2.17.42 Problems 4101 to 4200

Problem 4101

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

Problem 4102

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

Problem 4103

ODE	\[ \boxed {x {y^{\prime }}^{2}-2 y^{\prime } y=-a} \]
program solution	\[ x = \frac {-3 c_{1} \left (a x -4 y^{2}\right ) \sqrt {y^{2}-a x}+\left (-9 x a y+12 y^{3}\right ) c_{1} +a \,x^{3}}{3 \left (y+\sqrt {y^{2}-a x}\right )^{2} x} \] Verified OK. \[ x = \frac {3 c_{1} \left (a x -4 y^{2}\right ) \sqrt {y^{2}-a x}+\left (-9 x a y+12 y^{3}\right ) c_{1} +a \,x^{3}}{3 \left (y-\sqrt {y^{2}-a x}\right )^{2} x} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (\frac {4 x^{2}}{\left (-36 c_{1}^{2} a +8 x^{3}+12 \sqrt {a \left (9 c_{1}^{2} a -4 x^{3}\right )}\, c_{1} \right )^{\frac {1}{3}}}+2 x +\left (-36 c_{1}^{2} a +8 x^{3}+12 \sqrt {a \left (9 c_{1}^{2} a -4 x^{3}\right )}\, c_{1} \right )^{\frac {1}{3}}\right ) x}{12 c_{1}}+\frac {3 a c_{1} \left (-36 c_{1}^{2} a +8 x^{3}+12 \sqrt {a \left (9 c_{1}^{2} a -4 x^{3}\right )}\, c_{1} \right )^{\frac {1}{3}}}{\left (-36 c_{1}^{2} a +8 x^{3}+12 \sqrt {a \left (9 c_{1}^{2} a -4 x^{3}\right )}\, c_{1} \right )^{\frac {2}{3}}+2 x \left (-36 c_{1}^{2} a +8 x^{3}+12 \sqrt {a \left (9 c_{1}^{2} a -4 x^{3}\right )}\, c_{1} \right )^{\frac {1}{3}}+4 x^{2}} \\ y \left (x \right ) &= -\frac {x \left (\left (1+i \sqrt {3}\right ) \left (-36 c_{1}^{2} a +8 x^{3}+12 \sqrt {a \left (9 c_{1}^{2} a -4 x^{3}\right )}\, c_{1} \right )^{\frac {2}{3}}-4 x \left (-36 c_{1}^{2} a +8 x^{3}+12 \sqrt {a \left (9 c_{1}^{2} a -4 x^{3}\right )}\, c_{1} \right )^{\frac {1}{3}}-4 x^{2} \left (i \sqrt {3}-1\right )\right )}{24 \left (-36 c_{1}^{2} a +8 x^{3}+12 \sqrt {a \left (9 c_{1}^{2} a -4 x^{3}\right )}\, c_{1} \right )^{\frac {1}{3}} c_{1}}+\frac {6 a c_{1} \left (-36 c_{1}^{2} a +8 x^{3}+12 \sqrt {a \left (9 c_{1}^{2} a -4 x^{3}\right )}\, c_{1} \right )^{\frac {1}{3}}}{4 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (-36 c_{1}^{2} a +8 x^{3}+12 \sqrt {a \left (9 c_{1}^{2} a -4 x^{3}\right )}\, c_{1} \right )^{\frac {2}{3}}-4 x^{2}+4 x \left (-36 c_{1}^{2} a +8 x^{3}+12 \sqrt {a \left (9 c_{1}^{2} a -4 x^{3}\right )}\, c_{1} \right )^{\frac {1}{3}}-\left (-36 c_{1}^{2} a +8 x^{3}+12 \sqrt {a \left (9 c_{1}^{2} a -4 x^{3}\right )}\, c_{1} \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= \frac {\left (\left (i \sqrt {3}-1\right ) \left (-36 c_{1}^{2} a +8 x^{3}+12 \sqrt {a \left (9 c_{1}^{2} a -4 x^{3}\right )}\, c_{1} \right )^{\frac {2}{3}}+4 x \left (-36 c_{1}^{2} a +8 x^{3}+12 \sqrt {a \left (9 c_{1}^{2} a -4 x^{3}\right )}\, c_{1} \right )^{\frac {1}{3}}-4 x^{2} \left (1+i \sqrt {3}\right )\right ) x}{24 \left (-36 c_{1}^{2} a +8 x^{3}+12 \sqrt {a \left (9 c_{1}^{2} a -4 x^{3}\right )}\, c_{1} \right )^{\frac {1}{3}} c_{1}}-\frac {6 a c_{1} \left (-36 c_{1}^{2} a +8 x^{3}+12 \sqrt {a \left (9 c_{1}^{2} a -4 x^{3}\right )}\, c_{1} \right )^{\frac {1}{3}}}{4 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (-36 c_{1}^{2} a +8 x^{3}+12 \sqrt {a \left (9 c_{1}^{2} a -4 x^{3}\right )}\, c_{1} \right )^{\frac {2}{3}}+4 x^{2}-4 x \left (-36 c_{1}^{2} a +8 x^{3}+12 \sqrt {a \left (9 c_{1}^{2} a -4 x^{3}\right )}\, c_{1} \right )^{\frac {1}{3}}+\left (-36 c_{1}^{2} a +8 x^{3}+12 \sqrt {a \left (9 c_{1}^{2} a -4 x^{3}\right )}\, c_{1} \right )^{\frac {2}{3}}} \\ \end{align*}

Problem 4104

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

Problem 4105

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

Problem 4106

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

Problem 4107

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

Problem 4108

ODE	\[ \boxed {x {y^{\prime }}^{2}-a y y^{\prime }=-b} \]
program solution	\[ x = \frac {2 \left (a y \left (a 2^{\frac {a -2}{a -1}}-2^{-\frac {1}{a -1}}\right ) \sqrt {a^{2} y^{2}-4 b x}+\left (a^{3} y^{2}+b x \right ) 2^{\frac {a -2}{a -1}}-a 2^{-\frac {1}{a -1}} \left (y^{2} a +4 b x \right )\right ) c_{1} {\left (\frac {a y+\sqrt {a^{2} y^{2}-4 b x}}{x}\right )}^{\frac {1}{a -1}}+4 b \,x^{2}}{\left (2 a -1\right ) \left (a y+\sqrt {a^{2} y^{2}-4 b x}\right )^{2}} \] Warning, solution could not be verified \[ x = \frac {4 \left (a -\frac {1}{2}\right ) c_{1} \left (a^{2} y^{2}-a y \sqrt {a^{2} y^{2}-4 b x}-2 b x \right ) {\left (-\frac {-a y+\sqrt {a^{2} y^{2}-4 b x}}{2 x}\right )}^{\frac {1}{a -1}}+4 b \,x^{2}}{\left (2 a -1\right ) \left (a y-\sqrt {a^{2} y^{2}-4 b x}\right )^{2}} \] Verified OK.
Maple solution	\begin{align*} \frac {4 \left (2^{-\frac {1}{a -1}} y \left (x \right ) \left (a -\frac {1}{2}\right )^{2} a \sqrt {a^{2} y \left (x \right )^{2}-4 b x}-\frac {b x 2^{\frac {a -2}{a -1}}}{4}+2^{-\frac {1}{a -1}} \left (\left (a -\frac {1}{2}\right )^{2} a y \left (x \right )^{2}-2 b x \left (a -1\right )\right ) a \right ) c_{1} {\left (\frac {a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 b x}}{x}\right )}^{\frac {1}{a -1}}+4 x \left (y \left (x \right ) \left (a -\frac {1}{2}\right ) \sqrt {a^{2} y \left (x \right )^{2}-4 b x}+\left (a^{2}-\frac {1}{2} a \right ) y \left (x \right )^{2}-2 b x \right ) a}{\left (2 a -1\right ) \left (a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 b x}\right )^{2}} &= 0 \\ \frac {-4 c_{1} \left (-2^{-\frac {1}{a -1}} y \left (x \right ) \left (a -\frac {1}{2}\right )^{2} a \sqrt {a^{2} y \left (x \right )^{2}-4 b x}-\frac {b x 2^{\frac {a -2}{a -1}}}{4}+2^{-\frac {1}{a -1}} \left (\left (a -\frac {1}{2}\right )^{2} a y \left (x \right )^{2}-2 b x \left (a -1\right )\right ) a \right ) {\left (\frac {a y \left (x \right )-\sqrt {a^{2} y \left (x \right )^{2}-4 b x}}{x}\right )}^{\frac {1}{a -1}}+4 \left (-y \left (x \right ) \left (a -\frac {1}{2}\right ) \sqrt {a^{2} y \left (x \right )^{2}-4 b x}+\left (a^{2}-\frac {1}{2} a \right ) y \left (x \right )^{2}-2 b x \right ) x a}{\left (2 a -1\right ) \left (a y \left (x \right )-\sqrt {a^{2} y \left (x \right )^{2}-4 b x}\right )^{2}} &= 0 \\ \end{align*}

Problem 4109

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

Problem 4110

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

Problem 4111

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

Problem 4112

ODE	\[ \boxed {x {y^{\prime }}^{2}+\left (1-y x^{2}\right ) y^{\prime }-y x=0} \]
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 ) &= -\ln \left (x \right )+c_{1} \\ \end{align*}

Problem 4113

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

Problem 4114

ODE	\[ \boxed {\left (x +1\right ) {y^{\prime }}^{2}-\left (y+x \right ) y^{\prime }+y=0} \]
program solution	\[ y = c_{1} x +\frac {c_{1}^{2}}{c_{1} -1} \] Verified OK. \[ y = \frac {\left (x +2\right ) \sqrt {x +1}+2 x +2}{\sqrt {x +1}} \] Verified OK. \[ y = \frac {\left (x +2\right ) \sqrt {x +1}-2 x -2}{\sqrt {x +1}} \] 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 4115

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

Problem 4116

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

Problem 4117

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

Problem 4118

ODE	\[ \boxed {\left (1+3 x \right ) {y^{\prime }}^{2}-3 \left (2+y\right ) y^{\prime }=-9} \]
program solution	\[ y = c_{1} x +\frac {c_{1}^{2}-6 c_{1} +9}{3 c_{1}} \] 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 4119

ODE	\[ \boxed {\left (5+3 x \right ) {y^{\prime }}^{2}-\left (3+3 y\right ) y^{\prime }+y=0} \]
program solution	\[ y = 0 \] Verified OK. \[ x = \frac {-18 \left (c_{1} +3 \,\operatorname {expIntegral}_{1}\left (\frac {-9-9 y-3 \sqrt {9+9 y^{2}+\left (-12 x -2\right ) y}}{10+6 x}\right )\right ) \left (x -\frac {3 y}{2}-\frac {\sqrt {9+9 y^{2}+\left (-12 x -2\right ) y}}{2}+\frac {1}{6}\right ) {\mathrm e}^{\frac {-9-9 y-3 \sqrt {9+9 y^{2}+\left (-12 x -2\right ) y}}{10+6 x}}+24 x +40}{30+18 x} \] Verified OK. \[ x = \frac {-18 \left (x -\frac {3 y}{2}+\frac {\sqrt {9+9 y^{2}+\left (-12 x -2\right ) y}}{2}+\frac {1}{6}\right ) \left (c_{1} +3 \,\operatorname {expIntegral}_{1}\left (\frac {-9 y-9+3 \sqrt {9+9 y^{2}+\left (-12 x -2\right ) y}}{10+6 x}\right )\right ) {\mathrm e}^{\frac {-9 y-9+3 \sqrt {9+9 y^{2}+\left (-12 x -2\right ) y}}{10+6 x}}+24 x +40}{30+18 x} \] Verified OK.
Maple solution	\begin{align*} \frac {-108 \left (x -\frac {3 y \left (x \right )}{2}-\frac {\sqrt {9+9 y \left (x \right )^{2}+\left (-12 x -2\right ) y \left (x \right )}}{2}+\frac {1}{6}\right ) \left (c_{1} -\frac {\operatorname {expIntegral}_{1}\left (\frac {-9 y \left (x \right )-9-3 \sqrt {9+9 y \left (x \right )^{2}+\left (-12 x -2\right ) y \left (x \right )}}{10+6 x}\right )}{2}\right ) {\mathrm e}^{\frac {-9 y \left (x \right )-9-3 \sqrt {9+9 y \left (x \right )^{2}+\left (-12 x -2\right ) y \left (x \right )}}{10+6 x}}+18 x^{2}+6 x -40}{30+18 x} &= 0 \\ \frac {108 \left (x -\frac {3 y \left (x \right )}{2}+\frac {\sqrt {9+9 y \left (x \right )^{2}+\left (-12 x -2\right ) y \left (x \right )}}{2}+\frac {1}{6}\right ) \left (c_{1} +\frac {\operatorname {expIntegral}_{1}\left (\frac {-9 y \left (x \right )-9+3 \sqrt {9+9 y \left (x \right )^{2}+\left (-12 x -2\right ) y \left (x \right )}}{10+6 x}\right )}{2}\right ) {\mathrm e}^{\frac {-9 y \left (x \right )-9+3 \sqrt {9+9 y \left (x \right )^{2}+\left (-12 x -2\right ) y \left (x \right )}}{10+6 x}}+18 x^{2}+6 x -40}{30+18 x} &= 0 \\ \end{align*}

Problem 4120

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

Problem 4121

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

Problem 4122

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

Problem 4123

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

Problem 4124

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

Problem 4125

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

Problem 4126

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

Problem 4127

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

Problem 4128

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

Problem 4129

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

Problem 4130

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

Problem 4131

ODE	\[ \boxed {x^{2} {y^{\prime }}^{2}+y^{2}-y^{4}=0} \]
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 4132

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

Problem 4133

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

Problem 4134

ODE	\[ \boxed {x^{2} {y^{\prime }}^{2}-2 y y^{\prime } x +y \left (1+y\right )=x} \]
program solution	\[ 2 \,\operatorname {arctanh}\left (\sqrt {-y+x}\right )-\ln \left (y-x +1\right ) = -\ln \left (x \right )+c_{1} \] Verified OK. \[ -\ln \left (y-x +1\right )-2 \,\operatorname {arctanh}\left (\sqrt {-y+x}\right ) = -\ln \left (x \right )+c_{1} \] 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 4135

ODE	\[ \boxed {x^{2} {y^{\prime }}^{2}-2 y y^{\prime } x +\left (-x^{2}+1\right ) y^{2}=x^{4}} \]
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 4136

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

Problem 4137

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

Problem 4138

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

Problem 4139

ODE	\[ \boxed {x^{2} {y^{\prime }}^{2}+2 x \left (2 x +y\right ) y^{\prime }+y^{2}=4 a} \]
program solution	\[ -\frac {\operatorname {arctanh}\left (\frac {x +\sqrt {y x +x^{2}+a}}{\sqrt {a}}\right )+\operatorname {arctanh}\left (\frac {\sqrt {y x +x^{2}+a}-x}{\sqrt {a}}\right )+\operatorname {arctanh}\left (\frac {y}{2 \sqrt {a}}\right )}{2 \sqrt {a}} = c_{1} \] Verified OK. \[ \frac {\operatorname {arctanh}\left (\frac {x +\sqrt {y x +x^{2}+a}}{\sqrt {a}}\right )+\operatorname {arctanh}\left (\frac {\sqrt {y x +x^{2}+a}-x}{\sqrt {a}}\right )-\operatorname {arctanh}\left (\frac {y}{2 \sqrt {a}}\right )}{2 \sqrt {a}} = c_{1} \] 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 4140

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

Problem 4141

ODE	\[ \boxed {x^{2} {y^{\prime }}^{2}+3 y y^{\prime } x +2 y^{2}=0} \]
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 4142

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

Problem 4143

ODE	\[ \boxed {x^{2} {y^{\prime }}^{2}+4 y y^{\prime } x -5 y^{2}=0} \]
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 4144

ODE	\[ \boxed {x^{2} {y^{\prime }}^{2}-4 x \left (2+y\right ) y^{\prime }+4 \left (2+y\right ) y=0} \]
program solution	\[ y = \frac {{\mathrm e}^{2 c_{1}} x^{2} c_{2}^{2}}{2}-2 c_{2} {\mathrm e}^{c_{1}} x \] 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 \sqrt {2}\, c_{1} +x \right ) x}{\left (-4 \sqrt {2}\, c_{1} 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 \sqrt {2}\, c_{1} +x \right ) x}{\left (4 \sqrt {2}\, c_{1} x +8 c_{1}^{2}+x^{2}\right ) c_{1}^{2}} \\ \end{align*}

Problem 4145

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

Problem 4146

ODE	\[ \boxed {x^{2} {y^{\prime }}^{2}+x \left (x^{2}+y x -2 y\right ) y^{\prime }+\left (1-x \right ) \left (x^{2}-y\right ) y=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 4147

ODE	\[ \boxed {x^{2} {y^{\prime }}^{2}+\left (2 x +y\right ) y y^{\prime }+y^{2}=0} \]
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 (-\sqrt {2}\, c_{1} +x \right )}{-2 c_{1}^{2}+x^{2}} \\ y \left (x \right ) &= \frac {2 c_{1}^{2} \left (\sqrt {2}\, c_{1} +x \right )}{-2 c_{1}^{2}+x^{2}} \\ y \left (x \right ) &= \frac {c_{1}^{2} \left (\sqrt {2}\, c_{1} -2 x \right )}{2 c_{1}^{2}-4 x^{2}} \\ y \left (x \right ) &= \frac {c_{1}^{3} \sqrt {2}+2 x \,c_{1}^{2}}{-2 c_{1}^{2}+4 x^{2}} \\ \end{align*}

Problem 4148

ODE	\[ \boxed {x^{2} {y^{\prime }}^{2}+\left (2 x -y\right ) y y^{\prime }+y^{2}=0} \]
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 (-\sqrt {2}\, c_{1} +x \right )}{-2 c_{1}^{2}+x^{2}} \\ y \left (x \right ) &= -\frac {2 c_{1}^{2} \left (\sqrt {2}\, c_{1} +x \right )}{-2 c_{1}^{2}+x^{2}} \\ y \left (x \right ) &= \frac {c_{1}^{3} \sqrt {2}-2 x \,c_{1}^{2}}{-2 c_{1}^{2}+4 x^{2}} \\ y \left (x \right ) &= \frac {c_{1}^{2} \left (\sqrt {2}\, c_{1} +2 x \right )}{2 c_{1}^{2}-4 x^{2}} \\ \end{align*}

Problem 4149

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

Problem 4150

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 4151

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

Problem 4152

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

Problem 4153

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

Problem 4154

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

Problem 4155

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

Problem 4156

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

Problem 4157

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

Problem 4158

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

Problem 4159

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

Problem 4160

ODE	\[ \boxed {a \,x^{2} {y^{\prime }}^{2}-2 a x y y^{\prime }+y^{2}=-a \left (1-a \right ) x^{2}} \]
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. \[ x = c_{2} {\left (\frac {\sqrt {\left (a -1\right ) a}\, a y+\sqrt {\frac {\left (a^{2} x^{2}+2 y^{2} a +2 \sqrt {a \left (a -1\right ) \left (a \,x^{2}+y^{2}\right )}\, y-y^{2}\right ) \left (a -1\right )}{x^{2}}}\, a x +\sqrt {a \left (a -1\right ) \left (a \,x^{2}+y^{2}\right )}\, \sqrt {\left (a -1\right ) a}}{a x}\right )}^{-\frac {a}{\sqrt {\left (a -1\right ) a}}} \] Verified OK. \[ x = c_{2} {\left (\frac {\sqrt {\frac {\left (a^{2} x^{2}+2 y^{2} a -2 \sqrt {a \left (a -1\right ) \left (a \,x^{2}+y^{2}\right )}\, y-y^{2}\right ) \left (a -1\right )}{x^{2}}}\, a x +\sqrt {\left (a -1\right ) a}\, \left (a y-\sqrt {a \left (a -1\right ) \left (a \,x^{2}+y^{2}\right )}\right )}{a x}\right )}^{-\frac {a}{\sqrt {\left (a -1\right ) a}}} \] 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. \[ x = c_{4} {\left (\frac {\sqrt {\left (a -1\right ) a}\, a y+\sqrt {\frac {\left (a^{2} x^{2}+2 y^{2} a +2 \sqrt {a \left (a -1\right ) \left (a \,x^{2}+y^{2}\right )}\, y-y^{2}\right ) \left (a -1\right )}{x^{2}}}\, a x +\sqrt {a \left (a -1\right ) \left (a \,x^{2}+y^{2}\right )}\, \sqrt {\left (a -1\right ) a}}{a x}\right )}^{\frac {a}{\sqrt {\left (a -1\right ) a}}} \] Verified OK. \[ x = c_{4} {\left (\frac {\sqrt {\frac {\left (a^{2} x^{2}+2 y^{2} a -2 \sqrt {a \left (a -1\right ) \left (a \,x^{2}+y^{2}\right )}\, y-y^{2}\right ) \left (a -1\right )}{x^{2}}}\, a x +\sqrt {\left (a -1\right ) a}\, \left (a y-\sqrt {a \left (a -1\right ) \left (a \,x^{2}+y^{2}\right )}\right )}{a x}\right )}^{\frac {a}{\sqrt {\left (a -1\right ) a}}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= x \sqrt {-a} \\ y \left (x \right ) &= -x \sqrt {-a} \\ 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 4161

ODE	\[ \boxed {\left (-a^{2}+1\right ) x^{2} {y^{\prime }}^{2}-2 y y^{\prime } x +y^{2}=a^{2} x^{2}} \]
program solution	\[ y = -i x \] Verified OK. \[ y = i x \] Verified OK. \[ x = \frac {c_{2} a^{-\frac {1}{a}} {\left (\frac {\left (a^{2}-1\right ) x \sqrt {\frac {-2 y \sqrt {-a^{2} \left (x^{2} \left (a^{2}-1\right )-y^{2}\right )}+\left (y^{2}-x^{2}\right ) a^{2}+x^{2}+y^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}-y+\sqrt {-a^{2} \left (x^{2} \left (a^{2}-1\right )-y^{2}\right )}}{\left (a^{2}-1\right ) x}\right )}^{-\frac {1}{a}}}{\sqrt {\frac {-2 y \sqrt {-a^{2} \left (x^{2} \left (a^{2}-1\right )-y^{2}\right )}+\left (y^{2}-x^{2}\right ) a^{2}+x^{2}+y^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}} \] Warning, solution could not be verified \[ x = \frac {c_{2} a^{-\frac {1}{a}} {\left (\frac {\left (a^{2}-1\right ) x \sqrt {\frac {2 y \sqrt {-a^{2} \left (x^{2} \left (a^{2}-1\right )-y^{2}\right )}+\left (y^{2}-x^{2}\right ) a^{2}+x^{2}+y^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}-y-\sqrt {-a^{2} \left (x^{2} \left (a^{2}-1\right )-y^{2}\right )}}{\left (a^{2}-1\right ) x}\right )}^{-\frac {1}{a}}}{\sqrt {\frac {2 y \sqrt {-a^{2} \left (x^{2} \left (a^{2}-1\right )-y^{2}\right )}+\left (y^{2}-x^{2}\right ) a^{2}+x^{2}+y^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}} \] Warning, solution could not be verified \[ y = -i x \] Verified OK. \[ y = i x \] Verified OK. \[ x = \frac {c_{4} {\left (\frac {\left (a^{2}-1\right ) x \sqrt {-\frac {a^{2} \left (\left (x^{2}-y^{2}\right ) a^{2}+2 \sqrt {-a^{2} x^{2}+y^{2}+x^{2}}\, a y-x^{2}-y^{2}\right )}{\left (a^{2}-1\right )^{2} x^{2}}}+a \left (a \sqrt {-a^{2} x^{2}+y^{2}+x^{2}}-y\right )}{\left (a^{2}-1\right ) x}\right )}^{\frac {1}{a}}}{\sqrt {\frac {\left (y^{2}-x^{2}\right ) a^{2}-2 \sqrt {-a^{2} x^{2}+y^{2}+x^{2}}\, a y+x^{2}+y^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}} \] Warning, solution could not be verified \[ x = \frac {c_{4} {\left (\frac {\left (a^{2}-1\right ) x \sqrt {\frac {a^{2} \left (-a^{2} x^{2}+a^{2} y^{2}+2 \sqrt {-a^{2} x^{2}+y^{2}+x^{2}}\, a y+x^{2}+y^{2}\right )}{\left (a^{2}-1\right )^{2} x^{2}}}-\sqrt {-a^{2} x^{2}+y^{2}+x^{2}}\, a^{2}-a y}{\left (a^{2}-1\right ) x}\right )}^{\frac {1}{a}}}{\sqrt {\frac {\left (y^{2}-x^{2}\right ) a^{2}+2 \sqrt {-a^{2} x^{2}+y^{2}+x^{2}}\, a y+x^{2}+y^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}} \] 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 {-x^{2} a^{2}+x^{2}+y \left (x \right )^{2}}{x^{2}}}\, x}\right )+\ln \left (\frac {x^{2}+y \left (x \right )^{2}}{x^{2}}\right ) a -2 c_{1} a +2 \ln \left (\frac {\sqrt {\frac {-x^{2} a^{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 {-x^{2} a^{2}+x^{2}+y \left (x \right )^{2}}{x^{2}}}\, x}\right )+\ln \left (\frac {x^{2}+y \left (x \right )^{2}}{x^{2}}\right ) a -2 c_{1} a -2 \ln \left (\frac {\sqrt {\frac {-x^{2} a^{2}+x^{2}+y \left (x \right )^{2}}{x^{2}}}\, x +y \left (x \right )}{x}\right )}{2 a} &= 0 \\ \end{align*}

Problem 4162

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

Problem 4163

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

Problem 4164

ODE	\[ \boxed {x^{3} {y^{\prime }}^{2}+y^{\prime } y x^{2}=-a} \]
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 c_{1} x} \\ y \left (x \right ) &= \frac {4 a x +c_{1}^{2}}{2 c_{1} x} \\ \end{align*}

Problem 4165

ODE	\[ \boxed {x \left (-x^{2}+1\right ) {y^{\prime }}^{2}-2 \left (-x^{2}+1\right ) y y^{\prime }+x \left (1-y^{2}\right )=0} \]
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 4166

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

Problem 4167

ODE	\[ \boxed {x^{4} {y^{\prime }}^{2}-x y^{\prime }-y=0} \]
program solution	\[ -\frac {\ln \left (y\right )}{2}-\operatorname {arctanh}\left (\sqrt {1+4 y x^{2}}\right ) = c_{1} \] Verified OK. \[ -\frac {\ln \left (y\right )}{2}+\operatorname {arctanh}\left (\sqrt {1+4 y x^{2}}\right ) = c_{1} \] 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 4168

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

Problem 4169

ODE	\[ \boxed {x^{4} {y^{\prime }}^{2}+x y^{2} y^{\prime }-y^{3}=0} \]
program solution	\[ \frac {\ln \left (y\right )}{2}+\frac {\ln \left (2 x^{2}+y+\sqrt {y}\, \sqrt {4 x^{2}+y}\right )}{2}-\ln \left (x \right )-c_{1} = 0 \] Verified OK. \[ \ln \left (x \right )+4 \left (\int _{}^{\frac {\ln \left (y\right )}{2}-\frac {\ln \left (2 x^{2}+y+\sqrt {y}\, \sqrt {4 x^{2}+y}\right )}{2}}-\frac {\left ({\mathrm e}^{2 \textit {\_a}}-2\right ) \sqrt {2 \,{\mathrm e}^{2 \textit {\_a}}-1}\, {\mathrm e}^{2 \textit {\_a}}}{\left (\left (2 \,{\mathrm e}^{2 \textit {\_a}}-1\right ) {\operatorname {RootOf}\left (\left (2 \,{\mathrm e}^{2 \textit {\_a}}-1\right ) \textit {\_Z}^{4}-4 \,{\mathrm e}^{4 \textit {\_a}}+4 \textit {\_Z}^{2} {\mathrm e}^{2 \textit {\_a}}\right )}^{2}+6 \,{\mathrm e}^{2 \textit {\_a}}-4 \,{\mathrm e}^{4 \textit {\_a}}\right ) \sqrt {\left ({\mathrm e}^{2 \textit {\_a}}-1\right ) {\operatorname {RootOf}\left (\left (2 \,{\mathrm e}^{2 \textit {\_a}}-1\right ) \textit {\_Z}^{4}-4 \,{\mathrm e}^{4 \textit {\_a}}+4 \textit {\_Z}^{2} {\mathrm e}^{2 \textit {\_a}}\right )}^{2}+{\mathrm e}^{4 \textit {\_a}}}+5 \sqrt {2 \,{\mathrm e}^{2 \textit {\_a}}-1}\, \left (\left ({\mathrm e}^{2 \textit {\_a}}-\frac {2 \,{\mathrm e}^{4 \textit {\_a}}}{5}-\frac {2}{5}\right ) {\operatorname {RootOf}\left (\left (2 \,{\mathrm e}^{2 \textit {\_a}}-1\right ) \textit {\_Z}^{4}-4 \,{\mathrm e}^{4 \textit {\_a}}+4 \textit {\_Z}^{2} {\mathrm e}^{2 \textit {\_a}}\right )}^{2}+\frac {12 \,{\mathrm e}^{2 \textit {\_a}}}{5}-\frac {6 \,{\mathrm e}^{4 \textit {\_a}}}{5}\right )}d \textit {\_a} \right )+c_{1} = 0 \] Warning, solution could not be verified
Maple solution	\begin{align*} y \left (x \right ) &= -4 x^{2} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\left (\sqrt {2}\, c_{1} -2 x \right ) c_{1}^{2} x}{2 c_{1}^{2}-4 x^{2}} \\ y \left (x \right ) &= -\frac {\left (\sqrt {2}\, c_{1} +2 x \right ) c_{1}^{2} x}{2 c_{1}^{2}-4 x^{2}} \\ y \left (x \right ) &= -\frac {2 \left (-c_{1} x +\sqrt {2}\right ) x}{c_{1} \left (c_{1}^{2} x^{2}-2\right )} \\ y \left (x \right ) &= \frac {2 \left (c_{1} x +\sqrt {2}\right ) x}{c_{1} \left (c_{1}^{2} x^{2}-2\right )} \\ \end{align*}

Problem 4170

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 4171

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

Problem 4172

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

Problem 4173

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

Problem 4174

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

Problem 4175

ODE	\[ \boxed {y {y^{\prime }}^{2}=a} \]
program solution	\[ \frac {2 y^{2}}{3 \sqrt {a y}} = x +c_{1} \] Verified OK. \[ -\frac {2 y^{2}}{3 \sqrt {a y}} = x +c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {12^{\frac {2}{3}} \left (a^{2} \left (-c_{1} +x \right )\right )^{\frac {2}{3}}}{4 a} \\ y \left (x \right ) &= \frac {12^{\frac {2}{3}} \left (a^{2} \left (-c_{1} +x \right )\right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )^{2}}{16 a} \\ y \left (x \right ) &= \frac {12^{\frac {2}{3}} \left (a^{2} \left (-c_{1} +x \right )\right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )^{2}}{16 a} \\ y \left (x \right ) &= \frac {12^{\frac {2}{3}} \left (a^{2} \left (c_{1} -x \right )\right )^{\frac {2}{3}}}{4 a} \\ y \left (x \right ) &= \frac {12^{\frac {2}{3}} \left (a^{2} \left (c_{1} -x \right )\right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )^{2}}{16 a} \\ y \left (x \right ) &= \frac {12^{\frac {2}{3}} \left (a^{2} \left (c_{1} -x \right )\right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )^{2}}{16 a} \\ \end{align*}

Problem 4176

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

Problem 4177

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

Problem 4178

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

Problem 4179

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

Problem 4180

ODE	\[ \boxed {y {y^{\prime }}^{2}+a x y^{\prime }+b y=0} \]
program solution	\[ y = 0 \] Verified OK. \[ y = \sqrt {-a -b}\, x \] Verified OK. \[ y = -\sqrt {-a -b}\, x \] Verified OK. \[ y = -\frac {a x \operatorname {RootOf}\left (c_{2} x^{\frac {1}{a}} \textit {\_Z}^{\frac {1}{a +b}}-\left (\textit {\_Z}^{2}+b \right )^{\frac {1}{a}} \left (\textit {\_Z}^{2}+a +b \right )^{-\frac {a +2 b}{2 a \left (a +b \right )}}\right )}{\operatorname {RootOf}\left (c_{2} x^{\frac {1}{a}} \textit {\_Z}^{\frac {1}{a +b}}-\left (\textit {\_Z}^{2}+b \right )^{\frac {1}{a}} \left (\textit {\_Z}^{2}+a +b \right )^{-\frac {a +2 b}{2 a \left (a +b \right )}}\right )^{2}+b} \] Warning, solution could not be verified
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {2 \textit {\_a}^{2}+\sqrt {-4 \textit {\_a}^{2} b +a^{2}}+a}{\textit {\_a} \left (\textit {\_a}^{2}+a +b \right )}d \textit {\_a} \right )+2 c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {2 \textit {\_a}^{2}+a -\sqrt {-4 \textit {\_a}^{2} b +a^{2}}}{\textit {\_a} \left (\textit {\_a}^{2}+a +b \right )}d \textit {\_a} +2 c_{1} \right ) x \\ \end{align*}

Problem 4181

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

Problem 4182

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

Problem 4183

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

Problem 4184

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

Problem 4185

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

Problem 4186

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

Problem 4187

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

Problem 4188

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

Problem 4189

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

Problem 4190

ODE	\[ \boxed {2 y {y^{\prime }}^{2}+\left (5-4 x \right ) y^{\prime }+2 y=0} \]
program solution	\[ y = -x +\frac {5}{4} \] Verified OK. \[ y = 0 \] Verified OK. \[ y = x -\frac {5}{4} \] Verified OK. \[ x = \frac {\left (8 x -10\right ) c_{3} \sqrt {-16 y^{2}+16 x^{2}-40 x +25}+32 \left (x -\frac {5}{4}\right )^{2} c_{3} -20 y^{2}}{\left (-5+4 x +\sqrt {-16 y^{2}+16 x^{2}-40 x +25}\right )^{2}} \] Verified OK. \[ x = \frac {\left (-8 x +10\right ) c_{3} \sqrt {-16 y^{2}+16 x^{2}-40 x +25}+32 \left (x -\frac {5}{4}\right )^{2} c_{3} -20 y^{2}}{\left (-5+4 x -\sqrt {-16 y^{2}+16 x^{2}-40 x +25}\right )^{2}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= x -\frac {5}{4} \\ y \left (x \right ) &= -x +\frac {5}{4} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\sqrt {4 c_{1} +2 \sqrt {-c_{1} \left (-5+4 x \right )^{2}}}}{2} \\ y \left (x \right ) &= -\frac {\sqrt {4 c_{1} +2 \sqrt {-c_{1} \left (-5+4 x \right )^{2}}}}{2} \\ y \left (x \right ) &= \frac {\sqrt {4 c_{1} -2 \sqrt {-c_{1} \left (-5+4 x \right )^{2}}}}{2} \\ y \left (x \right ) &= -\frac {\sqrt {4 c_{1} -2 \sqrt {-c_{1} \left (-5+4 x \right )^{2}}}}{2} \\ \end{align*}

Problem 4191

ODE	\[ \boxed {9 y {y^{\prime }}^{2}+4 x^{3} y^{\prime }-4 y x^{2}=0} \]
program solution	\[ y = {\mathrm e}^{\frac {\ln \left (2\right )}{2}+\frac {\ln \left (2 \,{\mathrm e}^{2 c_{1}} {\mathrm e}^{-2 c_{1}} x^{2}+18 \,{\mathrm e}^{4 c_{1}} {\mathrm e}^{-2 c_{1}}\right )}{2}+c_{1}} \] Verified OK. \[ \frac {\ln \left (2\right )}{2}+\ln \left (x \right )+\frac {\ln \left (x^{2}+\sqrt {x^{4}+9 y^{2}}\right )}{2} = \ln \left (x \right )+c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {i x^{2}}{3} \\ y \left (x \right ) &= \frac {i x^{2}}{3} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= -\frac {2 \sqrt {c_{1} x^{2}+9}}{c_{1}} \\ y \left (x \right ) &= \frac {2 \sqrt {c_{1} x^{2}+9}}{c_{1}} \\ y \left (x \right ) &= -\frac {\sqrt {c_{1} \left (-4 x^{2}+c_{1} \right )}}{6} \\ y \left (x \right ) &= \frac {\sqrt {c_{1} \left (-4 x^{2}+c_{1} \right )}}{6} \\ \end{align*}

Problem 4192

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

Problem 4193

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

Problem 4194

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

Problem 4195

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

Problem 4196

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

Problem 4197

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

Problem 4198

ODE	\[ \boxed {x y {y^{\prime }}^{2}+\left (a +x^{2}-y^{2}\right ) y^{\prime }-y x=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 4199

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

Problem 4200

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