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

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

\[ {}x y^{\prime }-5 y-x \sqrt {y} = 0 \]

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

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

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

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

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

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

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

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

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

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

\[ \frac {5 y^{5} x^{3}-2}{10 y^{5} x^{5}} = {\frac {3}{10}} \] Verified OK.

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

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

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

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

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

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

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

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

\[ {}y^{\prime }+P \left (x \right ) y = Q \left (x \right ) \]

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

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

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

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

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

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

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

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

\[ -{\mathrm e}^{-y}-{\mathrm e}^{x} c_{2} -c_{3} = 0 \] Verified OK.

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

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

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

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

\[ -\frac {\sqrt {-4 y^{2}+30 y-36}}{4}-\frac {9 \arcsin \left (\frac {4 y}{9}-\frac {5}{3}\right )}{8} = x -\frac {3 \sqrt {2}}{4}+\frac {9 \arcsin \left (\frac {1}{3}\right )}{8} \] Verified OK.

\[ \frac {\sqrt {-4 y^{2}+30 y-36}}{4}+\frac {9 \arcsin \left (\frac {4 y}{9}-\frac {5}{3}\right )}{8} = x +\frac {3 \sqrt {2}}{4}-\frac {9 \arcsin \left (\frac {1}{3}\right )}{8} \] Verified OK.

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

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

\[ \arctan \left (\frac {1}{\sqrt {-1+4 y^{2}}}\right ) = x +\frac {\pi }{6} \] Warning, solution could not be verified

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

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

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

\[ y = 0 \] Verified OK.

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

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

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

\[ y = -x \] Verified OK.

\[ y = 0 \] Verified OK.

\[ y = x \] Verified OK.

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

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

\[ {}8 x +1 = y {y^{\prime }}^{2} \]

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

\[ y = \frac {8 x +1}{-2+2 i \sqrt {3}} \] Verified OK.

\[ y = \frac {8 x +1}{-2-2 i \sqrt {3}} \] Verified OK.

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

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

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

\[ y = x \] Verified OK.

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

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

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

\[ y = 0 \] Verified OK.

\[ y = 3 x \] Verified OK.

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

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

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

\[ y = 1+x \] 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.

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

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

\[ x = -\frac {54 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} 3^{\frac {1}{3}} 2^{\frac {2}{3}} \left (-\frac {\left (\sqrt {\frac {-4 y^{3}+27 x}{x}}\, c_{1} 3^{\frac {1}{6}}+2 \left (y+\frac {3 c_{1}}{2}\right ) 3^{\frac {2}{3}}\right ) 2^{\frac {1}{3}} x {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{3}-\frac {2^{\frac {2}{3}} 3^{\frac {5}{6}} x^{2} \sqrt {\frac {-4 y^{3}+27 x}{x}}}{3}-3 \,3^{\frac {1}{3}} \left (\frac {2 y^{2} c_{1}}{9}+x \right ) x 2^{\frac {2}{3}}+{\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} \left (-\frac {4 c_{1} y}{3}+x \right )\right ) x^{3}}{\left (2 \,2^{\frac {1}{3}} 3^{\frac {2}{3}} y x +2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}-6 x {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}\right )^{2} \left (2^{\frac {2}{3}} 3^{\frac {1}{3}} x y+{\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right )^{2}} \] Warning, solution could not be verified

\[ x = -\frac {36 \left (\left (\frac {8 c_{1} y}{9}-\frac {2 x}{3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+\left (-\frac {\left (\left (i 3^{\frac {2}{3}}+3^{\frac {1}{6}}\right ) c_{1} \sqrt {\frac {-4 y^{3}+27 x}{x}}+6 \left (y+\frac {3 c_{1}}{2}\right ) \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right )\right ) 2^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{9}+2^{\frac {2}{3}} \left (\frac {\left (i 3^{\frac {1}{3}}-\frac {3^{\frac {5}{6}}}{3}\right ) x \sqrt {\frac {-4 y^{3}+27 x}{x}}}{3}+\left (\frac {2 y^{2} c_{1}}{9}+x \right ) \left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right )\right )\right ) x \right ) 3^{\frac {1}{3}} 2^{\frac {2}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} x^{3}}{{\left (\left (\sqrt {3}+i\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+2^{\frac {2}{3}} y \left (i 3^{\frac {1}{3}}-3^{\frac {5}{6}}\right ) x \right )}^{2} \left (-\frac {2^{\frac {2}{3}} \left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}}{6}+\left (2 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+y \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right ) 2^{\frac {1}{3}}\right ) x \right )^{2}} \] Warning, solution could not be verified

\[ x = \frac {36 \left (\left (-\frac {8 c_{1} y}{9}+\frac {2 x}{3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+\left (-\frac {\left (\left (i 3^{\frac {2}{3}}-3^{\frac {1}{6}}\right ) c_{1} \sqrt {\frac {-4 y^{3}+27 x}{x}}+6 \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) \left (y+\frac {3 c_{1}}{2}\right )\right ) 2^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{9}+2^{\frac {2}{3}} \left (\frac {\left (i 3^{\frac {1}{3}}+\frac {3^{\frac {5}{6}}}{3}\right ) x \sqrt {\frac {-4 y^{3}+27 x}{x}}}{3}+\left (3^{\frac {1}{3}}+i 3^{\frac {5}{6}}\right ) \left (\frac {2 y^{2} c_{1}}{9}+x \right )\right )\right ) x \right ) 3^{\frac {1}{3}} 2^{\frac {2}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} x^{3}}{{\left (-\frac {\left (3^{\frac {1}{3}}+i 3^{\frac {5}{6}}\right ) 2^{\frac {2}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}}{6}+\left (-2 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+\left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) y 2^{\frac {1}{3}}\right ) x \right )}^{2} {\left (\left (i-\sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+\left (3^{\frac {5}{6}}+i 3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} y x \right )}^{2}} \] Warning, solution could not be verified

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

\[ y = -x \] Verified OK.

\[ y = 0 \] Verified OK.

\[ y = x \] Verified OK.

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

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

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

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

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

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

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

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

\[ y = 0 \] Verified OK.

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

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

\[ {}2 x {y^{\prime }}^{3}+1 = y {y^{\prime }}^{2} \]

\[ y = \infty \] Verified OK.

\[ x = \frac {36 \left (\left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {2}{3}} c_{1} +\left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {1}{3}} c_{1} y+c_{1} y^{2}-12 x \left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {1}{3}}\right ) x^{2} \left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {2}{3}}}{{\left (\left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {2}{3}}+y \left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {1}{3}}+y^{2}\right )}^{3}} \] Verified OK.

\[ x = \frac {144 \left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {2}{3}} \left (\left (-i \sqrt {3}\, c_{1} +c_{1} \right ) \left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {2}{3}}+\left (-2 c_{1} y+24 x \right ) \left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {1}{3}}+y^{2} \left (1+i \sqrt {3}\right ) c_{1} \right ) x^{2}}{{\left (-i \sqrt {3}\, \left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {2}{3}}+i \sqrt {3}\, y^{2}+\left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {2}{3}}-2 y \left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {1}{3}}+y^{2}\right )}^{3}} \] Verified OK.

\[ x = \frac {144 \left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {2}{3}} x^{2} \left (c_{1} \left (-i \sqrt {3}-1\right ) \left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {2}{3}}+2 \left (c_{1} y-12 x \right ) \left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {1}{3}}+y^{2} \left (i \sqrt {3}-1\right ) c_{1} \right )}{{\left (-i \sqrt {3}\, \left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {2}{3}}+i \sqrt {3}\, y^{2}-\left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {2}{3}}+2 y \left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {1}{3}}-y^{2}\right )}^{3}} \] Verified OK.

\[ {}3 {y^{\prime }}^{4} x = {y^{\prime }}^{3} y+1 \]

\[ x = \frac {5 \operatorname {RootOf}\left (3 x \,\textit {\_Z}^{4}-y \textit {\_Z}^{3}-1\right )^{3} c_{1} y+9 \operatorname {RootOf}\left (3 x \,\textit {\_Z}^{4}-y \textit {\_Z}^{3}-1\right )^{\frac {3}{2}} x +5 c_{1}}{15 x \operatorname {RootOf}\left (3 x \,\textit {\_Z}^{4}-y \textit {\_Z}^{3}-1\right )^{\frac {11}{2}}} \] Verified OK.

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

\[ y = 0 \] Verified OK.

\[ x = \frac {10 x \operatorname {RootOf}\left (2 \textit {\_Z}^{5}+2 x \textit {\_Z} -y\right )^{2}-5 y \operatorname {RootOf}\left (2 \textit {\_Z}^{5}+2 x \textit {\_Z} -y\right )+6 c_{1}}{6 \operatorname {RootOf}\left (2 \textit {\_Z}^{5}+2 x \textit {\_Z} -y\right )^{2}} \] Verified OK.

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

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

\[ x = \frac {108 x^{3}}{\left (\frac {16 y^{2}}{\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}}+4 y+\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}\right )^{3}}+\frac {c_{1} \left (\frac {16 y^{2}}{\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}}+4 y+\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}\right )}{6 x} \] Verified OK.

\[ x = \frac {-10368 x^{4} \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}-55296 y^{3} x^{3}+93312 x^{5}}{{\left (-i \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}} \sqrt {3}+16 i \sqrt {3}\, y^{2}+\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}}-8 y \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}+16 y^{2}\right )}^{3}}+\frac {\left (\left (i \sqrt {3}-1\right ) \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}}+8 y \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}-16 y^{2} \left (1+i \sqrt {3}\right )\right ) c_{1}}{12 \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}} x} \] Warning, solution could not be verified

\[ x = \frac {3456 \left (-3 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x -16 y^{3}+27 x^{2}\right ) x^{3}}{{\left (i \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}} \sqrt {3}-16 i \sqrt {3}\, y^{2}+\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}}-8 y \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}+16 y^{2}\right )}^{3}}-\frac {\left (\left (1+i \sqrt {3}\right ) \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}}-8 y \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}-16 y^{2} \left (i \sqrt {3}-1\right )\right ) c_{1}}{12 \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}} x} \] Warning, solution could not be verified

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

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

\[ x = -\frac {3 x \left (-9 c_{1} x^{2}+4 \operatorname {LambertW}\left (\frac {3 x \,{\mathrm e}^{y-2}}{2}\right )^{2}\right )}{8 \operatorname {LambertW}\left (\frac {3 x \,{\mathrm e}^{y-2}}{2}\right )^{3}} \] Verified OK.

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

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

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

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

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

\[ \frac {1}{2 y^{2}} = t -\frac {17}{18} \] Verified OK.

\[ {}y^{\prime } = \frac {{\mathrm e}^{t}}{y} \]

\[ -{\mathrm e}^{t}+\frac {y^{2}}{2} = 30 \] Verified OK.

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

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

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

\[ \int _{}^{y}\frac {1}{\textit {\_a}^{3}-\textit {\_a}^{2}}d \textit {\_a} = t +c_{1} \] Verified OK.

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

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

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

\[ \frac {\ln \left (y\right )}{2} = t -\frac {\ln \left (3\right )}{2} \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {\mathrm e}^{y}+\left (-1+c_{3} \right ) {\mathrm e}^{-x}-c_{3} = 0 \] Verified OK.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[ {}y^{\prime } = \frac {2 \sqrt {-1+y}}{3} \]

\[ 3 \sqrt {y-1}-x +1 = 0 \] Verified OK.

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

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

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

\[ \ln \left (\frac {y}{x}\right )+\frac {3 x}{y}+\ln \left (x \right )-c_{2} = 0 \] Verified OK.

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

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

\[ {}x y^{\prime } = \sqrt {16 x^{2}-y^{2}}+y \]

\[ \arcsin \left (\frac {y}{4 x}\right )-\ln \left (x \right )-c_{2} = 0 \] Verified OK. {0 < x}

\[ {}-y+x y^{\prime } = \sqrt {9 x^{2}+y^{2}} \]

\[ \operatorname {arcsinh}\left (\frac {y}{3 x}\right )-\ln \left (x \right )-c_{2} = 0 \] Verified OK. {0 < x}

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

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

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

\[ \frac {{\mathrm e}^{\frac {y^{2}}{x^{2}}}}{2}-\frac {\ln \left (x \right )}{2}-c_{2} = 0 \] Verified OK.

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

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

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

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

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

\[ -\frac {2 x}{y}+\ln \left (\frac {y}{x}\right )+\frac {3 \ln \left (x \right )}{2}-c_{2} = 0 \] Verified OK.

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

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

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

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

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

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

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

\[ \operatorname {arcsinh}\left (\frac {y}{x}\right )+\ln \left (x \right )-\operatorname {arcsinh}\left (\frac {4}{3}\right )-\ln \left (3\right ) = 0 \] Verified OK.

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

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

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

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

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

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

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

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

\[ {}y^{\prime }+\frac {2 y}{x} = 6 \sqrt {x^{2}+1}\, \sqrt {y} \]

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

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

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

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

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

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

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

\[ {}2 y^{\prime }+y \cot \left (x \right ) = \frac {8 \cos \left (x \right )^{3}}{y} \]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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