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

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

\[ {}y^{\prime }-\operatorname {a3} y^{3}-\operatorname {a2} y^{2}-\operatorname {a1} y-\operatorname {a0} = 0 \]

\[ \int _{}^{y}\frac {1}{\textit {\_a}^{3} \operatorname {a3} +\textit {\_a}^{2} \operatorname {a2} +\textit {\_a} \operatorname {a1} +\operatorname {a0}}d \textit {\_a} = x +c_{1} \] Verified OK.

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

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

\[ {}y^{\prime }-x^{a} y^{3}+3 y^{2}-x^{-a} y-x^{-2 a}+a \,x^{-a -1} = 0 \]

\[ \frac {-32 \left (y+x^{-a}\right )^{2} {\mathrm e}^{\frac {i \pi +2 x^{1-a}}{a -1}} \left (a -1\right )^{\frac {-2+a}{a -1}} \left (\left (x -\frac {x^{-1+2 a}}{4}\right ) 2^{\frac {-3 a +5}{a -1}}+\frac {x^{-1+2 a} 4^{\frac {1}{a -1}}}{32}\right ) \operatorname {WhittakerM}\left (-\frac {1}{a -1}, \frac {a -3}{2 a -2}, -\frac {4 x^{1-a}}{a -1}\right )-\left (a -3\right ) \left (\left (4 \left (y+x^{-a}\right )^{2} x^{2}+2 x^{a +1} \left (y+x^{-a}\right )^{2}+a +1\right ) {\mathrm e}^{\frac {4 x^{1-a}}{a -1}}+2 c_{1} \left (y+x^{-a}\right )^{2} \left (a +1\right )\right )}{2 \left (a +1\right ) \left (a -3\right ) \left (y+x^{-a}\right )^{2}} = 0 \] Warning, solution could not be verified

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

\[ \left \{\begin {array}{cc} -2 \sqrt {-y} & y\le 0 \\ 2 \sqrt {y} & 0

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

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

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

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

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

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

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

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

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

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

\[ {}y^{\prime }-\sqrt {\frac {y^{3}+1}{x^{3}+1}} = 0 \]

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

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

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

\[ {}y^{\prime }-\frac {\sqrt {1-y^{4}}}{\sqrt {-x^{4}+1}} = 0 \]

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

\[ {}y^{\prime }-\sqrt {\frac {a y^{4}+b y^{2}+1}{a \,x^{4}+b \,x^{2}+1}} = 0 \]

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

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

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

\[ {}y^{\prime }-\sqrt {\frac {a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}{b_{4} y^{4}+b_{3} y^{3}+b_{2} y^{2}+b_{1} y+b_{0}}} = 0 \]

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

\[ {}y^{\prime }-\sqrt {\frac {b_{4} y^{4}+b_{3} y^{3}+b_{2} y^{2}+b_{1} y+b_{0}}{a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}} = 0 \]

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

\[ {}y^{\prime }-\operatorname {R1} \left (x , \sqrt {a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}\right ) \operatorname {R2} \left (y, \sqrt {b_{4} y^{4}+b_{3} y^{3}+b_{2} y^{2}+b_{1} y+b_{0}}\right ) = 0 \]

\[ \int _{}^{x}-\operatorname {R1} \left (\textit {\_a} , \sqrt {\textit {\_a}^{4} a_{4} +\textit {\_a}^{3} a_{3} +\textit {\_a}^{2} a_{2} +\textit {\_a} a_{1} +a_{0}}\right )d \textit {\_a} +\int _{0}^{y}\frac {1}{\operatorname {R2} \left (\textit {\_a} , \sqrt {b_{4} \textit {\_a}^{4}+b_{3} \textit {\_a}^{3}+b_{2} \textit {\_a}^{2}+b_{1} \textit {\_a} +b_{0}}\right )}d \textit {\_a} = c_{1} \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ \frac {{\mathrm e}^{2 x a} \left (4 y^{2} a^{2}-4 b \cos \left (x +c \right ) a -2 b \sin \left (x +c \right )+y^{2}\right )}{8 a^{2}+2} = c_{1} \] Verified OK.

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

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

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

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

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

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

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

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

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

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

\[ {}\left (4 y-3 x -5\right ) y^{\prime }-3 y+7 x +2 = 0 \]

\[ \frac {x \left (7 x -6 y+4\right )}{2}+2 y^{2}-5 y = c_{1} \] Verified OK.

\[ {}\left (4 y+11 x -11\right ) y^{\prime }-25 y-8 x +62 = 0 \]

\[ -\frac {2 \ln \left (2 y-5+x \right )}{279}+\frac {2 \ln \left (y-2-4 x \right )}{93} = c_{1} \] Verified OK.

\[ {}\left (12 y-5 x -8\right ) y^{\prime }-5 y+2 x +3 = 0 \]

\[ x \left (x -5 y+3\right )+6 y^{2}-8 y = c_{1} \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}\left (2+3 x \right ) \left (y-2 x -1\right ) y^{\prime }-y^{2}+y x -7 x^{2}-9 x -3 = 0 \]

\[ \frac {\ln \left (x +y+1\right )}{3}+\frac {\ln \left (2 y-4-7 x \right )}{6} = \frac {\ln \left (3 x +2\right )}{6}+c_{1} \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}2 x^{3}+y y^{\prime }+3 y^{2} x^{2}+7 = 0 \]

\[ -\frac {10 \left (-\frac {3 \Gamma \left (\frac {2}{3}\right ) \left (\left (\frac {3 y^{2}}{2}+x \right ) {\mathrm e}^{2 x^{3}}-\frac {3 y^{2}}{2}\right ) \left (-x^{3}\right )^{\frac {1}{3}}}{10}+2^{\frac {2}{3}} \left (\Gamma \left (\frac {1}{3}, -2 x^{3}\right ) \Gamma \left (\frac {2}{3}\right )-\frac {2 \pi \sqrt {3}}{3}\right ) x \right )}{9 \left (-x^{3}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}\right )}+\frac {y^{2}}{2} = c_{1} \] Verified OK.

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

\[ \frac {2 a \arctan \left (\frac {a \tan \left (\frac {\arctan \left (x \right )}{2}-\frac {\arctan \left (y\right )}{2}\right )+1}{\sqrt {a^{2}-1}}\right )}{\sqrt {a^{2}-1}}+\arctan \left (y\right )-c_{1} = 0 \] Warning, solution could not be verified

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ -\sin \left (x \right ) {\mathrm e}^{-4 y}-\frac {\left (8 y^{2}+4 y+1\right ) {\mathrm e}^{-4 y}}{32} = c_{1} \] Verified OK.

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

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

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

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

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

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

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

\[ x = -\frac {4 y}{7}+\frac {8 x}{7}+\frac {2 \ln \left (7 \left (y-2 x \right )^{2}+8 y-16 x +2\right )}{49}+\frac {9 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (14 y-28 x +8\right ) \sqrt {2}}{4}\right )}{98}+c_{1} \] Verified OK.

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

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

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

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

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

\[ y^{2} b x +y c \,x^{2}+\frac {d \,x^{3}}{3}+\frac {y^{3} a}{3} = c_{1} \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}\left (10 x^{3} y^{2}+x^{2} y+2 x \right ) y^{\prime }+5 x^{2} y^{3}+x y^{2} = 0 \]

\[ \frac {\ln \left (5 x^{2} y^{2}+2\right )}{2}+\frac {\sqrt {10}\, \arctan \left (\frac {x y \sqrt {10}}{2}\right )}{10}+\ln \left (y\right ) = c_{1} \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

\[ \frac {\left (x^{2}+5 y^{2}\right )^{2}}{4}-\frac {23 y^{4}}{4} = c_{1} \] Verified OK.

\[ {}\left (20 y^{3}-3 x y^{2}+6 x^{2} y+3 x^{3}\right ) y^{\prime }-y^{3}+6 x y^{2}+9 x^{2} y+4 x^{3} = 0 \]

\[ 5 y^{4}-x y^{3}+3 x^{2} y^{2}+3 x^{3} y+x^{4} = c_{1} \] Verified OK.

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

\[ \left (x^{4}-12 x^{2}+24\right ) \cos \left (x \right )+4 \left (-x^{3}+6 x \right ) \sin \left (x \right )+\frac {y^{4} x^{4}}{4} = c_{1} \] Verified OK.

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

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

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

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

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

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

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

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

\[ {}\left (7 x y^{3}+y-5 x \right ) y^{\prime }+y^{4}-5 y = 0 \]

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

\[ {}\left (10 x^{2} y^{3}-3 y^{2}-2\right ) y^{\prime }+5 y^{4} x +x = 0 \]

\[ \frac {x^{2} \left (5 y^{4}+1\right )}{2}-y^{3}-2 y = c_{1} \] Verified OK.

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

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

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

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

\[ {}y^{m} x^{n} \left (a x y^{\prime }+b y\right )+\alpha x y^{\prime }+\beta y = 0 \]

\[ -\frac {n \left (\left (-a \beta +\alpha b \right ) \ln \left (y^{m} \left (a n -b m \right ) x^{n}-m \beta +\alpha n \right )+\alpha \ln \left (y\right ) \left (a n -b m \right )\right )}{\left (a n -b m \right ) \left (-\alpha n +m \beta \right )} = -\frac {n \beta \ln \left (x \right )}{\alpha n -m \beta }+c_{1} \] Verified OK.

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

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

\[ {}\frac {y^{\prime } f_{\nu }\left (x \right ) \left (-y+y^{p +1}\right )}{-1+y}-\frac {g_{\nu }\left (x \right ) \left (-y+y^{q +1}\right )}{-1+y} = 0 \]

\[ \int _{}^{x}-\frac {g_{\nu }\left (\textit {\_a} \right )}{f_{\nu }\left (\textit {\_a} \right )}d \textit {\_a} +\frac {y \left (-y^{p} \operatorname {LerchPhi}\left (y^{q}, 1, \frac {p +1}{q}\right )+\operatorname {LerchPhi}\left (y^{q}, 1, \frac {1}{q}\right )\right )}{q} = c_{1} \] Verified OK. {(I*y^q)::positive}