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

\[ -\ln \left (t \right )+\ln \left (s \left (s-1\right )\right ) = c_{1} \] Verified OK.

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

\[ x^{2} y+2 y^{2}-3 x = 7 \] Verified OK.

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

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

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

\[ -y \left (y \cos \left (x \right )-\sin \left (x \right )^{2}\right ) = -9 \] Verified OK.

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

\[ {\mathrm e}^{x} \left (y+2\right )+y^{2} x = 8 \] Verified OK.

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

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

\[ {}\frac {1+8 x y^{\frac {2}{3}}}{x^{\frac {2}{3}} y^{\frac {1}{3}}}+\frac {\left (2 x^{\frac {4}{3}} y^{\frac {2}{3}}-x^{\frac {1}{3}}\right ) y^{\prime }}{y^{\frac {4}{3}}} = 0 \]

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

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

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

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

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

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

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

\[ {}\tan \left (\theta \right )+2 r \theta ^{\prime } = 0 \]

\[ -\ln \left (r \right )-2 \ln \left (\sin \left (\theta \right )\right ) = c_{1} \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

\[ \frac {\sin \left (2 x \right )}{4}-\frac {x}{2}-\frac {\tan \left (y\right )}{8} = -\frac {\pi }{24} \] Verified OK.

\[ {}\left (3 x +8\right ) \left (y^{2}+4\right )-4 y \left (x^{2}+5 x +6\right ) y^{\prime } = 0 \]

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

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

\[ \frac {-x^{2}-y^{2}}{x^{3}} = -5 \] Verified OK.

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

\[ 2 \ln \left (2 x +y\right )-\ln \left (y-x \right ) = 2 \ln \left (2\right )+\ln \left (3\right ) \] Verified OK.

\[ {}3 x^{2}+9 x y+5 y^{2}-\left (6 x^{2}+4 x y\right ) y^{\prime } = 0 \]

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}y^{\prime }+\left (4 y-\frac {8}{y^{3}}\right ) x = 0 \]

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

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

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

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

\[ \frac {y^{4} x^{2}}{2}-\frac {x^{4}}{2} = {\frac {15}{2}} \] Verified OK.

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

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

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

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

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

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

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

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

\[ {}y^{\prime } = \frac {4 x^{3} y^{2}-3 x^{2} y}{x^{3}-2 x^{4} y} \]

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

\[ {}y^{\prime } = \frac {2 x -7 y}{3 y-8 x} \]

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

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

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

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

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

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

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

\[ {}2 y^{2}+8+\left (-x^{2}+1\right ) y y^{\prime } = 0 \]

\[ \operatorname {arctanh}\left (x \right )+\frac {\ln \left (y^{2}+4\right )}{4} = \operatorname {arccoth}\left (3\right )-\frac {i \pi }{2}+\frac {\ln \left (2\right )}{2} \] Verified OK.

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

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

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

\[ x^{2} y^{2}+x^{3}+2 y^{3} = 21 \] Verified OK.

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

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

\[ {}y^{\prime } = \frac {2 x +7 y}{2 x -2 y} \]

\[ -\ln \left (x +2 y\right )+2 \ln \left (2 x +y\right ) = -\ln \left (5\right )+4 \ln \left (2\right ) \] Verified OK.

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

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

\[ {}5 x y+4 y^{2}+1+\left (2 x y+x^{2}\right ) y^{\prime } = 0 \]

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

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

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

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

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

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

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

\[ {}4 x y^{2}+6 y+\left (5 x^{2} y+8 x \right ) y^{\prime } = 0 \]

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

\[ {}5 x +2 y+1+\left (2 x +y+1\right ) y^{\prime } = 0 \]

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

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

\[ y^{2}+\left (4 x +2\right ) y+3 x^{2}+x = {\frac {89}{4}} \] Verified OK.

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

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

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

\[ \int _{}^{x}\frac {1}{-\textit {\_a}^{4}+\textit {\_a}^{2}}d \textit {\_a} = t +c_{1} \] Verified OK.

\[ {}x^{\prime } = k x-x^{2} \]

\[ \frac {\ln \left (x\right )-\ln \left (x-k \right )}{k} = t +\frac {\ln \left (x_{0} \right )-\ln \left (-k +x_{0} \right )}{k} \] Verified OK.

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

\[ \frac {-2 \ln \left (x\right )+\ln \left (k^{2}+x^{2}\right )}{2 k^{2}} = t +\frac {-2 \ln \left (x_{0} \right )+\ln \left (k^{2}+x_{0}^{2}\right )}{2 k^{2}} \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}y = {y^{\prime }}^{4}-{y^{\prime }}^{3}-2 \]

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

\[ {}y^{\prime } = \frac {2 y-x -4}{2 x -y+5} \]

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

\[ {}y = 5 x y^{\prime }-{y^{\prime }}^{2} \]

\[ y = 0 \] Verified OK.

\[ x = \frac {5 x}{9}+\frac {\sqrt {25 x^{2}-4 y}}{9}+\frac {4 c_{1} \sqrt {2}}{\left (10 x +2 \sqrt {25 x^{2}-4 y}\right )^{\frac {5}{4}}} \] Verified OK.

\[ x = \frac {5 x}{9}-\frac {\sqrt {25 x^{2}-4 y}}{9}+\frac {4 c_{1} \sqrt {2}}{\left (10 x -2 \sqrt {25 x^{2}-4 y}\right )^{\frac {5}{4}}} \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}x^{3} x^{\prime \prime }+1 = 0 \]

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

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

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

\[ \frac {y^{2}}{2}-c_{2} \left (\frac {x^{2}}{2}+\frac {x \sqrt {x^{2}+1}}{2}+\frac {\operatorname {arcsinh}\left (x \right )}{2}\right )-c_{3} = 0 \] Verified OK.

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

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

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

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

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

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

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

\[ \int _{}^{x}\left (y \,{\mathrm e}^{\textit {\_a}^{2}}-\sinh \left (\textit {\_a} \right )\right ) {\mathrm e}^{-i \textit {\_a} +2 \left (\int \frac {{\mathrm e}^{\textit {\_a} \left (i+\textit {\_a} \right )}+i}{{\mathrm e}^{2 i \textit {\_a}}+1}d \textit {\_a} \right )}d \textit {\_a} +\int _{0}^{y}\left (\cos \left (x \right ) {\mathrm e}^{-i x +2 \left (\int \frac {{\mathrm e}^{x \left (i+x \right )}+i}{{\mathrm e}^{2 i x}+1}d x \right )}-\left (\int _{}^{x}{\mathrm e}^{\textit {\_a}^{2}-i \textit {\_a} +2 \left (\int \frac {{\mathrm e}^{\textit {\_a} \left (i+\textit {\_a} \right )}+i}{{\mathrm e}^{2 i \textit {\_a}}+1}d \textit {\_a} \right )}d \textit {\_a} \right )\right )d \textit {\_a} = c_{1} \] Verified OK.

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

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

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

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

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

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

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

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

\[ -y-2 y \cos \left (y\right )+c_{1} x -\sin \left (x \right )-c_{2} = 0 \] Verified OK.

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

\[ y = 0 \] Verified OK.

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

\[ y = i x \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ -\ln \left (2\right )-\ln \left (\frac {y \left (y+\sqrt {y^{2}+x^{2}}\right )}{x}\right )-\ln \left (x \right ) m +\int _{0}^{y}\frac {\textit {\_a} \left (\textit {\_a} \sqrt {\textit {\_a}^{2}+x^{2}}+\textit {\_a}^{2}+x^{2}\right ) \ln \left (\frac {\textit {\_a} \left (\sqrt {\textit {\_a}^{2}+x^{2}}+\textit {\_a} \right )}{x}\right )+\textit {\_a} \left (\textit {\_a} \ln \left (2\right )+1\right ) \sqrt {\textit {\_a}^{2}+x^{2}}+\left (1+\textit {\_a} \left (\ln \left (2\right )+1\right )\right ) \left (\textit {\_a}^{2}+x^{2}\right )}{\sqrt {\textit {\_a}^{2}+x^{2}}\, \textit {\_a} \left (\sqrt {\textit {\_a}^{2}+x^{2}}+\textit {\_a} \right )}d \textit {\_a} = c_{1} \] Verified OK. {1::positive, _a::positive}

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

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

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

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

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

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

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

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

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

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