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

\[ \sqrt {\frac {y \left (y+2 x \right )}{x^{2}}} = \frac {c_{6} {\mathrm e}^{c_{5}}}{x} \] Verified OK.

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

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

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

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

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

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

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

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

\[ \ln \left (-\frac {y^{2}}{x^{2}}+1\right )+\operatorname {arctanh}\left (\frac {y}{x}\right )+2 \ln \left (x \right )-c_{5} = 0 \] Verified OK.

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

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

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

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

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

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

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

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

\[ \frac {3^{\frac {5}{6}} \left (\frac {3 y^{2}+x^{2}}{x^{2}}\right )^{\frac {1}{6}}}{3} = \frac {c_{6} {\mathrm e}^{c_{5}}}{\sqrt {x}} \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ y = 0 \] Verified OK.

\[ x = -\frac {{\left (\left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {2}{3}}+12 x \right )}^{2}}{\left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {2}{3}} \left (24 a -36\right )}+c_{1} 6^{-\frac {1}{a -1}} {\left (\frac {\left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {2}{3}}+12 x}{\left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {1}{3}}}\right )}^{\frac {1}{a -1}} \] Warning, solution could not be verified

\[ x = \frac {{\left (\left (\sqrt {3}+i\right ) \left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {2}{3}}+12 x \left (i-\sqrt {3}\right )\right )}^{2}}{\left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {2}{3}} \left (96 a -144\right )}+c_{1} 12^{-\frac {1}{a -1}} {\left (\frac {i \sqrt {3}\, \left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {2}{3}}-12 i \sqrt {3}\, x -\left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {2}{3}}-12 x}{\left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {1}{3}}}\right )}^{\frac {1}{a -1}} \] Warning, solution could not be verified

\[ x = -\frac {{\left (i \sqrt {3}\, \left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {2}{3}}-12 i \sqrt {3}\, x +\left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {2}{3}}+12 x \right )}^{2}}{\left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {2}{3}} \left (96 a -144\right )}+c_{1} {\left (\frac {-i \sqrt {3}\, \left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {2}{3}}+12 i \sqrt {3}\, x -\left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {2}{3}}-12 x}{12 \left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {1}{3}}}\right )}^{\frac {1}{a -1}} \] Warning, solution could not be verified

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

\[ y = 0 \] Verified OK.

\[ x = -\frac {{\left (\left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-24 x \right )}^{2}}{48 \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}+\frac {36 c_{1} \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}{{\left (\left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-24 x \right )}^{2}} \] Verified OK.

\[ x = \frac {3 {\left (\frac {\left (\sqrt {3}+i\right ) \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}{24}+x \left (-i+\sqrt {3}\right )\right )}^{2}}{\left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}+\frac {144 c_{1} \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}{{\left (i \sqrt {3}\, \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 i \sqrt {3}\, x -\left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 x \right )}^{2}} \] Verified OK.

\[ x = \frac {3 {\left (\frac {\left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}} \left (-i+\sqrt {3}\right )}{24}+x \left (\sqrt {3}+i\right )\right )}^{2}}{\left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}+\frac {144 c_{1} \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}{{\left (i \sqrt {3}\, \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 i \sqrt {3}\, x +\left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-24 x \right )}^{2}} \] Verified OK.

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

\[ y = 0 \] Verified OK.

\[ x = \frac {{\left (\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 x \right )}^{2}}{96 \left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}+\frac {c_{1} 6^{\frac {2}{3}}}{{\left (\frac {\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 x}{\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {1}{3}}}\right )}^{\frac {2}{3}}} \] Verified OK.

\[ x = \frac {{\left (-i \sqrt {3}\, \left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 i \sqrt {3}\, x +\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 x \right )}^{2}}{384 \left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}+\frac {c_{1} 12^{\frac {2}{3}}}{{\left (\frac {i \sqrt {3}\, \left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-24 i \sqrt {3}\, x -\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-24 x}{\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {1}{3}}}\right )}^{\frac {2}{3}}} \] Verified OK.

\[ x = \frac {{\left (i \sqrt {3}\, \left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-24 i \sqrt {3}\, x +\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 x \right )}^{2}}{384 \left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}+\frac {2 c_{1} 18^{\frac {1}{3}}}{{\left (\frac {-i \sqrt {3}\, \left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 i \sqrt {3}\, x -\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-24 x}{\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {1}{3}}}\right )}^{\frac {2}{3}}} \] Warning, solution could not be verified

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

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

\[ x = \frac {-30 a^{2} \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {2}{3}} \ln \left (\frac {\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {2}{3}}+8 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {1}{3}} a +4 a^{2}}{\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {1}{3}}}\right )+\left (30 \ln \left (2\right ) a^{2}+30 \ln \left (3\right ) a^{2}+7 a^{2}+6 c_{2} b \right ) \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {2}{3}}+\left (14 a^{3}+27 a b x -3 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}+27 b y\right ) \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {1}{3}}+28 a^{4}-432 b \,a^{2} x +48 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}\, a -432 a b y}{6 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {2}{3}} b} \] Warning, solution could not be verified

\[ x = \frac {-1728 i \sqrt {3}\, a^{2} b x -1728 i \sqrt {3}\, a b y+576 i \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}\, a +112 i \sqrt {3}\, a^{4}+i \sqrt {3}\, \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {4}{3}}-64 i \sqrt {3}\, \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {1}{3}} a^{3}+240 a^{2} \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {2}{3}} \ln \left (12\right )-240 a^{2} \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {2}{3}} \ln \left (\frac {i \left (\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {2}{3}}-4 a^{2}\right ) \sqrt {3}-\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {2}{3}}+16 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {1}{3}} a -4 a^{2}}{\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {1}{3}}}\right )+\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {4}{3}}+48 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {2}{3}} b c_{2} +56 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {2}{3}} a^{2}-64 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {1}{3}} a^{3}-112 a^{4}+1728 b \,a^{2} x -192 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}\, a +1728 a b y}{48 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {2}{3}} b} \] Warning, solution could not be verified

\[ x = \frac {-576 \left (i+\frac {\sqrt {3}}{3}\right ) a \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}+i \left (-\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {4}{3}}+64 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {1}{3}} a^{3}-112 a^{4}+1728 b \,a^{2} x +1728 a b y\right ) \sqrt {3}-112 a^{4}-64 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {1}{3}} a^{3}+8 \left (216 b x -\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {2}{3}} \left (30 \ln \left (\frac {i \left (-\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {2}{3}}+4 a^{2}\right ) \sqrt {3}-\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {2}{3}}+16 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {1}{3}} a -4 a^{2}}{12 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {1}{3}}}\right )-7\right )\right ) a^{2}+1728 a b y+\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {2}{3}} \left (\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {2}{3}}+48 c_{2} b \right )}{48 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (x a +y\right ) b}-108 b y\right )^{\frac {2}{3}} b} \] Warning, solution could not be verified

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

\[ y = 0 \] Verified OK.

\[ x = \frac {\left (\left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}-6 x \right ) \left (\left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}+c_{1} \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {1}{3}}-6 x \right )}{6 \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}} \] Verified OK.

\[ x = \frac {\left (i \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}} \sqrt {3}+6 i \sqrt {3}\, x -\left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}+6 x \right ) \left (i \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}} \sqrt {3}+6 i \sqrt {3}\, x -\left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}+2 c_{1} \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {1}{3}}+6 x \right )}{24 \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}} \] Warning, solution could not be verified

\[ x = \frac {\left (i \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}} \sqrt {3}+6 i \sqrt {3}\, x +\left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}-6 x \right ) \left (i \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}} \sqrt {3}+6 i \sqrt {3}\, x +\left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}-2 c_{1} \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {1}{3}}-6 x \right )}{24 \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}} \] Warning, solution could not be verified

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

\[ y = x \] 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.

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

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

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

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

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

\[ y = 0 \] Verified OK.

\[ x = \frac {6 c_{2} y 2^{\frac {1}{3}} \left (-3 y^{3}+y^{2} \sqrt {\frac {9 y^{3}-4 x^{3}}{y}}\right )^{\frac {2}{3}} x \,{\mathrm e}^{-\frac {4 y^{5} \left (-3 y+\sqrt {\frac {9 y^{3}-4 x^{3}}{y}}\right )}{\left (2^{\frac {1}{3}} \left (-3 y^{3}+y^{2} \sqrt {\frac {9 y^{3}-4 x^{3}}{y}}\right )^{\frac {2}{3}}+2 x y\right )^{3}}}}{\left (2^{\frac {1}{3}} \left (-3 y^{3}+y^{2} \sqrt {\frac {9 y^{3}-4 x^{3}}{y}}\right )^{\frac {2}{3}}+2 x y\right )^{2}} \] Warning, solution could not be verified

\[ x = \frac {6 c_{2} y 2^{\frac {1}{3}} \left (-3 y^{3}+y^{2} \sqrt {\frac {9 y^{3}-4 x^{3}}{y}}\right )^{\frac {2}{3}} x \,{\mathrm e}^{-\frac {32 y^{5} \left (-3 y+\sqrt {\frac {9 y^{3}-4 x^{3}}{y}}\right )}{\left (i 2^{\frac {1}{3}} \left (-3 y^{3}+y^{2} \sqrt {\frac {9 y^{3}-4 x^{3}}{y}}\right )^{\frac {2}{3}} \sqrt {3}-2 i \sqrt {3}\, x y-2^{\frac {1}{3}} \left (-3 y^{3}+y^{2} \sqrt {\frac {9 y^{3}-4 x^{3}}{y}}\right )^{\frac {2}{3}}-2 x y\right )^{3}}}}{\left (-\frac {\left (i \sqrt {3}-1\right ) 2^{\frac {1}{3}} \left (-3 y^{3}+y^{2} \sqrt {\frac {9 y^{3}-4 x^{3}}{y}}\right )^{\frac {2}{3}}}{2}+y x \left (1+i \sqrt {3}\right )\right )^{2}} \] Warning, solution could not be verified

\[ x = \frac {6 c_{2} y 2^{\frac {1}{3}} \left (-3 y^{3}+y^{2} \sqrt {\frac {9 y^{3}-4 x^{3}}{y}}\right )^{\frac {2}{3}} x \,{\mathrm e}^{\frac {96 y^{6}-32 y^{5} \sqrt {\frac {9 y^{3}-4 x^{3}}{y}}}{8 \left (-\frac {\left (1+i \sqrt {3}\right ) 2^{\frac {1}{3}} \left (-3 y^{3}+y^{2} \sqrt {\frac {9 y^{3}-4 x^{3}}{y}}\right )^{\frac {2}{3}}}{2}+y \left (i \sqrt {3}-1\right ) x \right )^{3}}}}{\left (-\frac {\left (1+i \sqrt {3}\right ) 2^{\frac {1}{3}} \left (-3 y^{3}+y^{2} \sqrt {\frac {9 y^{3}-4 x^{3}}{y}}\right )^{\frac {2}{3}}}{2}+y \left (i \sqrt {3}-1\right ) x \right )^{2}} \] Warning, solution could not be verified

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

\[ y = 0 \] Verified OK.

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

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

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

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

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

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

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

\[ y^{2}+x y+x^{2} = c_{5} {\mathrm e}^{c_{4}} \] Verified OK.

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

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

\[ \int _{}^{y}-\frac {i}{\left (\left (b -y \right )^{2} \left (a -y \right )^{3}\right )^{\frac {1}{4}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{4}}d x +c_{1} \] Verified OK. {0 < (b-y)^2*(a-y)^3, 0 < f(x)}

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

\[ \int _{}^{y}\frac {i}{\left (\left (b -y \right )^{2} \left (a -y \right )^{3}\right )^{\frac {1}{4}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{4}}d x +c_{1} \] Verified OK. {0 < (b-y)^2*(a-y)^3, 0 < f(x)}

\[ {}{y^{\prime }}^{4}+f \left (x \right ) \left (y-a \right )^{3} \left (y-b \right )^{3} = 0 \]

\[ \int _{}^{y}\frac {1}{\left (\left (b -y \right )^{3} \left (a -y \right )^{3}\right )^{\frac {1}{4}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{4}}d x +c_{1} \] Verified OK. {0 < (b-y)^3*(a-y)^3, 0 < -f(x)}

\[ \int _{}^{y}-\frac {i}{\left (\left (b -y \right )^{3} \left (a -y \right )^{3}\right )^{\frac {1}{4}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{4}}d x +c_{1} \] Verified OK. {0 < (b-y)^3*(a-y)^3, 0 < -f(x)}

\[ \int _{}^{y}-\frac {1}{\left (\left (b -y \right )^{3} \left (a -y \right )^{3}\right )^{\frac {1}{4}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{4}}d x +c_{1} \] Verified OK. {0 < (b-y)^3*(a-y)^3, 0 < -f(x)}

\[ \int _{}^{y}\frac {i}{\left (\left (b -y \right )^{3} \left (a -y \right )^{3}\right )^{\frac {1}{4}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{4}}d x +c_{1} \] Verified OK. {0 < (b-y)^3*(a-y)^3, 0 < -f(x)}

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

\[ \int _{}^{y}\frac {1}{\left (\left (c -y \right )^{2} \left (b -y \right )^{3} \left (a -y \right )^{3}\right )^{\frac {1}{4}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{4}}d x +c_{1} \] Verified OK. {0 < (c-y)^2*(b-y)^3*(a-y)^3, 0 < -f(x)}

\[ \int _{}^{y}-\frac {i}{\left (\left (c -y \right )^{2} \left (b -y \right )^{3} \left (a -y \right )^{3}\right )^{\frac {1}{4}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{4}}d x +c_{1} \] Verified OK. {0 < (c-y)^2*(b-y)^3*(a-y)^3, 0 < -f(x)}

\[ \int _{}^{y}-\frac {1}{\left (\left (c -y \right )^{2} \left (b -y \right )^{3} \left (a -y \right )^{3}\right )^{\frac {1}{4}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{4}}d x +c_{1} \] Verified OK. {0 < (c-y)^2*(b-y)^3*(a-y)^3, 0 < -f(x)}

\[ \int _{}^{y}\frac {i}{\left (\left (c -y \right )^{2} \left (b -y \right )^{3} \left (a -y \right )^{3}\right )^{\frac {1}{4}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{4}}d x +c_{1} \] Verified OK. {0 < (c-y)^2*(b-y)^3*(a-y)^3, 0 < -f(x)}

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

\[ y = 0 \] Verified OK.

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

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

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

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

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

\[ {}3 {y^{\prime }}^{5}-y y^{\prime }+1 = 0 \]

\[ \int _{}^{y}\frac {1}{\operatorname {RootOf}\left (3 \textit {\_Z}^{5}-\textit {\_Z} \textit {\_a} +1\right )}d \textit {\_a} = x +c_{1} \] Verified OK.

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

\[ \int _{}^{y}\frac {1}{\left (\left (b -y \right )^{3} \left (a -y \right )^{4}\right )^{\frac {1}{6}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{6}}d x +c_{1} \] Verified OK. {0 < (b-y)^3*(a-y)^4, 0 < f(x)}

\[ \int _{}^{y}\frac {2}{\left (1+i \sqrt {3}\right ) \left (\left (b -y \right )^{3} \left (a -y \right )^{4}\right )^{\frac {1}{6}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{6}}d x +c_{1} \] Verified OK. {0 < (b-y)^3*(a-y)^4, 0 < f(x)}

\[ \int _{}^{y}\frac {2}{\left (i \sqrt {3}-1\right ) \left (\left (b -y \right )^{3} \left (a -y \right )^{4}\right )^{\frac {1}{6}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{6}}d x +c_{1} \] Verified OK. {0 < (b-y)^3*(a-y)^4, 0 < f(x)}

\[ \int _{}^{y}-\frac {1}{\left (\left (b -y \right )^{3} \left (a -y \right )^{4}\right )^{\frac {1}{6}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{6}}d x +c_{1} \] Verified OK. {0 < (b-y)^3*(a-y)^4, 0 < f(x)}

\[ \int _{}^{y}-\frac {2}{\left (1+i \sqrt {3}\right ) \left (\left (b -y \right )^{3} \left (a -y \right )^{4}\right )^{\frac {1}{6}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{6}}d x +c_{1} \] Verified OK. {0 < (b-y)^3*(a-y)^4, 0 < f(x)}

\[ \int _{}^{y}-\frac {2}{\left (i \sqrt {3}-1\right ) \left (\left (b -y \right )^{3} \left (a -y \right )^{4}\right )^{\frac {1}{6}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{6}}d x +c_{1} \] Verified OK. {0 < (b-y)^3*(a-y)^4, 0 < f(x)}

\[ {}{y^{\prime }}^{6}+f \left (x \right ) \left (y-a \right )^{5} \left (y-b \right )^{3} = 0 \]

\[ \int _{}^{y}\frac {1}{\left (\left (b -y \right )^{3} \left (a -y \right )^{5}\right )^{\frac {1}{6}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{6}}d x +c_{1} \] Verified OK. {0 < (b-y)^3*(a-y)^5, 0 < -f(x)}

\[ \int _{}^{y}\frac {2}{\left (1+i \sqrt {3}\right ) \left (\left (b -y \right )^{3} \left (a -y \right )^{5}\right )^{\frac {1}{6}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{6}}d x +c_{1} \] Verified OK. {0 < (b-y)^3*(a-y)^5, 0 < -f(x)}

\[ \int _{}^{y}\frac {2}{\left (i \sqrt {3}-1\right ) \left (\left (b -y \right )^{3} \left (a -y \right )^{5}\right )^{\frac {1}{6}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{6}}d x +c_{1} \] Verified OK. {0 < (b-y)^3*(a-y)^5, 0 < -f(x)}

\[ \int _{}^{y}-\frac {1}{\left (\left (b -y \right )^{3} \left (a -y \right )^{5}\right )^{\frac {1}{6}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{6}}d x +c_{1} \] Verified OK. {0 < (b-y)^3*(a-y)^5, 0 < -f(x)}

\[ \int _{}^{y}-\frac {2}{\left (1+i \sqrt {3}\right ) \left (\left (b -y \right )^{3} \left (a -y \right )^{5}\right )^{\frac {1}{6}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{6}}d x +c_{1} \] Verified OK. {0 < (b-y)^3*(a-y)^5, 0 < -f(x)}

\[ \int _{}^{y}-\frac {2}{\left (i \sqrt {3}-1\right ) \left (\left (b -y \right )^{3} \left (a -y \right )^{5}\right )^{\frac {1}{6}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{6}}d x +c_{1} \] Verified OK. {0 < (b-y)^3*(a-y)^5, 0 < -f(x)}

\[ {}{y^{\prime }}^{6}+f \left (x \right ) \left (y-a \right )^{5} \left (y-b \right )^{4} = 0 \]

\[ \int _{}^{y}\frac {1}{\left (\left (b -y \right )^{4} \left (a -y \right )^{5}\right )^{\frac {1}{6}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{6}}d x +c_{1} \] Verified OK. {0 < (b-y)^4*(a-y)^5, 0 < f(x)}

\[ \int _{}^{y}\frac {2}{\left (1+i \sqrt {3}\right ) \left (\left (b -y \right )^{4} \left (a -y \right )^{5}\right )^{\frac {1}{6}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{6}}d x +c_{1} \] Verified OK. {0 < (b-y)^4*(a-y)^5, 0 < f(x)}

\[ \int _{}^{y}\frac {2}{\left (i \sqrt {3}-1\right ) \left (\left (b -y \right )^{4} \left (a -y \right )^{5}\right )^{\frac {1}{6}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{6}}d x +c_{1} \] Verified OK. {0 < (b-y)^4*(a-y)^5, 0 < f(x)}

\[ \int _{}^{y}-\frac {1}{\left (\left (b -y \right )^{4} \left (a -y \right )^{5}\right )^{\frac {1}{6}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{6}}d x +c_{1} \] Verified OK. {0 < (b-y)^4*(a-y)^5, 0 < f(x)}

\[ \int _{}^{y}-\frac {2}{\left (1+i \sqrt {3}\right ) \left (\left (b -y \right )^{4} \left (a -y \right )^{5}\right )^{\frac {1}{6}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{6}}d x +c_{1} \] Verified OK. {0 < (b-y)^4*(a-y)^5, 0 < f(x)}

\[ \int _{}^{y}-\frac {2}{\left (i \sqrt {3}-1\right ) \left (\left (b -y \right )^{4} \left (a -y \right )^{5}\right )^{\frac {1}{6}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{6}}d x +c_{1} \] Verified OK. {0 < (b-y)^4*(a-y)^5, 0 < f(x)}

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

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

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

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

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

\[ -\ln \left (x \right )-\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{3}+3 \textit {\_Z}^{2}+12 \textit {\_Z} +4\right )}{\sum }\frac {\textit {\_R} \ln \left (y-\textit {\_R} \right )}{2 \textit {\_R}^{2}+\textit {\_R} +2}\right )}{2}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{3}+3 \textit {\_Z}^{2}-3\right )}{\sum }\frac {\textit {\_R} \left (2 \textit {\_R} +3\right ) \ln \left (\sqrt {1+y}-\textit {\_R} \right )}{\textit {\_R}^{2}+\textit {\_R}}\right )}{6}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{3}-3 \textit {\_Z}^{2}+3\right )}{\sum }\frac {\textit {\_R} \left (2 \textit {\_R} -3\right ) \ln \left (\sqrt {1+y}-\textit {\_R} \right )}{\textit {\_R}^{2}-\textit {\_R}}\right )}{6}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{3}+3 \textit {\_Z}^{2}-3\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\sqrt {1+y}-\textit {\_R} \right )}{\textit {\_R}^{2}+\textit {\_R}}\right )}{3}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{3}-3 \textit {\_Z}^{2}+3\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\sqrt {1+y}-\textit {\_R} \right )}{\textit {\_R}^{2}-\textit {\_R}}\right )}{3} = c_{1} \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

\[ y = 0 \] Verified OK.

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

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

\[ {}{\mathrm e}^{y^{\prime }-y}-{y^{\prime }}^{2}+1 = 0 \]

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

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

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

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

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

\[ y = -\infty \] Warning, solution could not be verified

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ \int _{}^{x}-\frac {\left (\sqrt {-\textit {\_a}^{2}+1}\, \textit {\_a} +\left (y-1\right ) \textit {\_a}^{2}-y+1\right ) {\mathrm e}^{\frac {\textit {\_a}}{\sqrt {-\textit {\_a}^{2}+1}}}}{\left (-\textit {\_a}^{2}+1\right )^{\frac {5}{2}}}d \textit {\_a} +\left (-\left (\int _{}^{x}\frac {{\mathrm e}^{\frac {\textit {\_a}}{\sqrt {-\textit {\_a}^{2}+1}}}}{\left (-\textit {\_a}^{2}+1\right )^{\frac {3}{2}}}d \textit {\_a} \right )+{\mathrm e}^{\frac {x}{\sqrt {-x^{2}+1}}}\right ) y = c_{1} \] Verified OK.

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

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

\[ {}3 z^{2} z^{\prime }-a z^{3} = 1+x \]

\[ \frac {\left (z^{3} a^{2}+a x +a +1\right ) {\mathrm e}^{-a x}}{a^{2}} = c_{1} \] Verified OK.

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

\[ z^{-n +1} = \frac {\left (n -1\right ) c_{1} {\mathrm e}^{\left (n -1\right ) \sin \left (x \right )}+2+\left (2 n -2\right ) \sin \left (x \right )}{n -1} \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}\frac {\sqrt {f \,x^{4}+c \,x^{3}+c \,x^{2}+b x +a}\, y^{\prime }}{\sqrt {a +b y+c y^{2}+c y^{3}+f y^{4}}} = -1 \]

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

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

\[ y = 0 \] 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.

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

\[ y = a -x \] 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

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

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

\[ y = i x \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ \frac {x \left (-4 y+3 x +8\right )}{2}+y-\frac {7 y^{2}}{2} = c_{1} \] Verified OK.

\[ {}y+7+\left (2 x +y+3\right ) y^{\prime } = 0 \]

\[ \left (y+7\right )^{2} x +\frac {y^{3}}{3}+5 y^{2}+21 y = {\frac {79}{3}} \] Verified OK.

\[ {}3 x^{2} y+8 x y^{2}+\left (x^{3}+8 x^{2} y+12 y^{2}\right ) y^{\prime } = 0 \]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}{\mathrm e}^{x} \left (y^{3}+x y^{3}+1\right )+3 y^{2} \left (x \,{\mathrm e}^{x}-6\right ) y^{\prime } = 0 \]

\[ {\mathrm e}^{x} y^{3} x -6 y^{3}+{\mathrm e}^{x} = -5 \] Verified OK.

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

\[ \ln \left (\cos \left (x \right )\right )+\ln \left (\cos \left (y\right )\right ) = -\ln \left (2\right ) \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ \frac {x^{4} y+3}{3 y x}+\frac {y^{3}}{3} = c_{1} \] Verified OK.

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

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

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

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

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

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