\[ {}4 y y^{\prime \prime }-5 {y^{\prime }}^{2}+a y^{2} = 0 \]

\[ -\frac {4 y \sqrt {{\mathrm e}^{2 c_{2}} c_{3}^{2} \sqrt {y}+a}\, \operatorname {arctanh}\left (\frac {\sqrt {{\mathrm e}^{2 c_{2}} c_{3}^{2} \sqrt {y}+a}}{\sqrt {a}}\right )}{\sqrt {y^{\frac {5}{2}} {\mathrm e}^{2 c_{2}} c_{3}^{2}+a y^{2}}\, \sqrt {a}} = x +c_{4} \] Verified OK.

\[ \frac {4 y \sqrt {{\mathrm e}^{2 c_{2}} c_{3}^{2} \sqrt {y}+a}\, \operatorname {arctanh}\left (\frac {\sqrt {{\mathrm e}^{2 c_{2}} c_{3}^{2} \sqrt {y}+a}}{\sqrt {a}}\right )}{\sqrt {y^{\frac {5}{2}} {\mathrm e}^{2 c_{2}} c_{3}^{2}+a y^{2}}\, \sqrt {a}} = x +c_{5} \] Verified OK.

\[ {}a y y^{\prime \prime }+b {y^{\prime }}^{2}+\operatorname {c4} y^{4}+\operatorname {c3} y^{3}+\operatorname {c2} y^{2}+\operatorname {c1} y+\operatorname {c0} = 0 \]

\[ \int _{}^{y}\frac {\textit {\_a}^{\frac {2 b}{a}} b \left (6 a^{4}+25 a^{3} b +35 b^{2} a^{2}+20 a \,b^{3}+4 b^{4}\right )}{\sqrt {-\textit {\_a}^{\frac {2 b}{a}} b \left (6 a^{4}+25 a^{3} b +35 b^{2} a^{2}+20 a \,b^{3}+4 b^{4}\right ) \left (12 a \operatorname {c4} \,b^{3} \textit {\_a}^{\frac {4 a +2 b}{a}}-8 c_{1} b^{5}+12 b \,\textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,a^{3}+3 b \operatorname {c4} \,a^{3} \textit {\_a}^{\frac {4 a +2 b}{a}}+11 \operatorname {c4} \,a^{2} b^{2} \textit {\_a}^{\frac {4 a +2 b}{a}}-12 c_{1} a^{4} b -50 c_{1} a^{3} b^{2}-70 c_{1} a^{2} b^{3}-40 c_{1} a \,b^{4}+6 b \operatorname {c2} \,a^{3} \textit {\_a}^{\frac {2 a +2 b}{a}}+20 \textit {\_a}^{\frac {2 b}{a}} a \,b^{3} \operatorname {c0} +35 \textit {\_a}^{\frac {2 b}{a}} a^{2} b^{2} \operatorname {c0} +4 \operatorname {c4} \,b^{4} \textit {\_a}^{\frac {4 a +2 b}{a}}+4 \operatorname {c2} \,b^{4} \textit {\_a}^{\frac {2 a +2 b}{a}}+4 \operatorname {c3} \,b^{4} \textit {\_a}^{\frac {3 a +2 b}{a}}+6 \textit {\_a}^{\frac {2 b}{a}} a^{4} \operatorname {c0} +4 \textit {\_a}^{\frac {2 b}{a}} b^{4} \operatorname {c0} +4 \textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,b^{4}+18 a \,\textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,b^{3}+4 b \operatorname {c3} \,a^{3} \textit {\_a}^{\frac {3 a +2 b}{a}}+26 \textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,a^{2} b^{2}+19 \operatorname {c2} \,a^{2} b^{2} \textit {\_a}^{\frac {2 a +2 b}{a}}+16 a \operatorname {c2} \,b^{3} \textit {\_a}^{\frac {2 a +2 b}{a}}+14 \operatorname {c3} \,a^{2} b^{2} \textit {\_a}^{\frac {3 a +2 b}{a}}+25 \textit {\_a}^{\frac {2 b}{a}} a^{3} b \operatorname {c0} +14 a \operatorname {c3} \,b^{3} \textit {\_a}^{\frac {3 a +2 b}{a}}\right )}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {\textit {\_a}^{\frac {2 b}{a}} b \left (6 a^{4}+25 a^{3} b +35 b^{2} a^{2}+20 a \,b^{3}+4 b^{4}\right )}{\sqrt {-\textit {\_a}^{\frac {2 b}{a}} b \left (6 a^{4}+25 a^{3} b +35 b^{2} a^{2}+20 a \,b^{3}+4 b^{4}\right ) \left (-8 c_{1} b^{5}-12 c_{1} a^{4} b -50 c_{1} a^{3} b^{2}-70 c_{1} a^{2} b^{3}-40 c_{1} a \,b^{4}+4 \operatorname {c3} \,b^{4} \textit {\_a}^{\frac {3 a +2 b}{a}}+6 \textit {\_a}^{\frac {2 b}{a}} a^{4} \operatorname {c0} +4 \textit {\_a}^{\frac {2 b}{a}} b^{4} \operatorname {c0} +4 \textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,b^{4}+4 \operatorname {c2} \,b^{4} \textit {\_a}^{\frac {2 a +2 b}{a}}+4 \operatorname {c4} \,b^{4} \textit {\_a}^{\frac {4 a +2 b}{a}}+6 b \operatorname {c2} \,a^{3} \textit {\_a}^{\frac {2 a +2 b}{a}}+19 \operatorname {c2} \,a^{2} b^{2} \textit {\_a}^{\frac {2 a +2 b}{a}}+16 a \operatorname {c2} \,b^{3} \textit {\_a}^{\frac {2 a +2 b}{a}}+3 b \operatorname {c4} \,a^{3} \textit {\_a}^{\frac {4 a +2 b}{a}}+11 \operatorname {c4} \,a^{2} b^{2} \textit {\_a}^{\frac {4 a +2 b}{a}}+4 b \operatorname {c3} \,a^{3} \textit {\_a}^{\frac {3 a +2 b}{a}}+14 \operatorname {c3} \,a^{2} b^{2} \textit {\_a}^{\frac {3 a +2 b}{a}}+25 \textit {\_a}^{\frac {2 b}{a}} a^{3} b \operatorname {c0} +35 \textit {\_a}^{\frac {2 b}{a}} a^{2} b^{2} \operatorname {c0} +20 \textit {\_a}^{\frac {2 b}{a}} a \,b^{3} \operatorname {c0} +12 b \,\textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,a^{3}+26 \textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,a^{2} b^{2}+18 a \,\textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,b^{3}+14 a \operatorname {c3} \,b^{3} \textit {\_a}^{\frac {3 a +2 b}{a}}+12 a \operatorname {c4} \,b^{3} \textit {\_a}^{\frac {4 a +2 b}{a}}\right )}}d \textit {\_a} = c_{3} +x \] Verified OK.

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

\[ \frac {y a y^{\frac {b}{a}}}{a +b}-\left (\int c_{2} \left (x +\sqrt {c^{2}+x^{2}}\right )^{\frac {1}{a}}d x \right )-c_{3} = 0 \] Verified OK.

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

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

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

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

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

\[ \frac {y^{3}}{3}+\frac {x c_{2} x^{-a}}{-1+a}-c_{3} = 0 \] Verified OK.

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

\[ -\frac {1}{y}+\frac {x c_{2} x^{-a}}{-1+a}-c_{3} = 0 \] Verified OK.

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

\[ -\frac {1}{3 y^{3}}+\frac {c_{2}}{3 x^{3}}-c_{3} = 0 \] Verified OK.

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

\[ \frac {-2 c_{3} x^{2}+y^{2}+2 x y+c_{1}}{2 x^{2}} = 0 \] Verified OK.

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

\[ 2 \sqrt {y}-2 \sqrt {x}\, c_{2} -c_{3} = 0 \] Verified OK.

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

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

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

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

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

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

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

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

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

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

\[ {}2 \left (y-a \right ) \left (y-b \right ) \left (y-c \right ) y^{\prime \prime }-\left (\left (y-a \right )^{2} \left (y-b \right ) \left (y-c \right )+\left (y-b \right ) \left (y-c \right )\right ) {y^{\prime }}^{2}+\left (y-a \right )^{2} \left (y-b \right )^{2} \left (y-c \right )^{2} \left (A_{0} +\frac {B_{0}}{\left (y-a \right )^{2}}+\frac {C_{1}}{\left (y-b \right )^{2}}+\frac {D_{0}}{\left (y-c \right )^{2}}\right ) = 0 \]

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

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

\[ {}\left (4 y^{3}-a y-b \right ) y^{\prime \prime }-\left (6 y^{2}-\frac {a}{2}\right ) {y^{\prime }}^{2} = 0 \]

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

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

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

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

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

\[ {}\sqrt {y}\, y^{\prime \prime }-a = 0 \]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ \int _{}^{y}-\frac {\sqrt {-a^{2} \textit {\_a}^{2}+b^{2}}\, b a}{a^{2} \textit {\_a}^{2}-\sqrt {-\frac {b^{2}}{a^{2} \textit {\_a}^{2}-b^{2}}}\, \sqrt {-a^{2} \textit {\_a}^{2}+b^{2}}\, b}d \textit {\_a} = x +c_{5} \] Verified OK.

\[ y = -\frac {c_{6} {\mathrm e}^{c_{2} x}-\sqrt {b^{2} c_{2}^{2}+1}}{c_{2} a} \] Verified OK.

\[ \int _{}^{y}-\frac {\sqrt {-a^{2} \textit {\_a}^{2}+b^{2}}\, b a}{a^{2} \textit {\_a}^{2}+\sqrt {-\frac {b^{2}}{a^{2} \textit {\_a}^{2}-b^{2}}}\, \sqrt {-a^{2} \textit {\_a}^{2}+b^{2}}\, b}d \textit {\_a} = x +c_{7} \] Verified OK.

\[ \int _{}^{y}\frac {\sqrt {-a^{2} \textit {\_a}^{2}+b^{2}}\, b a}{a^{2} \textit {\_a}^{2}-\sqrt {-\frac {b^{2}}{a^{2} \textit {\_a}^{2}-b^{2}}}\, \sqrt {-a^{2} \textit {\_a}^{2}+b^{2}}\, b}d \textit {\_a} = x +c_{8} \] Verified OK.

\[ {}y^{\prime } = f \left (y\right ) \]

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

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

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

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

\[ \int _{}^{x}-\left (f_{1} \left (\textit {\_a} \right ) y+f_{0} \left (\textit {\_a} \right )\right ) {\mathrm e}^{-\left (\int \frac {f_{1} \left (\textit {\_a} \right )+\frac {d}{d \textit {\_a}}g \left (\textit {\_a} \right )}{g \left (\textit {\_a} \right )}d \textit {\_a} \right )}d \textit {\_a} +\left (g \left (x \right ) {\mathrm e}^{-\left (\int \frac {f_{1} \left (x \right )+g^{\prime }\left (x \right )}{g \left (x \right )}d x \right )}+\int _{}^{x}f_{1} \left (\textit {\_a} \right ) {\mathrm e}^{-\left (\int \frac {f_{1} \left (\textit {\_a} \right )+\frac {d}{d \textit {\_a}}g \left (\textit {\_a} \right )}{g \left (\textit {\_a} \right )}d \textit {\_a} \right )}d \textit {\_a} \right ) y = c_{1} \] Verified OK.

\[ {}g \left (x \right ) y^{\prime } = f_{1} \left (x \right ) y+f_{n} \left (x \right ) y^{n} \]

\[ y^{1-n} = -\left (\left (-1+n \right ) \left (\int \frac {f_{n} \left (x \right ) {\mathrm e}^{\left (-1+n \right ) \left (\int \frac {f_{1} \left (x \right )}{g \left (x \right )}d x \right )}}{g \left (x \right )}d x \right )-c_{1} \right ) {\mathrm e}^{-\left (-1+n \right ) \left (\int \frac {f_{1} \left (x \right )}{g \left (x \right )}d x \right )} \] Verified OK.

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

\[ \int _{}^{x}\left (a y \left (-1+b \textit {\_a} \ln \left (\textit {\_a} \right )\right ) \ln \left (\textit {\_a} \right )^{n}-b \left (1+\ln \left (\textit {\_a} \right )\right )\right ) {\mathrm e}^{a \left (\int \ln \left (\textit {\_a} \right )^{n} \left (-1+b \textit {\_a} \ln \left (\textit {\_a} \right )\right )d \textit {\_a} \right )}d \textit {\_a} +\left (-{\mathrm e}^{a \left (\int _{}^{x}\ln \left (\textit {\_a} \right )^{n} \left (\ln \left (\textit {\_a} \right ) \textit {\_a} b -1\right )d \textit {\_a} \right )}+{\mathrm e}^{a \left (\int \ln \left (x \right )^{n} \left (-1+b x \ln \left (x \right )\right )d x \right )}\right ) y = c_{1} \] Verified OK.

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

\[ \frac {k \,x^{2}}{2}+\left (B y+b \right ) x +\frac {A y^{2}}{2}+a y = c_{1} \] Verified OK.

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

\[ b x +\frac {k \,x^{1+m}}{1+m}+y A \,x^{n}+a y+\frac {y^{2}}{2} = c_{1} \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}x y^{\prime }+y+x^{4} y^{4} {\mathrm e}^{x} = 0 \]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ y = 0 \] Verified OK.

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

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

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

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

\[ y = i x \] Verified OK.

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

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

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

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

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

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

\[ y = 0 \] Verified OK.

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

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

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

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

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

\[ y = 0 \] Verified OK.

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

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

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

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

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

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

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

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

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

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

\[ \int _{}^{y}\frac {1}{\sqrt {-1+\frac {{\mathrm e}^{-2 \textit {\_a}}}{c_{1}^{2}}}}d \textit {\_a} = x +c_{2} \] Verified OK.

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

\[ {}x^{\prime } = -\frac {t}{x} \]

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

\[ {}x^{\prime } = \sqrt {x} \]

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

\[ {}x^{\prime } = {\mathrm e}^{-2 x} \]

\[ \frac {{\mathrm e}^{2 x}}{2} = t +\frac {{\mathrm e}^{2}}{2} \] Verified OK.

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

\[ \int _{}^{x}{\mathrm e}^{-\textit {\_a}^{2}}d \textit {\_a} = t +c_{1} \] Verified OK.

\[ {}\theta ^{\prime } = t \sqrt {t^{2}+1}\, \sec \left (\theta \right ) \]

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

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

\[ u^{2}-\frac {t^{2}}{2}+u-t = c_{1} \] Verified OK.

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

\[ y \left (y+1\right ) = t +2 \] Verified OK.

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

\[ \frac {\ln \left (x\right )}{4}-\frac {\ln \left (4+x\right )}{4} = t -\frac {\ln \left (5\right )}{4} \] Verified OK.

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

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

\[ {}x^{\prime } = \frac {t^{2}}{1-x^{2}} \]

\[ -\frac {t^{3}}{3}-\frac {x^{3}}{3}+x = {\frac {1}{3}} \] Verified OK.

\[ {}x^{\prime } = \frac {4 t^{2}+3 x^{2}}{2 x t} \]

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

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

\[ -\ln \left (\sec \left (t \right )+\tan \left (t \right )\right )+\int _{0}^{x}\frac {\csc \left (\textit {\_a} \right )}{2 \textit {\_a}}d \textit {\_a} = c_{1} \] Verified OK.

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

\[ \frac {\frac {3 x^{2}}{2}-9 t^{2}}{t^{\frac {4}{3}}} = c_{1} \] Verified OK.

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

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

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

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

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

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

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

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

\[ {}t \cot \left (x\right ) x^{\prime } = -2 \]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ 3 x^{2} y+2 y^{2} x -5 x -6 y = c_{1} \] Verified OK.

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

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

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

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