	\[ \frac {3 \ln \left (y x -2\right )}{4}+\frac {\ln \left (y\right )}{4}-\frac {\ln \left (1+\sqrt {1+4 y x}\right )}{4}-\frac {3 \ln \left (\sqrt {1+4 y x}-3\right )}{4}+\frac {\ln \left (-1+\sqrt {1+4 y x}\right )}{4}+\frac {3 \ln \left (\sqrt {1+4 y x}+3\right )}{4} = \frac {3 \ln \left (x \right )}{4}+c_{1} \] Verified OK. \[ \frac {3 \ln \left (y x -2\right )}{4}+\frac {\ln \left (y\right )}{4}+\frac {\ln \left (1+\sqrt {1+4 y x}\right )}{4}+\frac {3 \ln \left (\sqrt {1+4 y x}-3\right )}{4}-\frac {\ln \left (-1+\sqrt {1+4 y x}\right )}{4}-\frac {3 \ln \left (\sqrt {1+4 y x}+3\right )}{4} = \frac {3 \ln \left (x \right )}{4}+c_{1} \] Verified OK.
Maple solution	\[ \text {No solution found} \]

Problem 6821

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

Problem 6822

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

Problem 6823

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

Problem 6824

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

Problem 6825

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

Problem 6826

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

Problem 6827

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

Problem 6828

ODE	\[ \boxed {x y^{\prime \prime }-y^{\prime }=x^{5}} \] With initial conditions \begin {align*} \left [y \left (1\right ) = {\frac {1}{2}}, y^{\prime }\left (1\right ) = 1\right ] \end {align*}
program solution	\[ y = \frac {1}{24} x^{6}+\frac {1}{12}+\frac {3}{8} x^{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {1}{24} x^{6}+\frac {3}{8} x^{2}+\frac {1}{12} \]

Problem 6829

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

Problem 6830

ODE	\[ \boxed {y^{\prime \prime }-2 y {y^{\prime }}^{3}=0} \]
program solution	\[ y = \frac {\left (-12 c_{2} -12 x +4 \sqrt {4 c_{1}^{3}+9 c_{2}^{2}+18 c_{2} x +9 x^{2}}\right )^{\frac {1}{3}}}{2}-\frac {2 c_{1}}{\left (-12 c_{2} -12 x +4 \sqrt {4 c_{1}^{3}+9 c_{2}^{2}+18 c_{2} x +9 x^{2}}\right )^{\frac {1}{3}}} \] Verified OK. \[ y = -\frac {\left (-12 c_{2} -12 x +4 \sqrt {4 c_{1}^{3}+9 c_{2}^{2}+18 c_{2} x +9 x^{2}}\right )^{\frac {1}{3}}}{4}+\frac {c_{1}}{\left (-12 c_{2} -12 x +4 \sqrt {4 c_{1}^{3}+9 c_{2}^{2}+18 c_{2} x +9 x^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-12 c_{2} -12 x +4 \sqrt {4 c_{1}^{3}+9 c_{2}^{2}+18 c_{2} x +9 x^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 c_{1}}{\left (-12 c_{2} -12 x +4 \sqrt {4 c_{1}^{3}+9 c_{2}^{2}+18 c_{2} x +9 x^{2}}\right )^{\frac {1}{3}}}\right )}{2} \] Verified OK. \[ y = -\frac {\left (-12 c_{2} -12 x +4 \sqrt {4 c_{1}^{3}+9 c_{2}^{2}+18 c_{2} x +9 x^{2}}\right )^{\frac {1}{3}}}{4}+\frac {c_{1}}{\left (-12 c_{2} -12 x +4 \sqrt {4 c_{1}^{3}+9 c_{2}^{2}+18 c_{2} x +9 x^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-12 c_{2} -12 x +4 \sqrt {4 c_{1}^{3}+9 c_{2}^{2}+18 c_{2} x +9 x^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 c_{1}}{\left (-12 c_{2} -12 x +4 \sqrt {4 c_{1}^{3}+9 c_{2}^{2}+18 c_{2} x +9 x^{2}}\right )^{\frac {1}{3}}}\right )}{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= c_{1} \\ y \left (x \right ) &= \frac {\left (-12 c_{2} -12 x +4 \sqrt {-4 c_{1}^{3}+9 c_{2}^{2}+18 c_{2} x +9 x^{2}}\right )^{\frac {2}{3}}+4 c_{1}}{2 \left (-12 c_{2} -12 x +4 \sqrt {-4 c_{1}^{3}+9 c_{2}^{2}+18 c_{2} x +9 x^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {-i \sqrt {3}\, \left (-12 c_{2} -12 x +4 \sqrt {-4 c_{1}^{3}+9 c_{2}^{2}+18 c_{2} x +9 x^{2}}\right )^{\frac {2}{3}}+4 i \sqrt {3}\, c_{1} -\left (-12 c_{2} -12 x +4 \sqrt {-4 c_{1}^{3}+9 c_{2}^{2}+18 c_{2} x +9 x^{2}}\right )^{\frac {2}{3}}-4 c_{1}}{4 \left (-12 c_{2} -12 x +4 \sqrt {-4 c_{1}^{3}+9 c_{2}^{2}+18 c_{2} x +9 x^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {-i \sqrt {3}\, \left (-12 c_{2} -12 x +4 \sqrt {-4 c_{1}^{3}+9 c_{2}^{2}+18 c_{2} x +9 x^{2}}\right )^{\frac {2}{3}}+4 i \sqrt {3}\, c_{1} +\left (-12 c_{2} -12 x +4 \sqrt {-4 c_{1}^{3}+9 c_{2}^{2}+18 c_{2} x +9 x^{2}}\right )^{\frac {2}{3}}+4 c_{1}}{4 \left (-12 c_{2} -12 x +4 \sqrt {-4 c_{1}^{3}+9 c_{2}^{2}+18 c_{2} x +9 x^{2}}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 6831

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

Problem 6832

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

Problem 6833

ODE	\[ \boxed {y y^{\prime \prime }+{y^{\prime }}^{3}=0} \]
program solution	\[ y = {\mathrm e}^{\operatorname {LambertW}\left (\left (x +c_{2} \right ) {\mathrm e}^{-1-c_{1}}\right )+1+c_{1}} \] 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 6834

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

Problem 6835

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

Problem 6836

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

Problem 6837

ODE	\[ \boxed {y^{\prime \prime }+{\mathrm e}^{-2 y}=0} \] With initial conditions \begin {align*} [y \left (3\right ) = 0, y^{\prime }\left (3\right ) = 1] \end {align*}
program solution	\[ y = \ln \left (x -2\right ) \] Verified OK. \[ y = \ln \left (-x +4\right ) \] Warning, solution could not be verified
Maple solution	\[ y \left (x \right ) = \frac {\ln \left (\left (-2+x \right )^{2}\right )}{2} \]

Problem 6838

ODE	\[ \boxed {y^{\prime \prime }+{\mathrm e}^{-2 y}=0} \] With initial conditions \begin {align*} [y \left (3\right ) = 0, y^{\prime }\left (3\right ) = -1] \end {align*}
program solution	\[ y = \ln \left (x -2\right ) \] Warning, solution could not be verified \[ y = \ln \left (-x +4\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\ln \left (\left (x -4\right )^{2}\right )}{2} \]

Problem 6839

ODE	\[ \boxed {2 y^{\prime \prime }-\sin \left (2 y\right )=0} \] With initial conditions \begin {align*} \left [y \left (0\right ) = \frac {\pi }{2}, y^{\prime }\left (0\right ) = 1\right ] \end {align*}
program solution	\[ -\operatorname {arctanh}\left (\cos \left (y\right )\right ) = x \] Verified OK. \[ \operatorname {arctanh}\left (\cos \left (y\right )\right ) = x \] Warning, solution could not be verified
Maple solution	\[ \text {Expression too large to display} \]

Problem 6840

ODE	\[ \boxed {2 y^{\prime \prime }-\sin \left (2 y\right )=0} \] With initial conditions \begin {align*} \left [y \left (0\right ) = -\frac {\pi }{2}, y^{\prime }\left (0\right ) = 1\right ] \end {align*}
program solution	\[ -\operatorname {arctanh}\left (\cos \left (y\right )\right ) = x \] Verified OK. \[ \operatorname {arctanh}\left (\cos \left (y\right )\right ) = x \] Verified OK.
Maple solution	\[ \text {Expression too large to display} \]

Problem 6841

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

Problem 6842

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

Problem 6843

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

Problem 6844

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

Problem 6845

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

Problem 6846

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

Problem 6847

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

Problem 6848

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

Problem 6849

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

Problem 6850

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

Problem 6851

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

Problem 6852

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

Problem 6853

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

Problem 6854

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

Problem 6855

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

Problem 6856

ODE	\[ \boxed {{y^{\prime \prime }}^{2}-2 y^{\prime \prime }+{y^{\prime }}^{2}-2 y^{\prime } x=-x^{2}} \] With initial conditions \begin {align*} \left [y \left (0\right ) = {\frac {1}{2}}, y^{\prime }\left (0\right ) = 1\right ] \end {align*}
program solution	\[ y = \frac {1}{2} x^{2}+x +\frac {1}{2} \] Verified OK. \[ y = \frac {7}{6}+\frac {x^{2}}{2}+\frac {\left (i \cos \left (x \right ) \sqrt {5}+3 \sin \left (x \right )-2\right )^{2} \left (i \cos \left (x \right ) \sqrt {5}+3 \sin \left (x \right )+2\right )}{\sqrt {2-3 i \sqrt {5}\, \sin \left (2 x \right )+7 \cos \left (2 x \right )}\, \left (3 i \sin \left (x \right ) \sqrt {5}-9 \cos \left (x \right )\right )}-\frac {i \sqrt {5}}{3} \] Warning, solution could not be verified \[ y = \frac {1}{2} x^{2}-x +\frac {1}{2} \] Warning, solution could not be verified \[ y = -\frac {2 \sin \left (x \right )}{3}+\frac {x^{2}}{2}+\frac {1}{2} \] Warning, solution could not be verified
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (x +1\right )^{2}}{2} \\ y \left (x \right ) &= \frac {x^{2}}{2}+\sin \left (x \right )+\frac {1}{2} \\ \end{align*}

Problem 6857

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

Problem 6858

ODE	\[ \boxed {{y^{\prime \prime }}^{3}-12 y^{\prime } \left (x y^{\prime \prime }-2 y^{\prime }\right )=0} \]
program solution	\[ y = \int \operatorname {RootOf}\left (-i \ln \left (x \right ) \sqrt {3}\, 2^{\frac {1}{3}}+i c_{1} \sqrt {3}\, 2^{\frac {1}{3}}+\ln \left (x \right ) 2^{\frac {1}{3}}-c_{1} 2^{\frac {1}{3}}+8 \left (\int _{}^{\frac {\textit {\_Z}}{x^{3}}}-\frac {i \left (-\sqrt {9 \textit {\_a} -4}+3 \sqrt {\textit {\_a}}\right )^{\frac {1}{3}}}{\left (3 \sqrt {\textit {\_a}}\, \sqrt {3}\, 2^{\frac {2}{3}} \left (-\sqrt {9 \textit {\_a} -4}+3 \sqrt {\textit {\_a}}\right )^{\frac {1}{3}}-2 \sqrt {3}\, 2^{\frac {1}{3}} \left (\sqrt {9 \textit {\_a} -4}-3 \sqrt {\textit {\_a}}\right )^{\frac {2}{3}}-3 i \sqrt {\textit {\_a}}\, 2^{\frac {2}{3}} \left (-\sqrt {9 \textit {\_a} -4}+3 \sqrt {\textit {\_a}}\right )^{\frac {1}{3}}-2 i 2^{\frac {1}{3}} \left (\sqrt {9 \textit {\_a} -4}-3 \sqrt {\textit {\_a}}\right )^{\frac {2}{3}}+8 i\right ) \sqrt {\textit {\_a}}}d \textit {\_a} \right )\right )d x +c_{4} \] Warning, solution could not be verified
Maple solution	\begin{align*} y \left (x \right ) &= \frac {x^{4}}{9}+c_{1} \\ y \left (x \right ) &= c_{1} \\ y \left (x \right ) &= \int \operatorname {RootOf}\left (-6 \ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {3 \textit {\_f} \sqrt {\frac {1}{\textit {\_f} \left (9 \textit {\_f} -4\right )}}\, 2^{\frac {1}{3}} \left (\left (3 \sqrt {\frac {1}{\textit {\_f} \left (9 \textit {\_f} -4\right )}}\, \textit {\_f} +1\right )^{2} \left (9 \textit {\_f} -4\right )^{4}\right )^{\frac {1}{3}}-2 \,2^{\frac {2}{3}} \left (\left (3 \sqrt {\frac {1}{\textit {\_f} \left (9 \textit {\_f} -4\right )}}\, \textit {\_f} +1\right ) \left (9 \textit {\_f} -4\right )^{2}\right )^{\frac {1}{3}}-2^{\frac {1}{3}} \left (\left (3 \sqrt {\frac {1}{\textit {\_f} \left (9 \textit {\_f} -4\right )}}\, \textit {\_f} +1\right )^{2} \left (9 \textit {\_f} -4\right )^{4}\right )^{\frac {1}{3}}+18 \textit {\_f} -8}{\textit {\_f} \left (9 \textit {\_f} -4\right )}d \textit {\_f} \right )+6 c_{1} \right ) x^{3}d x +c_{2} \\ \end{align*}

Problem 6859

ODE	\[ \boxed {3 y y^{\prime } y^{\prime \prime }-{y^{\prime }}^{3}=-1} \]
program solution	\[ y = -\frac {\left (-4 \left ({\mathrm e}^{3 c_{1}} c_{2}^{3} \left (x +c_{3} \right )\right )^{\frac {3}{2}}+3 \sqrt {6}\right ) \sqrt {6}\, {\mathrm e}^{-3 c_{1}}}{18 c_{2}^{3}} \] Verified OK. \[ y = \frac {\left (\left (3 i \sqrt {3}\, {\mathrm e}^{3 c_{1}} c_{2}^{3} c_{4} +3 i \sqrt {3}\, {\mathrm e}^{3 c_{1}} c_{2}^{3} x -3 c_{4} {\mathrm e}^{3 c_{1}} c_{2}^{3}-3 \,{\mathrm e}^{3 c_{1}} c_{2}^{3} x \right )^{\frac {3}{2}}-27\right ) {\mathrm e}^{-3 c_{1}}}{27 c_{2}^{3}} \] Verified OK. \[ y = \frac {\left (\left (-3 i \sqrt {3}\, {\mathrm e}^{3 c_{1}} c_{2}^{3} c_{5} -3 i \sqrt {3}\, {\mathrm e}^{3 c_{1}} c_{2}^{3} x -3 c_{5} {\mathrm e}^{3 c_{1}} c_{2}^{3}-3 \,{\mathrm e}^{3 c_{1}} c_{2}^{3} x \right )^{\frac {3}{2}}-27\right ) {\mathrm e}^{-3 c_{1}}}{27 c_{2}^{3}} \] Verified OK.
Maple solution	\begin{align*} \frac {3 \left (c_{1} y \left (x \right )+1\right )^{\frac {2}{3}}+\left (-2 x -2 c_{2} \right ) c_{1}}{2 c_{1}} &= 0 \\ \frac {-i \left (x +c_{2} \right ) c_{1} \sqrt {3}+\left (-x -c_{2} \right ) c_{1} -3 \left (c_{1} y \left (x \right )+1\right )^{\frac {2}{3}}}{c_{1} \left (1+i \sqrt {3}\right )} &= 0 \\ \frac {-3 i \left (c_{1} y \left (x \right )+1\right )^{\frac {2}{3}}+\left (-x -c_{2} \right ) c_{1} \sqrt {3}-i \left (x +c_{2} \right ) c_{1}}{c_{1} \left (\sqrt {3}+i\right )} &= 0 \\ \end{align*}

Problem 6860

ODE	\[ \boxed {4 y {y^{\prime }}^{2} y^{\prime \prime }-{y^{\prime }}^{4}=3} \]
program solution	\[ y = \frac {\left (3 \left ({\mathrm e}^{4 c_{1}} c_{2}^{4} \left (x +c_{3} \right )\right )^{\frac {4}{3}}+4 \,6^{\frac {2}{3}}\right ) 6^{\frac {1}{3}} {\mathrm e}^{-4 c_{1}}}{8 c_{2}^{4}} \] Verified OK. \[ y = \frac {\left (\frac {3 \,3^{\frac {1}{3}} 4^{\frac {2}{3}} \left (i {\mathrm e}^{4 c_{1}} c_{2}^{4} \left (x +c_{4} \right )\right )^{\frac {4}{3}}}{16}+3\right ) {\mathrm e}^{-4 c_{1}}}{c_{2}^{4}} \] Verified OK. \[ y = \frac {\left (\left (-\frac {3 c_{5} {\mathrm e}^{4 c_{1}} c_{2}^{4}}{4}-\frac {3 \,{\mathrm e}^{4 c_{1}} c_{2}^{4} x}{4}\right )^{\frac {4}{3}}+3\right ) {\mathrm e}^{-4 c_{1}}}{c_{2}^{4}} \] Verified OK. \[ y = \frac {\left (\left (-\frac {3 i {\mathrm e}^{4 c_{1}} c_{2}^{4} \left (x +c_{6} \right )}{4}\right )^{\frac {4}{3}}+3\right ) {\mathrm e}^{-4 c_{1}}}{c_{2}^{4}} \] Verified OK.
Maple solution	\begin{align*} \frac {-4 \left (c_{1} y \left (x \right )-3\right )^{\frac {3}{4}}+\left (-3 x -3 c_{2} \right ) c_{1}}{3 c_{1}} &= 0 \\ \frac {4 \left (c_{1} y \left (x \right )-3\right )^{\frac {3}{4}}+\left (-3 x -3 c_{2} \right ) c_{1}}{3 c_{1}} &= 0 \\ \frac {-4 i \left (c_{1} y \left (x \right )-3\right )^{\frac {3}{4}}+\left (-3 x -3 c_{2} \right ) c_{1}}{3 c_{1}} &= 0 \\ \frac {4 i \left (c_{1} y \left (x \right )-3\right )^{\frac {3}{4}}+\left (-3 x -3 c_{2} \right ) c_{1}}{3 c_{1}} &= 0 \\ \end{align*}

Problem 6861

ODE	\[ \boxed {y^{\prime \prime }+y=-\cos \left (x \right )} \]
program solution	\[ y = c_{1} \cos \left (x \right )+c_{2} \sin \left (x \right )-\frac {\sin \left (x \right ) x}{2} \] 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 6862

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

Problem 6863

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }+2 y=12 x^{2}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+{\mathrm e}^{-x} c_{2} +6 x^{2}-18 x +21 \] Verified OK.
Maple solution	\[ y \left (x \right ) = -{\mathrm e}^{-2 x} c_{1} +c_{2} {\mathrm e}^{-x}+6 x^{2}-18 x +21 \]

Problem 6864

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

Problem 6865

ODE	\[ \boxed {x^{3} {y^{\prime }}^{2}+y^{\prime } x^{2} y=-4} \]
program solution	\[ y = \frac {\left ({\mathrm e}^{2 c_{1}} x +16\right ) {\mathrm e}^{-c_{1}}}{2 x} \] Verified OK. \[ y = \frac {\left (16 \,{\mathrm e}^{2 c_{1}} x +1\right ) {\mathrm e}^{-c_{1}}}{2 x} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {4}{\sqrt {x}} \\ y \left (x \right ) &= \frac {4}{\sqrt {x}} \\ y \left (x \right ) &= \frac {c_{1}^{2} x +16}{2 x c_{1}} \\ y \left (x \right ) &= \frac {c_{1}^{2}+16 x}{2 x c_{1}} \\ \end{align*}

Problem 6866

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

Problem 6867

ODE	\[ \boxed {9 {y^{\prime }}^{2}+3 y^{4} y^{\prime } x +y^{5}=0} \]
program solution	\[ \ln \left (y\right )-\frac {\ln \left (-x y^{2}+\sqrt {y}\, \sqrt {y^{3} x^{2}-4}\right )}{3}+\frac {\ln \left (x y^{2}+\sqrt {y}\, \sqrt {y^{3} x^{2}-4}\right )}{3} = c_{1} \] Verified OK. \[ \ln \left (y\right )+\frac {\ln \left (-x y^{2}+\sqrt {y}\, \sqrt {y^{3} x^{2}-4}\right )}{3}-\frac {\ln \left (x y^{2}+\sqrt {y}\, \sqrt {y^{3} x^{2}-4}\right )}{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {2^{\frac {2}{3}}}{x^{\frac {2}{3}}} \\ y \left (x \right ) &= -\frac {2^{\frac {2}{3}} \left (1+i \sqrt {3}\right )}{2 x^{\frac {2}{3}}} \\ y \left (x \right ) &= \frac {2^{\frac {2}{3}} \left (i \sqrt {3}-1\right )}{2 x^{\frac {2}{3}}} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\operatorname {RootOf}\left (-2 \ln \left (x \right )+3 \left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{3}+\sqrt {\textit {\_a}^{3} \left (\textit {\_a}^{3}-4\right )}-4}{\textit {\_a} \left (\textit {\_a}^{3}-4\right )}d \textit {\_a} \right )+2 c_{1} \right )}{x^{\frac {2}{3}}} \\ \end{align*}

Problem 6868

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

Problem 6869

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

Problem 6870

ODE	\[ \boxed {5 {y^{\prime }}^{2}+6 y^{\prime } x -2 y=0} \]
program solution	\[ y = 0 \] Verified OK. \[ x = \frac {3 x}{5}-\frac {\sqrt {9 x^{2}+10 y}}{5}+\frac {125 c_{1}}{\left (-15 x +5 \sqrt {9 x^{2}+10 y}\right )^{\frac {3}{2}}} \] Verified OK. \[ x = \frac {3 x}{5}+\frac {\sqrt {9 x^{2}+10 y}}{5}+\frac {125 c_{1}}{\left (-15 x -5 \sqrt {9 x^{2}+10 y}\right )^{\frac {3}{2}}} \] Verified OK.
Maple solution	\begin{align*} \frac {c_{1}}{\left (-15 x -5 \sqrt {9 x^{2}+10 y \left (x \right )}\right )^{\frac {3}{2}}}+\frac {2 x}{5}-\frac {\sqrt {9 x^{2}+10 y \left (x \right )}}{5} &= 0 \\ \frac {c_{1}}{\left (-15 x +5 \sqrt {9 x^{2}+10 y \left (x \right )}\right )^{\frac {3}{2}}}+\frac {2 x}{5}+\frac {\sqrt {9 x^{2}+10 y \left (x \right )}}{5} &= 0 \\ \end{align*}

Problem 6871

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

Problem 6872

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

Problem 6873

ODE	\[ \boxed {4 y^{2} {y^{\prime }}^{3}-2 y^{\prime } x +y=0} \]
program solution	\[ 9 \left (\int _{}^{\frac {y}{x^{\frac {3}{4}}}}\frac {\textit {\_a} \left (3 \sqrt {3}\, \textit {\_a}^{2}-\sqrt {27 \textit {\_a}^{4}-8}\right )^{\frac {1}{3}}}{\left (2 \,3^{\frac {2}{3}}-6 i 3^{\frac {1}{6}}\right ) \left (3 \sqrt {3}\, \textit {\_a}^{2}-\sqrt {27 \textit {\_a}^{4}-8}\right )^{\frac {2}{3}}+9 \left (i 3^{\frac {2}{3}}+3^{\frac {1}{6}}\right ) \textit {\_a}^{2} \left (3 \sqrt {3}\, \textit {\_a}^{2}-\sqrt {27 \textit {\_a}^{4}-8}\right )^{\frac {1}{3}}-8 \,3^{\frac {2}{3}}}d \textit {\_a} \right ) \left (i 3^{\frac {2}{3}}+3^{\frac {1}{6}}\right )-c_{1} +\frac {3 \ln \left (x \right )}{4} = 0 \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {2^{\frac {3}{4}} 3^{\frac {1}{4}} x^{\frac {3}{4}}}{3} \\ y \left (x \right ) &= \frac {2^{\frac {3}{4}} 3^{\frac {1}{4}} x^{\frac {3}{4}}}{3} \\ y \left (x \right ) &= -\frac {i 2^{\frac {3}{4}} 3^{\frac {1}{4}} x^{\frac {3}{4}}}{3} \\ y \left (x \right ) &= \frac {i 2^{\frac {3}{4}} 3^{\frac {1}{4}} x^{\frac {3}{4}}}{3} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \sqrt {2}\, \sqrt {c_{1} \left (-2 c_{1}^{2}+x \right )} \\ y \left (x \right ) &= -\sqrt {2}\, \sqrt {c_{1} \left (-2 c_{1}^{2}+x \right )} \\ \end{align*}

Problem 6874

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

Problem 6875

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

Problem 6876

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

Problem 6877

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

Problem 6878

ODE	\[ \boxed {{y^{\prime }}^{3}-2 y^{\prime } x -y=0} \]
program solution	\[ y = 0 \] Verified OK. \[ x = \frac {{\left (\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 x \right )}^{2}}{96 \left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}+\frac {c_{1} 6^{\frac {2}{3}}}{{\left (\frac {\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 x}{\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {1}{3}}}\right )}^{\frac {2}{3}}} \] Verified OK. \[ x = \frac {{\left (-i \sqrt {3}\, \left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 i x \sqrt {3}+\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 x \right )}^{2}}{384 \left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}+\frac {c_{1} 12^{\frac {2}{3}}}{{\left (\frac {i \sqrt {3}\, \left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-24 i x \sqrt {3}-\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-24 x}{\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {1}{3}}}\right )}^{\frac {2}{3}}} \] Verified OK. \[ x = \frac {{\left (i \sqrt {3}\, \left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-24 i x \sqrt {3}+\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 x \right )}^{2}}{384 \left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}+\frac {2 c_{1} 18^{\frac {1}{3}}}{{\left (\frac {-i \sqrt {3}\, \left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 i x \sqrt {3}-\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-24 x}{\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {1}{3}}}\right )}^{\frac {2}{3}}} \] Warning, solution could not be verified
Maple solution	\begin{align*} -\frac {c_{1}}{{\left (\frac {\left (108 y \left (x \right )+12 \sqrt {-96 x^{3}+81 y \left (x \right )^{2}}\right )^{\frac {2}{3}}+24 x}{\left (108 y \left (x \right )+12 \sqrt {-96 x^{3}+81 y \left (x \right )^{2}}\right )^{\frac {1}{3}}}\right )}^{\frac {2}{3}}}+x -\frac {{\left (\left (108 y \left (x \right )+12 \sqrt {-96 x^{3}+81 y \left (x \right )^{2}}\right )^{\frac {2}{3}}+24 x \right )}^{2}}{96 \left (108 y \left (x \right )+12 \sqrt {-96 x^{3}+81 y \left (x \right )^{2}}\right )^{\frac {2}{3}}} &= 0 \\ -\frac {c_{1}}{{\left (\frac {i \sqrt {3}\, \left (108 y \left (x \right )+12 \sqrt {-96 x^{3}+81 y \left (x \right )^{2}}\right )^{\frac {2}{3}}-24 i \sqrt {3}\, x -\left (108 y \left (x \right )+12 \sqrt {-96 x^{3}+81 y \left (x \right )^{2}}\right )^{\frac {2}{3}}-24 x}{\left (108 y \left (x \right )+12 \sqrt {-96 x^{3}+81 y \left (x \right )^{2}}\right )^{\frac {1}{3}}}\right )}^{\frac {2}{3}}}+x +\frac {3 {\left (-\frac {\left (\sqrt {3}+i\right ) \left (108 y \left (x \right )+12 \sqrt {-96 x^{3}+81 y \left (x \right )^{2}}\right )^{\frac {2}{3}}}{24}+x \left (-i+\sqrt {3}\right )\right )}^{2}}{2 \left (108 y \left (x \right )+12 \sqrt {-96 x^{3}+81 y \left (x \right )^{2}}\right )^{\frac {2}{3}}} &= 0 \\ -\frac {12^{\frac {2}{3}} c_{1}}{{\left (\frac {-i \sqrt {3}\, \left (108 y \left (x \right )+12 \sqrt {-96 x^{3}+81 y \left (x \right )^{2}}\right )^{\frac {2}{3}}+24 i \sqrt {3}\, x -\left (108 y \left (x \right )+12 \sqrt {-96 x^{3}+81 y \left (x \right )^{2}}\right )^{\frac {2}{3}}-24 x}{\left (108 y \left (x \right )+12 \sqrt {-96 x^{3}+81 y \left (x \right )^{2}}\right )^{\frac {1}{3}}}\right )}^{\frac {2}{3}}}+x +\frac {3 {\left (\frac {\left (i-\sqrt {3}\right ) \left (108 y \left (x \right )+12 \sqrt {-96 x^{3}+81 y \left (x \right )^{2}}\right )^{\frac {2}{3}}}{24}+x \left (\sqrt {3}+i\right )\right )}^{2}}{2 \left (108 y \left (x \right )+12 \sqrt {-96 x^{3}+81 y \left (x \right )^{2}}\right )^{\frac {2}{3}}} &= 0 \\ \end{align*}

Problem 6879

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

Problem 6880

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

Problem 6881

ODE	\[ \boxed {x^{6} {y^{\prime }}^{2}-16 y-8 y^{\prime } x=0} \]
program solution	\[ -\frac {\ln \left (y\right )}{4}-\frac {\operatorname {arctanh}\left (\sqrt {x^{4} y+1}\right )}{2} = c_{1} \] Verified OK. \[ -\frac {\ln \left (y\right )}{4}+\frac {\operatorname {arctanh}\left (\sqrt {x^{4} y+1}\right )}{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {1}{x^{4}} \\ y \left (x \right ) &= \frac {2 c_{1} i-x^{2}}{x^{2} c_{1}^{2}} \\ y \left (x \right ) &= \frac {-2 c_{1} i-x^{2}}{x^{2} c_{1}^{2}} \\ y \left (x \right ) &= \frac {-2 c_{1} i-x^{2}}{x^{2} c_{1}^{2}} \\ y \left (x \right ) &= \frac {2 c_{1} i-x^{2}}{x^{2} c_{1}^{2}} \\ \end{align*}

Problem 6882

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

Problem 6883

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

Problem 6884

ODE	\[ \boxed {{y^{\prime }}^{3}-{y^{\prime }}^{2}+y^{\prime } x -y=0} \]
program solution	\[ y = c_{1}^{3}-c_{1}^{2}+c_{1} x \] Verified OK. \[ y = \frac {\left (6 x -2\right ) \sqrt {-3 x +1}}{27}+\frac {x}{3}-\frac {2}{27} \] Verified OK. \[ y = \frac {\left (-6 x +2\right ) \sqrt {-3 x +1}}{27}+\frac {x}{3}-\frac {2}{27} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {x}{3}-\frac {2}{27}-\frac {2 \sqrt {-\left (3 x -1\right )^{3}}}{27} \\ y \left (x \right ) &= \frac {x}{3}-\frac {2}{27}+\frac {2 \sqrt {-\left (3 x -1\right )^{3}}}{27} \\ y \left (x \right ) &= c_{1} \left (c_{1}^{2}-c_{1} +x \right ) \\ \end{align*}

Problem 6885

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

Problem 6886

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

Problem 6887

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

Problem 6888

ODE	\[ \boxed {x {y^{\prime }}^{3}-2 y {y^{\prime }}^{2}=-4 x^{2}} \]
program solution	\[ \frac {4 \ln \left (x \right )}{3}-4 \left (\int _{}^{\frac {y}{x^{\frac {4}{3}}}}\frac {\left (-54+8 \textit {\_a}^{3}+6 i \sqrt {24 \textit {\_a}^{3}-81}\right )^{\frac {1}{3}}}{\left (-54+8 \textit {\_a}^{3}+6 i \sqrt {24 \textit {\_a}^{3}-81}\right )^{\frac {2}{3}}-2 \textit {\_a} \left (-54+8 \textit {\_a}^{3}+6 i \sqrt {24 \textit {\_a}^{3}-81}\right )^{\frac {1}{3}}+4 \textit {\_a}^{2}}d \textit {\_a} \right )-c_{1} = 0 \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {3 x^{\frac {4}{3}}}{2} \\ y \left (x \right ) &= -\frac {3 x^{\frac {4}{3}} \left (1+i \sqrt {3}\right )}{4} \\ y \left (x \right ) &= \frac {3 x^{\frac {4}{3}} \left (i \sqrt {3}-1\right )}{4} \\ y \left (x \right ) &= \frac {c_{1}^{3}-128 x^{2}}{32 c_{1}} \\ y \left (x \right ) &= \frac {c_{1}^{3}+128 x^{2}}{32 c_{1}} \\ y \left (x \right ) &= \frac {c_{1} \left (c_{1}^{3}-1728 x^{2}+24 \sqrt {6}\, \sqrt {-x^{2} \left (c_{1}^{3}-864 x^{2}\right )}\right )^{\frac {1}{3}}}{96}+\frac {c_{1}^{3}}{96 \left (c_{1}^{3}-1728 x^{2}+24 \sqrt {6}\, \sqrt {-x^{2} \left (c_{1}^{3}-864 x^{2}\right )}\right )^{\frac {1}{3}}}+\frac {c_{1}^{2}}{96} \\ y \left (x \right ) &= \frac {c_{1} \left (c_{1}^{3}+24 \sqrt {6}\, \sqrt {x^{2} \left (c_{1}^{3}+864 x^{2}\right )}+1728 x^{2}\right )^{\frac {1}{3}}}{96}+\frac {c_{1}^{3}}{96 \left (c_{1}^{3}+24 \sqrt {6}\, \sqrt {x^{2} \left (c_{1}^{3}+864 x^{2}\right )}+1728 x^{2}\right )^{\frac {1}{3}}}+\frac {c_{1}^{2}}{96} \\ y \left (x \right ) &= \frac {\left (c_{1} -\left (c_{1}^{3}-1728 x^{2}+24 \sqrt {6}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{\frac {1}{3}}\right ) c_{1} \left (i \left (\left (c_{1}^{3}-1728 x^{2}+24 \sqrt {6}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{\frac {1}{3}}+c_{1} \right ) \sqrt {3}-c_{1} +\left (c_{1}^{3}-1728 x^{2}+24 \sqrt {6}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{\frac {1}{3}}\right )}{192 \left (c_{1}^{3}-1728 x^{2}+24 \sqrt {6}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (c_{1}^{3}-1728 x^{2}+24 \sqrt {2}\, \sqrt {3}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right ) c_{1}}{192}-\frac {\left (i \sqrt {3}\, c_{1} +c_{1} -2 \left (c_{1}^{3}-1728 x^{2}+24 \sqrt {2}\, \sqrt {3}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{\frac {1}{3}}\right ) c_{1}^{2}}{192 \left (c_{1}^{3}-1728 x^{2}+24 \sqrt {2}\, \sqrt {3}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (c_{1} -\left (c_{1}^{3}+24 \sqrt {6}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}+1728 x^{2}\right )^{\frac {1}{3}}\right ) c_{1} \left (i \left (\left (c_{1}^{3}+24 \sqrt {6}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}+1728 x^{2}\right )^{\frac {1}{3}}+c_{1} \right ) \sqrt {3}-c_{1} +\left (c_{1}^{3}+24 \sqrt {6}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}+1728 x^{2}\right )^{\frac {1}{3}}\right )}{192 \left (c_{1}^{3}+24 \sqrt {6}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}+1728 x^{2}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (c_{1}^{3}+24 \sqrt {2}\, \sqrt {3}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}+1728 x^{2}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right ) c_{1}}{192}-\frac {\left (i \sqrt {3}\, c_{1} +c_{1} -2 \left (c_{1}^{3}+24 \sqrt {2}\, \sqrt {3}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}+1728 x^{2}\right )^{\frac {1}{3}}\right ) c_{1}^{2}}{192 \left (c_{1}^{3}+24 \sqrt {2}\, \sqrt {3}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}+1728 x^{2}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 6889

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

Problem 6890

ODE	\[ \boxed {y^{\prime \prime }-9 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {9}{2} x^{2}+\frac {27}{8} x^{4}+\frac {81}{80} x^{6}+\frac {729}{4480} x^{8}\right ) y \left (0\right )+\left (x +\frac {3}{2} x^{3}+\frac {27}{40} x^{5}+\frac {81}{560} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1+\frac {9}{2} x^{2}+\frac {27}{8} x^{4}+\frac {81}{80} x^{6}\right ) c_{1} +\left (x +\frac {3}{2} x^{3}+\frac {27}{40} x^{5}+\frac {81}{560} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {9}{2} x^{2}+\frac {27}{8} x^{4}+\frac {81}{80} x^{6}\right ) y \left (0\right )+\left (x +\frac {3}{2} x^{3}+\frac {27}{40} x^{5}+\frac {81}{560} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6891

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime } x +3 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {3}{2} x^{2}+\frac {9}{8} x^{4}-\frac {9}{16} x^{6}+\frac {27}{128} x^{8}\right ) y \left (0\right )+\left (x -x^{3}+\frac {3}{5} x^{5}-\frac {9}{35} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {3}{2} x^{2}+\frac {9}{8} x^{4}-\frac {9}{16} x^{6}\right ) c_{1} +\left (x -x^{3}+\frac {3}{5} x^{5}-\frac {9}{35} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {3}{2} x^{2}+\frac {9}{8} x^{4}-\frac {9}{16} x^{6}\right ) y \left (0\right )+\left (x -x^{3}+\frac {3}{5} x^{5}-\frac {9}{35} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6892

ODE	\[ \boxed {\left (4 x^{2}+1\right ) y^{\prime \prime }-8 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (4 x^{2}+1\right ) y \left (0\right )+\left (x +\frac {4}{3} x^{3}-\frac {16}{15} x^{5}+\frac {64}{35} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (4 x^{2}+1\right ) c_{1} +\left (x +\frac {4}{3} x^{3}-\frac {16}{15} x^{5}+\frac {64}{35} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (4 x^{2}+1\right ) y \left (0\right )+\left (x +\frac {4}{3} x^{3}-\frac {16}{15} x^{5}+\frac {64}{35} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6893

ODE	\[ \boxed {\left (-4 x^{2}+1\right ) y^{\prime \prime }+8 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (-4 x^{2}+1\right ) y \left (0\right )+\left (x -\frac {4}{3} x^{3}-\frac {16}{15} x^{5}-\frac {64}{35} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (-4 x^{2}+1\right ) c_{1} +\left (x -\frac {4}{3} x^{3}-\frac {16}{15} x^{5}-\frac {64}{35} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (-4 x^{2}+1\right ) y \left (0\right )+\left (x -\frac {4}{3} x^{3}-\frac {16}{15} x^{5}-\frac {64}{35} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6894

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

Problem 6895

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+10 y^{\prime } x +20 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (165 x^{8}-84 x^{6}+35 x^{4}-10 x^{2}+1\right ) y \left (0\right )+\left (-30 x^{7}+14 x^{5}-5 x^{3}+x \right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (-84 x^{6}+35 x^{4}-10 x^{2}+1\right ) c_{1} +\left (-30 x^{7}+14 x^{5}-5 x^{3}+x \right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (-84 x^{6}+35 x^{4}-10 x^{2}+1\right ) y \left (0\right )+\left (-30 x^{7}+14 x^{5}-5 x^{3}+x \right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6896

ODE	\[ \boxed {\left (x^{2}+4\right ) y^{\prime \prime }+2 y^{\prime } x -12 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {3}{2} x^{2}+\frac {3}{16} x^{4}-\frac {1}{80} x^{6}+\frac {3}{1792} x^{8}\right ) y \left (0\right )+\left (x +\frac {5}{12} x^{3}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1+\frac {3}{2} x^{2}+\frac {3}{16} x^{4}-\frac {1}{80} x^{6}\right ) c_{1} +\left (x +\frac {5}{12} x^{3}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {3}{2} x^{2}+\frac {3}{16} x^{4}-\frac {1}{80} x^{6}\right ) y \left (0\right )+\left (x +\frac {5}{12} x^{3}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6897

ODE	\[ \boxed {\left (x^{2}-9\right ) y^{\prime \prime }+3 y^{\prime } x -3 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{6} x^{2}-\frac {5}{648} x^{4}-\frac {7}{11664} x^{6}-\frac {5}{93312} x^{8}\right ) y \left (0\right )+x y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{6} x^{2}-\frac {5}{648} x^{4}-\frac {7}{11664} x^{6}\right ) c_{1} +c_{2} x +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{6} x^{2}-\frac {5}{648} x^{4}-\frac {7}{11664} x^{6}\right ) y \left (0\right )+D\left (y \right )\left (0\right ) x +O\left (x^{8}\right ) \]

Problem 6898

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime } x +5 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {5}{2} x^{2}+\frac {15}{8} x^{4}-\frac {13}{16} x^{6}+\frac {221}{896} x^{8}\right ) y \left (0\right )+\left (x -\frac {7}{6} x^{3}+\frac {77}{120} x^{5}-\frac {11}{48} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {5}{2} x^{2}+\frac {15}{8} x^{4}-\frac {13}{16} x^{6}\right ) c_{1} +\left (x -\frac {7}{6} x^{3}+\frac {77}{120} x^{5}-\frac {11}{48} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {5}{2} x^{2}+\frac {15}{8} x^{4}-\frac {13}{16} x^{6}\right ) y \left (0\right )+\left (x -\frac {7}{6} x^{3}+\frac {77}{120} x^{5}-\frac {11}{48} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6899

ODE	\[ \boxed {\left (x^{2}+4\right ) y^{\prime \prime }+6 y^{\prime } x +4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2}+\frac {3}{16} x^{4}-\frac {1}{16} x^{6}+\frac {5}{256} x^{8}\right ) y \left (0\right )+\left (x -\frac {5}{12} x^{3}+\frac {7}{48} x^{5}-\frac {3}{64} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}+\frac {3}{16} x^{4}-\frac {1}{16} x^{6}\right ) c_{1} +\left (x -\frac {5}{12} x^{3}+\frac {7}{48} x^{5}-\frac {3}{64} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}+\frac {3}{16} x^{4}-\frac {1}{16} x^{6}\right ) y \left (0\right )+\left (x -\frac {5}{12} x^{3}+\frac {7}{48} x^{5}-\frac {3}{64} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6900

ODE	\[ \boxed {\left (2 x^{2}+1\right ) y^{\prime \prime }-5 y^{\prime } x +3 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {3}{2} x^{2}-\frac {3}{8} x^{4}+\frac {7}{80} x^{6}-\frac {33}{640} x^{8}\right ) y \left (0\right )+\left (x +\frac {1}{3} x^{3}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {3}{2} x^{2}-\frac {3}{8} x^{4}+\frac {7}{80} x^{6}\right ) c_{1} +\left (x +\frac {1}{3} x^{3}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {3}{2} x^{2}-\frac {3}{8} x^{4}+\frac {7}{80} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{3} x^{3}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]