Problems 8701 to 8800

Problem 8701


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

program solution	\[ \int _{}^{x}\frac {y+\textit {\_a} f \left (\textit {\_a}^{2}+y^{2}\right )}{\textit {\_a}^{2}+y^{2}}d \textit {\_a} +\int _{0}^{y}\frac {\left (\textit {\_a}^{2}+x^{2}\right ) \left (\int _{}^{x}\frac {-4 D\left (f \right )\left (2 \textit {\_a}^{2}\right ) \textit {\_a}^{4}+2 f \left (2 \textit {\_a}^{2}\right ) \textit {\_a}^{2}}{4 \textit {\_a}^{4}}d \textit {\_a} \right )+\textit {\_a} f \left (\textit {\_a}^{2}+x^{2}\right )-x}{\textit {\_a}^{2}+x^{2}}d \textit {\_a} = c_{1} \] Veriﬁed OK.

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

Problem 8702


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

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

Maple solution	\[ -\frac {a y \left (x \right )^{2} x}{\sqrt {a^{2} y \left (x \right )^{2}}}-\left (\int _{}^{-\frac {a y \left (x \right )^{2}}{2}-\frac {x^{2}}{2}}f \left (-2 \textit {\_a} \right )d \textit {\_a} \right )+c_{1} = 0 \]

Problem 8703


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

program solution

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

Problem 8704


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

program solution	\[ \ln \left (x \right ) = -\frac {\ln \left (-\frac {2 y a}{x^{2}}+a \sqrt {-\frac {y a}{x^{2}}-b}\right )}{4}-\frac {a \,\operatorname {arctanh}\left (\frac {a +4 \sqrt {-\frac {y a}{x^{2}}-b}}{\sqrt {a^{2}-16 b}}\right )}{2 \sqrt {a^{2}-16 b}}+\frac {\ln \left (-\frac {2 y a}{x^{2}}-a \sqrt {-\frac {y a}{x^{2}}-b}\right )}{4}-\frac {a \,\operatorname {arctanh}\left (\frac {4 \sqrt {-\frac {y a}{x^{2}}-b}-a}{\sqrt {a^{2}-16 b}}\right )}{2 \sqrt {a^{2}-16 b}}-\frac {\ln \left (\frac {4 y^{2}}{x^{4}}+\frac {y a}{x^{2}}+b \right )}{4}-\frac {a \,\operatorname {arctanh}\left (\frac {\frac {8 y}{x^{2}}+a}{\sqrt {a^{2}-16 b}}\right )}{2 \sqrt {a^{2}-16 b}}+c_{1} \] Veriﬁed OK.

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

Problem 8705


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

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

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

Problem 8706


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

program solution

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

Problem 8707


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

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

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

Problem 8708


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

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

Maple solution	\begin{align} y \left (x \right ) &= \frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}+3 a}{6 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {-i \sqrt {3}\, \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}+3 i \sqrt {3}\, a -\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}-3 a}{12 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {-i \sqrt {3}\, \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}+3 i \sqrt {3}\, a +\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}+3 a}{12 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \operatorname {WeierstrassP}\left (x +c_{1} , a , b\right ) \\ \end{align}

Problem 8709


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

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

Maple solution	\begin{align} y \left (x \right ) &= {\mathrm e}^{\operatorname {RootOf}\left (a^{2} {\mathrm e}^{2 \textit {\_Z}} \left (\textit {\_Z}^{2}-1\right )\right )} \\ y \left (x \right ) &= {\mathrm e}^{-\sin \left (a \left (-x +c_{1} \right )\right )} \\ y \left (x \right ) &= {\mathrm e}^{\sin \left (a \left (-x +c_{1} \right )\right )} \\ \end{align}

Problem 8710


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

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

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

Problem 8711


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

program solution	\[ y = -\frac {\left (a^{2}-4 b x \right )^{\frac {3}{2}}}{12 b}-\frac {a x}{2}+c_{1} \] Veriﬁed OK. \[ y = -\frac {a x}{2}+\frac {\left (a^{2}-4 b x \right )^{\frac {3}{2}}}{12 b}+c_{2} \] Veriﬁed OK.

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

Problem 8712


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

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

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

Problem 8713


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

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

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

Problem 8714


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

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

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

Problem 8715


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

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

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

Problem 8716


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

program solution	\[ y = 0 \] Veriﬁed OK. \[ x = \frac {\left (-8 x^{2}-2 y\right ) \sqrt {x^{2}+y}+8 x^{3}+6 y x +3 c_{1}}{3 \left (x -\sqrt {x^{2}+y}\right )^{2}} \] Veriﬁed OK. \[ x = \frac {\left (8 x^{2}+2 y\right ) \sqrt {x^{2}+y}+8 x^{3}+6 y x +3 c_{1}}{3 \left (x +\sqrt {x^{2}+y}\right )^{2}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \frac {\left (x^{2}-x \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {1}{3}}+\left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}\right ) \left (x^{2}+3 x \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {1}{3}}+\left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}\right )}{4 \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= \frac {\left (i \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}} \sqrt {3}-i \sqrt {3}\, x^{2}+\left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}+2 x \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {1}{3}}+x^{2}\right ) \left (i \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}} \sqrt {3}-i \sqrt {3}\, x^{2}+\left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}-6 x \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {1}{3}}+x^{2}\right )}{16 \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}\, x^{2}-i \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}} \sqrt {3}+x^{2}+2 x \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {1}{3}}+\left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}\right ) \left (i \sqrt {3}\, x^{2}-i \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}} \sqrt {3}+x^{2}-6 x \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {1}{3}}+\left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}\right )}{16 \left (-x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}+6 c_{1} \right )^{\frac {2}{3}}} \\ \end{align}

Problem 8717


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

program solution	\[ y = 0 \] Veriﬁed OK. \[ x = \frac {\left (8 x^{2}-2 y\right ) \sqrt {x^{2}-y}+8 x^{3}-6 y x +3 c_{1}}{3 \left (x +\sqrt {x^{2}-y}\right )^{2}} \] Veriﬁed OK. \[ x = \frac {\left (-8 x^{2}+2 y\right ) \sqrt {x^{2}-y}+8 x^{3}-6 y x +3 c_{1}}{3 \left (x -\sqrt {x^{2}-y}\right )^{2}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -\frac {\left (x^{2}+x \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {1}{3}}+\left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}\right ) \left (x^{2}-3 x \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {1}{3}}+\left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}\right )}{4 \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= -\frac {\left (i \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}} \sqrt {3}-i \sqrt {3}\, x^{2}+\left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}-2 x \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {1}{3}}+x^{2}\right ) \left (i \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}} \sqrt {3}-i \sqrt {3}\, x^{2}+\left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}+6 x \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {1}{3}}+x^{2}\right )}{16 \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= -\frac {\left (i \sqrt {3}\, x^{2}-i \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}} \sqrt {3}+x^{2}-2 x \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {1}{3}}+\left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}\right ) \left (i \sqrt {3}\, x^{2}-i \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}} \sqrt {3}+x^{2}+6 x \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {1}{3}}+\left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}\right )}{16 \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}} \\ \end{align}

Problem 8718


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

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

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

Problem 8719


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

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

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

Problem 8720


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

program solution	\[ y = x c_{1} +\frac {b c_{1} +c_{1}^{2}+c}{a} \] Veriﬁed OK. \[ y = \frac {-a^{2} x^{2}-2 a b x -b^{2}+4 c}{4 a} \] Veriﬁed OK.

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

Problem 8721


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

program solution	\[ \frac {\ln \left (y\right )}{4}+\frac {\ln \left (-4 x^{3}+9 y\right )}{12}-\frac {\ln \left (-x^{2}+\sqrt {x}\, \sqrt {x^{3}-2 y}\right )}{4}-\frac {\ln \left (x^{2}+3 \sqrt {x}\, \sqrt {x^{3}-2 y}\right )}{12}+\frac {\ln \left (x^{2}+\sqrt {x}\, \sqrt {x^{3}-2 y}\right )}{4}+\frac {\ln \left (-x^{2}+3 \sqrt {x}\, \sqrt {x^{3}-2 y}\right )}{12} = c_{1} \] Veriﬁed OK. \[ \frac {\ln \left (-4 x^{3}+9 y\right )}{12}+\frac {\ln \left (y\right )}{4}+\frac {\ln \left (-x^{2}+\sqrt {x}\, \sqrt {x^{3}-2 y}\right )}{4}+\frac {\ln \left (x^{2}+3 \sqrt {x}\, \sqrt {x^{3}-2 y}\right )}{12}-\frac {\ln \left (x^{2}+\sqrt {x}\, \sqrt {x^{3}-2 y}\right )}{4}-\frac {\ln \left (-x^{2}+3 \sqrt {x}\, \sqrt {x^{3}-2 y}\right )}{12} = c_{1} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \frac {x^{4}-\operatorname {RootOf}\left (x^{16}-12 \textit {\_Z}^{2} x^{12}+16 \textit {\_Z}^{3} x^{10}+30 \textit {\_Z}^{4} x^{8}-96 \textit {\_Z}^{5} x^{6}+100 \textit {\_Z}^{6} x^{4}-48 \textit {\_Z}^{7} x^{2}+9 \textit {\_Z}^{8}-16 c_{1} x^{4}\right )^{2}}{2 x} \\ y \left (x \right ) &= \frac {x^{4}-\operatorname {RootOf}\left (x^{16}-12 \textit {\_Z}^{2} x^{12}-16 \textit {\_Z}^{3} x^{10}+30 \textit {\_Z}^{4} x^{8}+96 \textit {\_Z}^{5} x^{6}+100 \textit {\_Z}^{6} x^{4}+48 \textit {\_Z}^{7} x^{2}+9 \textit {\_Z}^{8}-16 c_{1} x^{4}\right )^{2}}{2 x} \\ \end{align}

Problem 8722


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

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

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

Problem 8723


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

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

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

Problem 8724


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

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

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

Problem 8725


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

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

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

Problem 8726


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

program solution	\[ y = \frac {a b x +a c -b}{\sqrt {b a}\, a} \] Veriﬁed OK. \[ y = \frac {-a b x -a c +b}{\sqrt {b a}\, a} \] Veriﬁed OK. \[ x = \frac {b \left (a y-\sqrt {y^{2} a^{2}+4 b x +4 c}\right ) \ln \left (-a^{\frac {3}{2}} y+\sqrt {a}\, \sqrt {y^{2} a^{2}+4 b x +4 c}+\sqrt {-2 a^{2} y \sqrt {y^{2} a^{2}+4 b x +4 c}+2 y^{2} a^{3}+\left (4 b x +4 c \right ) a -4 b}\right )-c \sqrt {-2 a^{2} y \sqrt {y^{2} a^{2}+4 b x +4 c}+2 y^{2} a^{3}+\left (4 b x +4 c \right ) a -4 b}\, \sqrt {a}-b \left (\left (-c_{1} \sqrt {a}-\ln \left (2\right )\right ) \sqrt {y^{2} a^{2}+4 b x +4 c}+y \left (a \ln \left (2\right )+a^{\frac {3}{2}} c_{1} \right )\right )}{\sqrt {-2 a^{2} y \sqrt {y^{2} a^{2}+4 b x +4 c}+2 y^{2} a^{3}+\left (4 b x +4 c \right ) a -4 b}\, \sqrt {a}\, b} \] Warning, solution could not be veriﬁed \[ x = \frac {b \left (a y+\sqrt {y^{2} a^{2}+4 b x +4 c}\right ) \ln \left (-a^{\frac {3}{2}} y-\sqrt {a}\, \sqrt {y^{2} a^{2}+4 b x +4 c}+\sqrt {2 a^{2} y \sqrt {y^{2} a^{2}+4 b x +4 c}+2 y^{2} a^{3}+\left (4 b x +4 c \right ) a -4 b}\right )-c \sqrt {2 a^{2} y \sqrt {y^{2} a^{2}+4 b x +4 c}+2 y^{2} a^{3}+\left (4 b x +4 c \right ) a -4 b}\, \sqrt {a}-b \left (\left (c_{1} \sqrt {a}+\ln \left (2\right )\right ) \sqrt {y^{2} a^{2}+4 b x +4 c}+y \left (a \ln \left (2\right )+a^{\frac {3}{2}} c_{1} \right )\right )}{\sqrt {2 a^{2} y \sqrt {y^{2} a^{2}+4 b x +4 c}+2 y^{2} a^{3}+\left (4 b x +4 c \right ) a -4 b}\, \sqrt {a}\, b} \] Warning, solution could not be veriﬁed

Maple solution	\[ y \left (x \right ) = \frac {4 \left (a x b +a c -\frac {1}{2} b \right ) {\mathrm e}^{\operatorname {RootOf}\left (\sqrt {a}\, c_{1} b \,{\mathrm e}^{2 \textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} a b x -{\mathrm e}^{2 \textit {\_Z}} \textit {\_Z} b -{\mathrm e}^{2 \textit {\_Z}} a c +\sqrt {a}\, c_{1} b^{2}+a \,b^{2} x -\textit {\_Z} \,b^{2}+a c b \right )}-b^{2} {\mathrm e}^{-\operatorname {RootOf}\left (\sqrt {a}\, c_{1} b \,{\mathrm e}^{2 \textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} a b x -{\mathrm e}^{2 \textit {\_Z}} \textit {\_Z} b -{\mathrm e}^{2 \textit {\_Z}} a c +\sqrt {a}\, c_{1} b^{2}+a \,b^{2} x -\textit {\_Z} \,b^{2}+a c b \right )}-{\mathrm e}^{3 \operatorname {RootOf}\left (\sqrt {a}\, c_{1} b \,{\mathrm e}^{2 \textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} a b x -{\mathrm e}^{2 \textit {\_Z}} \textit {\_Z} b -{\mathrm e}^{2 \textit {\_Z}} a c +\sqrt {a}\, c_{1} b^{2}+a \,b^{2} x -\textit {\_Z} \,b^{2}+a c b \right )}}{a^{\frac {3}{2}} \left (2 \,{\mathrm e}^{2 \operatorname {RootOf}\left (\sqrt {a}\, c_{1} b \,{\mathrm e}^{2 \textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} a b x -{\mathrm e}^{2 \textit {\_Z}} \textit {\_Z} b -{\mathrm e}^{2 \textit {\_Z}} a c +\sqrt {a}\, c_{1} b^{2}+a \,b^{2} x -\textit {\_Z} \,b^{2}+a c b \right )}+2 b \right )} \]

Problem 8727


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

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

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

Problem 8728


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

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

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

Problem 8729


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

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

Maple solution	\begin{align} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\operatorname {csgn}\left (\sin \left (x \right )\right ) c_{1}}{\cos \left (x \right )+\operatorname {csgn}\left (\sec \left (x \right )\right )} \\ y \left (x \right ) &= \csc \left (x \right )^{2} \left (\cos \left (x \right )+\operatorname {csgn}\left (\sec \left (x \right )\right )\right ) \operatorname {csgn}\left (\sin \left (x \right )\right ) c_{1} \\ \end{align}

Problem 8730


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

program solution

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

Problem 8731


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

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

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

Problem 8732


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

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

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

Problem 8733


ODE	\[ \boxed {{y^{\prime }}^{2}-3 x y^{\frac {2}{3}} y^{\prime }+9 y^{\frac {5}{3}}=0} \]

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

Maple solution	\begin{align} y \left (x \right ) &= \frac {x^{6}}{64} \\ y \left (x \right ) &= 0 \\ \ln \left (x \right )+\frac {\ln \left (\frac {64 y \left (x \right )}{x^{6}}-1\right )}{6}-\frac {\ln \left (4 \left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {1}{3}}-1\right )}{6}-\frac {\ln \left (16 \left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {2}{3}}+4 \left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {1}{3}}+1\right )}{6}+\frac {\ln \left (\frac {y \left (x \right )}{x^{6}}\right )}{6}-\frac {\sqrt {-\frac {y \left (x \right ) \left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {1}{3}} \left (4 \left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {1}{3}}-1\right )}{x^{6}}}\, \operatorname {arctanh}\left (\sqrt {-4 \left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {1}{3}}+1}\right )}{\left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {2}{3}} \sqrt {-4 \left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {1}{3}}+1}}-c_{1} &= 0 \\ \end{align}

Problem 8734


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

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

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

Problem 8735


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

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

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

Problem 8736


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

program solution	\[ y = 0 \] Veriﬁed OK. \[ x = \frac {\left (8 x^{2}-6 y\right ) \sqrt {x^{2}-3 y}+8 x^{3}-18 y x +27 c_{1}}{3 \left (x +\sqrt {x^{2}-3 y}\right )^{2}} \] Veriﬁed OK. \[ x = \frac {\left (-8 x^{2}+6 y\right ) \sqrt {x^{2}-3 y}+8 x^{3}-18 y x +27 c_{1}}{3 \left (x -\sqrt {x^{2}-3 y}\right )^{2}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -\frac {\left (x^{2}+x \left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}+\left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}\right ) \left (x^{2}-3 x \left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}+\left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}\right )}{12 \left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= -\frac {\left (i \left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}} \sqrt {3}-i \sqrt {3}\, x^{2}+\left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}-2 x \left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}+x^{2}\right ) \left (i \left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}} \sqrt {3}-i \sqrt {3}\, x^{2}+\left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}+6 x \left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}+x^{2}\right )}{48 \left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= -\frac {\left (i \sqrt {3}\, x^{2}-i \left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}} \sqrt {3}+x^{2}-2 x \left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}+\left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}\right ) \left (i \sqrt {3}\, x^{2}-i \left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}} \sqrt {3}+x^{2}+6 x \left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}+\left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}\right )}{48 \left (x^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}} \\ \end{align}

Problem 8737


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

program solution	\[ y = \frac {3}{16} c_{1}^{2}+\frac {1}{4} x c_{1} -\frac {1}{4} x^{2} \] Veriﬁed OK. \[ y = \frac {3}{16} c_{1}^{2}+\frac {1}{4} x c_{1} -\frac {1}{4} x^{2} \] Veriﬁed OK.

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

Problem 8738


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

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

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

Problem 8739


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

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

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

Problem 8740


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

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

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

Problem 8741


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

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

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

Problem 8742


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

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

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

Problem 8743


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

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

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

Problem 8744


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

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

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

Problem 8745


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

program solution	\[ y = 0 \] Veriﬁed OK. \[ y = 4+2 x \] Veriﬁed OK. \[ x = \frac {\left (8 \ln \left (\frac {-2+\sqrt {4+2 y x}}{x}\right ) x +x c_{1} -4 \sqrt {4+2 y x}+8\right ) x}{\left (2 x +2-\sqrt {4+2 y x}\right )^{2}} \] Veriﬁed OK. \[ x = \frac {\left (8 \ln \left (\frac {-2-\sqrt {4+2 y x}}{x}\right ) x +x c_{1} +4 \sqrt {4+2 y x}+8\right ) x}{\left (2 x +2+\sqrt {4+2 y x}\right )^{2}} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = 2 x \,{\mathrm e}^{\operatorname {RootOf}\left (-x \,{\mathrm e}^{2 \textit {\_Z}}+4 x \,{\mathrm e}^{\textit {\_Z}}-4 \,{\mathrm e}^{\textit {\_Z}}+c_{1} +8 \textit {\_Z} -4 x \right )}+4 \operatorname {RootOf}\left (-x \,{\mathrm e}^{2 \textit {\_Z}}+4 x \,{\mathrm e}^{\textit {\_Z}}-4 \,{\mathrm e}^{\textit {\_Z}}+c_{1} +8 \textit {\_Z} -4 x \right )+\frac {c_{1}}{2}-2 x \]

Problem 8746


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

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

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

Problem 8747


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

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

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

Problem 8748


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

program solution	\[ \frac {2 \ln \left (y+\sqrt {y^{2}+4 x^{3}}\right )}{15}-\frac {4 \ln \left (-4 \sqrt {15}\, x^{\frac {3}{2}} \sqrt {y^{2}+4 x^{3}}-y \sqrt {15}\, x^{\frac {3}{2}}-30 x^{3}\right )}{15}+\frac {4 \ln \left (2 \sqrt {15}\, x^{\frac {3}{2}}-15 y\right )}{15}+\frac {4 \ln \left (4 \sqrt {15}\, x^{\frac {3}{2}} \sqrt {y^{2}+4 x^{3}}-y \sqrt {15}\, x^{\frac {3}{2}}+30 x^{3}\right )}{15}-\frac {4 \ln \left (2 \sqrt {15}\, x^{\frac {3}{2}}+15 y\right )}{15}+\frac {4 \ln \left (-4 x^{3}+15 y^{2}\right )}{15} = c_{1} \] Veriﬁed OK.

Maple solution	\begin{align} -\left (\int _{\textit {\_b}}^{x}\frac {y \left (x \right )-\sqrt {4 \textit {\_a}^{3}+y \left (x \right )^{2}}}{\textit {\_a} \left (4 y \left (x \right )-\sqrt {4 \textit {\_a}^{3}+y \left (x \right )^{2}}\right )}d \textit {\_a} \right )+2 \left (\int _{}^{y \left (x \right )}\frac {-1+\left (24 \textit {\_f} -6 \sqrt {4 x^{3}+\textit {\_f}^{2}}\right ) \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{2}}{\left (-4 \textit {\_f} +\sqrt {4 \textit {\_a}^{3}+\textit {\_f}^{2}}\right )^{2} \sqrt {4 \textit {\_a}^{3}+\textit {\_f}^{2}}}d \textit {\_a} \right )}{4 \textit {\_f} -\sqrt {4 x^{3}+\textit {\_f}^{2}}}d \textit {\_f} \right )+c_{1} &= 0 \\ -\left (\int _{\textit {\_b}}^{x}\frac {y \left (x \right )+\sqrt {4 \textit {\_a}^{3}+y \left (x \right )^{2}}}{\left (\sqrt {4 \textit {\_a}^{3}+y \left (x \right )^{2}}+4 y \left (x \right )\right ) \textit {\_a}}d \textit {\_a} \right )-2 \left (\int _{}^{y \left (x \right )}\frac {1+6 \left (\sqrt {4 x^{3}+\textit {\_f}^{2}}+4 \textit {\_f} \right ) \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{2}}{\left (\sqrt {4 \textit {\_a}^{3}+\textit {\_f}^{2}}+4 \textit {\_f} \right )^{2} \sqrt {4 \textit {\_a}^{3}+\textit {\_f}^{2}}}d \textit {\_a} \right )}{\sqrt {4 x^{3}+\textit {\_f}^{2}}+4 \textit {\_f}}d \textit {\_f} \right )+c_{1} &= 0 \\ \end{align}

Problem 8749


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

program solution	\[ \frac {\ln \left (y+\sqrt {y^{2}-4 x^{4}}\right )}{12}-\frac {5 \ln \left (-5 \sqrt {6}\, \sqrt {y^{2}-4 x^{4}}-\sqrt {6}\, y-24 i x^{2}\right )}{24}+\frac {5 \ln \left (x^{2} \sqrt {6}+6 i y\right )}{24}+\frac {5 \ln \left (-5 \sqrt {6}\, \sqrt {y^{2}-4 x^{4}}+\sqrt {6}\, y-24 i x^{2}\right )}{24}-\frac {5 \ln \left (-x^{2} \sqrt {6}+6 i y\right )}{24}+\frac {5 \ln \left (x^{4}+6 y^{2}\right )}{24} = c_{1} \] Veriﬁed OK.

Maple solution	\begin{align} -\left (\int _{\textit {\_b}}^{x}\frac {y \left (x \right )-\sqrt {-4 \textit {\_a}^{4}+y \left (x \right )^{2}}}{\textit {\_a} \left (5 y \left (x \right )-\sqrt {-4 \textit {\_a}^{4}+y \left (x \right )^{2}}\right )}d \textit {\_a} \right )-2 \left (\int _{}^{y \left (x \right )}\frac {1+\left (40 \textit {\_f} -8 \sqrt {-4 x^{4}+\textit {\_f}^{2}}\right ) \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{3}}{\left (-5 \textit {\_f} +\sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}\right )^{2} \sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}}d \textit {\_a} \right )}{5 \textit {\_f} -\sqrt {-4 x^{4}+\textit {\_f}^{2}}}d \textit {\_f} \right )+c_{1} &= 0 \\ -\left (\int _{\textit {\_b}}^{x}\frac {y \left (x \right )+\sqrt {-4 \textit {\_a}^{4}+y \left (x \right )^{2}}}{\left (\sqrt {-4 \textit {\_a}^{4}+y \left (x \right )^{2}}+5 y \left (x \right )\right ) \textit {\_a}}d \textit {\_a} \right )+2 \left (\int _{}^{y \left (x \right )}\frac {-1+8 \left (\sqrt {-4 x^{4}+\textit {\_f}^{2}}+5 \textit {\_f} \right ) \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{3}}{\left (\sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}+5 \textit {\_f} \right )^{2} \sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}}d \textit {\_a} \right )}{\sqrt {-4 x^{4}+\textit {\_f}^{2}}+5 \textit {\_f}}d \textit {\_f} \right )+c_{1} &= 0 \\ \end{align}

Problem 8750


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

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

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

Problem 8751


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

program solution	\[ y = 0 \] Veriﬁed OK. \[ y = x \] Veriﬁed OK. \[ x = \frac {c_{3} \left (-y+5 x +\sqrt {y^{2}-10 y x +9 x^{2}}\right ) \sqrt {2}}{\left (-y+3 x +\sqrt {y^{2}-10 y x +9 x^{2}}\right ) \sqrt {\frac {-y+3 x +\sqrt {y^{2}-10 y x +9 x^{2}}}{x}}} \] Veriﬁed OK. \[ x = \frac {c_{3} \left (y-5 x +\sqrt {y^{2}-10 y x +9 x^{2}}\right ) \sqrt {2}}{\left (y-3 x +\sqrt {y^{2}-10 y x +9 x^{2}}\right ) \sqrt {\frac {-y+3 x -\sqrt {y^{2}-10 y x +9 x^{2}}}{x}}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= x \\ \frac {c_{1} \left (-5 x +y \left (x \right )-\sqrt {9 x^{2}-10 x y \left (x \right )+y \left (x \right )^{2}}\right )}{x {\left (\frac {3 x -y \left (x \right )+\sqrt {9 x^{2}-10 x y \left (x \right )+y \left (x \right )^{2}}}{x}\right )}^{\frac {3}{2}}}+x &= 0 \\ \frac {\left (-5 x +y \left (x \right )+\sqrt {9 x^{2}-10 x y \left (x \right )+y \left (x \right )^{2}}\right ) c_{1} \sqrt {2}}{4 x {\left (\frac {-y \left (x \right )+3 x -\sqrt {9 x^{2}-10 x y \left (x \right )+y \left (x \right )^{2}}}{x}\right )}^{\frac {3}{2}}}+x &= 0 \\ \end{align}

Problem 8752


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

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

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

Problem 8753


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

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

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

Problem 8754


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

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

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

Problem 8755


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

program solution	\[ x = \frac {-3 c_{1} \left (a x -4 y^{2}\right ) \sqrt {y^{2}-a x}+\left (-9 y a x +12 y^{3}\right ) c_{1} +a \,x^{3}}{3 \left (y+\sqrt {y^{2}-a x}\right )^{2} x} \] Veriﬁed OK. \[ x = \frac {3 c_{1} \left (a x -4 y^{2}\right ) \sqrt {y^{2}-a x}+\left (-9 y a x +12 y^{3}\right ) c_{1} +a \,x^{3}}{3 \left (y-\sqrt {y^{2}-a x}\right )^{2} x} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \frac {x \left (\frac {4 x^{2}}{\left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{\frac {1}{3}}}+2 x +\left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{\frac {1}{3}}\right )}{12 c_{1}}+\frac {3 a c_{1} \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{\frac {1}{3}}}{\left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{\frac {2}{3}}+2 x \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{\frac {1}{3}}+4 x^{2}} \\ y \left (x \right ) &= -\frac {\left (\left (1+i \sqrt {3}\right ) \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{\frac {2}{3}}-4 x \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{\frac {1}{3}}-4 x^{2} \left (i \sqrt {3}-1\right )\right ) x}{24 \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{\frac {1}{3}} c_{1}}+\frac {6 a c_{1} \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{\frac {1}{3}}}{4 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{\frac {2}{3}}-4 x^{2}+4 x \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{\frac {1}{3}}-\left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= \frac {x \left (\left (i \sqrt {3}-1\right ) \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{\frac {2}{3}}+4 x \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{\frac {1}{3}}-4 x^{2} \left (1+i \sqrt {3}\right )\right )}{24 \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{\frac {1}{3}} c_{1}}-\frac {6 a c_{1} \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{\frac {1}{3}}}{4 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{\frac {2}{3}}+4 x^{2}-4 x \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{\frac {1}{3}}+\left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{\frac {2}{3}}} \\ \end{align}

Problem 8756


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

program solution	\[ y = -i x \] Veriﬁed OK. \[ y = i x \] Veriﬁed OK. \[ y = \frac {c_{1}^{2} x^{2}-1}{2 c_{1}} \] Veriﬁed OK.

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

Problem 8757


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

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

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

Problem 8758


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

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

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

Problem 8759


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

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

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

Problem 8760


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

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

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

Problem 8761


ODE	\[ \boxed {\left (1+3 x \right ) {y^{\prime }}^{2}-3 \left (y+2\right ) y^{\prime }=-9} \]

program solution	\[ y = x c_{1} +\frac {c_{1}^{2}-6 c_{1} +9}{3 c_{1}} \] Veriﬁed OK. \[ y = \frac {-2 \sqrt {1+3 x}+6 x +2}{\sqrt {1+3 x}} \] Veriﬁed OK. \[ y = \frac {-2 \sqrt {1+3 x}-6 x -2}{\sqrt {1+3 x}} \] Veriﬁed OK.

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

Problem 8762


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

program solution	\[ y = x c_{1} +\frac {5 c_{1}^{2}}{3 c_{1} -1} \] Veriﬁed OK. \[ y = \frac {\left (3 x +10\right ) \sqrt {15 x +25}+30 x +50}{9 \sqrt {15 x +25}} \] Veriﬁed OK. \[ y = \frac {\left (3 x +10\right ) \sqrt {15 x +25}-30 x -50}{9 \sqrt {15 x +25}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \frac {x}{3}+\frac {10}{9}-\frac {2 \sqrt {15 x +25}}{9} \\ y \left (x \right ) &= \frac {x}{3}+\frac {10}{9}+\frac {2 \sqrt {15 x +25}}{9} \\ y \left (x \right ) &= \frac {\left (3 x +5\right ) c_{1}^{2}-c_{1} x}{3 c_{1} -1} \\ \end{align}

Problem 8763


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

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

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

Problem 8764


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

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

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

Problem 8765


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

program solution	\[ y = \frac {-\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a1} \operatorname {b1} x +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a2} \operatorname {b0} x -2 \operatorname {a0} \operatorname {a2} \operatorname {b1} x -2 \operatorname {a0} \,\operatorname {b1}^{2} x +\operatorname {a1}^{2} \operatorname {b1} x -\operatorname {a1} \operatorname {a2} \operatorname {b0} x +\operatorname {a1} \operatorname {b0} \operatorname {b1} x -\operatorname {a2} \,\operatorname {b0}^{2} x +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a1} \operatorname {c2} -\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a2} \operatorname {c1} +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b0} \operatorname {c2} -\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b1} \operatorname {c1} +2 \operatorname {a0} \operatorname {a2} \operatorname {c2} +2 \operatorname {a0} \operatorname {b1} \operatorname {c2} -\operatorname {a1}^{2} \operatorname {c2} +\operatorname {a1} \operatorname {a2} \operatorname {c1} -2 \operatorname {a1} \operatorname {b0} \operatorname {c2} +\operatorname {a1} \operatorname {b1} \operatorname {c1} -2 \operatorname {a2}^{2} \operatorname {c0} +\operatorname {a2} \operatorname {b0} \operatorname {c1} -4 \operatorname {a2} \operatorname {b1} \operatorname {c0} -\operatorname {b0}^{2} \operatorname {c2} +\operatorname {b0} \operatorname {b1} \operatorname {c1} -2 \operatorname {b1}^{2} \operatorname {c0}}{\operatorname {a2} \sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b1} -\operatorname {a2} \operatorname {a1} \operatorname {b1} +2 \operatorname {a2}^{2} \operatorname {b0} +3 \operatorname {a2} \operatorname {b0} \operatorname {b1} +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b1}^{2}-\operatorname {a1} \,\operatorname {b1}^{2}+\operatorname {b0} \,\operatorname {b1}^{2}} \] Veriﬁed OK. \[ y = \frac {-\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a1} \operatorname {b1} x +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a2} \operatorname {b0} x +2 \operatorname {a0} \operatorname {a2} \operatorname {b1} x +2 \operatorname {a0} \,\operatorname {b1}^{2} x -\operatorname {a1}^{2} \operatorname {b1} x +\operatorname {a1} \operatorname {a2} \operatorname {b0} x -\operatorname {a1} \operatorname {b0} \operatorname {b1} x +\operatorname {a2} \,\operatorname {b0}^{2} x +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a1} \operatorname {c2} -\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a2} \operatorname {c1} +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b0} \operatorname {c2} -\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b1} \operatorname {c1} -2 \operatorname {a0} \operatorname {a2} \operatorname {c2} -2 \operatorname {a0} \operatorname {b1} \operatorname {c2} +\operatorname {a1}^{2} \operatorname {c2} -\operatorname {a1} \operatorname {a2} \operatorname {c1} +2 \operatorname {a1} \operatorname {b0} \operatorname {c2} -\operatorname {a1} \operatorname {b1} \operatorname {c1} +2 \operatorname {a2}^{2} \operatorname {c0} -\operatorname {a2} \operatorname {b0} \operatorname {c1} +4 \operatorname {a2} \operatorname {b1} \operatorname {c0} +\operatorname {b0}^{2} \operatorname {c2} -\operatorname {b0} \operatorname {b1} \operatorname {c1} +2 \operatorname {b1}^{2} \operatorname {c0}}{\operatorname {a2} \sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b1} +\operatorname {a2} \operatorname {a1} \operatorname {b1} -2 \operatorname {a2}^{2} \operatorname {b0} -3 \operatorname {a2} \operatorname {b0} \operatorname {b1} +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b1}^{2}+\operatorname {a1} \,\operatorname {b1}^{2}-\operatorname {b0} \,\operatorname {b1}^{2}} \] 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

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

Problem 8766


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

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

Maple solution	\begin{align} y \left (x \right ) &= -1 \\ y \left (x \right ) &= 1 \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \csc \left (-\ln \left (x \right )+c_{1} \right ) \operatorname {csgn}\left (\sec \left (-\ln \left (x \right )+c_{1} \right )\right ) \\ y \left (x \right ) &= -\csc \left (-\ln \left (x \right )+c_{1} \right ) \operatorname {csgn}\left (\sec \left (-\ln \left (x \right )+c_{1} \right )\right ) \\ \end{align}

Problem 8767


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

program solution

Maple solution	\[ y \left (x \right )-\operatorname {RootOf}\left (-a \,\operatorname {arcsinh}\left (\frac {\operatorname {RootOf}\left (-2 a y \left (x \right )+a^{2}+x^{2}+2 a \textit {\_Z} +\textit {\_Z}^{2}\right )}{x}\right )-x \sqrt {\frac {a \left (-2 \operatorname {RootOf}\left (-2 a y \left (x \right )+a^{2}+x^{2}+2 a \textit {\_Z} +\textit {\_Z}^{2}\right )+2 \textit {\_Z} -a \right )}{x^{2}}}+c_{1} \right ) = 0 \]

Problem 8768


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

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

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

Problem 8769


ODE	\[ \boxed {y^{\prime }=1} \]

program solution	\[ y = x +c_{1} \] Veriﬁed OK.

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

Problem 8770


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

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

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

Problem 8771


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

program solution

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

Problem 8772


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

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

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

Problem 8773


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

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

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

Problem 8774


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

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

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

Problem 8775


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

program solution	\[ y = c_{2} x \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= c_{1} x \\ y \left (x \right ) &= \frac {c_{1}}{x^{5}} \\ \end{align}

Problem 8776


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

program solution	\[ y = \frac {{\mathrm e}^{2 c_{1}} c_{2}^{2} x^{2}}{2}-2 c_{2} {\mathrm e}^{c_{1}} x \] Veriﬁed OK. \[ y = \frac {{\mathrm e}^{2 c_{3}} x^{2} c_{4}^{2}}{2}+2 c_{4} {\mathrm e}^{c_{3}} x \] Veriﬁed OK.

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

Problem 8777


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

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

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

Problem 8778


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

program solution	\[ y = 0 \] Veriﬁed OK. \[ y = 4 x \] Veriﬁed OK. \[ y = 0 \] Veriﬁed OK.

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

Problem 8779


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

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

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

Problem 8780


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

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

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

Problem 8781


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

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

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

Problem 8782


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

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

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

Problem 8783


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

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

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

Problem 8784


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

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

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

Problem 8785


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

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

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

Problem 8786


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

program solution

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

Problem 8787


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

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

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

Problem 8788


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

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

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

Problem 8789


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

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

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

Problem 8790


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

program solution

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

Problem 8791


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

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

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

Problem 8792


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

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

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

Problem 8793


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

program solution

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

Problem 8794


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

program solution

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

Problem 8795


ODE	\[ \boxed {\operatorname {d0} \left (x \right ) {y^{\prime }}^{2}+2 \operatorname {b0} \left (x \right ) y y^{\prime }+\operatorname {c0} \left (x \right ) y^{2}+2 \operatorname {d0} \left (x \right ) y^{\prime }+2 \operatorname {e0} \left (x \right ) y=-\operatorname {f0} \left (x \right )} \]

program solution

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

Problem 8796


ODE	\[ \boxed {y {y^{\prime }}^{2}=1} \]

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

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

Problem 8797


ODE	\[ \boxed {y {y^{\prime }}^{2}={\mathrm e}^{2 x}} \]

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

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

Problem 8798


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

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

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

Problem 8799


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

program solution	\[ y = 0 \] Veriﬁed OK. \[ y = \sqrt {7}\, x \] Veriﬁed OK. \[ y = -\sqrt {7}\, x \] 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 {9 \textit {\_a}^{2}+1}+1}{\textit {\_a} \left (\textit {\_a}^{2}-7\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 {9 \textit {\_a}^{2}+1}+1}{\textit {\_a} \left (\textit {\_a}^{2}-7\right )}d \textit {\_a} +c_{1} \right ) x \\ \end{align}

Problem 8800


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

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

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

2.17.88 Problems 8701 to 8800