Problems 8801 to 8900

Problem 8801


ODE	\[ \boxed {y {y^{\prime }}^{2}-4 y^{\prime } x +y=0} \]

program solution	\[ y = 0 \] Veriﬁed OK. \[ y = \sqrt {3}\, x \] Veriﬁed OK. \[ y = -\sqrt {3}\, x \] Veriﬁed OK. \[ y = \frac {4 {\operatorname {RootOf}\left (\left (c_{2}^{12}-x^{3}\right ) \textit {\_Z}^{18}+\left (3 c_{2}^{12}+3 x^{3}\right ) \textit {\_Z}^{12}+3 c_{2}^{12} \textit {\_Z}^{6}+c_{2}^{12}\right )}^{3} x}{{\operatorname {RootOf}\left (\left (c_{2}^{12}-x^{3}\right ) \textit {\_Z}^{18}+\left (3 c_{2}^{12}+3 x^{3}\right ) \textit {\_Z}^{12}+3 c_{2}^{12} \textit {\_Z}^{6}+c_{2}^{12}\right )}^{6}+1} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{2}+\sqrt {-\textit {\_a}^{2}+4}-2}{\textit {\_a} \left (\textit {\_a}^{2}-3\right )}d \textit {\_a} \right )+c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {\textit {\_a}^{2}-\sqrt {-\textit {\_a}^{2}+4}-2}{\textit {\_a} \left (\textit {\_a}^{2}-3\right )}d \textit {\_a} +c_{1} \right ) x \\ \end{align}

Problem 8802


ODE	\[ \boxed {y {y^{\prime }}^{2}-4 a^{2} x y^{\prime }+a^{2} y=0} \]

program solution	\[ y = 0 \] Veriﬁed OK. \[ y = \sqrt {3}\, a x \] Veriﬁed OK. \[ y = -\sqrt {3}\, a x \] Veriﬁed OK. \[ x = \frac {2 c_{3} a^{2} x 2^{\frac {1}{3}}}{y {\left (\frac {\left (2 a x +\sqrt {4 a^{2} x^{2}-y^{2}}\right ) a}{y}\right )}^{\frac {1}{3}} \left (\frac {2 a^{4} x^{2}+a^{3} x \sqrt {4 a^{2} x^{2}-y^{2}}-y^{2} a^{2}}{y^{2}}\right )^{\frac {1}{3}}} \] Veriﬁed OK. \[ x = \frac {2 c_{3} a^{2} x 2^{\frac {1}{3}}}{y {\left (\frac {\left (2 a x -\sqrt {4 a^{2} x^{2}-y^{2}}\right ) a}{y}\right )}^{\frac {1}{3}} \left (\frac {2 a^{4} x^{2}-a^{3} x \sqrt {4 a^{2} x^{2}-y^{2}}-y^{2} a^{2}}{y^{2}}\right )^{\frac {1}{3}}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{2}-2 a^{2}+\sqrt {-\textit {\_a}^{2} a^{2}+4 a^{4}}}{\textit {\_a} \left (\textit {\_a}^{2}-3 a^{2}\right )}d \textit {\_a} \right )+c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {\textit {\_a}^{2}-2 a^{2}-\sqrt {-\textit {\_a}^{2} a^{2}+4 a^{4}}}{\textit {\_a} \left (\textit {\_a}^{2}-3 a^{2}\right )}d \textit {\_a} +c_{1} \right ) x \\ \end{align}

Problem 8803


ODE	\[ \boxed {y {y^{\prime }}^{2}+a x y^{\prime }+b y=0} \]

program solution	\[ y = 0 \] Veriﬁed OK. \[ y = \sqrt {-a -b}\, x \] Veriﬁed OK. \[ y = -\sqrt {-a -b}\, x \] Veriﬁed OK. \[ y = -\frac {a x \operatorname {RootOf}\left (c_{2} x^{\frac {1}{a}} \textit {\_Z}^{\frac {1}{a +b}}-\left (\textit {\_Z}^{2}+a +b \right )^{-\frac {a +2 b}{2 a \left (a +b \right )}} \left (\textit {\_Z}^{2}+b \right )^{\frac {1}{a}}\right )}{\operatorname {RootOf}\left (c_{2} x^{\frac {1}{a}} \textit {\_Z}^{\frac {1}{a +b}}-\left (\textit {\_Z}^{2}+a +b \right )^{-\frac {a +2 b}{2 a \left (a +b \right )}} \left (\textit {\_Z}^{2}+b \right )^{\frac {1}{a}}\right )^{2}+b} \] Warning, solution could not be veriﬁed

Maple solution	\begin{align} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {2 \textit {\_a}^{2}+\sqrt {-4 b \,\textit {\_a}^{2}+a^{2}}+a}{\textit {\_a} \left (\textit {\_a}^{2}+a +b \right )}d \textit {\_a} \right )+2 c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {2 \textit {\_a}^{2}+a -\sqrt {-4 b \,\textit {\_a}^{2}+a^{2}}}{\textit {\_a} \left (\textit {\_a}^{2}+a +b \right )}d \textit {\_a} +2 c_{1} \right ) x \\ \end{align}

Problem 8804


ODE	\[ \boxed {y {y^{\prime }}^{2}+x^{3} y^{\prime }-x^{2} y=0} \]

program solution	\[ y = {\mathrm e}^{\frac {\ln \left (2\right )}{2}+\frac {\ln \left (2 \,{\mathrm e}^{-2 c_{1}} {\mathrm e}^{2 c_{1}} x^{2}+8 \,{\mathrm e}^{-2 c_{1}} {\mathrm e}^{4 c_{1}}\right )}{2}+c_{1}} \] Veriﬁed OK. \[ \frac {\ln \left (2\right )}{2}+\ln \left (x \right )+\frac {\ln \left (x^{2}+\sqrt {x^{4}+4 y^{2}}\right )}{2} = \ln \left (x \right )+c_{1} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -\frac {i x^{2}}{2} \\ y \left (x \right ) &= \frac {i x^{2}}{2} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= -\frac {\sqrt {c_{1} \left (-4 x^{2}+c_{1} \right )}}{4} \\ y \left (x \right ) &= \frac {\sqrt {c_{1} \left (-4 x^{2}+c_{1} \right )}}{4} \\ y \left (x \right ) &= -\frac {2 \sqrt {c_{1} x^{2}+4}}{c_{1}} \\ y \left (x \right ) &= \frac {2 \sqrt {c_{1} x^{2}+4}}{c_{1}} \\ \end{align}

Problem 8805


ODE	\[ \boxed {y {y^{\prime }}^{2}-\left (y-x \right ) y^{\prime }=x} \]

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 {-x^{2}+c_{1}} \\ y \left (x \right ) &= -\sqrt {-x^{2}+c_{1}} \\ y \left (x \right ) &= x +c_{1} \\ \end{align}

Problem 8806


ODE	\[ \boxed {\left (y+x \right ) {y^{\prime }}^{2}+2 y^{\prime } x -y=0} \]

program solution	\[ y = 0 \] Veriﬁed OK. \[ y = -\frac {i \sqrt {3}\, x}{2}-\frac {x}{2} \] Veriﬁed OK. \[ y = \frac {i \sqrt {3}\, x}{2}-\frac {x}{2} \] Veriﬁed OK. \[ x = \frac {x \left (-2 \sqrt {y^{2}+x y+x^{2}}+x -y\right ) c_{3}}{\left (x -\sqrt {y^{2}+x y+x^{2}}\right )^{2}} \] Veriﬁed OK. \[ x = \frac {x \left (2 \sqrt {y^{2}+x y+x^{2}}+x -y\right ) c_{3}}{\left (x +\sqrt {y^{2}+x y+x^{2}}\right )^{2}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -\frac {x \left (1+i \sqrt {3}\right )}{2} \\ y \left (x \right ) &= \frac {x \left (i \sqrt {3}-1\right )}{2} \\ \ln \left (x \right )-\operatorname {arctanh}\left (\frac {y \left (x \right )+2 x}{2 x \sqrt {\frac {y \left (x \right )^{2}+x y \left (x \right )+x^{2}}{x^{2}}}}\right )+\ln \left (\frac {y \left (x \right )}{x}\right )-c_{1} &= 0 \\ \ln \left (x \right )+\operatorname {arctanh}\left (\frac {y \left (x \right )+2 x}{2 x \sqrt {\frac {y \left (x \right )^{2}+x y \left (x \right )+x^{2}}{x^{2}}}}\right )+\ln \left (\frac {y \left (x \right )}{x}\right )-c_{1} &= 0 \\ \end{align}

Problem 8807


ODE	\[ \boxed {\left (y-2 x \right ) {y^{\prime }}^{2}-2 \left (x -1\right ) y^{\prime }+y=2} \]

program solution	\[ y = 2 \] Veriﬁed OK. \[ y = \frac {3 \sqrt {2}\, x -\sqrt {2}-4 x}{\sqrt {2}-2} \] Veriﬁed OK. \[ y = \frac {4 x +3 \sqrt {2}\, x -\sqrt {2}}{2+\sqrt {2}} \] Veriﬁed OK. \[ x = \frac {-2 c_{3} \left (x -1\right ) \sqrt {x^{2}+\left (2 y-6\right ) x -y^{2}+2 y+1}+6 \left (x -\frac {y}{3}-\frac {1}{3}\right ) \left (x -1\right ) c_{3} -4 \left (-\frac {y}{2}+x \right )^{2}}{\left (-1+x -\sqrt {x^{2}+\left (2 y-6\right ) x -y^{2}+2 y+1}\right )^{2}} \] Veriﬁed OK. \[ x = \frac {2 c_{3} \left (x -1\right ) \sqrt {x^{2}+\left (2 y-6\right ) x -y^{2}+2 y+1}+6 \left (x -\frac {y}{3}-\frac {1}{3}\right ) \left (x -1\right ) c_{3} -4 \left (-\frac {y}{2}+x \right )^{2}}{\left (-1+x +\sqrt {x^{2}+\left (2 y-6\right ) x -y^{2}+2 y+1}\right )^{2}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -\sqrt {2}\, x +\sqrt {2}+x +1 \\ y \left (x \right ) &= \left (x -1\right ) \sqrt {2}+x +1 \\ y \left (x \right ) &= 2+\frac {c_{1}}{2}-\frac {\sqrt {c_{1} \left (-c_{1} +4 x -4\right )}}{2} \\ y \left (x \right ) &= 2+c_{1} -\sqrt {c_{1} \left (-c_{1} +2 x -2\right )} \\ \end{align}

Problem 8808


ODE	\[ \boxed {2 y {y^{\prime }}^{2}-\left (4 x -5\right ) y^{\prime }+2 y=0} \]

program solution	\[ y = -x +\frac {5}{4} \] Veriﬁed OK. \[ y = 0 \] Veriﬁed OK. \[ y = x -\frac {5}{4} \] Veriﬁed OK. \[ x = \frac {\left (8 x -10\right ) c_{3} \sqrt {-16 y^{2}+16 x^{2}-40 x +25}+32 \left (x -\frac {5}{4}\right )^{2} c_{3} -20 y^{2}}{\left (4 x -5+\sqrt {-16 y^{2}+16 x^{2}-40 x +25}\right )^{2}} \] Veriﬁed OK. \[ x = \frac {\left (-8 x +10\right ) c_{3} \sqrt {-16 y^{2}+16 x^{2}-40 x +25}+32 \left (x -\frac {5}{4}\right )^{2} c_{3} -20 y^{2}}{\left (4 x -5-\sqrt {-16 y^{2}+16 x^{2}-40 x +25}\right )^{2}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= x -\frac {5}{4} \\ y \left (x \right ) &= -x +\frac {5}{4} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\sqrt {4 c_{1} +2 \sqrt {-c_{1} \left (4 x -5\right )^{2}}}}{2} \\ y \left (x \right ) &= -\frac {\sqrt {4 c_{1} +2 \sqrt {-c_{1} \left (4 x -5\right )^{2}}}}{2} \\ y \left (x \right ) &= \frac {\sqrt {4 c_{1} -2 \sqrt {-c_{1} \left (4 x -5\right )^{2}}}}{2} \\ y \left (x \right ) &= -\frac {\sqrt {4 c_{1} -2 \sqrt {-c_{1} \left (4 x -5\right )^{2}}}}{2} \\ \end{align}

Problem 8809


ODE	\[ \boxed {4 y {y^{\prime }}^{2}+2 y^{\prime } x -y=0} \]

program solution	\[ y = 0 \] Veriﬁed OK. \[ y = -\frac {i x}{2} \] Veriﬁed OK. \[ y = \frac {i x}{2} \] Veriﬁed OK. \[ x = -\frac {8 c_{3} x}{-x +\sqrt {x^{2}+4 y^{2}}} \] Veriﬁed OK. \[ x = \frac {8 c_{3} x}{x +\sqrt {x^{2}+4 y^{2}}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -\frac {i x}{2} \\ y \left (x \right ) &= \frac {i x}{2} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \sqrt {c_{1} \left (-x +c_{1} \right )} \\ y \left (x \right ) &= \sqrt {c_{1} \left (x +c_{1} \right )} \\ y \left (x \right ) &= -\sqrt {c_{1} \left (-x +c_{1} \right )} \\ y \left (x \right ) &= -\sqrt {c_{1} \left (x +c_{1} \right )} \\ \end{align}

Problem 8810


ODE	\[ \boxed {9 y {y^{\prime }}^{2}+4 x^{3} y^{\prime }-4 x^{2} y=0} \]

program solution	\[ y = {\mathrm e}^{\frac {\ln \left (2\right )}{2}+\frac {\ln \left (2 \,{\mathrm e}^{-2 c_{1}} {\mathrm e}^{2 c_{1}} x^{2}+18 \,{\mathrm e}^{-2 c_{1}} {\mathrm e}^{4 c_{1}}\right )}{2}+c_{1}} \] Veriﬁed OK. \[ \frac {\ln \left (2\right )}{2}+\ln \left (x \right )+\frac {\ln \left (x^{2}+\sqrt {x^{4}+9 y^{2}}\right )}{2} = \ln \left (x \right )+c_{1} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -\frac {i x^{2}}{3} \\ y \left (x \right ) &= \frac {i x^{2}}{3} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= -\frac {2 \sqrt {c_{1} x^{2}+9}}{c_{1}} \\ y \left (x \right ) &= \frac {2 \sqrt {c_{1} x^{2}+9}}{c_{1}} \\ y \left (x \right ) &= -\frac {\sqrt {c_{1} \left (-4 x^{2}+c_{1} \right )}}{6} \\ y \left (x \right ) &= \frac {\sqrt {c_{1} \left (-4 x^{2}+c_{1} \right )}}{6} \\ \end{align}

Problem 8811


ODE	\[ \boxed {a y {y^{\prime }}^{2}+\left (2 x -b \right ) y^{\prime }-y=0} \]

program solution	\[ y = 0 \] Veriﬁed OK. \[ y = \frac {2 x -b}{2 \sqrt {-a}} \] Veriﬁed OK. \[ y = \frac {-2 x +b}{2 \sqrt {-a}} \] Veriﬁed OK. \[ x = \frac {2 \left (c_{3} \left (-2 x +b \right ) \sqrt {4 y^{2} a +b^{2}-4 x b +4 x^{2}}+\left (-2 x +b \right )^{2} c_{3} +y^{2} b \right ) a}{\left (-2 x +b +\sqrt {4 y^{2} a +b^{2}-4 x b +4 x^{2}}\right )^{2}} \] Veriﬁed OK. \[ x = \frac {2 a \left (-c_{3} \left (-2 x +b \right ) \sqrt {4 y^{2} a +b^{2}-4 x b +4 x^{2}}+\left (-2 x +b \right )^{2} c_{3} +y^{2} b \right )}{\left (-2 x +b -\sqrt {4 y^{2} a +b^{2}-4 x b +4 x^{2}}\right )^{2}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -\frac {-2 x +b}{2 \sqrt {-a}} \\ y \left (x \right ) &= \frac {-2 x +b}{2 \sqrt {-a}} \\ y \left (x \right ) &= 0 \\ -4 a \left (\int _{}^{y \left (x \right )}\frac {16 \textit {\_f} \left (\frac {1}{16}+\left (\left (\frac {b}{4}-\frac {x}{2}\right ) \sqrt {4 a \,\textit {\_f}^{2}+b^{2}-4 b x +4 x^{2}}+a \,\textit {\_f}^{2}+\frac {b^{2}}{4}-b x +x^{2}\right ) \left (\int _{\textit {\_b}}^{x}\frac {-2 a \,\textit {\_f}^{2}+2 \sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 \textit {\_a} b +b^{2}}\, \textit {\_a} -\sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 \textit {\_a} b +b^{2}}\, b -4 \textit {\_a}^{2}+4 \textit {\_a} b -b^{2}}{\left (-4 a \,\textit {\_f}^{2}+2 \sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 \textit {\_a} b +b^{2}}\, \textit {\_a} -\sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 \textit {\_a} b +b^{2}}\, b -4 \textit {\_a}^{2}+4 \textit {\_a} b -b^{2}\right )^{2} \sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 \textit {\_a} b +b^{2}}}d \textit {\_a} \right )\right )}{\left (-2 x +b \right ) \sqrt {4 a \,\textit {\_f}^{2}+b^{2}-4 b x +4 x^{2}}+4 a \,\textit {\_f}^{2}+b^{2}-4 b x +4 x^{2}}d \textit {\_f} \right )+2 \left (\int _{\textit {\_b}}^{x}\frac {-2 \textit {\_a} +b +\sqrt {4 a y \left (x \right )^{2}+\left (-2 \textit {\_a} +b \right )^{2}}}{\left (-2 \textit {\_a} +b \right ) \sqrt {4 a y \left (x \right )^{2}+\left (-2 \textit {\_a} +b \right )^{2}}+4 a y \left (x \right )^{2}+\left (-2 \textit {\_a} +b \right )^{2}}d \textit {\_a} \right )+c_{1} &= 0 \\ -4 a \left (\int _{}^{y \left (x \right )}\frac {16 \textit {\_f} \left (\frac {1}{16}+\left (\left (-\frac {b}{4}+\frac {x}{2}\right ) \sqrt {4 a \,\textit {\_f}^{2}+b^{2}-4 b x +4 x^{2}}+a \,\textit {\_f}^{2}+\frac {b^{2}}{4}-b x +x^{2}\right ) \left (\int _{\textit {\_b}}^{x}\frac {2 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 \textit {\_a} b +b^{2}+2 \sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 \textit {\_a} b +b^{2}}\, \textit {\_a} -\sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 \textit {\_a} b +b^{2}}\, b}{\left (4 a \,\textit {\_f}^{2}+2 \sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 \textit {\_a} b +b^{2}}\, \textit {\_a} -\sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 \textit {\_a} b +b^{2}}\, b +4 \textit {\_a}^{2}-4 \textit {\_a} b +b^{2}\right )^{2} \sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 \textit {\_a} b +b^{2}}}d \textit {\_a} \right )\right )}{\left (2 x -b \right ) \sqrt {4 a \,\textit {\_f}^{2}+b^{2}-4 b x +4 x^{2}}+4 a \,\textit {\_f}^{2}+b^{2}-4 b x +4 x^{2}}d \textit {\_f} \right )-2 \left (\int _{\textit {\_b}}^{x}\frac {2 \textit {\_a} -b +\sqrt {4 a y \left (x \right )^{2}+\left (-2 \textit {\_a} +b \right )^{2}}}{\left (2 \textit {\_a} -b \right ) \sqrt {4 a y \left (x \right )^{2}+\left (-2 \textit {\_a} +b \right )^{2}}+4 a y \left (x \right )^{2}+\left (-2 \textit {\_a} +b \right )^{2}}d \textit {\_a} \right )+c_{1} &= 0 \\ \end{align}

Problem 8812


ODE	\[ \boxed {\left (y a +b \right ) \left ({y^{\prime }}^{2}+1\right )=c} \]

program solution	\[ \frac {\left (y a +b \right ) \left (y a +b -c \right )}{a \sqrt {-\left (y a +b \right ) \left (y a +b -c \right )}}+\frac {c \arctan \left (\frac {\sqrt {a^{2}}\, \left (y-\frac {-a b -a \left (b -c \right )}{2 a^{2}}\right )}{\sqrt {-y^{2} a^{2}+\left (-a b -a \left (b -c \right )\right ) y-b \left (b -c \right )}}\right )}{2 \sqrt {a^{2}}} = x +c_{1} \] Veriﬁed OK. \[ -\frac {\left (y a +b \right ) \left (y a +b -c \right )}{a \sqrt {-\left (y a +b \right ) \left (y a +b -c \right )}}-\frac {c \arctan \left (\frac {\sqrt {a^{2}}\, \left (y-\frac {-a b -a \left (b -c \right )}{2 a^{2}}\right )}{\sqrt {-y^{2} a^{2}+\left (-a b -a \left (b -c \right )\right ) y-b \left (b -c \right )}}\right )}{2 \sqrt {a^{2}}} = x +c_{2} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \frac {-b +c}{a} \\ \frac {-\arctan \left (\frac {2 a y \left (x \right )+2 b -c}{2 \sqrt {-\left (a y \left (x \right )+b \right ) \left (a y \left (x \right )+b -c \right )}}\right ) c +2 \sqrt {-\left (a y \left (x \right )+b \right ) \left (a y \left (x \right )+b -c \right )}+\left (2 x -2 c_{1} \right ) a}{2 a} &= 0 \\ \frac {\arctan \left (\frac {2 a y \left (x \right )+2 b -c}{2 \sqrt {-\left (a y \left (x \right )+b \right ) \left (a y \left (x \right )+b -c \right )}}\right ) c -2 \sqrt {-\left (a y \left (x \right )+b \right ) \left (a y \left (x \right )+b -c \right )}+\left (2 x -2 c_{1} \right ) a}{2 a} &= 0 \\ \end{align}

Problem 8813


ODE	\[ \boxed {\left (b_{2} y+a_{2} x +c_{2} \right ) {y^{\prime }}^{2}+\left (a_{1} x +b_{1} y+c_{1} \right ) y^{\prime }+b_{0} y=-a_{0} x -c_{0}} \]

program solution	\[ \text {Expression too large to display} \] Warning, solution could not be veriﬁed \[ \text {Expression too large to display} \] Warning, solution could not be veriﬁed \[ \text {Expression too large to display} \] Warning, solution could not be veriﬁed

Maple solution	\begin{align} x -{\mathrm e}^{\int _{}^{\frac {-a_{1} x -b_{1} y \left (x \right )-c_{1} -\sqrt {\left (-4 b_{0} b_{2} +b_{1}^{2}\right ) y \left (x \right )^{2}+\left (\left (-4 a_{0} b_{2} +2 a_{1} b_{1} -4 a_{2} b_{0} \right ) x -4 b_{2} c_{0} +2 c_{1} b_{1} -4 c_{2} b_{0} \right ) y \left (x \right )+\left (-4 a_{0} a_{2} +a_{1}^{2}\right ) x^{2}+\left (-4 a_{0} c_{2} +2 a_{1} c_{1} -4 c_{0} a_{2} \right ) x -4 c_{2} c_{0} +c_{1}^{2}}}{2 b_{2} y \left (x \right )+2 c_{2} +2 a_{2} x}}\frac {\left (a_{1} b_{2} -a_{2} b_{1} \right ) \textit {\_a}^{2}+\left (2 a_{0} b_{2} -2 a_{2} b_{0} \right ) \textit {\_a} +a_{0} b_{1} -b_{0} a_{1}}{\left (\textit {\_a}^{2} b_{2} +\textit {\_a} b_{1} +b_{0} \right ) \left (\textit {\_a}^{3} b_{2} +\left (a_{2} +b_{1} \right ) \textit {\_a}^{2}+\left (a_{1} +b_{0} \right ) \textit {\_a} +a_{0} \right )}d \textit {\_a}} \left (\int _{}^{\frac {-a_{1} x -b_{1} y \left (x \right )-c_{1} -\sqrt {\left (-4 b_{0} b_{2} +b_{1}^{2}\right ) y \left (x \right )^{2}+\left (\left (-4 a_{0} b_{2} +2 a_{1} b_{1} -4 a_{2} b_{0} \right ) x -4 b_{2} c_{0} +2 c_{1} b_{1} -4 c_{2} b_{0} \right ) y \left (x \right )+\left (-4 a_{0} a_{2} +a_{1}^{2}\right ) x^{2}+\left (-4 a_{0} c_{2} +2 a_{1} c_{1} -4 c_{0} a_{2} \right ) x -4 c_{2} c_{0} +c_{1}^{2}}}{2 b_{2} y \left (x \right )+2 c_{2} +2 a_{2} x}}\frac {\left (-\textit {\_a}^{2} b_{1} c_{2} +\textit {\_a}^{2} b_{2} c_{1} -2 \textit {\_a} b_{0} c_{2} +2 \textit {\_a} b_{2} c_{0} -b_{0} c_{1} +c_{0} b_{1} \right ) {\mathrm e}^{-\left (\int \frac {\left (a_{1} b_{2} -a_{2} b_{1} \right ) \textit {\_a}^{2}+\left (2 a_{0} b_{2} -2 a_{2} b_{0} \right ) \textit {\_a} +a_{0} b_{1} -b_{0} a_{1}}{\left (\textit {\_a}^{2} b_{2} +\textit {\_a} b_{1} +b_{0} \right ) \left (\textit {\_a}^{3} b_{2} +\left (a_{2} +b_{1} \right ) \textit {\_a}^{2}+\left (a_{1} +b_{0} \right ) \textit {\_a} +a_{0} \right )}d \textit {\_a} \right )}}{\left (\textit {\_a}^{3} b_{2} +\left (a_{2} +b_{1} \right ) \textit {\_a}^{2}+\left (a_{1} +b_{0} \right ) \textit {\_a} +a_{0} \right ) \left (\textit {\_a}^{2} b_{2} +\textit {\_a} b_{1} +b_{0} \right )}d \textit {\_a} +c_{3} \right ) &= 0 \\ x -{\mathrm e}^{\int _{}^{\frac {-a_{1} x -b_{1} y \left (x \right )-c_{1} +\sqrt {\left (-4 b_{0} b_{2} +b_{1}^{2}\right ) y \left (x \right )^{2}+\left (\left (-4 a_{0} b_{2} +2 a_{1} b_{1} -4 a_{2} b_{0} \right ) x -4 b_{2} c_{0} +2 c_{1} b_{1} -4 c_{2} b_{0} \right ) y \left (x \right )+\left (-4 a_{0} a_{2} +a_{1}^{2}\right ) x^{2}+\left (-4 a_{0} c_{2} +2 a_{1} c_{1} -4 c_{0} a_{2} \right ) x -4 c_{2} c_{0} +c_{1}^{2}}}{2 b_{2} y \left (x \right )+2 c_{2} +2 a_{2} x}}\frac {\left (a_{1} b_{2} -a_{2} b_{1} \right ) \textit {\_a}^{2}+\left (2 a_{0} b_{2} -2 a_{2} b_{0} \right ) \textit {\_a} +a_{0} b_{1} -b_{0} a_{1}}{\left (\textit {\_a}^{2} b_{2} +\textit {\_a} b_{1} +b_{0} \right ) \left (\textit {\_a}^{3} b_{2} +\left (a_{2} +b_{1} \right ) \textit {\_a}^{2}+\left (a_{1} +b_{0} \right ) \textit {\_a} +a_{0} \right )}d \textit {\_a}} \left (\int _{}^{\frac {-a_{1} x -b_{1} y \left (x \right )-c_{1} +\sqrt {\left (-4 b_{0} b_{2} +b_{1}^{2}\right ) y \left (x \right )^{2}+\left (\left (-4 a_{0} b_{2} +2 a_{1} b_{1} -4 a_{2} b_{0} \right ) x -4 b_{2} c_{0} +2 c_{1} b_{1} -4 c_{2} b_{0} \right ) y \left (x \right )+\left (-4 a_{0} a_{2} +a_{1}^{2}\right ) x^{2}+\left (-4 a_{0} c_{2} +2 a_{1} c_{1} -4 c_{0} a_{2} \right ) x -4 c_{2} c_{0} +c_{1}^{2}}}{2 b_{2} y \left (x \right )+2 c_{2} +2 a_{2} x}}\frac {\left (-\textit {\_a}^{2} b_{1} c_{2} +\textit {\_a}^{2} b_{2} c_{1} -2 \textit {\_a} b_{0} c_{2} +2 \textit {\_a} b_{2} c_{0} -b_{0} c_{1} +c_{0} b_{1} \right ) {\mathrm e}^{-\left (\int \frac {\left (a_{1} b_{2} -a_{2} b_{1} \right ) \textit {\_a}^{2}+\left (2 a_{0} b_{2} -2 a_{2} b_{0} \right ) \textit {\_a} +a_{0} b_{1} -b_{0} a_{1}}{\left (\textit {\_a}^{2} b_{2} +\textit {\_a} b_{1} +b_{0} \right ) \left (\textit {\_a}^{3} b_{2} +\left (a_{2} +b_{1} \right ) \textit {\_a}^{2}+\left (a_{1} +b_{0} \right ) \textit {\_a} +a_{0} \right )}d \textit {\_a} \right )}}{\left (\textit {\_a}^{3} b_{2} +\left (a_{2} +b_{1} \right ) \textit {\_a}^{2}+\left (a_{1} +b_{0} \right ) \textit {\_a} +a_{0} \right ) \left (\textit {\_a}^{2} b_{2} +\textit {\_a} b_{1} +b_{0} \right )}d \textit {\_a} +c_{3} \right ) &= 0 \\ \end{align}

Problem 8814


ODE	\[ \boxed {\left (y a -x^{2}\right ) {y^{\prime }}^{2}+2 x y {y^{\prime }}^{2}-y^{2}=0} \]

program solution

Maple solution	\[ \text {No solution found} \]

Problem 8815


ODE	\[ \boxed {x y {y^{\prime }}^{2}+\left (y^{2}+x^{2}\right ) y^{\prime }+x y=0} \]

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 ) &= \frac {c_{1}}{x} \\ y \left (x \right ) &= \sqrt {-x^{2}+c_{1}} \\ y \left (x \right ) &= -\sqrt {-x^{2}+c_{1}} \\ \end{align}

Problem 8816


ODE	\[ \boxed {x y {y^{\prime }}^{2}+\left (x^{22}-y^{2}+a \right ) y^{\prime }-x y=0} \]

program solution

Maple solution	\[ \text {No solution found} \]

Problem 8817


ODE	\[ \boxed {\left (2 x y-x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+2 x y-y^{2}=0} \]

program solution	\[ y = -i x \] Veriﬁed OK. \[ y = i x \] Veriﬁed OK. \[ y = 0 \] Veriﬁed OK. \[ y = -i x \] Veriﬁed OK. \[ y = i x \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {2 \textit {\_a}^{2}+\sqrt {2}\, \sqrt {\textit {\_a} \left (\textit {\_a} -1\right )^{2}}}{\textit {\_a} \left (\textit {\_a}^{2}+1\right )}d \textit {\_a} \right )+2 c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {\sqrt {2}\, \sqrt {\textit {\_a} \left (\textit {\_a} -1\right )^{2}}-2 \textit {\_a}^{2}}{\textit {\_a} \left (\textit {\_a}^{2}+1\right )}d \textit {\_a} +2 c_{1} \right ) x \\ \end{align}

Problem 8818


ODE	\[ \boxed {\left (2 x y-x^{2}\right ) {y^{\prime }}^{2}-6 x y y^{\prime }-y^{2}+2 x y=0} \]

program solution	\[ y = -\sqrt {3}\, x +2 x \] Veriﬁed OK. \[ y = \sqrt {3}\, x +2 x \] Veriﬁed OK. \[ x = \frac {c_{2} {\left (\frac {x \left (-2 y+x \right ) \sqrt {\frac {-4 \left (-\frac {y}{2}+x \right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 x^{2} y-7 y^{2} x +2 y^{3}}{x \left (-2 y+x \right )^{2}}}-2 x^{2}+x y+\sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}}{x \left (-2 y+x \right )}\right )}^{\frac {\sqrt {\frac {\left (-4 \left (-\frac {y}{2}+x \right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 x^{2} y-7 y^{2} x +2 y^{3}\right ) \left (-\sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x \left (y+x \right )\right )^{2}}{x^{3} \left (-2 y+x \right )^{4}}}\, x \left (-2 y+x \right )}{\sqrt {\frac {-4 \left (-\frac {y}{2}+x \right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 x^{2} y-7 y^{2} x +2 y^{3}}{x \left (-2 y+x \right )^{2}}}\, \left (-\sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x \left (y+x \right )\right )}}}{\sqrt {\frac {-4 \left (-\frac {y}{2}+x \right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 x^{2} y-7 y^{2} x +2 y^{3}}{x \left (-2 y+x \right )^{2}}}} \] Warning, solution could not be veriﬁed \[ x = \frac {c_{2} {\left (\frac {x \left (-2 y+x \right ) \sqrt {\frac {4 \left (-\frac {y}{2}+x \right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 x^{2} y-7 y^{2} x +2 y^{3}}{x \left (-2 y+x \right )^{2}}}-2 x^{2}+x y-\sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}}{x \left (-2 y+x \right )}\right )}^{\frac {\sqrt {\frac {\left (4 \left (-\frac {y}{2}+x \right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 x^{2} y-7 y^{2} x +2 y^{3}\right ) \left (\sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x \left (y+x \right )\right )^{2}}{x^{3} \left (-2 y+x \right )^{4}}}\, x \left (-2 y+x \right )}{\left (\sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{2}+x y\right ) \sqrt {\frac {4 \left (-\frac {y}{2}+x \right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 x^{2} y-7 y^{2} x +2 y^{3}}{x \left (-2 y+x \right )^{2}}}}}}{\sqrt {\frac {4 \left (-\frac {y}{2}+x \right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 x^{2} y-7 y^{2} x +2 y^{3}}{x \left (-2 y+x \right )^{2}}}} \] Warning, solution could not be veriﬁed \[ y = 0 \] Veriﬁed OK. \[ y = -\sqrt {3}\, x +2 x \] Veriﬁed OK. \[ y = \sqrt {3}\, x +2 x \] Veriﬁed OK. \[ x = \frac {c_{4} {\left (\frac {x \left (-2 y+x \right ) \sqrt {\frac {-4 \left (-\frac {y}{2}+x \right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 x^{2} y-7 y^{2} x +2 y^{3}}{x \left (-2 y+x \right )^{2}}}-2 x^{2}+x y+\sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}}{x \left (-2 y+x \right )}\right )}^{-\frac {\sqrt {\frac {\left (-4 \left (-\frac {y}{2}+x \right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 x^{2} y-7 y^{2} x +2 y^{3}\right ) \left (-\sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x \left (y+x \right )\right )^{2}}{x^{3} \left (-2 y+x \right )^{4}}}\, x \left (-2 y+x \right )}{\sqrt {\frac {-4 \left (-\frac {y}{2}+x \right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 x^{2} y-7 y^{2} x +2 y^{3}}{x \left (-2 y+x \right )^{2}}}\, \left (-\sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x \left (y+x \right )\right )}}}{\sqrt {\frac {-4 \left (-\frac {y}{2}+x \right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 x^{2} y-7 y^{2} x +2 y^{3}}{x \left (-2 y+x \right )^{2}}}} \] Warning, solution could not be veriﬁed \[ x = \frac {c_{4} {\left (\frac {x \left (-2 y+x \right ) \sqrt {\frac {4 \left (-\frac {y}{2}+x \right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 x^{2} y-7 y^{2} x +2 y^{3}}{x \left (-2 y+x \right )^{2}}}-2 x^{2}+x y-\sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}}{x \left (-2 y+x \right )}\right )}^{-\frac {\sqrt {\frac {\left (4 \left (-\frac {y}{2}+x \right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 x^{2} y-7 y^{2} x +2 y^{3}\right ) \left (\sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x \left (y+x \right )\right )^{2}}{x^{3} \left (-2 y+x \right )^{4}}}\, x \left (-2 y+x \right )}{\left (\sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{2}+x y\right ) \sqrt {\frac {4 \left (-\frac {y}{2}+x \right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 x^{2} y-7 y^{2} x +2 y^{3}}{x \left (-2 y+x \right )^{2}}}}}}{\sqrt {\frac {4 \left (-\frac {y}{2}+x \right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 x^{2} y-7 y^{2} x +2 y^{3}}{x \left (-2 y+x \right )^{2}}}} \] Warning, solution could not be veriﬁed

Maple solution	\begin{align} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {2 \textit {\_a}^{2}+\sqrt {2}\, \sqrt {\textit {\_a} \left (\textit {\_a} +1\right )^{2}}-4 \textit {\_a}}{\textit {\_a} \left (\textit {\_a}^{2}-4 \textit {\_a} +1\right )}d \textit {\_a} \right )+2 c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {\sqrt {2}\, \sqrt {\textit {\_a} \left (\textit {\_a} +1\right )^{2}}-2 \textit {\_a}^{2}+4 \textit {\_a}}{\textit {\_a} \left (\textit {\_a}^{2}-4 \textit {\_a} +1\right )}d \textit {\_a} +2 c_{1} \right ) x \\ \end{align}

Problem 8819


ODE	\[ \boxed {a x y {y^{\prime }}^{2}-\left (y^{2} a +b \,x^{2}+c \right ) y^{\prime }+b x y=0} \]

program solution	\[ -\frac {\ln \left (\left (b \,x^{2}+c \right ) \sqrt {y^{4} a^{2}-2 y^{2} \left (b \,x^{2}-c \right ) a +\left (b \,x^{2}+c \right )^{2}}+b^{2} x^{4}-b \left (y^{2} a -2 c \right ) x^{2}+c \left (y^{2} a +c \right )\right )}{4}+\ln \left (y\right )-\frac {\ln \left (y^{2} a -b \,x^{2}+\sqrt {y^{4} a^{2}-2 y^{2} \left (b \,x^{2}-c \right ) a +\left (b \,x^{2}+c \right )^{2}}+c \right )}{4}-\frac {\ln \left (2\right )}{4} = c_{1} \] Veriﬁed OK.

Maple solution	\[ \text {No solution found} \]

Problem 8820


ODE	\[ \boxed {y^{2} {y^{\prime }}^{2}+y^{2}=a^{2}} \]

program solution	\[ -\sqrt {-y^{2}+a^{2}} = x +c_{1} \] Veriﬁed OK. \[ \sqrt {-y^{2}+a^{2}} = x +c_{2} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -a \\ y \left (x \right ) &= a \\ y \left (x \right ) &= \sqrt {a^{2}-c_{1}^{2}+2 c_{1} x -x^{2}} \\ y \left (x \right ) &= -\sqrt {\left (a +x -c_{1} \right ) \left (c_{1} +a -x \right )} \\ \end{align}

Problem 8821


ODE	\[ \boxed {y^{2} {y^{\prime }}^{2}-6 x^{3} y^{\prime }+4 x^{2} y=0} \]

program solution	\[ \frac {3 \ln \left (y\right )}{4}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {9 x^{4}-4 y^{3}}}{3 x^{2}}\right )}{2} = c_{1} \] Veriﬁed OK. \[ \frac {3 \ln \left (y\right )}{4}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {9 x^{4}-4 y^{3}}}{3 x^{2}}\right )}{2} = c_{1} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \frac {18^{\frac {1}{3}} x^{\frac {4}{3}}}{2} \\ y \left (x \right ) &= -\frac {18^{\frac {1}{3}} x^{\frac {4}{3}} \left (1+i \sqrt {3}\right )}{4} \\ y \left (x \right ) &= \frac {18^{\frac {1}{3}} x^{\frac {4}{3}} \left (i \sqrt {3}-1\right )}{4} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-4 \ln \left (x \right )-3 \left (\int _{}^{\textit {\_Z}}\frac {4 \textit {\_a}^{3}+3 \sqrt {-4 \textit {\_a}^{3}+9}-9}{\textit {\_a} \left (4 \textit {\_a}^{3}-9\right )}d \textit {\_a} \right )+4 c_{1} \right ) x^{\frac {4}{3}} \\ \end{align}

Problem 8822


ODE	\[ \boxed {y^{2} {y^{\prime }}^{2}-4 a y y^{\prime }+y^{2}=-4 a^{2}+4 a x} \]

program solution

Maple solution	\begin{align} y \left (x \right ) &= -2 \sqrt {a x} \\ y \left (x \right ) &= 2 \sqrt {a x} \\ y \left (x \right ) &= \sqrt {4 a x -c_{1}^{2}+2 c_{1} x -x^{2}} \\ y \left (x \right ) &= -\sqrt {-x^{2}+\left (4 a +2 c_{1} \right ) x -c_{1}^{2}} \\ \end{align}

Problem 8823


ODE	\[ \boxed {y^{2} {y^{\prime }}^{2}+2 x y y^{\prime }+y^{2} a=-x b -c} \]

program solution

Maple solution	\begin{align} y \left (x \right ) &= -\frac {2 \sqrt {a \left (a \left (a x -\frac {1}{2} b +x \right )^{2} \left (a +1\right )^{2} \operatorname {RootOf}\left (-2 b \ln \left (2 a x -b +2 x \right )+b \left (\int _{}^{\textit {\_Z}}-\frac {4 \textit {\_a} \,a^{2}-\sqrt {-\left (4 \textit {\_a} \,a^{3}+8 \textit {\_a} \,a^{2}+4 \textit {\_a} a -1\right ) {\mathrm e}^{\frac {4 a}{b}} {\mathrm e}^{\frac {4}{b}}}\, {\mathrm e}^{-\frac {2 \left (a +1\right )}{b}}+8 \textit {\_a} a +4 \textit {\_a} +1}{\textit {\_a} \left (4 \textit {\_a} \,a^{2}+8 \textit {\_a} a +4 \textit {\_a} +a +2\right )}d \textit {\_a} \right )+4 c_{1} a +4 c_{1} \right )+\frac {\left (-b x -c \right ) a^{2}}{4}+\frac {\left (-\frac {b x}{2}-c \right ) a}{2}-\frac {b^{2}}{16}-\frac {c}{4}\right )}}{a \left (a +1\right )} \\ y \left (x \right ) &= \frac {2 \sqrt {a \left (a \left (a x -\frac {1}{2} b +x \right )^{2} \left (a +1\right )^{2} \operatorname {RootOf}\left (-2 b \ln \left (2 a x -b +2 x \right )+b \left (\int _{}^{\textit {\_Z}}-\frac {4 \textit {\_a} \,a^{2}-\sqrt {-\left (4 \textit {\_a} \,a^{3}+8 \textit {\_a} \,a^{2}+4 \textit {\_a} a -1\right ) {\mathrm e}^{\frac {4 a}{b}} {\mathrm e}^{\frac {4}{b}}}\, {\mathrm e}^{-\frac {2 \left (a +1\right )}{b}}+8 \textit {\_a} a +4 \textit {\_a} +1}{\textit {\_a} \left (4 \textit {\_a} \,a^{2}+8 \textit {\_a} a +4 \textit {\_a} +a +2\right )}d \textit {\_a} \right )+4 c_{1} a +4 c_{1} \right )+\frac {\left (-b x -c \right ) a^{2}}{4}+\frac {\left (-\frac {b x}{2}-c \right ) a}{2}-\frac {b^{2}}{16}-\frac {c}{4}\right )}}{a \left (a +1\right )} \\ \end{align}

Problem 8824


ODE	\[ \boxed {y^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+2 y^{2}=x^{2}-a} \]

program solution

Maple solution	\begin{align} y \left (x \right ) &= -\frac {\sqrt {4 x^{2}-2 a}}{2} \\ y \left (x \right ) &= \frac {\sqrt {4 x^{2}-2 a}}{2} \\ y \left (x \right ) &= -\frac {\sqrt {-8 c_{1}^{2}+16 c_{1} x -4 x^{2}-2 a}}{2} \\ y \left (x \right ) &= \frac {\sqrt {-8 c_{1}^{2}+16 c_{1} x -4 x^{2}-2 a}}{2} \\ \end{align}

Problem 8825


ODE	\[ \boxed {y^{2} {y^{\prime }}^{2}+2 a x y y^{\prime }+\left (-a +1\right ) y^{2}=-a \,x^{2}-\left (a -1\right ) b} \]

program solution

Maple solution	\begin{align} y \left (x \right ) &= \sqrt {-a \,x^{2}+b} \\ y \left (x \right ) &= -\sqrt {-a \,x^{2}+b} \\ y \left (x \right ) &= \sqrt {a \,c_{1}^{2}-2 a c_{1} x -c_{1}^{2}+2 c_{1} x -x^{2}+b} \\ y \left (x \right ) &= -\sqrt {\left (a -1\right ) c_{1}^{2}-2 x \left (a -1\right ) c_{1} -x^{2}+b} \\ \end{align}

Problem 8826


ODE	\[ \boxed {\left (y^{2}-a^{2}\right ) {y^{\prime }}^{2}+y^{2}=0} \]

program solution	\[ \sqrt {-y^{2}+a^{2}}+\frac {a \ln \left (-a +\sqrt {-y^{2}+a^{2}}\right )}{2}-\frac {a \ln \left (\sqrt {-y^{2}+a^{2}}+a \right )}{2} = x +c_{1} \] Veriﬁed OK. \[ -\sqrt {-y^{2}+a^{2}}-\frac {a \ln \left (-a +\sqrt {-y^{2}+a^{2}}\right )}{2}+\frac {a \ln \left (\sqrt {-y^{2}+a^{2}}+a \right )}{2} = x +c_{2} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= 0 \\ a \,\operatorname {csgn}\left (a \right ) \ln \left (2\right )+a \,\operatorname {csgn}\left (a \right ) \ln \left (\frac {a \left (\operatorname {csgn}\left (a \right ) \sqrt {-y \left (x \right )^{2}+a^{2}}+a \right )}{y \left (x \right )}\right )-\sqrt {-y \left (x \right )^{2}+a^{2}}-c_{1} +x &= 0 \\ -a \,\operatorname {csgn}\left (a \right ) \ln \left (2\right )-a \,\operatorname {csgn}\left (a \right ) \ln \left (\frac {a \left (\operatorname {csgn}\left (a \right ) \sqrt {-y \left (x \right )^{2}+a^{2}}+a \right )}{y \left (x \right )}\right )+\sqrt {-y \left (x \right )^{2}+a^{2}}-c_{1} +x &= 0 \\ \end{align}

Problem 8827


ODE	\[ \boxed {\left (y^{2}-2 a x +a^{2}\right ) {y^{\prime }}^{2}+2 a y y^{\prime }+y^{2}=0} \]

program solution

Maple solution	\begin{align} y \left (x \right ) &= 0 \\ \left [x \left (\textit {\_T} \right ) &= \frac {\sqrt {\textit {\_T}^{2}+1}\, {\operatorname {arctanh}\left (\frac {1}{\sqrt {\textit {\_T}^{2}+1}}\right )}^{2} a^{2}+\left (-2 a c_{1} \sqrt {\textit {\_T}^{2}+1}-2 a^{2}\right ) \operatorname {arctanh}\left (\frac {1}{\sqrt {\textit {\_T}^{2}+1}}\right )+\left (a^{2}+c_{1}^{2}\right ) \sqrt {\textit {\_T}^{2}+1}+2 c_{1} a}{2 \sqrt {\textit {\_T}^{2}+1}\, a}, y \left (\textit {\_T} \right ) &= \frac {\left (-a \,\operatorname {arctanh}\left (\frac {1}{\sqrt {\textit {\_T}^{2}+1}}\right )+c_{1} \right ) \textit {\_T}}{\sqrt {\textit {\_T}^{2}+1}}\right ] \\ \end{align}

Problem 8828


ODE	\[ \boxed {\left (y^{2}-a^{2} x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }=-\left (-a^{2}+1\right ) x^{2}} \]

program solution	\[ y = -i x \] Veriﬁed OK. \[ y = i x \] Veriﬁed OK. \[ y = \frac {-x +x \sqrt {a^{4}}}{\sqrt {a^{2}-1}} \] Veriﬁed OK. \[ y = \frac {x -x \sqrt {a^{4}}}{\sqrt {a^{2}-1}} \] Veriﬁed OK. \[ y = \frac {-1+\sqrt {a^{2} \left (-\frac {c_{2}^{2} x^{2}}{c_{2}^{2} x^{2}-1}+1\right )}}{c_{2} \sqrt {-\frac {1}{c_{2}^{2} x^{2}-1}}} \] Veriﬁed OK. \[ y = -i x \] Veriﬁed OK. \[ y = i x \] Veriﬁed OK. \[ y = \frac {-x -x \sqrt {a^{4}}}{\sqrt {a^{2}-1}} \] Veriﬁed OK. \[ y = \frac {x +x \sqrt {a^{4}}}{\sqrt {a^{2}-1}} \] Veriﬁed OK. \[ y = \frac {-1-\sqrt {a^{2} \left (-\frac {c_{4}^{2} x^{2}}{c_{4}^{2} x^{2}-1}+1\right )}}{c_{4} \sqrt {-\frac {1}{c_{4}^{2} x^{2}-1}}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \sqrt {a^{2}-1}\, x \\ y \left (x \right ) &= -\sqrt {a^{2}-1}\, x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {\textit {\_a}^{3}-\textit {\_a} \,a^{2}-\sqrt {a^{2} \left (\textit {\_a}^{2}-a^{2}+1\right )}+\textit {\_a}}{\left (\textit {\_a}^{2}+1\right ) \left (\textit {\_a}^{2}-a^{2}+1\right )}d \textit {\_a} +c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{3}-\textit {\_a} \,a^{2}+\sqrt {a^{2} \left (\textit {\_a}^{2}-a^{2}+1\right )}+\textit {\_a}}{\left (\textit {\_a}^{2}+1\right ) \left (\textit {\_a}^{2}-a^{2}+1\right )}d \textit {\_a} \right )+c_{1} \right ) x \\ \end{align}

Problem 8829


ODE	\[ \boxed {\left (y^{2}+\left (-a +1\right ) x^{2}\right ) {y^{\prime }}^{2}+2 a x y y^{\prime }+\left (-a +1\right ) y^{2}=-x^{2}} \]

program solution	\[ y = -i x \] Veriﬁed OK. \[ y = i x \] Veriﬁed OK. \[ y = -i x \] Veriﬁed OK. \[ y = i x \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -i x \\ y \left (x \right ) &= i x \\ y \left (x \right ) &= \tan \left (\operatorname {RootOf}\left (-2 \textit {\_Z} \sqrt {a -1}-\ln \left (\sec \left (\textit {\_Z} \right )^{2} x^{2}\right )+2 c_{1} \right )\right ) x \\ y \left (x \right ) &= \tan \left (\operatorname {RootOf}\left (2 \textit {\_Z} \sqrt {a -1}-\ln \left (\sec \left (\textit {\_Z} \right )^{2} x^{2}\right )+2 c_{1} \right )\right ) x \\ \end{align}

Problem 8830


ODE	\[ \boxed {\left (y-x \right )^{2} \left ({y^{\prime }}^{2}+1\right )-a^{2} \left (y^{\prime }+1\right )^{2}=0} \]

program solution	\[ y = a \sqrt {2}+x \] Veriﬁed OK. \[ x = -\frac {a^{2} \sqrt {2}\, \left (a^{2}-\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 x y-y^{2}\right )}\right ) \left (-\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 x y-y^{2}\right )}+\left (-y+x \right )^{2}\right )}{2 \sqrt {-\frac {a^{4} \left (-a^{2}+\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 x y-y^{2}\right )}\right ) \left (-x^{2}+2 x y-y^{2}+\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 x y-y^{2}\right )}\right )^{2}}{\left (a^{2}-x^{2}+2 x y-y^{2}\right )^{4}}}\, \left (a +x -y\right )^{2} \left (a -x +y\right )^{2}}+c_{2} \] Veriﬁed OK. \[ x = -\frac {\left (a^{2}+\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 x y-y^{2}\right )}\right ) a^{2} \left (\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 x y-y^{2}\right )}+\left (-y+x \right )^{2}\right ) \sqrt {2}}{2 \sqrt {\frac {a^{4} \left (a^{2}+\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 x y-y^{2}\right )}\right ) \left (x^{2}-2 x y+y^{2}+\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 x y-y^{2}\right )}\right )^{2}}{\left (a^{2}-x^{2}+2 x y-y^{2}\right )^{4}}}\, \left (a +x -y\right )^{2} \left (a -x +y\right )^{2}}+c_{2} \] Veriﬁed OK. \[ y = -a \sqrt {2}+x \] Veriﬁed OK. \[ x = \frac {a^{2} \sqrt {2}\, \left (a^{2}-\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 x y-y^{2}\right )}\right ) \left (-\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 x y-y^{2}\right )}+\left (-y+x \right )^{2}\right )}{2 \sqrt {-\frac {a^{4} \left (-a^{2}+\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 x y-y^{2}\right )}\right ) \left (-x^{2}+2 x y-y^{2}+\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 x y-y^{2}\right )}\right )^{2}}{\left (a^{2}-x^{2}+2 x y-y^{2}\right )^{4}}}\, \left (a +x -y\right )^{2} \left (a -x +y\right )^{2}}+c_{4} \] Veriﬁed OK. \[ x = \frac {\left (a^{2}+\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 x y-y^{2}\right )}\right ) a^{2} \left (\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 x y-y^{2}\right )}+\left (-y+x \right )^{2}\right ) \sqrt {2}}{2 \sqrt {\frac {a^{4} \left (a^{2}+\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 x y-y^{2}\right )}\right ) \left (x^{2}-2 x y+y^{2}+\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 x y-y^{2}\right )}\right )^{2}}{\left (a^{2}-x^{2}+2 x y-y^{2}\right )^{4}}}\, \left (a +x -y\right )^{2} \left (a -x +y\right )^{2}}+c_{4} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= x -\sqrt {2}\, a \\ y \left (x \right ) &= x +\sqrt {2}\, a \\ y \left (x \right ) &= x +\operatorname {RootOf}\left (-2 x -\left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{2}-2 a^{2}+\sqrt {-\textit {\_a}^{4}+2 \textit {\_a}^{2} a^{2}}}{\textit {\_a}^{2}-2 a^{2}}d \textit {\_a} \right )+2 c_{1} \right ) \\ y \left (x \right ) &= x +\operatorname {RootOf}\left (-2 x +\int _{}^{\textit {\_Z}}-\frac {-2 a^{2}+\textit {\_a}^{2}-\sqrt {-\textit {\_a}^{4}+2 \textit {\_a}^{2} a^{2}}}{\textit {\_a}^{2}-2 a^{2}}d \textit {\_a} +2 c_{1} \right ) \\ \end{align}

Problem 8831


ODE	\[ \boxed {3 y^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+4 y^{2}=x^{2}} \]

program solution	\[ y = -i x \] Veriﬁed OK. \[ y = i x \] Veriﬁed OK. \[ y = \frac {\sqrt {3}\, x}{3} \] Veriﬁed OK. \[ y = -i x \] Veriﬁed OK. \[ y = i x \] Veriﬁed OK. \[ y = -\frac {\sqrt {3}\, x}{3} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -\frac {\sqrt {3}\, x}{3} \\ y \left (x \right ) &= \frac {\sqrt {3}\, x}{3} \\ \ln \left (x \right )-\operatorname {arctanh}\left (\frac {\sqrt {\frac {x^{2}-3 y \left (x \right )^{2}}{x^{2}}}}{2}\right )+\frac {\ln \left (\frac {y \left (x \right )^{2}+x^{2}}{x^{2}}\right )}{2}-c_{1} &= 0 \\ \ln \left (x \right )+\operatorname {arctanh}\left (\frac {\sqrt {\frac {x^{2}-3 y \left (x \right )^{2}}{x^{2}}}}{2}\right )+\frac {\ln \left (\frac {y \left (x \right )^{2}+x^{2}}{x^{2}}\right )}{2}-c_{1} &= 0 \\ \end{align}

Problem 8832


ODE	\[ \boxed {\left (3 y-2\right ) {y^{\prime }}^{2}+4 y=4} \]

program solution	\[ \frac {\sqrt {3}\, \arcsin \left (6 y-5\right )}{12}-\frac {\sqrt {-3 y^{2}+5 y-2}}{2} = x +c_{1} \] Veriﬁed OK. \[ -\frac {\sqrt {3}\, \arcsin \left (6 y-5\right )}{12}+\frac {\sqrt {-3 y^{2}+5 y-2}}{2} = x +c_{2} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= 1 \\ y \left (x \right ) &= \frac {\sin \left (\operatorname {RootOf}\left (8 \sqrt {3}\, c_{1} \textit {\_Z} -8 \sqrt {3}\, x \textit {\_Z} +\cos \left (\textit {\_Z} \right )^{2}-48 c_{1}^{2}+96 c_{1} x -48 x^{2}-\textit {\_Z}^{2}\right )\right )}{6}+\frac {5}{6} \\ y \left (x \right ) &= \frac {\sin \left (\operatorname {RootOf}\left (8 \sqrt {3}\, c_{1} \textit {\_Z} -8 \sqrt {3}\, x \textit {\_Z} -\cos \left (\textit {\_Z} \right )^{2}+48 c_{1}^{2}-96 c_{1} x +48 x^{2}+\textit {\_Z}^{2}\right )\right )}{6}+\frac {5}{6} \\ \end{align}

Problem 8833


ODE	\[ \boxed {\left (-a^{2}+1\right ) y^{2} {y^{\prime }}^{2}-2 a^{2} x y y^{\prime }+y^{2}=a^{2} x^{2}} \]

program solution	\[ y = -i x \] Veriﬁed OK. \[ y = i x \] Veriﬁed OK. \[ y = \frac {-a^{3} x \sqrt {-a^{2}+1}-a^{3} x \sqrt {-\frac {1}{a^{2}-1}}+a x \sqrt {-\frac {1}{a^{2}-1}}}{a^{4}-1} \] Veriﬁed OK. \[ y = \frac {a^{3} x \sqrt {-a^{2}+1}-a^{3} x \sqrt {-\frac {1}{a^{2}-1}}+a x \sqrt {-\frac {1}{a^{2}-1}}}{a^{4}-1} \] Veriﬁed OK. \[ y = -i x \] Veriﬁed OK. \[ y = i x \] Veriﬁed OK. \[ y = a x \sqrt {-\frac {1}{a^{2}-1}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \frac {x a}{\sqrt {-a^{2}+1}} \\ y \left (x \right ) &= -\frac {x a}{\sqrt {-a^{2}+1}} \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {\left (\textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}-\sqrt {\textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}}\right ) \textit {\_a}}{\left (\textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}\right ) \left (\textit {\_a}^{2}+1\right )}d \textit {\_a} +c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {\left (\textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}+\sqrt {\textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}}\right ) \textit {\_a}}{\left (\textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}\right ) \left (\textit {\_a}^{2}+1\right )}d \textit {\_a} \right )+c_{1} \right ) x \\ \end{align}

Problem 8834


ODE	\[ \boxed {\left (a -b \right ) y^{2} {y^{\prime }}^{2}-2 b x y y^{\prime }+y^{2} a=b \,x^{2}+a b} \]

program solution

Maple solution	\begin{align} y \left (x \right ) &= \frac {\sqrt {b \left (x^{2}+a -b \right ) \left (a -b \right )}}{a -b} \\ y \left (x \right ) &= -\frac {\sqrt {b \left (x^{2}+a -b \right ) \left (a -b \right )}}{a -b} \\ \int _{\textit {\_b}}^{x}\frac {-\textit {\_a} b -\sqrt {a \left (\left (-a +b \right ) y \left (x \right )^{2}+b \left (\textit {\_a}^{2}+a -b \right )\right )}}{\sqrt {a \left (\left (-a +b \right ) y \left (x \right )^{2}+b \left (\textit {\_a}^{2}+a -b \right )\right )}\, \textit {\_a} +\left (-a +b \right ) y \left (x \right )^{2}+b \left (\textit {\_a}^{2}+a -b \right )}d \textit {\_a} +\int _{}^{y \left (x \right )}\frac {\textit {\_f} \left (\left (\sqrt {a \left (-b^{2}+\left (\textit {\_f}^{2}+x^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}\, x +\left (-a +b \right ) \textit {\_f}^{2}+b \left (x^{2}+a -b \right )\right ) \left (\int _{\textit {\_b}}^{x}\frac {\left (a -b \right ) \left (2 b \textit {\_a} \sqrt {a \left (-b^{2}+\left (\textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}+\left (-b^{2}+\left (2 \textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right ) a \right )}{\sqrt {a \left (-b^{2}+\left (\textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}\, {\left (\sqrt {a \left (-b^{2}+\left (\textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}\, \textit {\_a} -b^{2}+\left (\textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}^{2}}d \textit {\_a} \right )+a -b \right )}{\sqrt {a \left (-b^{2}+\left (\textit {\_f}^{2}+x^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}\, x +\left (-a +b \right ) \textit {\_f}^{2}+b \left (x^{2}+a -b \right )}d \textit {\_f} +c_{1} &= 0 \\ -\left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a} b -\sqrt {a \left (\left (-a +b \right ) y \left (x \right )^{2}+b \left (\textit {\_a}^{2}+a -b \right )\right )}}{-\sqrt {a \left (\left (-a +b \right ) y \left (x \right )^{2}+b \left (\textit {\_a}^{2}+a -b \right )\right )}\, \textit {\_a} +\left (-a +b \right ) y \left (x \right )^{2}+b \left (\textit {\_a}^{2}+a -b \right )}d \textit {\_a} \right )+\int _{}^{y \left (x \right )}\frac {\left (\left (-\sqrt {a \left (-b^{2}+\left (\textit {\_f}^{2}+x^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}\, x +\left (-a +b \right ) \textit {\_f}^{2}+b \left (x^{2}+a -b \right )\right ) \left (\int _{\textit {\_b}}^{x}-\frac {\left (-2 b \textit {\_a} \sqrt {a \left (-b^{2}+\left (\textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}+\left (-b^{2}+\left (2 \textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right ) a \right ) \left (a -b \right )}{\sqrt {a \left (-b^{2}+\left (\textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}\, {\left (-\sqrt {a \left (-b^{2}+\left (\textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}\, \textit {\_a} -b^{2}+\left (\textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}^{2}}d \textit {\_a} \right )+a -b \right ) \textit {\_f}}{-\sqrt {a \left (-b^{2}+\left (\textit {\_f}^{2}+x^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}\, x +\left (-a +b \right ) \textit {\_f}^{2}+b \left (x^{2}+a -b \right )}d \textit {\_f} +c_{1} &= 0 \\ \end{align}

Problem 8835


ODE	\[ \boxed {\left (y^{2} a +x b +c \right ) {y^{\prime }}^{2}-y^{\prime } y b +d y^{2}=0} \]

program solution

Maple solution	\begin{align} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\left (-b x -c \right ) \sqrt {-a d}}{a b} \\ y \left (x \right ) &= \frac {\sqrt {-a d}\, \left (b x +c \right )}{a b} \\ \end{align}

Problem 8836


ODE	\[ \boxed {\left (y a -x b \right )^{2} \left (a^{2} {y^{\prime }}^{2}+b^{2}\right )-c^{2} \left (a y^{\prime }+b \right )^{2}=0} \]

program solution	\[ y = \frac {b^{3} x +\sqrt {b^{4} c^{2}}\, \sqrt {2}}{b^{2} a} \] Veriﬁed OK. \[ x = -\frac {\sqrt {2}\, \left (c^{2}+\sqrt {-\left (y a -x b \right )^{2} \left (y^{2} a^{2}-2 y a b x +b^{2} x^{2}-2 c^{2}\right )}\right ) c^{2} b \left (\sqrt {-\left (y a -x b \right )^{2} \left (y^{2} a^{2}-2 y a b x +b^{2} x^{2}-2 c^{2}\right )}+\left (-y a +x b \right )^{2}\right )}{2 \sqrt {\frac {c^{4} b^{4} \left (c^{2}+\sqrt {-\left (y a -x b \right )^{2} \left (y^{2} a^{2}-2 y a b x +b^{2} x^{2}-2 c^{2}\right )}\right ) \left (y^{2} a^{2}-2 y a b x +b^{2} x^{2}+\sqrt {-\left (y a -x b \right )^{2} \left (y^{2} a^{2}-2 y a b x +b^{2} x^{2}-2 c^{2}\right )}\right )^{2}}{\left (y^{2} a^{2}-2 y a b x +b^{2} x^{2}-c^{2}\right )^{4}}}\, \left (y^{2} a^{2}-2 y a b x +b^{2} x^{2}-c^{2}\right )^{2}}+c_{2} \] Veriﬁed OK. \[ x = \frac {\sqrt {2}\, c^{2} b \left (-c^{2}+\sqrt {-\left (y a -x b \right )^{2} \left (y^{2} a^{2}-2 y a b x +b^{2} x^{2}-2 c^{2}\right )}\right ) \left (-\sqrt {-\left (y a -x b \right )^{2} \left (y^{2} a^{2}-2 y a b x +b^{2} x^{2}-2 c^{2}\right )}+\left (-y a +x b \right )^{2}\right )}{2 \sqrt {-\frac {c^{4} b^{4} \left (-c^{2}+\sqrt {-\left (y a -x b \right )^{2} \left (y^{2} a^{2}-2 y a b x +b^{2} x^{2}-2 c^{2}\right )}\right ) \left (-y^{2} a^{2}+2 y a b x -b^{2} x^{2}+\sqrt {-\left (y a -x b \right )^{2} \left (y^{2} a^{2}-2 y a b x +b^{2} x^{2}-2 c^{2}\right )}\right )^{2}}{\left (y^{2} a^{2}-2 y a b x +b^{2} x^{2}-c^{2}\right )^{4}}}\, \left (y^{2} a^{2}-2 y a b x +b^{2} x^{2}-c^{2}\right )^{2}}+c_{2} \] Veriﬁed OK. \[ y = \frac {b^{3} x -\sqrt {b^{4} c^{2}}\, \sqrt {2}}{a \,b^{2}} \] Veriﬁed OK. \[ x = \frac {\sqrt {2}\, \left (c^{2}+\sqrt {-\left (y a -x b \right )^{2} \left (y^{2} a^{2}-2 y a b x +b^{2} x^{2}-2 c^{2}\right )}\right ) c^{2} b \left (\sqrt {-\left (y a -x b \right )^{2} \left (y^{2} a^{2}-2 y a b x +b^{2} x^{2}-2 c^{2}\right )}+\left (-y a +x b \right )^{2}\right )}{2 \sqrt {\frac {c^{4} b^{4} \left (c^{2}+\sqrt {-\left (y a -x b \right )^{2} \left (y^{2} a^{2}-2 y a b x +b^{2} x^{2}-2 c^{2}\right )}\right ) \left (y^{2} a^{2}-2 y a b x +b^{2} x^{2}+\sqrt {-\left (y a -x b \right )^{2} \left (y^{2} a^{2}-2 y a b x +b^{2} x^{2}-2 c^{2}\right )}\right )^{2}}{\left (y^{2} a^{2}-2 y a b x +b^{2} x^{2}-c^{2}\right )^{4}}}\, \left (y^{2} a^{2}-2 y a b x +b^{2} x^{2}-c^{2}\right )^{2}}+c_{4} \] Veriﬁed OK. \[ x = -\frac {\sqrt {2}\, c^{2} b \left (-c^{2}+\sqrt {-\left (y a -x b \right )^{2} \left (y^{2} a^{2}-2 y a b x +b^{2} x^{2}-2 c^{2}\right )}\right ) \left (-\sqrt {-\left (y a -x b \right )^{2} \left (y^{2} a^{2}-2 y a b x +b^{2} x^{2}-2 c^{2}\right )}+\left (-y a +x b \right )^{2}\right )}{2 \sqrt {-\frac {c^{4} b^{4} \left (-c^{2}+\sqrt {-\left (y a -x b \right )^{2} \left (y^{2} a^{2}-2 y a b x +b^{2} x^{2}-2 c^{2}\right )}\right ) \left (-y^{2} a^{2}+2 y a b x -b^{2} x^{2}+\sqrt {-\left (y a -x b \right )^{2} \left (y^{2} a^{2}-2 y a b x +b^{2} x^{2}-2 c^{2}\right )}\right )^{2}}{\left (y^{2} a^{2}-2 y a b x +b^{2} x^{2}-c^{2}\right )^{4}}}\, \left (y^{2} a^{2}-2 y a b x +b^{2} x^{2}-c^{2}\right )^{2}}+c_{4} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \frac {b x -\sqrt {2}\, c}{a} \\ y \left (x \right ) &= \frac {b x +\sqrt {2}\, c}{a} \\ y \left (x \right ) &= \frac {\operatorname {RootOf}\left (-a \left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{2} a^{2}-2 c^{2}+\sqrt {-a^{2} \textit {\_a}^{2} \left (\textit {\_a}^{2} a^{2}-2 c^{2}\right )}}{\textit {\_a}^{2} a^{2}-2 c^{2}}d \textit {\_a} \right )+2 c_{1} b -2 b x \right ) a +b x}{a} \\ y \left (x \right ) &= \frac {\operatorname {RootOf}\left (a \left (\int _{}^{\textit {\_Z}}-\frac {\textit {\_a}^{2} a^{2}-2 c^{2}-\sqrt {-a^{2} \textit {\_a}^{2} \left (\textit {\_a}^{2} a^{2}-2 c^{2}\right )}}{\textit {\_a}^{2} a^{2}-2 c^{2}}d \textit {\_a} \right )+2 c_{1} b -2 b x \right ) a +b x}{a} \\ \end{align}

Problem 8837


ODE	\[ \boxed {\left (\operatorname {b2} y+\operatorname {a2} x +\operatorname {c2} \right )^{2} {y^{\prime }}^{2}+\left (\operatorname {a1} x +\operatorname {b1} y+\operatorname {c1} \right ) y^{\prime }+\operatorname {b0} y=-\operatorname {a0} -\operatorname {c0}} \]

program solution

Maple solution	\[ \text {No solution found} \]

Problem 8838


ODE	\[ \boxed {x y^{2} {y^{\prime }}^{2}-\left (y^{3}+x^{3}-a \right ) y^{\prime }+x^{2} y=0} \]

program solution

Maple solution	\begin{align} y \left (x \right ) &= \left (x^{3}+a -2 x \sqrt {a x}\right )^{\frac {1}{3}} \\ y \left (x \right ) &= \left (x^{3}+a +2 x \sqrt {a x}\right )^{\frac {1}{3}} \\ y \left (x \right ) &= -\frac {\left (x^{3}+a -2 x \sqrt {a x}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{2} \\ y \left (x \right ) &= \frac {\left (x^{3}+a -2 x \sqrt {a x}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{2} \\ y \left (x \right ) &= -\frac {\left (x^{3}+a +2 x \sqrt {a x}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{2} \\ y \left (x \right ) &= \frac {\left (x^{3}+a +2 x \sqrt {a x}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{2} \\ y \left (x \right ) &= 0 \\ \int _{\textit {\_b}}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{6}+\left (-2 x^{3}-2 a \right ) \textit {\_a}^{3}+\left (-x^{3}+a \right )^{2}}}d \textit {\_a} +\frac {\ln \left (x \right )}{2}-c_{1} &= 0 \\ \int _{\textit {\_b}}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{6}+\left (-2 x^{3}-2 a \right ) \textit {\_a}^{3}+\left (-x^{3}+a \right )^{2}}}d \textit {\_a} -\frac {\ln \left (x \right )}{2}-c_{1} &= 0 \\ \end{align}

Problem 8839


ODE	\[ \boxed {x y^{2} {y^{\prime }}^{2}-2 y^{3} y^{\prime }+2 y^{2} x=x^{3}} \]

program solution	\[ \sqrt {\frac {-x^{2}+y^{2}}{x^{2}}} = c_{4} {\mathrm e}^{c_{3}} x \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \sqrt {x^{2}+c_{1}} \\ y \left (x \right ) &= -\sqrt {x^{2}+c_{1}} \\ y \left (x \right ) &= \sqrt {c_{1} x^{2}+1}\, x \\ y \left (x \right ) &= -\sqrt {c_{1} x^{2}+1}\, x \\ \end{align}

Problem 8840


ODE	\[ \boxed {x^{2} \left (y^{2} x -1\right ) {y^{\prime }}^{2}+2 x^{2} y^{2} \left (y-x \right ) y^{\prime }-y^{2} \left (x^{2} y-1\right )=0} \]

program solution

Maple solution	\[ \text {No solution found} \]

Problem 8841


ODE	\[ \boxed {\left (y^{4}-a^{2} x^{2}\right ) {y^{\prime }}^{2}+2 a^{2} x y y^{\prime }+y^{2} \left (y^{2}-a^{2}\right )=0} \]

program solution

Maple solution	\begin{align} y \left (x \right ) &= 0 \\ y \left (x \right )-\operatorname {RootOf}\left (c_{1} \sqrt {\operatorname {RootOf}\left (\left (-y \left (x \right )^{4}+a^{2} x^{2}\right ) \textit {\_Z}^{2}-y \left (x \right )^{2}+a^{2}-2 \textit {\_Z} \,a^{2} x \right ) \textit {\_Z}}+a \operatorname {hypergeom}\left (\left [-\frac {1}{4}, \frac {1}{4}\right ], \left [\frac {3}{4}\right ], \frac {\textit {\_Z}^{2} \left (2 \operatorname {RootOf}\left (\left (-y \left (x \right )^{4}+a^{2} x^{2}\right ) \textit {\_Z}^{2}-y \left (x \right )^{2}+a^{2}-2 \textit {\_Z} \,a^{2} x \right ) a^{2} x +\textit {\_Z}^{2}-a^{2}\right )}{\textit {\_Z}^{4}-a^{2} x^{2}}\right )+\textit {\_Z} \left (-\frac {a^{2} \left (2 \operatorname {RootOf}\left (\left (-y \left (x \right )^{4}+a^{2} x^{2}\right ) \textit {\_Z}^{2}-y \left (x \right )^{2}+a^{2}-2 \textit {\_Z} \,a^{2} x \right ) \textit {\_Z}^{2} x -\textit {\_Z}^{2}+x^{2}\right )}{\textit {\_Z}^{4}-a^{2} x^{2}}\right )^{\frac {1}{4}}\right ) &= 0 \\ \end{align}

Problem 8842


ODE	\[ \boxed {\left (y^{4}+y^{2} x^{2}-x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }-y^{2}=0} \]

program solution

Maple solution	\begin{align} y \left (x \right ) &= -i x \\ y \left (x \right ) &= i x \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= -\operatorname {arctanh}\left (\operatorname {RootOf}\left (\operatorname {arctanh}\left (\textit {\_Z} \right )^{2} \textit {\_Z}^{2}-2 \,\operatorname {arctanh}\left (\textit {\_Z} \right ) c_{1} \textit {\_Z}^{2}+c_{1}^{2} \textit {\_Z}^{2}+x^{2} \textit {\_Z}^{2}-x^{2}\right )\right )+c_{1} \\ \end{align}

Problem 8843


ODE	\[ \boxed {9 y^{4} \left (x^{2}-1\right ) {y^{\prime }}^{2}-6 x y^{5} y^{\prime }=4 x^{2}} \]

program solution

Maple solution	\begin{align} y \left (x \right ) &= 2^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {1}{6}} \\ y \left (x \right ) &= -2^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {1}{6}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) 2^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {1}{6}}}{2} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) 2^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {1}{6}}}{2} \\ y \left (x \right ) &= -\frac {\left (i \sqrt {3}-1\right ) 2^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {1}{6}}}{2} \\ y \left (x \right ) &= \frac {\left (1+i \sqrt {3}\right ) 2^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {1}{6}}}{2} \\ y \left (x \right ) &= \frac {2^{\frac {2}{3}} {\left (\left (-4 c_{1}^{2}+x^{2}-1\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}{2 c_{1}} \\ y \left (x \right ) &= -\frac {2^{\frac {2}{3}} {\left (\left (-4 c_{1}^{2}+x^{2}-1\right ) c_{1}^{2}\right )}^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{4 c_{1}} \\ y \left (x \right ) &= \frac {2^{\frac {2}{3}} {\left (\left (-4 c_{1}^{2}+x^{2}-1\right ) c_{1}^{2}\right )}^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{4 c_{1}} \\ \end{align}

Problem 8844


ODE	\[ \boxed {x^{2} \left (y^{4} x^{2}-1\right ) {y^{\prime }}^{2}+2 x^{3} y^{3} \left (-x^{2}+y^{2}\right ) y^{\prime }-y^{2} \left (y^{2} x^{4}-1\right )=0} \]

program solution

Maple solution	\[ \text {No solution found} \]

Problem 8845


ODE	\[ \boxed {\left (a^{2} \sqrt {y^{2}+x^{2}}-x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+a^{2} \sqrt {y^{2}+x^{2}}-y^{2}=0} \]

program solution	\[ -\arctan \left (\frac {x}{y}\right ) = -2 \arctan \left (\frac {\sqrt {-a^{2}+\sqrt {y^{2}+x^{2}}}}{a}\right )+c_{1} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -i x \\ y \left (x \right ) &= i x \\ \frac {2 \sqrt {-a^{2}+\sqrt {y \left (x \right )^{2}+x^{2}}}\, \sqrt {a^{2} \left (y \left (x \right )^{2}+x^{2}\right )^{2} \left (-a^{2}+\sqrt {y \left (x \right )^{2}+x^{2}}\right )}\, \arctan \left (\frac {\sqrt {-a^{2}+\sqrt {y \left (x \right )^{2}+x^{2}}}}{a}\right )-a \left (y \left (x \right )^{2}+x^{2}\right ) \left (a^{2}-\sqrt {y \left (x \right )^{2}+x^{2}}\right ) \left (c_{1} -\arctan \left (\frac {x}{y \left (x \right )}\right )\right )}{a \left (y \left (x \right )^{2}+x^{2}\right ) \left (a^{2}-\sqrt {y \left (x \right )^{2}+x^{2}}\right )} &= 0 \\ \frac {-2 \sqrt {-a^{2}+\sqrt {y \left (x \right )^{2}+x^{2}}}\, \sqrt {a^{2} \left (y \left (x \right )^{2}+x^{2}\right )^{2} \left (-a^{2}+\sqrt {y \left (x \right )^{2}+x^{2}}\right )}\, \arctan \left (\frac {\sqrt {-a^{2}+\sqrt {y \left (x \right )^{2}+x^{2}}}}{a}\right )-a \left (y \left (x \right )^{2}+x^{2}\right ) \left (a^{2}-\sqrt {y \left (x \right )^{2}+x^{2}}\right ) \left (c_{1} -\arctan \left (\frac {x}{y \left (x \right )}\right )\right )}{a \left (y \left (x \right )^{2}+x^{2}\right ) \left (a^{2}-\sqrt {y \left (x \right )^{2}+x^{2}}\right )} &= 0 \\ \end{align}

Problem 8846


ODE	\[ \boxed {\left (a \left (y^{2}+x^{2}\right )^{\frac {3}{2}}-x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+a \left (y^{2}+x^{2}\right )^{\frac {3}{2}}-y^{2}=0} \]

program solution

Maple solution	\begin{align} y \left (x \right ) &= -i x \\ y \left (x \right ) &= i x \\ y \left (x \right ) &= \cot \left (\operatorname {RootOf}\left (-4 \textit {\_Z} -2 \left (\int _{}^{\csc \left (\textit {\_Z} \right )^{2} x^{2}}\frac {\sqrt {-\textit {\_a}^{\frac {17}{2}} \left (\sqrt {\textit {\_a}}\, a -1\right ) \left (2 \sqrt {\textit {\_a}}\, a +\cos \left (2\right )-1\right )^{2} a}}{\left (2 \textit {\_a} \,a^{2}-3 \sqrt {\textit {\_a}}\, a +1+\sqrt {\textit {\_a}}\, a \cos \left (2\right )-\cos \left (2\right )\right ) \textit {\_a}^{5}}d \textit {\_a} \right )+4 c_{1} \right )\right ) x \\ \end{align}

Problem 8847


ODE	\[ \boxed {{y^{\prime }}^{2} \sin \left (y\right )+2 x y^{\prime } \cos \left (y\right )^{3}-\sin \left (y\right ) \cos \left (y\right )^{4}=0} \]

program solution

Maple solution	\[ \text {No solution found} \]

Problem 8848


ODE	\[ \boxed {{y^{\prime }}^{2} \left (a \cos \left (y\right )+b \right )-c \cos \left (y\right )=-d} \]

program solution	\[ \int _{}^{y}\frac {a \cos \left (\textit {\_a} \right )+b}{\sqrt {\left (a \cos \left (\textit {\_a} \right )+b \right ) \left (c \cos \left (\textit {\_a} \right )-d \right )}}d \textit {\_a} = x +c_{1} \] Veriﬁed OK. \[ \int _{}^{y}-\frac {a \cos \left (\textit {\_a} \right )+b}{\sqrt {\left (a \cos \left (\textit {\_a} \right )+b \right ) \left (c \cos \left (\textit {\_a} \right )-d \right )}}d \textit {\_a} = x +c_{2} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \arccos \left (\frac {d}{c}\right ) \\ x -\left (\int _{}^{y \left (x \right )}\frac {a \cos \left (\textit {\_a} \right )+b}{\sqrt {\left (a \cos \left (\textit {\_a} \right )+b \right ) \left (c \cos \left (\textit {\_a} \right )-d \right )}}d \textit {\_a} \right )-c_{1} &= 0 \\ x +\int _{}^{y \left (x \right )}\frac {a \cos \left (\textit {\_a} \right )+b}{\sqrt {\left (a \cos \left (\textit {\_a} \right )+b \right ) \left (c \cos \left (\textit {\_a} \right )-d \right )}}d \textit {\_a} -c_{1} &= 0 \\ \end{align}

Problem 8849


ODE	\[ \boxed {f \left (y^{2}+x^{2}\right ) \left ({y^{\prime }}^{2}+1\right )-\left (y^{\prime } x -y\right )^{2}=0} \]

program solution	\[ -\arctan \left (\frac {x}{y}\right )-\frac {\left (\int _{}^{y^{2}+x^{2}}\frac {i x \sqrt {f \left (\textit {\_a} \right )-\textit {\_a}}\, \sqrt {f \left (\textit {\_a} \right )}+f \left (\textit {\_a} \right ) y}{\textit {\_a} \left (i \sqrt {f \left (\textit {\_a} \right )-\textit {\_a}}\, \sqrt {f \left (\textit {\_a} \right )}\, y+x \left (-f \left (\textit {\_a} \right )+\textit {\_a} \right )\right )}d \textit {\_a} \right )}{2}-c_{1} = 0 \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -i x \\ y \left (x \right ) &= i x \\ y \left (x \right ) &= \operatorname {RootOf}\left (x^{2}+\textit {\_Z}^{2}-f \left (\textit {\_Z}^{2}+x^{2}\right )\right ) \\ y \left (x \right ) &= \cot \left (\operatorname {RootOf}\left (-2 \textit {\_Z} -\left (\int _{}^{\csc \left (\textit {\_Z} \right )^{2} x^{2}}\frac {\sqrt {-f \left (\textit {\_a} \right ) \left (f \left (\textit {\_a} \right )-\textit {\_a} \right )}}{\textit {\_a} \left (f \left (\textit {\_a} \right )-\textit {\_a} \right )}d \textit {\_a} \right )+2 c_{1} \right )\right ) x \\ y \left (x \right ) &= \cot \left (\operatorname {RootOf}\left (-2 \textit {\_Z} +\int _{}^{\csc \left (\textit {\_Z} \right )^{2} x^{2}}\frac {\sqrt {-f \left (\textit {\_a} \right ) \left (f \left (\textit {\_a} \right )-\textit {\_a} \right )}}{\textit {\_a} \left (f \left (\textit {\_a} \right )-\textit {\_a} \right )}d \textit {\_a} +2 c_{1} \right )\right ) x \\ \end{align}

Problem 8850


ODE	\[ \boxed {\left (y^{2}+x^{2}\right ) f \left (\frac {x}{\sqrt {y^{2}+x^{2}}}\right ) \left ({y^{\prime }}^{2}+1\right )-\left (y^{\prime } x -y\right )^{2}=0} \]

program solution	\[ \ln \left (x \right ) = \int _{}^{\frac {y}{x}}\frac {f \left (\frac {1}{\sqrt {\textit {\_a}^{2}+1}}\right ) \textit {\_a}^{2}+f \left (\frac {1}{\sqrt {\textit {\_a}^{2}+1}}\right )-1}{\left (i \sqrt {f \left (\frac {1}{\sqrt {\textit {\_a}^{2}+1}}\right )}\, \sqrt {f \left (\frac {1}{\sqrt {\textit {\_a}^{2}+1}}\right )-1}-f \left (\frac {1}{\sqrt {\textit {\_a}^{2}+1}}\right ) \textit {\_a} \right ) \left (\textit {\_a}^{2}+1\right )}d \textit {\_a} +c_{1} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \operatorname {RootOf}\left (-\ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a} f \left (\frac {1}{\sqrt {\textit {\_a}^{2}+1}}\right )-\sqrt {-f \left (\frac {1}{\sqrt {\textit {\_a}^{2}+1}}\right ) \left (f \left (\frac {1}{\sqrt {\textit {\_a}^{2}+1}}\right )-1\right )}}{\left (\textit {\_a}^{2}+1\right ) f \left (\frac {1}{\sqrt {\textit {\_a}^{2}+1}}\right )}d \textit {\_a} \right )+c_{1} \right ) x \]

Problem 8851


ODE	\[ \boxed {\left (y^{2}+x^{2}\right ) f \left (\frac {y}{\sqrt {y^{2}+x^{2}}}\right ) \left ({y^{\prime }}^{2}+1\right )-\left (y^{\prime } x -y\right )^{2}=0} \]

program solution	\[ \ln \left (x \right ) = \int _{}^{\frac {y}{x}}\frac {f \left (\frac {\textit {\_a}}{\sqrt {\textit {\_a}^{2}+1}}\right ) \textit {\_a}^{2}+f \left (\frac {\textit {\_a}}{\sqrt {\textit {\_a}^{2}+1}}\right )-1}{\left (i \sqrt {f \left (\frac {\textit {\_a}}{\sqrt {\textit {\_a}^{2}+1}}\right )}\, \sqrt {f \left (\frac {\textit {\_a}}{\sqrt {\textit {\_a}^{2}+1}}\right )-1}-f \left (\frac {\textit {\_a}}{\sqrt {\textit {\_a}^{2}+1}}\right ) \textit {\_a} \right ) \left (\textit {\_a}^{2}+1\right )}d \textit {\_a} +c_{1} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {\textit {\_a} f \left (\frac {\textit {\_a}}{\sqrt {\textit {\_a}^{2}+1}}\right )-\sqrt {-f \left (\frac {\textit {\_a}}{\sqrt {\textit {\_a}^{2}+1}}\right ) \left (f \left (\frac {\textit {\_a}}{\sqrt {\textit {\_a}^{2}+1}}\right )-1\right )}}{\left (\textit {\_a}^{2}+1\right ) f \left (\frac {\textit {\_a}}{\sqrt {\textit {\_a}^{2}+1}}\right )}d \textit {\_a} +c_{1} \right ) x \]

Problem 8852


ODE	\[ \boxed {{y^{\prime }}^{3}-\left (y-a \right )^{2} \left (y-b \right )^{2}=0} \]

program solution	\[ \int _{}^{y}\frac {1}{\left (\textit {\_a}^{4}-2 \textit {\_a}^{3} a -2 \textit {\_a}^{3} b +\textit {\_a}^{2} a^{2}+4 \textit {\_a}^{2} a b +\textit {\_a}^{2} b^{2}-2 \textit {\_a} \,a^{2} b -2 \textit {\_a} a \,b^{2}+a^{2} b^{2}\right )^{\frac {1}{3}}}d \textit {\_a} = x +c_{1} \] Veriﬁed OK. \[ \int _{}^{y}\frac {1}{-\frac {\left (\textit {\_a}^{4}-2 \textit {\_a}^{3} a -2 \textit {\_a}^{3} b +\textit {\_a}^{2} a^{2}+4 \textit {\_a}^{2} a b +\textit {\_a}^{2} b^{2}-2 \textit {\_a} \,a^{2} b -2 \textit {\_a} a \,b^{2}+a^{2} b^{2}\right )^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, \left (\textit {\_a}^{4}-2 \textit {\_a}^{3} a -2 \textit {\_a}^{3} b +\textit {\_a}^{2} a^{2}+4 \textit {\_a}^{2} a b +\textit {\_a}^{2} b^{2}-2 \textit {\_a} \,a^{2} b -2 \textit {\_a} a \,b^{2}+a^{2} b^{2}\right )^{\frac {1}{3}}}{2}}d \textit {\_a} = x +c_{2} \] Veriﬁed OK. \[ \int _{}^{y}\frac {1}{-\frac {\left (\textit {\_a}^{4}-2 \textit {\_a}^{3} a -2 \textit {\_a}^{3} b +\textit {\_a}^{2} a^{2}+4 \textit {\_a}^{2} a b +\textit {\_a}^{2} b^{2}-2 \textit {\_a} \,a^{2} b -2 \textit {\_a} a \,b^{2}+a^{2} b^{2}\right )^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, \left (\textit {\_a}^{4}-2 \textit {\_a}^{3} a -2 \textit {\_a}^{3} b +\textit {\_a}^{2} a^{2}+4 \textit {\_a}^{2} a b +\textit {\_a}^{2} b^{2}-2 \textit {\_a} \,a^{2} b -2 \textit {\_a} a \,b^{2}+a^{2} b^{2}\right )^{\frac {1}{3}}}{2}}d \textit {\_a} = x +c_{3} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= a \\ y \left (x \right ) &= b \\ x -\left (\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{2} \left (\textit {\_a} -b \right )^{2}\right )^{\frac {1}{3}}}d \textit {\_a} \right )-c_{1} &= 0 \\ \frac {2 \left (\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{2} \left (\textit {\_a} -b \right )^{2}\right )^{\frac {1}{3}}}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}+x -c_{1}}{1+i \sqrt {3}} &= 0 \\ \frac {-2 \left (\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{2} \left (\textit {\_a} -b \right )^{2}\right )^{\frac {1}{3}}}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}-x +c_{1}}{i \sqrt {3}-1} &= 0 \\ \end{align}

Problem 8853


ODE	\[ \boxed {{y^{\prime }}^{3}-f \left (x \right ) \left (y^{2} a +b y+c \right )^{2}=0} \]

program solution	\[ \int _{}^{y}\frac {1}{{\left (\left (y^{2} a +y b +c \right )^{2}\right )}^{\frac {1}{3}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{3}}d x +c_{1} \] Veriﬁed OK. `{0 < (ay^2+by+c)^2, 0 < f(x)}` \[ \int _{}^{y}\frac {2}{{\left (\left (y^{2} a +y b +c \right )^{2}\right )}^{\frac {1}{3}} \left (-1+i \sqrt {3}\right )}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{3}}d x +c_{1} \] Veriﬁed OK. `{0 < (ay^2+by+c)^2, 0 < f(x)}` \[ \int _{}^{y}-\frac {2}{{\left (\left (y^{2} a +y b +c \right )^{2}\right )}^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{3}}d x +c_{1} \] Veriﬁed OK. `{0 < (ay^2+by+c)^2, 0 < f(x)}`

Maple solution	\begin{align} \int _{}^{y \left (x \right )}\frac {1}{\left (\textit {\_a}^{2} a +\textit {\_a} b +c \right )^{\frac {2}{3}}}d \textit {\_a} -\frac {\int _{}^{x}{\left (f \left (\textit {\_a} \right ) \left (a y \left (x \right )^{2}+b y \left (x \right )+c \right )^{2}\right )}^{\frac {1}{3}}d \textit {\_a}}{\left (a y \left (x \right )^{2}+b y \left (x \right )+c \right )^{\frac {2}{3}}}+c_{1} &= 0 \\ \int _{}^{y \left (x \right )}\frac {1}{\left (\textit {\_a}^{2} a +\textit {\_a} b +c \right )^{\frac {2}{3}}}d \textit {\_a} +\frac {\left (1+i \sqrt {3}\right ) \left (\int _{}^{x}{\left (f \left (\textit {\_a} \right ) \left (a y \left (x \right )^{2}+b y \left (x \right )+c \right )^{2}\right )}^{\frac {1}{3}}d \textit {\_a} \right )}{2 \left (a y \left (x \right )^{2}+b y \left (x \right )+c \right )^{\frac {2}{3}}}+c_{1} &= 0 \\ \int _{}^{y \left (x \right )}\frac {1}{\left (\textit {\_a}^{2} a +\textit {\_a} b +c \right )^{\frac {2}{3}}}d \textit {\_a} -\frac {\left (i \sqrt {3}-1\right ) \left (\int _{}^{x}{\left (f \left (\textit {\_a} \right ) \left (a y \left (x \right )^{2}+b y \left (x \right )+c \right )^{2}\right )}^{\frac {1}{3}}d \textit {\_a} \right )}{2 \left (a y \left (x \right )^{2}+b y \left (x \right )+c \right )^{\frac {2}{3}}}+c_{1} &= 0 \\ \end{align}

Problem 8854


ODE	\[ \boxed {{y^{\prime }}^{3}+y^{\prime }-y=0} \]

program solution	\[ \int _{}^{y}\frac {6 \left (108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}+12}\right )^{\frac {1}{3}}}{\left (108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}+12}\right )^{\frac {2}{3}}-12}d \textit {\_a} = x +c_{1} \] Veriﬁed OK. \[ \int _{}^{y}\frac {12 \left (108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}+12}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}+12}\right )^{\frac {2}{3}}+12 i \sqrt {3}-\left (108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}+12}\right )^{\frac {2}{3}}+12}d \textit {\_a} = x +c_{2} \] Veriﬁed OK. \[ \int _{}^{y}-\frac {12 \left (108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}+12}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}+12}\right )^{\frac {2}{3}}+12 i \sqrt {3}+\left (108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}+12}\right )^{\frac {2}{3}}-12}d \textit {\_a} = x +c_{3} \] Veriﬁed OK.

Maple solution	\begin{align} x -6 \left (\int _{}^{y \left (x \right )}\frac {\left (108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}+12}\right )^{\frac {1}{3}}}{\left (108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}+12}\right )^{\frac {2}{3}}-12}d \textit {\_a} \right )-c_{1} &= 0 \\ \frac {-12 \left (\int _{}^{y \left (x \right )}\frac {\left (108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}+12}\right )^{\frac {1}{3}}}{-6-6 i \sqrt {3}-\left (108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}+12}\right )^{\frac {2}{3}}}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}-c_{1} +x}{1+i \sqrt {3}} &= 0 \\ \frac {12 \left (\int _{}^{y \left (x \right )}\frac {\left (108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}+12}\right )^{\frac {1}{3}}}{-\left (108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}+12}\right )^{\frac {2}{3}}+\left (\sqrt {3}+3 i\right )^{2}}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}+c_{1} -x}{i \sqrt {3}-1} &= 0 \\ \end{align}

Problem 8855


ODE	\[ \boxed {{y^{\prime }}^{3}+y^{\prime } x -y=0} \]

program solution	\[ y = c_{1}^{3}+c_{1} x \] Veriﬁed OK. \[ y = -\frac {2 \sqrt {3}\, \left (-x \right )^{\frac {3}{2}}}{9} \] Veriﬁed OK. \[ y = \frac {2 \sqrt {3}\, \left (-x \right )^{\frac {3}{2}}}{9} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \frac {2 \sqrt {3}\, \left (-x \right )^{\frac {3}{2}}}{9} \\ y \left (x \right ) &= -\frac {2 \sqrt {3}\, \left (-x \right )^{\frac {3}{2}}}{9} \\ y \left (x \right ) &= c_{1} \left (c_{1}^{2}+x \right ) \\ \end{align}

Problem 8856


ODE	\[ \boxed {{y^{\prime }}^{3}-\left (x +5\right ) y^{\prime }+y=0} \]

program solution	\[ y = -c_{1}^{3}+c_{1} x +5 c_{1} \] Veriﬁed OK. \[ y = \frac {2 \sqrt {3 x +15}\, \left (x +5\right )}{9} \] Veriﬁed OK. \[ y = -\frac {2 \sqrt {3 x +15}\, \left (x +5\right )}{9} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -\frac {2 \sqrt {3 x +15}\, \left (x +5\right )}{9} \\ y \left (x \right ) &= \frac {2 \sqrt {3 x +15}\, \left (x +5\right )}{9} \\ y \left (x \right ) &= c_{1} \left (-c_{1}^{2}+x +5\right ) \\ \end{align}

Problem 8857


ODE	\[ \boxed {{y^{\prime }}^{3}-a x y^{\prime }=-x^{3}} \]

program solution	\[ y = \int \frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{\frac {2}{3}}+12 a x}{6 \left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{\frac {1}{3}}}d x +c_{1} \] Veriﬁed OK. \[ y = \int \frac {i \left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{\frac {2}{3}} \sqrt {3}-12 i \sqrt {3}\, a x -\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{\frac {2}{3}}-12 a x}{12 \left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{\frac {1}{3}}}d x +c_{2} \] Veriﬁed OK. \[ y = \int -\frac {i \left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{\frac {2}{3}} \sqrt {3}-12 i \sqrt {3}\, a x +\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{\frac {2}{3}}+12 a x}{12 \left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{\frac {1}{3}}}d x +c_{3} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \frac {\left (\int \left (\left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )-\frac {12 \left (1+i \sqrt {3}\right ) a x}{\left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{\frac {1}{3}}}\right )d x \right )}{12}+c_{1} \\ y \left (x \right ) &= -\frac {\left (\int \left (\left (1+i \sqrt {3}\right ) \left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{\frac {1}{3}}-\frac {12 \left (i \sqrt {3}-1\right ) a x}{\left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{\frac {1}{3}}}\right )d x \right )}{12}+c_{1} \\ y \left (x \right ) &= \frac {\left (\int \frac {\left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{\frac {2}{3}}+12 a x}{\left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{\frac {1}{3}}}d x \right )}{6}+c_{1} \\ \end{align}

Problem 8858


ODE	\[ \boxed {{y^{\prime }}^{3}-2 y y^{\prime }+y^{2}=0} \]

program solution	\[ \int _{}^{y}\frac {6 \left (-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-96 \textit {\_a}^{3}}\right )^{\frac {1}{3}}}{\left (-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-96 \textit {\_a}^{3}}\right )^{\frac {2}{3}}+24 \textit {\_a}}d \textit {\_a} = x +c_{1} \] Veriﬁed OK. \[ \int _{}^{y}-\frac {12 \left (-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-96 \textit {\_a}^{3}}\right )^{\frac {1}{3}}}{-i \sqrt {3}\, \left (-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-96 \textit {\_a}^{3}}\right )^{\frac {2}{3}}+24 i \sqrt {3}\, \textit {\_a} +\left (-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-96 \textit {\_a}^{3}}\right )^{\frac {2}{3}}+24 \textit {\_a}}d \textit {\_a} = x +c_{2} \] Veriﬁed OK. \[ \int _{}^{y}\frac {12 \left (-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-96 \textit {\_a}^{3}}\right )^{\frac {1}{3}}}{-i \sqrt {3}\, \left (-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-96 \textit {\_a}^{3}}\right )^{\frac {2}{3}}+24 i \sqrt {3}\, \textit {\_a} -\left (-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-96 \textit {\_a}^{3}}\right )^{\frac {2}{3}}-24 \textit {\_a}}d \textit {\_a} = x +c_{3} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= 0 \\ -\sqrt {3}\, 2^{\frac {1}{3}} \left (\int _{}^{y \left (x \right )}\frac {\left (-3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}-32 \textit {\_a}^{3}}\right )^{\frac {1}{3}}}{2 \,2^{\frac {2}{3}} \textit {\_a} +\left (-3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}-32 \textit {\_a}^{3}}\right )^{\frac {2}{3}}}d \textit {\_a} \right )+x -c_{1} &= 0 \\ \frac {2 i 2^{\frac {1}{3}} 3^{\frac {5}{6}} \left (\int _{}^{y \left (x \right )}\frac {\left (-3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}-32 \textit {\_a}^{3}}\right )^{\frac {1}{3}}}{-3^{\frac {1}{3}} \left (-3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}-32 \textit {\_a}^{3}}\right )^{\frac {2}{3}}+\textit {\_a} \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}}}d \textit {\_a} \right )+\left (x -c_{1} \right ) \left (-i+\sqrt {3}\right )}{-i+\sqrt {3}} &= 0 \\ \frac {2 i 2^{\frac {1}{3}} 3^{\frac {5}{6}} \left (\int _{}^{y \left (x \right )}\frac {\left (-3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}-32 \textit {\_a}^{3}}\right )^{\frac {1}{3}}}{3^{\frac {1}{3}} \left (-3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}-32 \textit {\_a}^{3}}\right )^{\frac {2}{3}}+\textit {\_a} \left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}}}d \textit {\_a} \right )+\left (x -c_{1} \right ) \left (\sqrt {3}+i\right )}{\sqrt {3}+i} &= 0 \\ \end{align}

Problem 8859


ODE	\[ \boxed {{y^{\prime }}^{2}-a x y y^{\prime }+2 y^{2} a=0} \]

program solution	\[ y = c_{1} {\mathrm e}^{\frac {x \sqrt {a^{2} x^{2}-8 a}}{4}-\frac {2 a \ln \left (\frac {x \,a^{2}}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-8 a}\right )}{\sqrt {a^{2}}}+\frac {a \,x^{2}}{4}} \] Veriﬁed OK. \[ y = c_{2} {\mathrm e}^{\frac {a \,x^{2}}{4}-\frac {x \sqrt {a^{2} x^{2}-8 a}}{4}+\frac {2 a \ln \left (\frac {x \,a^{2}}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-8 a}\right )}{\sqrt {a^{2}}}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= 0 \\ y \left (x \right ) &= c_{1} {\left (a x \,\operatorname {csgn}\left (a \right )+\sqrt {a \left (a \,x^{2}-8\right )}\right )}^{-2 \,\operatorname {csgn}\left (a \right )} {\mathrm e}^{\frac {x \left (a x +\sqrt {a \left (a \,x^{2}-8\right )}\right )}{4}} \\ y \left (x \right ) &= c_{1} {\left (a x \,\operatorname {csgn}\left (a \right )+\sqrt {a \left (a \,x^{2}-8\right )}\right )}^{2 \,\operatorname {csgn}\left (a \right )} {\mathrm e}^{-\frac {x \left (-a x +\sqrt {a \left (a \,x^{2}-8\right )}\right )}{4}} \\ \end{align}

Problem 8860


ODE	\[ \boxed {{y^{\prime }}^{3}-\left (y^{2}+x y+x^{2}\right ) {y^{\prime }}^{2}+\left (x y^{3}+y^{2} x^{2}+x^{3} y\right ) y^{\prime }-y^{3} x^{3}=0} \]

program solution	\[ y = c_{3} {\mathrm e}^{\frac {x^{2}}{2}} \] Veriﬁed OK.

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

Problem 8861


ODE	\[ \boxed {{y^{\prime }}^{3}-x y^{4} y^{\prime }-y^{5}=0} \]

program solution	\[ \ln \left (x \right ) = \int _{}^{y x^{\frac {3}{2}}}\frac {2 \left (i \sqrt {12 \textit {\_a}^{2}-81}+9\right )^{\frac {1}{3}} 18^{\frac {2}{3}}}{\left (2 \textit {\_a}^{\frac {4}{3}} \left (i \sqrt {12 \textit {\_a}^{2}-81}+9\right )^{\frac {2}{3}} 18^{\frac {1}{3}}+9 \left (i \sqrt {12 \textit {\_a}^{2}-81}+9\right )^{\frac {1}{3}} 12^{\frac {1}{3}} \left (\textit {\_a}^{2}\right )^{\frac {1}{3}}+12 \textit {\_a}^{2}\right ) \textit {\_a}^{\frac {1}{3}}}d \textit {\_a} +c_{1} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -\frac {3 \sqrt {3}}{2 x^{\frac {3}{2}}} \\ y \left (x \right ) &= \frac {3 \sqrt {3}}{2 x^{\frac {3}{2}}} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= c_{1} \sqrt {\frac {c_{1}^{10}}{\left (c_{1}^{4} x -1\right )^{2}}} \\ \end{align}

Problem 8862


ODE	\[ \boxed {{y^{\prime }}^{3}+a {y^{\prime }}^{2}+b y=-a b x} \]

program solution	\[ y = -a x \] Veriﬁed OK. \[ x = \frac {\left (10 a^{3}+27 a b x +27 b y-3 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}\right ) \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}+24 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}\, a +6 a^{2} \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}} \ln \left (2\right )+6 a^{2} \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}} \ln \left (3\right )-20 a^{4}+\left (-216 x b -\left (6 \ln \left (\frac {{\left (\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}+2 a \right )}^{2}}{\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}}\right )+5\right ) \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}}\right ) a^{2}-216 y a b +6 c_{2} \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}} b}{6 \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}} b} \] Veriﬁed OK. \[ x = \frac {8 \left (6 a^{2} \ln \left (12\right )+\left (-5-6 \ln \left (\left (-1+i \sqrt {3}\right ) \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}+\frac {4 \left (-i a \sqrt {3}-a +2 \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}\right ) a}{\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}}\right )\right ) a^{2}+6 c_{2} b \right ) \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}}+32 \left (-1-i \sqrt {3}\right ) a^{3} \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}+288 a \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}+i \left (\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {4}{3}}-80 a^{4}-864 a^{2} b x -864 y a b \right ) \sqrt {3}+\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {4}{3}}+80 a^{4}+864 a^{2} b x +864 y a b}{48 \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}} b} \] Warning, solution could not be veriﬁed \[ x = \frac {8 \left (\left (-5-6 \ln \left (\frac {-\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}}+8 \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}} a -4 a^{2}-i \sqrt {3}\, \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}}+4 i \sqrt {3}\, a^{2}}{12 \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}}\right )\right ) a^{2}+6 c_{2} b \right ) \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}}+32 \left (-1+i \sqrt {3}\right ) a^{3} \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}-288 a \left (i+\frac {\sqrt {3}}{3}\right ) \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}+i \left (-\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {4}{3}}+80 a^{4}+864 a^{2} b x +864 y a b \right ) \sqrt {3}+\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {4}{3}}+80 a^{4}+864 a^{2} b x +864 y a b}{48 \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}} b} \] Warning, solution could not be veriﬁed

Maple solution	\[ y \left (x \right ) = \frac {-{\mathrm e}^{3 \operatorname {RootOf}\left (-2 \textit {\_Z} \,a^{2}-3 \,{\mathrm e}^{2 \textit {\_Z}}+8 a \,{\mathrm e}^{\textit {\_Z}}+2 c_{1} b -5 a^{2}-2 b x \right )}+2 \,{\mathrm e}^{2 \operatorname {RootOf}\left (-2 \textit {\_Z} \,a^{2}-3 \,{\mathrm e}^{2 \textit {\_Z}}+8 a \,{\mathrm e}^{\textit {\_Z}}+2 c_{1} b -5 a^{2}-2 b x \right )} a -{\mathrm e}^{\operatorname {RootOf}\left (-2 \textit {\_Z} \,a^{2}-3 \,{\mathrm e}^{2 \textit {\_Z}}+8 a \,{\mathrm e}^{\textit {\_Z}}+2 c_{1} b -5 a^{2}-2 b x \right )} a^{2}-a x b}{b} \]

Problem 8863


ODE	\[ \boxed {{y^{\prime }}^{3}+x {y^{\prime }}^{2}-y=0} \]

program solution	\[ y = 0 \] Veriﬁed OK. \[ y = x +1 \] Veriﬁed OK. \[ x = \frac {24 \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}}{2}-\frac {27 y}{2}\right ) \left (x^{3}+\frac {3 x^{2}}{2}-3 y+3 c_{1} \right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}+96 \left (x +\frac {3}{2}\right ) \left (\left (\sqrt {3}\, \left (x^{3}-\frac {27 y}{4}\right ) \sqrt {27 y^{2}-4 x^{3} y}-\frac {x^{6}}{2}+\frac {27 x^{3} y}{2}-\frac {243 y^{2}}{4}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}+\left (-\frac {5 \sqrt {3}\, \left (x^{3}-\frac {54 y}{5}\right ) \sqrt {27 y^{2}-4 x^{3} y}}{2}+x^{6}-\frac {81 x^{3} y}{2}+243 y^{2}\right ) x \right )}{\left (2 x^{3}-3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}-27 y\right ) {\left (\left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}-2 x \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}+4 x^{2}-6 \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}\right )}^{2}} \] Warning, solution could not be veriﬁed \[ x = \frac {96 \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}}{2}-\frac {27 y}{2}\right ) \left (x^{3}+\frac {3 x^{2}}{2}-3 y+3 c_{1} \right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}+192 \left (\left (-3 \left (i+\frac {\sqrt {3}}{3}\right ) \left (x^{3}-\frac {27 y}{4}\right ) \sqrt {27 y^{2}-4 x^{3} y}+\frac {\left (1+i \sqrt {3}\right ) \left (x^{6}-27 x^{3} y+\frac {243 y^{2}}{2}\right )}{2}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}+x \left (-\frac {15 \left (x^{3}-\frac {54 y}{5}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {27 y^{2}-4 x^{3} y}}{2}+\left (-1+i \sqrt {3}\right ) \left (x^{6}-\frac {81 x^{3} y}{2}+243 y^{2}\right )\right )\right ) \left (x +\frac {3}{2}\right )}{\left (2 x^{3}-3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}-27 y\right ) \left (4 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}+4 x^{2}+4 x \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}+\left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}+12 \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}\right )^{2}} \] Warning, solution could not be veriﬁed \[ x = \frac {96 \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}}{2}-\frac {27 y}{2}\right ) \left (x^{3}+\frac {3 x^{2}}{2}-3 y+3 c_{1} \right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}-192 \left (\left (-3 \left (i-\frac {\sqrt {3}}{3}\right ) \left (x^{3}-\frac {27 y}{4}\right ) \sqrt {27 y^{2}-4 x^{3} y}+\frac {\left (-1+i \sqrt {3}\right ) \left (x^{6}-27 x^{3} y+\frac {243 y^{2}}{2}\right )}{2}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}+\left (-\frac {15 \left (i+\frac {\sqrt {3}}{3}\right ) \left (x^{3}-\frac {54 y}{5}\right ) \sqrt {27 y^{2}-4 x^{3} y}}{2}+\left (x^{6}-\frac {81 x^{3} y}{2}+243 y^{2}\right ) \left (1+i \sqrt {3}\right )\right ) x \right ) \left (x +\frac {3}{2}\right )}{\left (2 x^{3}-3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}-27 y\right ) \left (4 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}-4 x^{2}-4 x \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}-\left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}-12 \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}\right )^{2}} \] Warning, solution could not be veriﬁed

Maple solution	\begin{align} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\left (4 x^{2}-2 x \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+12 x +3 \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {2}{3}}+9\right )^{2} \left (4 x^{2}+4 x \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+12 x +3 \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {2}{3}}+9\right )}{-1728 x^{3}-7776 x^{2}-11664 x +23328 c_{1} +1296 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}+5832} \\ y \left (x \right ) &= \frac {\left (\frac {\left (-i \sqrt {3}-1\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {2}{3}}}{4}+\left (2 x +\frac {3}{2}\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+\left (i \sqrt {3}-1\right ) \left (x +\frac {3}{2}\right )^{2}\right ) {\left (\frac {\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {2}{3}} \left (i-\sqrt {3}\right )}{4}-i \left (-x +\frac {3}{2}\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+\left (x +\frac {3}{2}\right )^{2} \left (\sqrt {3}+i\right )\right )}^{2}}{216 x^{3}+972 x^{2}+1458 x -2916 c_{1} -162 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}-729} \\ y \left (x \right ) &= \frac {\left (\frac {\left (i \sqrt {3}-1\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {2}{3}}}{4}-\left (-2 x -\frac {3}{2}\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+\left (-i \sqrt {3}-1\right ) \left (x +\frac {3}{2}\right )^{2}\right ) {\left (\frac {\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )}{4}+i \left (x -\frac {3}{2}\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+\left (x +\frac {3}{2}\right )^{2} \left (i-\sqrt {3}\right )\right )}^{2}}{216 x^{3}+972 x^{2}+1458 x -2916 c_{1} -162 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}-729} \\ \end{align}

Problem 8864


ODE	\[ \boxed {{y^{\prime }}^{3}-y {y^{\prime }}^{2}+y^{2}=0} \]

program solution	\[ \int _{}^{y}\frac {6 \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{\left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}+2 \textit {\_a} \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}+4 \textit {\_a}^{2}}d \textit {\_a} = x +c_{1} \] Veriﬁed OK. \[ \int _{}^{y}-\frac {12 \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{4 i \textit {\_a}^{2} \sqrt {3}-i \sqrt {3}\, \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}+4 \textit {\_a}^{2}-4 \textit {\_a} \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}+\left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{2} \] Veriﬁed OK. \[ \int _{}^{y}\frac {12 \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{-i \sqrt {3}\, \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}-\left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}+4 \textit {\_a} \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}+4 i \textit {\_a}^{2} \sqrt {3}-4 \textit {\_a}^{2}}d \textit {\_a} = x +c_{3} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= 0 \\ x -6 \left (\int _{}^{y \left (x \right )}\frac {\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{\frac {2}{3}}+2 \left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{\frac {1}{3}} \textit {\_a} +4 \textit {\_a}^{2}}d \textit {\_a} \right )-c_{1} &= 0 \\ \frac {12 \left (\int _{}^{y \left (x \right )}\frac {\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{\left (\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{\frac {1}{3}}-2 \textit {\_a} \right ) \left (i \sqrt {3}\, \textit {\_a} +\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{\frac {1}{3}}+\textit {\_a} \right )}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}-c_{1} +x}{1+i \sqrt {3}} &= 0 \\ \frac {12 \left (\int _{}^{y \left (x \right )}\frac {\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{\left (-i \sqrt {3}\, \textit {\_a} +\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{\frac {1}{3}}+\textit {\_a} \right ) \left (-\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{\frac {1}{3}}+2 \textit {\_a} \right )}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}+c_{1} -x}{i \sqrt {3}-1} &= 0 \\ \end{align}

Problem 8865


ODE	\[ \boxed {{y^{\prime }}^{2}-\left (y^{4}+y^{2} x +x^{2}\right ) {y^{\prime }}^{2}+\left (y^{6} x +y^{4} x^{2}+x^{3} y^{2}\right ) y^{\prime }-y^{6} x^{3}=0} \]

program solution

Maple solution	\[ \text {No solution found} \]

Problem 8866


ODE	\[ \boxed {a {y^{\prime }}^{3}+b {y^{\prime }}^{2}+c y^{\prime }-y=d} \]

program solution	\[ \int _{}^{y}\frac {6 a \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 a b c -8 b^{3}\right )^{\frac {1}{3}}}{\left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 a b c -8 b^{3}\right )^{\frac {2}{3}}-2 b \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 a b c -8 b^{3}\right )^{\frac {1}{3}}-12 a c +4 b^{2}}d \textit {\_a} = x +c_{1} \] Veriﬁed OK. \[ \int _{}^{y}\frac {12 a \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 a b c -8 b^{3}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 a b c -8 b^{3}\right )^{\frac {2}{3}}+12 i \sqrt {3}\, a c -4 i \sqrt {3}\, b^{2}-\left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 a b c -8 b^{3}\right )^{\frac {2}{3}}-4 b \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 a b c -8 b^{3}\right )^{\frac {1}{3}}+12 a c -4 b^{2}}d \textit {\_a} = x +c_{2} \] Veriﬁed OK. \[ \int _{}^{y}-\frac {12 a \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 a b c -8 b^{3}\right )^{\frac {1}{3}}}{12 i \sqrt {3}\, a c -4 i \sqrt {3}\, b^{2}+i \sqrt {3}\, \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 a b c -8 b^{3}\right )^{\frac {2}{3}}-12 a c +4 b^{2}+4 b \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 a b c -8 b^{3}\right )^{\frac {1}{3}}+\left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 a b c -8 b^{3}\right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{3} \] Veriﬁed OK.

Maple solution	\begin{align} 3 \sqrt {3}\, 2^{\frac {1}{3}} a \left (\int _{}^{y \left (x \right )}\frac {\left (9 \sqrt {27 \left (d +\textit {\_a} \right )^{2} a^{2}+18 \left (\left (d +\textit {\_a} \right ) b +\frac {2 c^{2}}{9}\right ) c a +\left (-4 \textit {\_a} -4 d \right ) b^{3}-b^{2} c^{2}}\, a +27 \left (a^{2} \left (d +\textit {\_a} \right )+\frac {a c b}{3}-\frac {2 b^{3}}{27}\right ) \sqrt {3}\right )^{\frac {1}{3}}}{\sqrt {3}\, 2^{\frac {1}{3}} \left (9 \sqrt {27 \left (d +\textit {\_a} \right )^{2} a^{2}+18 \left (\left (d +\textit {\_a} \right ) b +\frac {2 c^{2}}{9}\right ) c a +\left (-4 \textit {\_a} -4 d \right ) b^{3}-b^{2} c^{2}}\, a +27 \left (a^{2} \left (d +\textit {\_a} \right )+\frac {a c b}{3}-\frac {2 b^{3}}{27}\right ) \sqrt {3}\right )^{\frac {1}{3}} b -3^{\frac {1}{3}} \left (9 \sqrt {27 \left (d +\textit {\_a} \right )^{2} a^{2}+18 \left (\left (d +\textit {\_a} \right ) b +\frac {2 c^{2}}{9}\right ) c a +\left (-4 \textit {\_a} -4 d \right ) b^{3}-b^{2} c^{2}}\, a +27 \left (a^{2} \left (d +\textit {\_a} \right )+\frac {a c b}{3}-\frac {2 b^{3}}{27}\right ) \sqrt {3}\right )^{\frac {2}{3}}+3 \,3^{\frac {2}{3}} 2^{\frac {2}{3}} \left (a c -\frac {b^{2}}{3}\right )}d \textit {\_a} \right )+x -c_{1} &= 0 \\ \frac {12 \,2^{\frac {1}{3}} \sqrt {3}\, a \left (\int _{}^{y \left (x \right )}\frac {\left (9 \sqrt {27 \left (d +\textit {\_a} \right )^{2} a^{2}+18 \left (\left (d +\textit {\_a} \right ) b +\frac {2 c^{2}}{9}\right ) c a +\left (-4 \textit {\_a} -4 d \right ) b^{3}-b^{2} c^{2}}\, a +27 \left (a^{2} \left (d +\textit {\_a} \right )+\frac {a c b}{3}-\frac {2 b^{3}}{27}\right ) \sqrt {3}\right )^{\frac {1}{3}}}{-3 \,2^{\frac {1}{3}} \left (i-\frac {\sqrt {3}}{3}\right ) b \left (9 \sqrt {27 \left (d +\textit {\_a} \right )^{2} a^{2}+18 \left (\left (d +\textit {\_a} \right ) b +\frac {2 c^{2}}{9}\right ) c a +\left (-4 \textit {\_a} -4 d \right ) b^{3}-b^{2} c^{2}}\, a +27 \left (a^{2} \left (d +\textit {\_a} \right )+\frac {a c b}{3}-\frac {2 b^{3}}{27}\right ) \sqrt {3}\right )^{\frac {1}{3}}+2 \,3^{\frac {1}{3}} \left (9 \sqrt {27 \left (d +\textit {\_a} \right )^{2} a^{2}+18 \left (\left (d +\textit {\_a} \right ) b +\frac {2 c^{2}}{9}\right ) c a +\left (-4 \textit {\_a} -4 d \right ) b^{3}-b^{2} c^{2}}\, a +27 \left (a^{2} \left (d +\textit {\_a} \right )+\frac {a c b}{3}-\frac {2 b^{3}}{27}\right ) \sqrt {3}\right )^{\frac {2}{3}}+9 \,2^{\frac {2}{3}} \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right ) \left (a c -\frac {b^{2}}{3}\right )}d \textit {\_a} \right )+\left (x -c_{1} \right ) \left (1+i \sqrt {3}\right )}{1+i \sqrt {3}} &= 0 \\ \frac {12 \,2^{\frac {1}{3}} \sqrt {3}\, a \left (\int _{}^{y \left (x \right )}\frac {\left (9 \sqrt {27 \left (d +\textit {\_a} \right )^{2} a^{2}+18 \left (\left (d +\textit {\_a} \right ) b +\frac {2 c^{2}}{9}\right ) c a +\left (-4 \textit {\_a} -4 d \right ) b^{3}-b^{2} c^{2}}\, a +27 \left (a^{2} \left (d +\textit {\_a} \right )+\frac {a c b}{3}-\frac {2 b^{3}}{27}\right ) \sqrt {3}\right )^{\frac {1}{3}}}{-3 \,2^{\frac {1}{3}} \left (i+\frac {\sqrt {3}}{3}\right ) b \left (9 \sqrt {27 \left (d +\textit {\_a} \right )^{2} a^{2}+18 \left (\left (d +\textit {\_a} \right ) b +\frac {2 c^{2}}{9}\right ) c a +\left (-4 \textit {\_a} -4 d \right ) b^{3}-b^{2} c^{2}}\, a +27 \left (a^{2} \left (d +\textit {\_a} \right )+\frac {a c b}{3}-\frac {2 b^{3}}{27}\right ) \sqrt {3}\right )^{\frac {1}{3}}-2 \,3^{\frac {1}{3}} \left (9 \sqrt {27 \left (d +\textit {\_a} \right )^{2} a^{2}+18 \left (\left (d +\textit {\_a} \right ) b +\frac {2 c^{2}}{9}\right ) c a +\left (-4 \textit {\_a} -4 d \right ) b^{3}-b^{2} c^{2}}\, a +27 \left (a^{2} \left (d +\textit {\_a} \right )+\frac {a c b}{3}-\frac {2 b^{3}}{27}\right ) \sqrt {3}\right )^{\frac {2}{3}}+9 \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) 2^{\frac {2}{3}} \left (a c -\frac {b^{2}}{3}\right )}d \textit {\_a} \right )+\left (x -c_{1} \right ) \left (i \sqrt {3}-1\right )}{i \sqrt {3}-1} &= 0 \\ \end{align}

Problem 8867


ODE	\[ \boxed {x {y^{\prime }}^{3}-y {y^{\prime }}^{2}=-a} \]

program solution	\[ y = c_{1} x +\frac {a}{c_{1}^{2}} \] Veriﬁed OK. \[ y = \frac {3 a \,x^{2} 2^{\frac {1}{3}}}{2 \left (a \,x^{2}\right )^{\frac {2}{3}}} \] Veriﬁed OK. \[ y = -\frac {3 a \,x^{2} 2^{\frac {1}{3}}}{\left (a \,x^{2}\right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )} \] Veriﬁed OK. \[ y = \frac {3 a \,x^{2} 2^{\frac {1}{3}}}{\left (a \,x^{2}\right )^{\frac {2}{3}} \left (-1+i \sqrt {3}\right )} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \frac {3 \,2^{\frac {1}{3}} \left (a \,x^{2}\right )^{\frac {1}{3}}}{2} \\ y \left (x \right ) &= -\frac {3 \,2^{\frac {1}{3}} \left (a \,x^{2}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{4} \\ y \left (x \right ) &= \frac {3 \,2^{\frac {1}{3}} \left (a \,x^{2}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{4} \\ y \left (x \right ) &= \frac {c_{1}^{3} x +a}{c_{1}^{2}} \\ \end{align}

Problem 8868


ODE	\[ \boxed {4 x {y^{\prime }}^{3}-6 y {y^{\prime }}^{2}+3 y=x} \]

program solution	\[ y = x \] Veriﬁed OK. \[ y = \frac {-\sqrt {3}\, x -2 x}{1+\sqrt {3}} \] Veriﬁed OK. \[ y = \frac {-\sqrt {3}\, x +2 x}{\sqrt {3}-1} \] Veriﬁed OK. \[ x = c_{3} \left (-1+\frac {{\left (\left (y^{3}-3 x^{2} y+x^{3}+\sqrt {-6 y^{4}+2 x y^{3}+9 y^{2} x^{2}-6 x^{3} y+x^{4}}\, x \right )^{\frac {1}{3}}+\frac {y^{2}}{\left (y^{3}-3 x^{2} y+x^{3}+\sqrt {-6 y^{4}+2 x y^{3}+9 y^{2} x^{2}-6 x^{3} y+x^{4}}\, x \right )^{\frac {1}{3}}}+y\right )}^{2}}{2 x^{2}}\right ) \] Veriﬁed OK. \[ x = \frac {c_{3} \left (\left (-2 x^{2}+3 y^{2}\right ) \left (y^{3}-3 x^{2} y+x^{3}+\sqrt {-6 y^{4}+2 x y^{3}+9 y^{2} x^{2}-6 x^{3} y+x^{4}}\, x \right )^{\frac {2}{3}}-\frac {\left (x^{3}-3 x^{2} y+3 y^{3}+\sqrt {-6 y^{4}+2 x y^{3}+9 y^{2} x^{2}-6 x^{3} y+x^{4}}\, x \right ) \left (1+i \sqrt {3}\right ) \left (y^{3}-3 x^{2} y+x^{3}+\sqrt {-6 y^{4}+2 x y^{3}+9 y^{2} x^{2}-6 x^{3} y+x^{4}}\, x \right )^{\frac {1}{3}}}{2}+\left (-1+i \sqrt {3}\right ) y \left (x^{3}-3 x^{2} y+\frac {3 y^{3}}{2}+\sqrt {-6 y^{4}+2 x y^{3}+9 y^{2} x^{2}-6 x^{3} y+x^{4}}\, x \right )\right )}{2 \left (y^{3}-3 x^{2} y+x^{3}+\sqrt {-6 y^{4}+2 x y^{3}+9 y^{2} x^{2}-6 x^{3} y+x^{4}}\, x \right )^{\frac {2}{3}} x^{2}} \] Veriﬁed OK. \[ x = -\frac {\left (\left (2 x^{2}-3 y^{2}\right ) \left (y^{3}-3 x^{2} y+x^{3}+\sqrt {-6 y^{4}+2 x y^{3}+9 y^{2} x^{2}-6 x^{3} y+x^{4}}\, x \right )^{\frac {2}{3}}-\frac {\left (-1+i \sqrt {3}\right ) \left (x^{3}-3 x^{2} y+3 y^{3}+\sqrt {-6 y^{4}+2 x y^{3}+9 y^{2} x^{2}-6 x^{3} y+x^{4}}\, x \right ) \left (y^{3}-3 x^{2} y+x^{3}+\sqrt {-6 y^{4}+2 x y^{3}+9 y^{2} x^{2}-6 x^{3} y+x^{4}}\, x \right )^{\frac {1}{3}}}{2}+y \left (x^{3}-3 x^{2} y+\frac {3 y^{3}}{2}+\sqrt {-6 y^{4}+2 x y^{3}+9 y^{2} x^{2}-6 x^{3} y+x^{4}}\, x \right ) \left (1+i \sqrt {3}\right )\right ) c_{3}}{2 \left (y^{3}-3 x^{2} y+x^{3}+\sqrt {-6 y^{4}+2 x y^{3}+9 y^{2} x^{2}-6 x^{3} y+x^{4}}\, x \right )^{\frac {2}{3}} x^{2}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -\frac {\left (1+\sqrt {3}\right ) x}{2} \\ y \left (x \right ) &= \frac {\left (\sqrt {3}-1\right ) x}{2} \\ y \left (x \right ) &= x \\ y \left (x \right ) &= \frac {-\left (x +c_{1} \right ) \sqrt {2}\, \sqrt {c_{1} \left (x +c_{1} \right )}-c_{1}^{2}}{3 c_{1}} \\ y \left (x \right ) &= \frac {\left (x +c_{1} \right ) \sqrt {2}\, \sqrt {c_{1} \left (x +c_{1} \right )}-c_{1}^{2}}{3 c_{1}} \\ \end{align}

Problem 8869


ODE	\[ \boxed {8 x {y^{\prime }}^{3}-12 y {y^{\prime }}^{2}+9 y=0} \]

program solution	\[ y = 0 \] Veriﬁed OK. \[ y = -\frac {3 x}{2} \] Veriﬁed OK. \[ y = \frac {3 x}{2} \] Veriﬁed OK. \[ x = c_{3} \left (\frac {\left (2^{\frac {2}{3}} {\left (y \left (2 y^{2}+3 \sqrt {-4 y^{2}+9 x^{2}}\, x -9 x^{2}\right )\right )}^{\frac {2}{3}}+2 y^{2} 2^{\frac {1}{3}}+2 y {\left (y \left (2 y^{2}+3 \sqrt {-4 y^{2}+9 x^{2}}\, x -9 x^{2}\right )\right )}^{\frac {1}{3}}\right )^{2}}{4 x^{2} {\left (y \left (2 y^{2}+3 \sqrt {-4 y^{2}+9 x^{2}}\, x -9 x^{2}\right )\right )}^{\frac {2}{3}}}-3\right ) \] Veriﬁed OK. \[ x = c_{3} \left (-3+\frac {{\left (-i {\left (y \left (2 y^{2}+3 \sqrt {-4 y^{2}+9 x^{2}}\, x -9 x^{2}\right )\right )}^{\frac {2}{3}} 2^{\frac {2}{3}} \sqrt {3}+2 i 2^{\frac {1}{3}} \sqrt {3}\, y^{2}+2^{\frac {2}{3}} {\left (y \left (2 y^{2}+3 \sqrt {-4 y^{2}+9 x^{2}}\, x -9 x^{2}\right )\right )}^{\frac {2}{3}}+2 y^{2} 2^{\frac {1}{3}}-4 y {\left (y \left (2 y^{2}+3 \sqrt {-4 y^{2}+9 x^{2}}\, x -9 x^{2}\right )\right )}^{\frac {1}{3}}\right )}^{2}}{16 x^{2} {\left (y \left (2 y^{2}+3 \sqrt {-4 y^{2}+9 x^{2}}\, x -9 x^{2}\right )\right )}^{\frac {2}{3}}}\right ) \] Veriﬁed OK. \[ x = -c_{3} \left (\frac {{\left (-{\left (y \left (2 y^{2}+3 \sqrt {-4 y^{2}+9 x^{2}}\, x -9 x^{2}\right )\right )}^{\frac {2}{3}} 2^{\frac {2}{3}} \sqrt {3}+2 \,2^{\frac {1}{3}} \sqrt {3}\, y^{2}+i {\left (y \left (2 y^{2}+3 \sqrt {-4 y^{2}+9 x^{2}}\, x -9 x^{2}\right )\right )}^{\frac {2}{3}} 2^{\frac {2}{3}}+2 i y^{2} 2^{\frac {1}{3}}-4 i y {\left (y \left (2 y^{2}+3 \sqrt {-4 y^{2}+9 x^{2}}\, x -9 x^{2}\right )\right )}^{\frac {1}{3}}\right )}^{2}}{16 x^{2} {\left (y \left (2 y^{2}+3 \sqrt {-4 y^{2}+9 x^{2}}\, x -9 x^{2}\right )\right )}^{\frac {2}{3}}}+3\right ) \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -\frac {3 x}{2} \\ y \left (x \right ) &= \frac {3 x}{2} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= -\frac {\left (3 c_{1} +x \right ) \sqrt {c_{1} \left (3 c_{1} +x \right )}}{3 c_{1}} \\ y \left (x \right ) &= \frac {\left (3 c_{1} +x \right ) \sqrt {c_{1} \left (3 c_{1} +x \right )}}{3 c_{1}} \\ \end{align}

Problem 8870


ODE	\[ \boxed {\left (-a^{2}+x^{2}\right ) {y^{\prime }}^{3}+b x \left (-a^{2}+x^{2}\right ) {y^{\prime }}^{2}+y^{\prime }=-x b} \]

program solution	\[ y = -\frac {b \,x^{2}}{2}+c_{3} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -\frac {b \,x^{2}}{2}+c_{1} \\ y \left (x \right ) &= \arctan \left (\frac {x}{\sqrt {a^{2}-x^{2}}}\right )+c_{1} \\ y \left (x \right ) &= -\arctan \left (\frac {x}{\sqrt {a^{2}-x^{2}}}\right )+c_{1} \\ \end{align}

Problem 8871


ODE	\[ \boxed {x^{3} {y^{\prime }}^{3}-3 x^{2} y {y^{\prime }}^{2}+\left (3 y^{2} x +x^{6}\right ) y^{\prime }-y^{3}-2 x^{5} y=0} \]

program solution	\[ \frac {5 \ln \left (x \right )}{2} = \int _{}^{\frac {y}{x^{\frac {5}{2}}}}\frac {\frac {5 \left (3 \sqrt {3}\, \textit {\_a} +\sqrt {27 \textit {\_a}^{2}+4}\right )^{\frac {2}{3}} \sqrt {3}\, 2^{\frac {1}{3}} \textit {\_a}}{2}-\frac {5 \left (3 \sqrt {3}\, \textit {\_a} +\sqrt {27 \textit {\_a}^{2}+4}\right )^{\frac {2}{3}} 2^{\frac {1}{3}} \sqrt {27 \textit {\_a}^{2}+4}}{6}-\frac {5 \,2^{\frac {2}{3}} \left (3 \sqrt {3}\, \textit {\_a} +\sqrt {27 \textit {\_a}^{2}+4}\right )^{\frac {1}{3}}}{3}-\frac {5 \sqrt {27 \textit {\_a}^{2}+4}}{3}}{\textit {\_a} \sqrt {27 \textit {\_a}^{2}+4}}d \textit {\_a} +c_{1} \] Veriﬁed OK.

Maple solution	\[ \text {No solution found} \]

Problem 8872


ODE	\[ \boxed {2 \left (y^{\prime } x +y\right )^{3}-y y^{\prime }=0} \]

program solution	\[ -\ln \left (x \right ) = \int _{}^{x y}-\frac {\left (3 i+\sqrt {3}\right ) \left (12 \sqrt {3}\, \sqrt {\textit {\_a}}-4 \sqrt {27 \textit {\_a} -2}\right )^{\frac {1}{3}}}{\left (i \left (3 \sqrt {3}\, \sqrt {\textit {\_a}}-\sqrt {27 \textit {\_a} -2}\right )^{\frac {2}{3}} \sqrt {3}-\left (3 \sqrt {3}\, \sqrt {\textit {\_a}}-\sqrt {27 \textit {\_a} -2}\right )^{\frac {2}{3}}+2 \,2^{\frac {1}{3}}\right ) \sqrt {\textit {\_a}}}d \textit {\_a} +c_{1} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= 0 \\ \text {Expression too large to display} \\ \text {Expression too large to display} \\ \text {Expression too large to display} \\ \end{align}

Problem 8873


ODE	\[ \boxed {{y^{\prime }}^{3} \sin \left (x \right )-\left (y \sin \left (x \right )-\cos \left (x \right )^{2}\right ) {y^{\prime }}^{2}-\left (y \cos \left (x \right )^{2}+\sin \left (x \right )\right ) y^{\prime }+y \sin \left (x \right )=0} \]

program solution	\[ y = -\cos \left (x \right )+c_{2} \] Veriﬁed OK. \[ y = -\ln \left (\tan \left (\frac {x}{2}\right )\right )+c_{3} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= c_{1} {\mathrm e}^{x} \\ y \left (x \right ) &= -\ln \left (\csc \left (x \right )-\cot \left (x \right )\right )+c_{1} \\ y \left (x \right ) &= -\cos \left (x \right )+c_{1} \\ \end{align}

Problem 8874


ODE	\[ \boxed {2 y {y^{\prime }}^{3}-y {y^{\prime }}^{2}+2 y^{\prime } x=x} \]

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

Maple solution	\begin{align} x \left (1+\frac {c_{1}}{\left (\frac {-\sqrt {-x y \left (x \right )}+y \left (x \right )}{y \left (x \right )}\right )^{\frac {2}{3}} \left (\frac {-x +\sqrt {-x y \left (x \right )}+y \left (x \right )}{y \left (x \right )}\right )^{\frac {2}{3}} y \left (x \right )}\right ) &= 0 \\ x \left (1+\frac {c_{1}}{\left (\frac {\sqrt {-x y \left (x \right )}+y \left (x \right )}{y \left (x \right )}\right )^{\frac {2}{3}} \left (\frac {-x -\sqrt {-x y \left (x \right )}+y \left (x \right )}{y \left (x \right )}\right )^{\frac {2}{3}} y \left (x \right )}\right ) &= 0 \\ y \left (x \right ) &= \frac {x}{2}+c_{1} \\ \end{align}

Problem 8875


ODE	\[ \boxed {y^{2} {y^{\prime }}^{3}+2 y^{\prime } x -y=0} \]

program solution	\[ \frac {3 \ln \left (x \right )}{4} = \int _{}^{\frac {y}{x^{\frac {3}{4}}}}-\frac {9 \textit {\_a} \left (3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}+32}\right )^{\frac {1}{3}} 3^{\frac {1}{6}}}{-2 \,36^{\frac {1}{3}} {\left (\left (3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}+32}\right )^{2}\right )}^{\frac {1}{3}}+9 \,3^{\frac {1}{6}} \left (3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}+32}\right )^{\frac {1}{3}} \textit {\_a}^{2}+8 \,18^{\frac {1}{3}}}d \textit {\_a} +c_{1} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -\frac {2 \left (-x^{3}\right )^{\frac {1}{4}} 6^{\frac {1}{4}}}{3} \\ y \left (x \right ) &= \frac {2 \left (-x^{3}\right )^{\frac {1}{4}} 6^{\frac {1}{4}}}{3} \\ y \left (x \right ) &= -\frac {2 i \left (-x^{3}\right )^{\frac {1}{4}} 6^{\frac {1}{4}}}{3} \\ y \left (x \right ) &= \frac {2 i \left (-x^{3}\right )^{\frac {1}{4}} 6^{\frac {1}{4}}}{3} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \sqrt {c_{1} \left (c_{1}^{2}+2 x \right )} \\ y \left (x \right ) &= -\sqrt {c_{1} \left (c_{1}^{2}+2 x \right )} \\ \end{align}

Problem 8876


ODE	\[ \boxed {16 y^{2} {y^{\prime }}^{3}+2 y^{\prime } x -y=0} \]

program solution	\[ \frac {3 \ln \left (x \right )}{4} = \int _{}^{\frac {y}{x^{\frac {3}{4}}}}\frac {9 \textit {\_a} \left (3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}+2}\right )^{\frac {1}{3}} 3^{\frac {1}{6}}}{18^{\frac {1}{3}} {\left (\left (3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}+2}\right )^{2}\right )}^{\frac {1}{3}}-9 \,3^{\frac {1}{6}} \left (3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}+2}\right )^{\frac {1}{3}} \textit {\_a}^{2}-6^{\frac {2}{3}}}d \textit {\_a} +c_{1} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -\frac {\left (-x^{3}\right )^{\frac {1}{4}} 6^{\frac {1}{4}}}{3} \\ y \left (x \right ) &= \frac {\left (-x^{3}\right )^{\frac {1}{4}} 6^{\frac {1}{4}}}{3} \\ y \left (x \right ) &= -\frac {i \left (-x^{3}\right )^{\frac {1}{4}} 6^{\frac {1}{4}}}{3} \\ y \left (x \right ) &= \frac {i \left (-x^{3}\right )^{\frac {1}{4}} 6^{\frac {1}{4}}}{3} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \sqrt {2}\, \sqrt {c_{1} \left (8 c_{1}^{2}+x \right )} \\ y \left (x \right ) &= -\sqrt {2}\, \sqrt {c_{1} \left (8 c_{1}^{2}+x \right )} \\ \end{align}

Problem 8877


ODE	\[ \boxed {x y^{2} {y^{\prime }}^{3}-y^{3} {y^{\prime }}^{2}+x \left (x^{2}+1\right ) y^{\prime }-x^{2} y=0} \]

program solution

Maple solution	\[ \text {No solution found} \]

Problem 8878


ODE	\[ \boxed {x^{7} y^{2} {y^{\prime }}^{3}-\left (3 x^{6} y^{3}-1\right ) {y^{\prime }}^{2}+3 x^{5} y^{4} y^{\prime }-x^{4} y^{5}=0} \]

program solution	\[ -3 \,2^{\frac {2}{3}} 3^{\frac {5}{6}} \left (\int _{}^{x^{2} y}\frac {\textit {\_a}^{2} \left (9 \sqrt {27 \textit {\_a}^{3}-4}\, \textit {\_a}^{\frac {9}{2}}+\left (-27 \textit {\_a}^{6}+18 \textit {\_a}^{3}-2\right ) \sqrt {3}\right )^{\frac {1}{3}}}{9 \,3^{\frac {5}{6}} \left (\textit {\_a}^{3}-\frac {1}{9}\right ) 2^{\frac {2}{3}} \left (9 \sqrt {27 \textit {\_a}^{3}-4}\, \textit {\_a}^{\frac {9}{2}}+\left (-27 \textit {\_a}^{6}+18 \textit {\_a}^{3}-2\right ) \sqrt {3}\right )^{\frac {1}{3}}-36 \textit {\_a}^{3}+2^{\frac {1}{3}} 3^{\frac {2}{3}} \left (9 \sqrt {27 \textit {\_a}^{3}-4}\, \textit {\_a}^{\frac {9}{2}}+\left (-27 \textit {\_a}^{6}+18 \textit {\_a}^{3}-2\right ) \sqrt {3}\right )^{\frac {2}{3}}+6}d \textit {\_a} \right )-c_{1} +\ln \left (x \right ) = 0 \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \frac {2^{\frac {2}{3}}}{3 x^{2}} \\ y \left (x \right ) &= -\frac {2^{\frac {2}{3}} \left (1+i \sqrt {3}\right )}{6 x^{2}} \\ y \left (x \right ) &= \frac {2^{\frac {2}{3}} \left (i \sqrt {3}-1\right )}{6 x^{2}} \\ y \left (x \right ) &= 0 \\ \text {Expression too large to display} \\ \text {Expression too large to display} \\ \text {Expression too large to display} \\ \end{align}

Problem 8879


ODE	\[ \boxed {{y^{\prime }}^{4}-\left (y-a \right )^{3} \left (y-b \right )^{2}=0} \]

program solution	\[ \int _{}^{y}\frac {1}{\left (\textit {\_a}^{5}-3 \textit {\_a}^{4} a -2 \textit {\_a}^{4} b +3 \textit {\_a}^{3} a^{2}+6 \textit {\_a}^{3} a b +\textit {\_a}^{3} b^{2}-\textit {\_a}^{2} a^{3}-6 \textit {\_a}^{2} a^{2} b -3 \textit {\_a}^{2} a \,b^{2}+2 \textit {\_a} \,a^{3} b +3 \textit {\_a} \,a^{2} b^{2}-a^{3} b^{2}\right )^{\frac {1}{4}}}d \textit {\_a} = x +c_{1} \] Veriﬁed OK. \[ \int _{}^{y}-\frac {i}{\left (\textit {\_a}^{5}-3 \textit {\_a}^{4} a -2 \textit {\_a}^{4} b +3 \textit {\_a}^{3} a^{2}+6 \textit {\_a}^{3} a b +\textit {\_a}^{3} b^{2}-\textit {\_a}^{2} a^{3}-6 \textit {\_a}^{2} a^{2} b -3 \textit {\_a}^{2} a \,b^{2}+2 \textit {\_a} \,a^{3} b +3 \textit {\_a} \,a^{2} b^{2}-a^{3} b^{2}\right )^{\frac {1}{4}}}d \textit {\_a} = x +c_{2} \] Veriﬁed OK. \[ \int _{}^{y}-\frac {1}{\left (\textit {\_a}^{5}-3 \textit {\_a}^{4} a -2 \textit {\_a}^{4} b +3 \textit {\_a}^{3} a^{2}+6 \textit {\_a}^{3} a b +\textit {\_a}^{3} b^{2}-\textit {\_a}^{2} a^{3}-6 \textit {\_a}^{2} a^{2} b -3 \textit {\_a}^{2} a \,b^{2}+2 \textit {\_a} \,a^{3} b +3 \textit {\_a} \,a^{2} b^{2}-a^{3} b^{2}\right )^{\frac {1}{4}}}d \textit {\_a} = x +c_{3} \] Veriﬁed OK. \[ \int _{}^{y}\frac {i}{\left (\textit {\_a}^{5}-3 \textit {\_a}^{4} a -2 \textit {\_a}^{4} b +3 \textit {\_a}^{3} a^{2}+6 \textit {\_a}^{3} a b +\textit {\_a}^{3} b^{2}-\textit {\_a}^{2} a^{3}-6 \textit {\_a}^{2} a^{2} b -3 \textit {\_a}^{2} a \,b^{2}+2 \textit {\_a} \,a^{3} b +3 \textit {\_a} \,a^{2} b^{2}-a^{3} b^{2}\right )^{\frac {1}{4}}}d \textit {\_a} = x +c_{4} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= a \\ y \left (x \right ) &= b \\ x -\left (\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{3} \left (\textit {\_a} -b \right )^{2}\right )^{\frac {1}{4}}}d \textit {\_a} \right )-c_{1} &= 0 \\ x -i \left (\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{3} \left (\textit {\_a} -b \right )^{2}\right )^{\frac {1}{4}}}d \textit {\_a} \right )-c_{1} &= 0 \\ x +i \left (\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{3} \left (\textit {\_a} -b \right )^{2}\right )^{\frac {1}{4}}}d \textit {\_a} \right )-c_{1} &= 0 \\ x +\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{3} \left (\textit {\_a} -b \right )^{2}\right )^{\frac {1}{4}}}d \textit {\_a} -c_{1} &= 0 \\ \end{align}

Problem 8880


ODE	\[ \boxed {{y^{\prime }}^{4}+3 \left (x -1\right ) {y^{\prime }}^{2}-3 \left (2 y-1\right ) y^{\prime }=-3 x} \]

program solution	\[ y = -x +\frac {5}{6} \] Veriﬁed OK. \[ y = \frac {1}{6}+x \] Veriﬁed OK. \[ x = \operatorname {RootOf}\left (\textit {\_Z}^{4}+\left (3 x -3\right ) \textit {\_Z}^{2}+\left (-6 y+3\right ) \textit {\_Z} +3 x \right ) \left (c_{1} +\operatorname {RootOf}\left (\textit {\_Z}^{4}+\left (3 x -3\right ) \textit {\_Z}^{2}+\left (-6 y+3\right ) \textit {\_Z} +3 x \right )\right ) \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= x +\frac {1}{6} \\ y \left (x \right ) &= -x +\frac {5}{6} \\ y \left (x \right ) &= \frac {\left (3-c_{1}^{3}+\left (-5 x +3\right ) c_{1} \right ) \sqrt {c_{1}^{2}+4 x}-c_{1}^{4}+\left (-7 x +3\right ) c_{1}^{2}+3 c_{1} -8 x^{2}}{6 c_{1} +6 \sqrt {c_{1}^{2}+4 x}} \\ y \left (x \right ) &= \frac {\left (-3+c_{1}^{3}+\left (5 x -3\right ) c_{1} \right ) \sqrt {c_{1}^{2}+4 x}-c_{1}^{4}+\left (-7 x +3\right ) c_{1}^{2}+3 c_{1} -8 x^{2}}{6 c_{1} -6 \sqrt {c_{1}^{2}+4 x}} \\ \end{align}

Problem 8881


ODE	\[ \boxed {{y^{\prime }}^{4}-4 y \left (y^{\prime } x -2 y\right )^{2}=0} \]

program solution

Maple solution	\begin{align} y \left (x \right ) &= \frac {x^{4}}{16} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) \left (\sqrt {x^{2}-4 \sqrt {y \left (x \right )}}-x \right )^{-\frac {2 \sqrt {x^{2} y \left (x \right )-4 y \left (x \right )^{\frac {3}{2}}}}{\sqrt {x^{2}-4 \sqrt {y \left (x \right )}}\, \sqrt {y \left (x \right )}}} \left (\sqrt {x^{2}-4 \sqrt {y \left (x \right )}}+x \right )^{\frac {2 \sqrt {x^{2} y \left (x \right )-4 y \left (x \right )^{\frac {3}{2}}}}{\sqrt {x^{2}-4 \sqrt {y \left (x \right )}}\, \sqrt {y \left (x \right )}}}-c_{1} &= 0 \\ \end{align}

Problem 8882


ODE	\[ \boxed {{y^{\prime }}^{6}-\left (y-a \right )^{4} \left (y-b \right )^{3}=0} \]

program solution	\[ \int _{}^{y}\frac {1}{\left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} = x +c_{1} \] Veriﬁed OK. \[ \int _{}^{y}\frac {2}{\left (1+i \sqrt {3}\right ) \left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} = x +c_{2} \] Veriﬁed OK. \[ \int _{}^{y}\frac {2}{\left (-1+i \sqrt {3}\right ) \left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} = x +c_{3} \] Veriﬁed OK. \[ \int _{}^{y}-\frac {1}{\left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} = x +c_{4} \] Veriﬁed OK. \[ \int _{}^{y}-\frac {2}{\left (1+i \sqrt {3}\right ) \left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} = x +c_{5} \] Veriﬁed OK. \[ \int _{}^{y}-\frac {2}{\left (-1+i \sqrt {3}\right ) \left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} = x +c_{6} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= a \\ y \left (x \right ) &= b \\ x -\left (\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{4} \left (\textit {\_a} -b \right )^{3}\right )^{\frac {1}{6}}}d \textit {\_a} \right )-c_{1} &= 0 \\ \frac {2 \left (\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{4} \left (\textit {\_a} -b \right )^{3}\right )^{\frac {1}{6}}}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}+x -c_{1}}{1+i \sqrt {3}} &= 0 \\ \frac {-2 \left (\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{4} \left (\textit {\_a} -b \right )^{3}\right )^{\frac {1}{6}}}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}-x +c_{1}}{i \sqrt {3}-1} &= 0 \\ \frac {2 \left (\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{4} \left (\textit {\_a} -b \right )^{3}\right )^{\frac {1}{6}}}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}-x +c_{1}}{i \sqrt {3}-1} &= 0 \\ \frac {-2 \left (\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{4} \left (\textit {\_a} -b \right )^{3}\right )^{\frac {1}{6}}}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}+x -c_{1}}{1+i \sqrt {3}} &= 0 \\ x +\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{4} \left (\textit {\_a} -b \right )^{3}\right )^{\frac {1}{6}}}d \textit {\_a} -c_{1} &= 0 \\ \end{align}

Problem 8883


ODE	\[ \boxed {x^{2} \left ({y^{\prime }}^{2}+1\right )^{3}=a^{2}} \]

program solution	\[ y = -\frac {\sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{a^{4}}}\, \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{\left (a^{2} x \right )^{\frac {2}{3}}}+c_{1} \] Veriﬁed OK. \[ y = \frac {\sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{a^{4}}}\, \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{\left (a^{2} x \right )^{\frac {2}{3}}}+c_{2} \] Veriﬁed OK. \[ y = -\frac {\sqrt {2}\, \sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (i \sqrt {3}\, a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{a^{4}}}\, \left (i \sqrt {3}\, a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{4 \left (a^{2} x \right )^{\frac {2}{3}}}+c_{3} \] Veriﬁed OK. \[ y = \frac {\sqrt {2}\, \sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (i \sqrt {3}\, a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{a^{4}}}\, \left (i \sqrt {3}\, a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{4 \left (a^{2} x \right )^{\frac {2}{3}}}+c_{4} \] Veriﬁed OK. \[ y = \frac {\sqrt {-\frac {2 \left (a^{2} x \right )^{\frac {4}{3}} \left (i \sqrt {3}\, a^{2}+2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}\right )}{a^{4}}}\, \left (i \sqrt {3}\, a^{2}+2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}\right )}{4 \left (a^{2} x \right )^{\frac {2}{3}}}+c_{5} \] Veriﬁed OK. \[ y = -\frac {\sqrt {-\frac {2 \left (a^{2} x \right )^{\frac {4}{3}} \left (i \sqrt {3}\, a^{2}+2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}\right )}{a^{4}}}\, \left (i \sqrt {3}\, a^{2}+2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}\right )}{4 \left (a^{2} x \right )^{\frac {2}{3}}}+c_{6} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \frac {-\sqrt {\frac {x \left (a^{2} x \right )^{\frac {1}{3}} \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{a^{2}}}\, a^{2}+c_{1} \left (a^{2} x \right )^{\frac {2}{3}}+\left (a^{2} x \right )^{\frac {2}{3}} \sqrt {\frac {x \left (a^{2} x \right )^{\frac {1}{3}} \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{a^{2}}}}{\left (a^{2} x \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= \frac {\left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right ) \sqrt {\frac {x \left (a^{2} x \right )^{\frac {1}{3}} \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{a^{2}}}+c_{1} \left (a^{2} x \right )^{\frac {2}{3}}}{\left (a^{2} x \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= -\frac {\sqrt {2}\, \sqrt {-x \left (i \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}+\left (a^{2} x \right )^{\frac {1}{3}}+2 x \right )}\, \sqrt {\frac {x \left (a^{2} x \right )^{\frac {1}{3}} \left (2 i \left (a^{2} x \right )^{\frac {2}{3}}+i a^{2}-\sqrt {3}\, a^{2}\right )}{a^{2}}}\, \left (2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}+i \sqrt {3}\, a^{2}\right )}{4 \sqrt {\left (i \left (a^{2} x \right )^{\frac {1}{3}}+2 i x -\sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}\right ) x}\, \left (a^{2} x \right )^{\frac {2}{3}}}+c_{1} \\ y \left (x \right ) &= \frac {\sqrt {2}\, \sqrt {-x \left (i \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}+\left (a^{2} x \right )^{\frac {1}{3}}+2 x \right )}\, \sqrt {\frac {x \left (a^{2} x \right )^{\frac {1}{3}} \left (2 i \left (a^{2} x \right )^{\frac {2}{3}}+i a^{2}-\sqrt {3}\, a^{2}\right )}{a^{2}}}\, \left (2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}+i \sqrt {3}\, a^{2}\right )}{4 \sqrt {\left (i \left (a^{2} x \right )^{\frac {1}{3}}+2 i x -\sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}\right ) x}\, \left (a^{2} x \right )^{\frac {2}{3}}}+c_{1} \\ y \left (x \right ) &= \frac {\left (\left (i \sqrt {3}-1\right ) a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}\right ) \sqrt {2}\, \sqrt {\frac {\left (\left (i \sqrt {3}-1\right ) a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}\right ) x \left (a^{2} x \right )^{\frac {1}{3}}}{a^{2}}}+4 c_{1} \left (a^{2} x \right )^{\frac {2}{3}}}{4 \left (a^{2} x \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= -\frac {\left (-2 \left (a^{2} x \right )^{\frac {2}{3}} \sqrt {2}+a^{2} \left (i \sqrt {6}-\sqrt {2}\right )\right ) \sqrt {\frac {\left (\left (i \sqrt {3}-1\right ) a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}\right ) x \left (a^{2} x \right )^{\frac {1}{3}}}{a^{2}}}-4 c_{1} \left (a^{2} x \right )^{\frac {2}{3}}}{4 \left (a^{2} x \right )^{\frac {2}{3}}} \\ \end{align}

Problem 8884


ODE	\[ \boxed {{y^{\prime }}^{r}-a y^{s}=b \,x^{\frac {r s}{r -s}}} \]

program solution	\[ \frac {r \ln \left (x \right )}{r -s}-r \left (\int _{}^{y x^{-\frac {r}{r -s}}}-\frac {1}{\left (-r +s \right ) \left (\textit {\_a}^{s} a +b \right )^{\frac {1}{r}}+\textit {\_a} r}d \textit {\_a} \right )-c_{1} = 0 \] Veriﬁed OK.

Maple solution	\[ -\left (\int _{\textit {\_b}}^{y \left (x \right )}\frac {1}{x \left (r -s \right ) \left (a \,\textit {\_a}^{s}+b \,x^{\frac {r s}{r -s}}\right )^{\frac {1}{r}}-r \textit {\_a}}d \textit {\_a} \right )+\frac {\ln \left (x \right )}{r -s}-c_{1} = 0 \]

Problem 8885


ODE	\[ \boxed {{y^{\prime }}^{n}-f \left (x \right )^{n} \left (y-a \right )^{n +1} \left (y-b \right )^{n -1}=0} \]

program solution	\[ -\frac {n \left (-y+a \right ) \left (-y+b \right ) \left (\left (y-a \right )^{n +1} \left (y-b \right )^{n -1}\right )^{-\frac {1}{n}}}{a -b} = \int \left (f \left (x \right )^{n}\right )^{\frac {1}{n}}d x +c_{1} \] Veriﬁed OK. `{0 < f(x)^n, 0 < (y-a)^(n+1)*(y-b)^(n-1)}`

Maple solution	\[ y \left (x \right ) = \frac {b \left (-\frac {n}{\left (a -b \right ) \left (\int f \left (x \right )d x +c_{1} \right )}\right )^{n}-a}{-1+\left (-\frac {n}{\left (a -b \right ) \left (\int f \left (x \right )d x +c_{1} \right )}\right )^{n}} \]

Problem 8886


ODE	\[ \boxed {{y^{\prime }}^{n}-f \left (x \right ) g \left (y\right )=0} \]

program solution	\[ \int _{}^{y}g \left (y \right )^{-\frac {1}{n}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{n}}d x +c_{1} \] Veriﬁed OK. `{0 < f(x), 0 < g(y)}`

Maple solution	\[ \int _{}^{y \left (x \right )}g \left (\textit {\_a} \right )^{-\frac {1}{n}}d \textit {\_a} -g \left (y \left (x \right )\right )^{-\frac {1}{n}} \left (\int _{}^{x}\left (f \left (\textit {\_a} \right ) g \left (y \left (x \right )\right )\right )^{\frac {1}{n}}d \textit {\_a} \right )+c_{1} = 0 \]

Problem 8887


ODE	\[ \boxed {a {y^{\prime }}^{m}+b {y^{\prime }}^{n}-y=0} \]

program solution	\[ \int _{}^{y}\frac {1}{\operatorname {RootOf}\left (a \,\textit {\_Z}^{m}+b \,\textit {\_Z}^{n}-\textit {\_a} \right )}d \textit {\_a} = x +c_{1} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= 0 \\ x -\left (\int _{}^{y \left (x \right )}\frac {1}{\operatorname {RootOf}\left (a \,\textit {\_Z}^{m}+b \,\textit {\_Z}^{n}-\textit {\_a} \right )}d \textit {\_a} \right )-c_{1} &= 0 \\ \end{align}

Problem 8888


ODE	\[ \boxed {x^{n -1} {y^{\prime }}^{n}-n x y^{\prime }+y=0} \]

program solution

Maple solution	\[ y \left (x \right ) = -x^{n -1} \left (\frac {c_{1} \left (\frac {x}{c_{1}}\right )^{\frac {1}{n}}}{x}\right )^{n}+n c_{1} \left (\frac {x}{c_{1}}\right )^{\frac {1}{n}} \]

Problem 8889


ODE	\[ \boxed {\sqrt {{y^{\prime }}^{2}+1}+y^{\prime } x -y=0} \]

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

Maple solution	\[ y \left (x \right ) = \sqrt {c_{1}^{2}+1}+c_{1} x \]

Problem 8890


ODE	\[ \boxed {\sqrt {{y^{\prime }}^{2}+1}+x {y^{\prime }}^{2}+y=0} \]

program solution	\[ y = -x -\sqrt {2} \] Veriﬁed OK. \[ y = -1 \] Veriﬁed OK. \[ x = \frac {4 \left (-\operatorname {arcsinh}\left (\frac {\sqrt {2}\, \sqrt {-2 x y+1+\sqrt {-4 x y+4 x^{2}+1}}}{2 x}\right )-\frac {\sqrt {2}\, \sqrt {\frac {2 x^{2}-2 x y+\sqrt {-4 x y+4 x^{2}+1}+1}{x^{2}}}}{2}+c_{1} \right ) x^{2}}{{\left (\sqrt {2}\, \sqrt {-2 x y+1+\sqrt {-4 x y+4 x^{2}+1}}+2 x \right )}^{2}} \] Warning, solution could not be veriﬁed \[ x = \frac {4 \left (\operatorname {arcsinh}\left (\frac {\sqrt {2}\, \sqrt {-2 x y+1+\sqrt {-4 x y+4 x^{2}+1}}}{2 x}\right )-\frac {\sqrt {2}\, \sqrt {\frac {2 x^{2}-2 x y+\sqrt {-4 x y+4 x^{2}+1}+1}{x^{2}}}}{2}+c_{1} \right ) x^{2}}{{\left (-\sqrt {2}\, \sqrt {-2 x y+1+\sqrt {-4 x y+4 x^{2}+1}}+2 x \right )}^{2}} \] Warning, solution could not be veriﬁed \[ x = -\frac {2 \left (2 \,\operatorname {arcsinh}\left (\frac {\sqrt {-4 x y-2 \sqrt {-4 x y+4 x^{2}+1}+2}}{2 x}\right )+\sqrt {\frac {4 x^{2}-4 x y-2 \sqrt {-4 x y+4 x^{2}+1}+2}{x^{2}}}-2 c_{1} \right ) x^{2}}{{\left (\sqrt {-4 x y-2 \sqrt {-4 x y+4 x^{2}+1}+2}+2 x \right )}^{2}} \] Warning, solution could not be veriﬁed \[ x = \frac {2 \left (2 \,\operatorname {arcsinh}\left (\frac {\sqrt {-4 x y-2 \sqrt {-4 x y+4 x^{2}+1}+2}}{2 x}\right )-\sqrt {\frac {4 x^{2}-4 x y-2 \sqrt {-4 x y+4 x^{2}+1}+2}{x^{2}}}+2 c_{1} \right ) x^{2}}{{\left (\sqrt {-4 x y-2 \sqrt {-4 x y+4 x^{2}+1}+2}-2 x \right )}^{2}} \] Warning, solution could not be veriﬁed

Maple solution	\begin{align} y \left (x \right ) &= -1 \\ \frac {x \left (2 \sqrt {2}\, \sqrt {\frac {2 x^{2}-2 x y \left (x \right )+\sqrt {4 x^{2}-4 x y \left (x \right )+1}+1}{x^{2}}}\, x -4 \,\operatorname {arcsinh}\left (\frac {\sqrt {-4 x y \left (x \right )+2+2 \sqrt {4 x^{2}-4 x y \left (x \right )+1}}}{2 x}\right ) x -4 x y \left (x \right )+2+2 \sqrt {4 x^{2}-4 x y \left (x \right )+1}-4 x \sqrt {-4 x y \left (x \right )+2+2 \sqrt {4 x^{2}-4 x y \left (x \right )+1}}+c_{1} x +4 x^{2}\right )}{{\left (\sqrt {-4 x y \left (x \right )+2+2 \sqrt {4 x^{2}-4 x y \left (x \right )+1}}-2 x \right )}^{2}} &= 0 \\ \frac {2 x^{2} \sqrt {2}\, \sqrt {\frac {2 x^{2}-2 x y \left (x \right )+\sqrt {4 x^{2}-4 x y \left (x \right )+1}+1}{x^{2}}}+4 x^{3}+c_{1} x^{2}-4 x^{2} y \left (x \right )+4 x^{2} \sqrt {-4 x y \left (x \right )+2+2 \sqrt {4 x^{2}-4 x y \left (x \right )+1}}+4 x^{2} \operatorname {arcsinh}\left (\frac {\sqrt {-4 x y \left (x \right )+2+2 \sqrt {4 x^{2}-4 x y \left (x \right )+1}}}{2 x}\right )+2 \sqrt {4 x^{2}-4 x y \left (x \right )+1}\, x +2 x}{{\left (\sqrt {-4 x y \left (x \right )+2+2 \sqrt {4 x^{2}-4 x y \left (x \right )+1}}+2 x \right )}^{2}} &= 0 \\ -\frac {x \left (4 x y \left (x \right )+4 x \sqrt {-4 x y \left (x \right )-2 \sqrt {4 x^{2}-4 x y \left (x \right )+1}+2}+4 x \,\operatorname {arcsinh}\left (\frac {\sqrt {-4 x y \left (x \right )-2 \sqrt {4 x^{2}-4 x y \left (x \right )+1}+2}}{2 x}\right )-2 x \sqrt {\frac {4 x^{2}-4 x y \left (x \right )-2 \sqrt {4 x^{2}-4 x y \left (x \right )+1}+2}{x^{2}}}-c_{1} x -4 x^{2}+2 \sqrt {4 x^{2}-4 x y \left (x \right )+1}-2\right )}{{\left (\sqrt {-4 x y \left (x \right )-2 \sqrt {4 x^{2}-4 x y \left (x \right )+1}+2}-2 x \right )}^{2}} &= 0 \\ \frac {4 x^{3}+c_{1} x^{2}-4 x^{2} y \left (x \right )+4 \sqrt {-4 x y \left (x \right )-2 \sqrt {4 x^{2}-4 x y \left (x \right )+1}+2}\, x^{2}+2 x^{2} \sqrt {\frac {4 x^{2}-4 x y \left (x \right )-2 \sqrt {4 x^{2}-4 x y \left (x \right )+1}+2}{x^{2}}}+4 \,\operatorname {arcsinh}\left (\frac {\sqrt {-4 x y \left (x \right )-2 \sqrt {4 x^{2}-4 x y \left (x \right )+1}+2}}{2 x}\right ) x^{2}-2 \sqrt {4 x^{2}-4 x y \left (x \right )+1}\, x +2 x}{{\left (\sqrt {-4 x y \left (x \right )-2 \sqrt {4 x^{2}-4 x y \left (x \right )+1}+2}+2 x \right )}^{2}} &= 0 \\ \end{align}

Problem 8891


ODE	\[ \boxed {x \left (\sqrt {{y^{\prime }}^{2}+1}+y^{\prime }\right )-y=0} \]

program solution	\[ y = -i x \] Veriﬁed OK. \[ y = i x \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \frac {x \left (\sqrt {-x \left (-2 c_{1} +x \right )}\, \sqrt {-\frac {c_{1}^{2}}{x \left (-2 c_{1} +x \right )}}-x +c_{1} \right )}{\sqrt {-x \left (-2 c_{1} +x \right )}} \\ y \left (x \right ) &= \frac {x \left (\sqrt {-x \left (-2 c_{1} +x \right )}\, \sqrt {-\frac {c_{1}^{2}}{x \left (-2 c_{1} +x \right )}}+x -c_{1} \right )}{\sqrt {-x \left (-2 c_{1} +x \right )}} \\ \end{align}

Problem 8892


ODE	\[ \boxed {a x \sqrt {{y^{\prime }}^{2}+1}+y^{\prime } x -y=0} \]

program solution	\[ y = -i x \] Veriﬁed OK. \[ y = i x \] Veriﬁed OK.

Maple solution	\begin{align} \frac {x \sqrt {\frac {-a^{2} x^{2}+a^{2} y \left (x \right )^{2}+2 \sqrt {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}\, a y \left (x \right )+x^{2}+y \left (x \right )^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}-{\mathrm e}^{\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}\, a +y \left (x \right )}{\left (a^{2}-1\right ) x}\right )}{a}} c_{1}}{\sqrt {\frac {-a^{2} x^{2}+a^{2} y \left (x \right )^{2}+2 \sqrt {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}\, a y \left (x \right )+x^{2}+y \left (x \right )^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}} &= 0 \\ \frac {x \sqrt {\frac {-a^{2} x^{2}+a^{2} y \left (x \right )^{2}-2 \sqrt {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}\, a y \left (x \right )+x^{2}+y \left (x \right )^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}-{\mathrm e}^{\frac {\operatorname {arcsinh}\left (\frac {-\sqrt {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}\, a +y \left (x \right )}{\left (a^{2}-1\right ) x}\right )}{a}} c_{1}}{\sqrt {\frac {-a^{2} x^{2}+a^{2} y \left (x \right )^{2}-2 \sqrt {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}\, a y \left (x \right )+x^{2}+y \left (x \right )^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}} &= 0 \\ \end{align}

Problem 8893


ODE	\[ \boxed {y \sqrt {{y^{\prime }}^{2}+1}-a y y^{\prime }=a x} \]

program solution	\[ \ln \left (x \right ) = \int _{}^{\frac {y}{x}}\frac {\left (a^{2}-1\right ) \textit {\_a}}{-\textit {\_a}^{2} a^{2}+\textit {\_a}^{2}-a^{2}+\sqrt {\textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}}}d \textit {\_a} +c_{1} \] Veriﬁed OK.

Maple solution	\begin{align} -{\mathrm e}^{a \left (\int _{}^{\frac {-a^{2} x +\sqrt {y \left (x \right )^{2} \left (a^{2}-1\right )+a^{2} x^{2}}}{\left (a^{2}-1\right ) y \left (x \right )}}\frac {a \sqrt {\textit {\_a}^{2}+1}-\textit {\_a}}{\sqrt {\textit {\_a}^{2}+1}\, \left (-\sqrt {\textit {\_a}^{2}+1}\, \textit {\_a} +a \left (\textit {\_a}^{2}+1\right )\right ) \left (\textit {\_a} a -\sqrt {\textit {\_a}^{2}+1}\right )}d \textit {\_a} \right )} c_{1} +x &= 0 \\ -{\mathrm e}^{a \left (\int _{}^{\frac {-a^{2} x -\sqrt {y \left (x \right )^{2} \left (a^{2}-1\right )+a^{2} x^{2}}}{\left (a^{2}-1\right ) y \left (x \right )}}\frac {a \sqrt {\textit {\_a}^{2}+1}-\textit {\_a}}{\sqrt {\textit {\_a}^{2}+1}\, \left (-\sqrt {\textit {\_a}^{2}+1}\, \textit {\_a} +a \left (\textit {\_a}^{2}+1\right )\right ) \left (\textit {\_a} a -\sqrt {\textit {\_a}^{2}+1}\right )}d \textit {\_a} \right )} c_{1} +x &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {\left (\textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}-\sqrt {\textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}}\right ) \textit {\_a}}{\left (\textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}\right ) \left (\textit {\_a}^{2}+1\right )}d \textit {\_a} +c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {\left (\textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}+\sqrt {\textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}}\right ) \textit {\_a}}{\left (\textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}\right ) \left (\textit {\_a}^{2}+1\right )}d \textit {\_a} \right )+c_{1} \right ) x \\ \end{align}

Problem 8894


ODE	\[ \boxed {a y \sqrt {{y^{\prime }}^{2}+1}-2 x y y^{\prime }+y^{2}=x^{2}} \]

program solution	\[ -\frac {16 x^{4} \left (-a x +\sqrt {-y^{2} a^{2}+x^{4}+2 y^{2} x^{2}+y^{4}}\right )}{a \left (y^{2}+x^{2}\right ) \left (\sqrt {a^{2}-4 x^{2}}+a \right )^{2} \left (\sqrt {a^{2}-4 x^{2}}-a \right )^{2}} = c_{1} \] Veriﬁed OK. \[ \frac {16 x^{4} \left (a x +\sqrt {-y^{2} a^{2}+x^{4}+2 y^{2} x^{2}+y^{4}}\right )}{a \left (y^{2}+x^{2}\right ) \left (\sqrt {a^{2}-4 x^{2}}+a \right )^{2} \left (\sqrt {a^{2}-4 x^{2}}-a \right )^{2}} = c_{1} \] Veriﬁed OK.

Maple solution	\begin{align} -\left (\int _{\textit {\_b}}^{x}\frac {-2 y \left (x \right )^{2} \textit {\_a} +2 \textit {\_a}^{3}+\sqrt {a^{2} \left (y \left (x \right )^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) y \left (x \right )^{2}+\textit {\_a}^{4}\right )}}{\left (-\textit {\_a}^{2}+y \left (x \right )^{2}\right ) \sqrt {a^{2} \left (y \left (x \right )^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) y \left (x \right )^{2}+\textit {\_a}^{4}\right )}-2 \textit {\_a} y \left (x \right )^{4}+\left (-4 \textit {\_a}^{3}+2 \textit {\_a} \,a^{2}\right ) y \left (x \right )^{2}-2 \textit {\_a}^{5}}d \textit {\_a} \right )+\int _{}^{y \left (x \right )}\frac {4 \textit {\_f} \left (\left (\frac {\left (\textit {\_f}^{2}-x^{2}\right ) \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (a^{2}-2 x^{2}\right ) \textit {\_f}^{2}+x^{4}\right )}}{2}-x^{5}-2 x^{3} \textit {\_f}^{2}+\left (-\textit {\_f}^{4}+\textit {\_f}^{2} a^{2}\right ) x \right ) \left (\int _{\textit {\_b}}^{x}\frac {\left (\left (-4 \textit {\_a}^{2}-a^{2}\right ) \textit {\_f}^{4}+\left (8 \textit {\_a}^{4}-2 \textit {\_a}^{2} a^{2}+a^{4}\right ) \textit {\_f}^{2}-5 \textit {\_a}^{4} a^{2}+12 \textit {\_a}^{6}\right ) \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{2}+\textit {\_a}^{4}\right )}+a^{2} \textit {\_a} \left (-2 \textit {\_a}^{2}-2 \textit {\_f}^{2}+a^{2}\right ) \left (-2 \textit {\_a}^{4}-2 \textit {\_f}^{4}+\textit {\_f}^{2} a^{2}\right )}{\sqrt {a^{2} \left (\textit {\_f}^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{2}+\textit {\_a}^{4}\right )}\, \left (2 \textit {\_a} \,\textit {\_f}^{2} a^{2}-2 \textit {\_a}^{5}-4 \textit {\_a}^{3} \textit {\_f}^{2}-2 \textit {\_a} \,\textit {\_f}^{4}-\textit {\_a}^{2} \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{2}+\textit {\_a}^{4}\right )}+\textit {\_f}^{2} \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{2}+\textit {\_a}^{4}\right )}\right )^{2}}d \textit {\_a} \right )+\frac {a^{2}}{4}-x^{2}\right )}{\left (\textit {\_f}^{2}-x^{2}\right ) \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (a^{2}-2 x^{2}\right ) \textit {\_f}^{2}+x^{4}\right )}-2 x^{5}-4 x^{3} \textit {\_f}^{2}+2 \left (-\textit {\_f}^{4}+\textit {\_f}^{2} a^{2}\right ) x}d \textit {\_f} +c_{1} &= 0 \\ -\left (\int _{\textit {\_b}}^{x}\frac {2 y \left (x \right )^{2} \textit {\_a} -2 \textit {\_a}^{3}+\sqrt {a^{2} \left (y \left (x \right )^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) y \left (x \right )^{2}+\textit {\_a}^{4}\right )}}{2 \textit {\_a} y \left (x \right )^{4}+4 \textit {\_a}^{3} y \left (x \right )^{2}-2 \textit {\_a} y \left (x \right )^{2} a^{2}+2 \textit {\_a}^{5}+y \left (x \right )^{2} \sqrt {a^{2} \left (y \left (x \right )^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) y \left (x \right )^{2}+\textit {\_a}^{4}\right )}-\textit {\_a}^{2} \sqrt {a^{2} \left (y \left (x \right )^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) y \left (x \right )^{2}+\textit {\_a}^{4}\right )}}d \textit {\_a} \right )+\int _{}^{y \left (x \right )}\frac {4 \textit {\_f} \left (\left (\frac {\left (-\textit {\_f}^{2}+x^{2}\right ) \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (a^{2}-2 x^{2}\right ) \textit {\_f}^{2}+x^{4}\right )}}{2}-x^{5}-2 x^{3} \textit {\_f}^{2}+\left (-\textit {\_f}^{4}+\textit {\_f}^{2} a^{2}\right ) x \right ) \left (\int _{\textit {\_b}}^{x}-\frac {\left (\left (4 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{4}+\left (-8 \textit {\_a}^{4}+2 \textit {\_a}^{2} a^{2}-a^{4}\right ) \textit {\_f}^{2}+5 \textit {\_a}^{4} a^{2}-12 \textit {\_a}^{6}\right ) \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{2}+\textit {\_a}^{4}\right )}+a^{2} \textit {\_a} \left (-2 \textit {\_a}^{2}-2 \textit {\_f}^{2}+a^{2}\right ) \left (-2 \textit {\_a}^{4}-2 \textit {\_f}^{4}+\textit {\_f}^{2} a^{2}\right )}{\sqrt {a^{2} \left (\textit {\_f}^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{2}+\textit {\_a}^{4}\right )}\, \left (2 \textit {\_a} \,\textit {\_f}^{2} a^{2}-2 \textit {\_a}^{5}-4 \textit {\_a}^{3} \textit {\_f}^{2}-2 \textit {\_a} \,\textit {\_f}^{4}+\textit {\_a}^{2} \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{2}+\textit {\_a}^{4}\right )}-\textit {\_f}^{2} \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{2}+\textit {\_a}^{4}\right )}\right )^{2}}d \textit {\_a} \right )+\frac {a^{2}}{4}-x^{2}\right )}{\left (-\textit {\_f}^{2}+x^{2}\right ) \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (a^{2}-2 x^{2}\right ) \textit {\_f}^{2}+x^{4}\right )}-2 x^{5}-4 x^{3} \textit {\_f}^{2}+2 \left (-\textit {\_f}^{4}+\textit {\_f}^{2} a^{2}\right ) x}d \textit {\_f} +c_{1} &= 0 \\ \end{align}

Problem 8895


ODE	\[ \boxed {f \left (y^{2}+x^{2}\right ) \sqrt {{y^{\prime }}^{2}+1}-y^{\prime } x +y=0} \]

program solution

Maple solution	\[ y \left (x \right ) = \cot \left (\operatorname {RootOf}\left (-2 \textit {\_Z} +\int _{}^{\csc \left (\textit {\_Z} \right )^{2} x^{2}}\frac {f \left (\textit {\_a} \right )}{\sqrt {-f \left (\textit {\_a} \right )^{2}+\textit {\_a}}\, \textit {\_a}}d \textit {\_a} +2 c_{1} \right )\right ) x \]

Problem 8896


ODE	\[ \boxed {a \left ({y^{\prime }}^{3}+1\right )^{\frac {1}{3}}+b x y^{\prime }-y=0} \]

program solution	\[ y = a \] Veriﬁed OK. \[ \text {Expression too large to display} \] Warning, solution could not be veriﬁed \[ \text {Expression too large to display} \] Warning, solution could not be veriﬁed \[ \text {Expression too large to display} \] Warning, solution could not be veriﬁed

Maple solution	\begin{align} \text {Expression too large to display} \\ \text {Expression too large to display} \\ \text {Expression too large to display} \\ \end{align}

Problem 8897


ODE	\[ \boxed {\ln \left (y^{\prime }\right )+y^{\prime } x +y a=-b} \]

program solution	\[ y = \frac {\infty }{\operatorname {signum}\left (a \right )} \] Warning, solution could not be veriﬁed \[ x = \frac {c_{1} {\left (\frac {\operatorname {LambertW}\left (x \,{\mathrm e}^{-y a -b}\right )}{x}\right )}^{-\frac {1}{a +1}} \operatorname {LambertW}\left (x \,{\mathrm e}^{-y a -b}\right ) a +x}{\operatorname {LambertW}\left (x \,{\mathrm e}^{-y a -b}\right ) a} \] Veriﬁed OK.

Maple solution	\[ \frac {-a \left ({\left (\frac {\operatorname {LambertW}\left (x \,{\mathrm e}^{-a y \left (x \right )-b}\right )}{x}\right )}^{-\frac {1}{a +1}} c_{1} -x \right ) \operatorname {LambertW}\left (x \,{\mathrm e}^{-a y \left (x \right )-b}\right )-x}{a \operatorname {LambertW}\left (x \,{\mathrm e}^{-a y \left (x \right )-b}\right )} = 0 \]

Problem 8898


ODE	\[ \boxed {\ln \left (y^{\prime }\right )+a \left (y^{\prime } x -y\right )=0} \]

program solution	\[ y = c_{1} x +\frac {\ln \left (c_{1} \right )}{a} \] Veriﬁed OK. \[ y = \frac {\ln \left (-\frac {1}{a x}\right )-1}{a} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \frac {\ln \left (-\frac {1}{a x}\right )-1}{a} \\ y \left (x \right ) &= c_{1} x +\frac {\ln \left (c_{1} \right )}{a} \\ \end{align}

Problem 8899


ODE	\[ \boxed {y \ln \left (y^{\prime }\right )+y^{\prime }-y \ln \left (y\right )-x y=0} \]

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

Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{\frac {\operatorname {LambertW}\left ({\mathrm e}^{x}\right ) \left (\operatorname {LambertW}\left ({\mathrm e}^{x}\right )+2\right )}{2}} \]

Problem 8900


ODE	\[ \boxed {\sin \left (y^{\prime }\right )+y^{\prime }=x} \]

program solution	\[ y = \int \operatorname {RootOf}\left (\sin \left (\textit {\_Z} \right )+\textit {\_Z} -x \right )d x +c_{1} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \int \operatorname {RootOf}\left (\sin \left (\textit {\_Z} \right )+\textit {\_Z} -x \right )d x +c_{1} \]

2.17.89 Problems 8801 to 8900