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

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

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

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

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

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

\[ {}\sqrt {y^{2}-1}\, y^{\prime }-\sqrt {x^{2}-1} = 0 \]

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

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

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

\[ {}\left (\frac {\operatorname {e1} \left (x +a \right )}{\left (\left (x +a \right )^{2}+y^{2}\right )^{\frac {3}{2}}}+\frac {\operatorname {e2} \left (x -a \right )}{\left (\left (x -a \right )^{2}+y^{2}\right )^{\frac {3}{2}}}\right ) y^{\prime }-y \left (\frac {\operatorname {e1}}{\left (\left (x +a \right )^{2}+y^{2}\right )^{\frac {3}{2}}}+\frac {\operatorname {e2}}{\left (\left (x -a \right )^{2}+y^{2}\right )^{\frac {3}{2}}}\right ) = 0 \]

\[ -\frac {\operatorname {e1} \left (x +a \right )}{\sqrt {y^{2}+a^{2}+2 a x +x^{2}}}+\frac {\operatorname {e2} \left (-x +a \right )}{\sqrt {y^{2}+a^{2}-2 a x +x^{2}}} = c_{1} \] Verified OK.

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

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

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

\[ -{\mathrm e}^{-y} x -{\mathrm e}^{-y} \ln \left (y\right )-\operatorname {expIntegral}_{1}\left (y\right ) = c_{1} \] Verified OK.

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

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

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

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

\[ {}y^{\prime } \left (\sin \left (x \right )+1\right ) \sin \left (y\right )+\cos \left (x \right ) \left (\cos \left (y\right )-1\right ) = 0 \]

\[ -\ln \left (1+\sin \left (x \right )\right )+\ln \left (\cos \left (y\right )-1\right ) = c_{1} \] Verified OK.

\[ {}\left (x \cos \left (y\right )+\sin \left (x \right )\right ) y^{\prime }+y \cos \left (x \right )+\sin \left (y\right ) = 0 \]

\[ y \sin \left (x \right )+x \sin \left (y\right ) = c_{1} \] Verified OK.

\[ {}y^{\prime } \left (\cos \left (y\right )-\sin \left (\alpha \right ) \sin \left (x \right )\right ) \cos \left (y\right )+\left (\cos \left (x \right )-\sin \left (\alpha \right ) \sin \left (y\right )\right ) \cos \left (x \right ) = 0 \]

\[ \frac {\sin \left (x \right ) \left (-2 \sin \left (\alpha \right ) \sin \left (y\right )+\cos \left (x \right )\right )}{2}+\frac {x}{2}+\frac {\cos \left (y\right ) \sin \left (y\right )}{2}+\frac {y}{2} = c_{1} \] Verified OK.

\[ {}x \cos \left (y\right ) y^{\prime }+\sin \left (y\right ) = 0 \]

\[ -\ln \left (x \right )-\ln \left (\sin \left (y\right )\right ) = c_{1} \] Verified OK.

\[ {}\left (\sin \left (y\right ) x -1\right ) y^{\prime }+\cos \left (y\right ) = 0 \]

\[ x \sec \left (y\right )-\tan \left (y\right ) = c_{1} \] Verified OK.

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

\[ y \cos \left (x \right )+x \sin \left (y\right ) = c_{1} \] Verified OK.

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

\[ y^{2} \sin \left (x \right )+x^{2} \sin \left (y\right ) = c_{1} \] Verified OK.

\[ {}\sin \left (x \right ) \cos \left (y\right )+\cos \left (x \right ) \sin \left (y\right ) y^{\prime } = 0 \]

\[ \ln \left (\cos \left (x \right )\right )+\ln \left (\cos \left (y\right )\right ) = c_{1} \] Verified OK.

\[ {}3 \sin \left (x \right ) \sin \left (y\right ) y^{\prime }+5 \cos \left (x \right )^{4} y = 0 \]

\[ -\frac {\cos \left (x \right )^{3}}{3}-\cos \left (x \right )-\ln \left (\csc \left (x \right )-\cot \left (x \right )\right )-\frac {3 \,\operatorname {Si}\left (y\right )}{5} = c_{1} \] Verified OK.

\[ {}y^{\prime } \cos \left (a y\right )-b \left (1-c \cos \left (a y\right )\right ) \sqrt {\cos \left (a y\right )^{2}-1+c \cos \left (a y\right )} = 0 \]

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

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

\[ -\cos \left (y x \right )+\sin \left (x \right )+\sin \left (x +y\right )+\cos \left (y\right ) = c_{1} \] Verified OK.

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

\[ -\cos \left (y x \right )-\ln \left (y x \right )-3 \ln \left (y\right ) = c_{1} \] Verified OK.

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

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

\[ {}\left (y \sin \left (\frac {y}{x}\right )-x \cos \left (\frac {y}{x}\right )\right ) x y^{\prime }-\left (x \cos \left (\frac {y}{x}\right )+y \sin \left (\frac {y}{x}\right )\right ) y = 0 \]

\[ -\cos \left (\frac {y}{x}\right ) y x = c_{1} \] Verified OK.

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

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

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

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

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

\[ y = 0 \] 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.

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

\[ y = 0 \] 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.

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

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

\[ y = \frac {1}{2} i+i x \] 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.

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

\[ y = \frac {a b x +a c -b}{\sqrt {a b}\, a} \] Verified OK.

\[ y = \frac {-a b x -a c +b}{\sqrt {a b}\, a} \] Verified OK.

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

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

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

\[ y = 0 \] Verified OK.

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

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

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

\[ y = a -x \] Verified OK.

\[ y = -a +x \] 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

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

\[ y = -i a -i x \] 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.

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

\[ y = x \] 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.

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

\[ y = 0 \] Verified OK.

\[ y = x -2 \] Verified OK.

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

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

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

\[ y = 0 \] Verified OK.

\[ y = 2 x +4 \] Verified OK.

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

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

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

\[ y = 0 \] Verified OK.

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

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

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

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

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

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

\[ y = 0 \] Verified OK.

\[ y = x \] Verified OK.

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

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

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

\[ y = 0 \] Verified OK.

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

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

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

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

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

\[ {}\left (\operatorname {a2} x +\operatorname {c2} \right ) {y^{\prime }}^{2}+\left (\operatorname {a1} x +\operatorname {b1} y+\operatorname {c1} \right ) y^{\prime }+\operatorname {a0} x +\operatorname {b0} y+\operatorname {c0} = 0 \]

\[ 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

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

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

\[ y = i x \] Verified OK.

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

\[ x = \frac {c_{2} a^{-\frac {1}{a}} {\left (\frac {\left (a^{2}-1\right ) x \sqrt {\frac {2 y \sqrt {-\left (\left (a^{2}-1\right ) x^{2}-y^{2}\right ) a^{2}}+\left (-x^{2}+y^{2}\right ) a^{2}+x^{2}+y^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}-y-\sqrt {-\left (\left (a^{2}-1\right ) x^{2}-y^{2}\right ) a^{2}}}{\left (a^{2}-1\right ) x}\right )}^{-\frac {1}{a}}}{\sqrt {\frac {2 y \sqrt {-\left (\left (a^{2}-1\right ) x^{2}-y^{2}\right ) a^{2}}+\left (-x^{2}+y^{2}\right ) a^{2}+x^{2}+y^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}} \] 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

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

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

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

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

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

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

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

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

\[ y = 0 \] 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.

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

\[ y = -x \] 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.

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

\[ y = 0 \] Verified OK.

\[ y = \sqrt {3}\, a x \] Verified OK.

\[ y = -\sqrt {3}\, a x \] Verified OK.

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

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

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

\[ y = 0 \] Verified OK.

\[ y = -\frac {i \sqrt {3}\, x}{2}-\frac {x}{2} \] Verified OK.

\[ y = \frac {i \sqrt {3}\, x}{2}-\frac {x}{2} \] Verified OK.

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

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

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

\[ y = 2 \] Verified OK.

\[ y = \frac {3 \sqrt {2}\, x -\sqrt {2}-4 x}{\sqrt {2}-2} \] Verified OK.

\[ y = \frac {4 x +3 \sqrt {2}\, x -\sqrt {2}}{2+\sqrt {2}} \] Verified OK.

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

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

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

\[ y = -x +\frac {5}{4} \] Verified OK.

\[ y = 0 \] Verified OK.

\[ y = x -\frac {5}{4} \] Verified OK.

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

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

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

\[ y = 0 \] Verified OK.

\[ y = -\frac {i x}{2} \] Verified OK.

\[ y = \frac {i x}{2} \] Verified OK.

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

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

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

\[ y = 0 \] Verified OK.

\[ y = \frac {2 x -b}{2 \sqrt {-a}} \] Verified OK.

\[ y = \frac {-2 x +b}{2 \sqrt {-a}} \] Verified OK.

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

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

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

\[ y = -x \sqrt {3}+2 x \] Verified OK.

\[ y = x \sqrt {3}+2 x \] Verified OK.

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

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

\[ y = 0 \] Verified OK.

\[ y = -x \sqrt {3}+2 x \] Verified OK.

\[ y = x \sqrt {3}+2 x \] Verified OK.

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

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

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

\[ y = a \sqrt {2}+x \] Verified OK.

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

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

\[ y = -a \sqrt {2}+x \] Verified OK.

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

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

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

\[ y = \frac {b^{3} x +\sqrt {b^{4} c^{2}}\, \sqrt {2}}{b^{2} a} \] Verified OK.

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

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

\[ y = \frac {b^{3} x -\sqrt {b^{4} c^{2}}\, \sqrt {2}}{a \,b^{2}} \] Verified OK.

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

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

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

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

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

\[ \int _{}^{y}\frac {1}{{\left (\left (a \,y^{2}+y b +c \right )^{2}\right )}^{\frac {1}{3}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{3}}d x +c_{1} \] Verified OK. {0 < (a*y^2+b*y+c)^2, 0 < f(x)}

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

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

\[ {}{y^{\prime }}^{3}+a {y^{\prime }}^{2}+b y+a b x = 0 \]

\[ y = -a x \] Verified OK.

\[ x = \frac {\left (10 a^{3}+27 a b x +27 b y-3 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}\right ) \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}+24 \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}\, \sqrt {3}\, a +6 a^{2} \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}} \ln \left (2\right )+6 a^{2} \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}} \ln \left (3\right )-20 a^{4}+\left (-216 b x -\left (6 \ln \left (\frac {{\left (\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}+2 a \right )}^{2}}{\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}}\right )+5\right ) \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}}\right ) a^{2}-216 y a b +6 c_{2} \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}} b}{6 \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}} b} \] Verified OK.

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

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

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

\[ y = 0 \] Verified OK.

\[ y = 1+x \] Verified OK.

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

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

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

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

\[ y = x \] Verified OK.

\[ y = \frac {-2 x -x \sqrt {3}}{1+\sqrt {3}} \] Verified OK.

\[ y = \frac {-x \sqrt {3}+2 x}{\sqrt {3}-1} \] Verified OK.

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

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

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

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

\[ y = 0 \] Verified OK.

\[ y = -\frac {3 x}{2} \] Verified OK.

\[ y = \frac {3 x}{2} \] Verified OK.

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

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

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

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

\[ y = -x +\frac {5}{6} \] Verified OK.

\[ y = \frac {1}{6}+x \] Verified OK.

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

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

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

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

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

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

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

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

\[ y = -x -\sqrt {2} \] Verified OK.

\[ y = -1 \] Verified OK.

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

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

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

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

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

\[ y = a \] Verified OK.

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

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

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

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

\[ y = \frac {\infty }{\operatorname {signum}\left (a \right )} \] Warning, solution could not be verified

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

\[ {}{y^{\prime }}^{2} \sin \left (y^{\prime }\right )-y = 0 \]

\[ \int _{}^{y}\frac {1}{\operatorname {RootOf}\left (\sin \left (\textit {\_Z} \right ) \textit {\_Z}^{2}-\textit {\_a} \right )}d \textit {\_a} = x +c_{1} \] Verified OK.

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

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

\[ {}y^{\prime } = -\frac {x a}{2}+F \left (y+\frac {a \,x^{2}}{4}+\frac {b x}{2}\right ) \]

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

\[ {}y^{\prime } = \frac {-2 x^{2}+x +F \left (y+x^{2}-x \right )}{x} \]

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

\[ {}y^{\prime } = \frac {\sqrt {y}}{\sqrt {y}+F \left (\frac {x -y}{\sqrt {y}}\right )} \]

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

\[ {}y^{\prime } = \frac {y}{x \left (-1+F \left (y x \right ) y\right )} \]

\[ \int _{}^{x}-\frac {1}{F \left (\textit {\_a} y\right ) \textit {\_a}}d \textit {\_a} +y = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {x \left (-2+3 \sqrt {x^{2}+3 y}\right )}{3} \]

\[ \frac {\ln \left (-9 x^{4}+16 x^{2}+48 y\right )}{6}-\frac {\ln \left (3 x^{2}+4 \sqrt {x^{2}+3 y}\right )}{6}+\frac {\ln \left (-3 x^{2}+4 \sqrt {x^{2}+3 y}\right )}{6} = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {x^{2} \left (1+2 \sqrt {x^{3}-6 y}\right )}{2} \]

\[ \frac {\ln \left (-x^{3}+\sqrt {x^{3}-6 y}\right )}{12}-\frac {\ln \left (x^{3}+\sqrt {x^{3}-6 y}\right )}{12}-\frac {\ln \left (x^{6}-x^{3}+6 y\right )}{12} = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {x \left (x +2 \sqrt {x^{3}-6 y}\right )}{2} \]

\[ -\frac {\ln \left (9 x^{4}-4 x^{3}+24 y\right )}{24}+\frac {\ln \left (-3 x^{2}+2 \sqrt {x^{3}-6 y}\right )}{24}-\frac {\ln \left (3 x^{2}+2 \sqrt {x^{3}-6 y}\right )}{24} = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {x \left (-2+3 x \sqrt {x^{2}+3 y}\right )}{3} \]

\[ -\frac {2 \ln \left (x^{3}+2 \sqrt {x^{2}+3 y}\right )}{9}+\frac {2 \ln \left (-x^{3}+2 \sqrt {x^{2}+3 y}\right )}{9}+\frac {2 \ln \left (-x^{6}+4 x^{2}+12 y\right )}{9} = c_{1} \] Verified OK.

\[ {}y^{\prime } = -\frac {x}{4}+\frac {1}{4}+x \sqrt {x^{2}-2 x +1+8 y} \]

\[ \frac {\ln \left (-4 x^{4}+x^{2}-2 x +8 y+1\right )}{16}+\frac {\ln \left (-2 x^{2}+\sqrt {x^{2}-2 x +1+8 y}\right )}{16}-\frac {\ln \left (2 x^{2}+\sqrt {x^{2}-2 x +1+8 y}\right )}{16} = c_{1} \] Verified OK.

\[ {}y^{\prime } = -\frac {x}{2}-\frac {a}{2}+x \sqrt {x^{2}+2 x a +a^{2}+4 y} \]

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

\[ {}y^{\prime } = -\frac {x}{2}+1+x \sqrt {x^{2}-4 x +4 y} \]

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

\[ {}y^{\prime } = -\frac {x}{4}+\frac {1}{4}+x^{2} \sqrt {x^{2}-2 x +1+8 y} \]

\[ \frac {\ln \left (-16 x^{6}+9 x^{2}-18 x +72 y+9\right )}{12}-\frac {\ln \left (4 x^{3}+3 \sqrt {x^{2}-2 x +1+8 y}\right )}{12}+\frac {\ln \left (-4 x^{3}+3 \sqrt {x^{2}-2 x +1+8 y}\right )}{12} = c_{1} \] Verified OK.

\[ {}y^{\prime } = -\frac {x a}{2}-\frac {b}{2}+x \sqrt {x^{2} a^{2}+2 a b x +b^{2}+4 a y-4 c} \]

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

\[ {}y^{\prime } = -\frac {x}{2}-\frac {a}{2}+x^{2} \sqrt {x^{2}+2 x a +a^{2}+4 y} \]

\[ \frac {\ln \left (-4 x^{6}+9 a^{2}+18 a x +9 x^{2}+36 y\right )}{6}+\frac {\ln \left (-2 x^{3}+3 \sqrt {x^{2}+2 a x +a^{2}+4 y}\right )}{6}-\frac {\ln \left (2 x^{3}+3 \sqrt {x^{2}+2 a x +a^{2}+4 y}\right )}{6} = c_{1} \] Verified OK.

\[ {}y^{\prime } = -\frac {x a}{2}-\frac {b}{2}+x^{2} \sqrt {x^{2} a^{2}+2 a b x +b^{2}+4 a y-4 c} \]

\[ \frac {\ln \left (\left (-4 x^{6}+9 x^{2}\right ) a^{2}+18 \left (x b +2 y\right ) a +9 b^{2}-36 c \right )}{6}+\frac {\ln \left (-2 a \,x^{3}+3 \sqrt {a^{2} x^{2}+\left (2 x b +4 y\right ) a +b^{2}-4 c}\right )}{6}-\frac {\ln \left (2 a \,x^{3}+3 \sqrt {a^{2} x^{2}+\left (2 x b +4 y\right ) a +b^{2}-4 c}\right )}{6} = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {x}{2}+\frac {1}{2}+x^{2} \sqrt {x^{2}+2 x +1-4 y} \]

\[ -\frac {\ln \left (4 x^{6}-9 x^{2}-18 x +36 y-9\right )}{6}+\frac {\ln \left (-2 x^{3}+3 \sqrt {x^{2}+2 x +1-4 y}\right )}{6}-\frac {\ln \left (2 x^{3}+3 \sqrt {x^{2}+2 x +1-4 y}\right )}{6} = c_{1} \] Verified OK.

\[ {}y^{\prime } = -\frac {x}{2}+1+x^{2} \sqrt {x^{2}-4 x +4 y} \]

\[ \frac {\ln \left (-4 x^{6}+9 x^{2}-36 x +36 y\right )}{6}+\frac {\ln \left (-2 x^{3}+3 \sqrt {x^{2}-4 x +4 y}\right )}{6}-\frac {\ln \left (2 x^{3}+3 \sqrt {x^{2}-4 x +4 y}\right )}{6} = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {y^{3} {\mathrm e}^{-2 b x}}{y \,{\mathrm e}^{-b x}+1} \]

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

\[ {}y^{\prime } = \frac {-\sin \left (2 y\right )+\cos \left (2 y\right ) x^{2}+x^{2}}{2 x} \]

\[ -\frac {x \left (\cos \left (2 y\right ) x^{2}+x^{2}-3 \sin \left (2 y\right )\right )}{3 \cos \left (2 y\right )+3} = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {2 a}{x \left (-y x +2 a x y^{2}-8 a^{2}\right )} \]

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

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

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

\[ {}y^{\prime } = \frac {2 a}{y+2 a y^{4}-16 a^{2} x y^{2}+32 a^{3} x^{2}} \]

\[ \frac {16 a^{2} x y-4 a y^{3}+1}{32 a^{3} x -8 y^{2} a^{2}} = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {y \left (x -y\right )}{x \left (x -y^{3}\right )} \]

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

\[ {}y^{\prime } = \frac {2 a}{-x^{2} y+2 a y^{4} x^{2}-16 a^{2} x y^{2}+32 a^{3}} \]

\[ \frac {4 a x y^{3}-16 y a^{2}+x}{-8 y^{2} a^{2} x +32 a^{3}} = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {y \left (x +y\right )}{x \left (x +y^{3}\right )} \]

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

\[ {}y^{\prime } = \frac {y x +x^{3}+x y^{2}+y^{3}}{x^{2}} \]

\[ 27 \left (\int _{}^{\frac {y-\frac {x}{3}}{x}}\frac {1}{27 \textit {\_a}^{3}-9 \textit {\_a} +29}d \textit {\_a} \right ) = x +c_{4} \] Verified OK.

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

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

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

\[ \frac {\ln \left (\frac {3 y+x}{x}\right )}{2}-\frac {\ln \left (\frac {9 \left (y-\frac {x}{3}\right )^{2}}{x^{2}}-\frac {6 \left (y-\frac {x}{3}\right )}{x}+10\right )}{4}-\frac {\arctan \left (-\frac {y-\frac {x}{3}}{x}+\frac {1}{3}\right )}{2}-\ln \left (x -1\right )+\ln \left (x \right )-c_{4} = 0 \] Verified OK.

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

\[ \int _{}^{x}-\frac {1}{\textit {\_a} \left (\textit {\_a}^{3} y^{3}+\textit {\_a}^{2} y^{2}+1\right )}d \textit {\_a} +y = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {-125+300 x -240 x^{2}+64 x^{3}-80 y^{2}+64 x y^{2}+64 y^{3}}{\left (4 x -5\right )^{3}} \]

\[ 27 \left (\int _{}^{\frac {y-\frac {\left (\frac {64 x}{64 x^{3}-240 x^{2}+300 x -125}-\frac {80}{64 x^{3}-240 x^{2}+300 x -125}\right ) \left (64 x^{3}-240 x^{2}+300 x -125\right )}{192}}{x -\frac {5}{4}}}\frac {1}{27 \textit {\_a}^{3}-36 \textit {\_a} +38}d \textit {\_a} \right )-\ln \left (x -\frac {5}{4}\right )-c_{4} = 0 \] Verified OK.