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ODE |
solution |
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\[ {}x \left (1-y x \right )^{2} y^{\prime }+\left (1+y^{2} x^{2}\right ) y = 0 \] |
\[ \frac {x^{2} y^{2}-1}{x y}-2 \ln \left (y\right ) = c_{1} \] Verified OK. |
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\[ {}\left (1-x^{4} y^{2}\right ) y^{\prime } = x^{3} y^{3} \] |
\[ -\frac {y^{4} x^{4}}{4}+\frac {y^{2}}{2} = c_{1} \] Verified OK. |
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\[ {}\left (3 x -y^{3}\right ) y^{\prime } = x^{2}-3 y \] |
\[ -\frac {y^{4}}{4}-\frac {x^{3}}{3}+3 x y = c_{1} \] Verified OK. |
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\[ {}\left (x^{3}-y^{3}\right ) y^{\prime }+x^{2} y = 0 \] |
\[ \frac {y^{3} x^{3}}{3}-\frac {y^{6}}{6} = c_{1} \] Verified OK. |
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\[ {}\left (x^{3}+y^{3}\right ) y^{\prime }+x^{2} \left (x a +3 y\right ) = 0 \] |
\[ \frac {a \,x^{4}}{4}+y x^{3}+\frac {y^{4}}{4} = c_{1} \] Verified OK. |
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\[ {}\left (x -x^{2} y-y^{3}\right ) y^{\prime } = x^{3}-y+x y^{2} \] |
\[ -\frac {x \left (x^{3}+2 x y^{2}-4 y\right )}{4}-\frac {y^{4}}{4} = c_{1} \] Verified OK. |
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\[ {}\left (x \,a^{2}+y \left (x^{2}-y^{2}\right )\right ) y^{\prime }+x \left (x^{2}-y^{2}\right ) = a^{2} y \] |
\[ \frac {x^{2}}{2}+\frac {a^{2} \ln \left (x +y\right )}{2}-\frac {a^{2} \ln \left (x -y\right )}{2}+\frac {y^{2}}{2} = c_{1} \] Verified OK. |
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\[ {}\left (a +x^{2}+y^{2}\right ) y y^{\prime } = x \left (a -x^{2}-y^{2}\right ) \] |
\[ \frac {\left (a -x^{2}-y^{2}\right )^{2}}{4}+a y^{2} = c_{1} \] Verified OK. |
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\[ {}\left (3 x^{2}+y^{2}\right ) y y^{\prime }+x \left (x^{2}+3 y^{2}\right ) = 0 \] |
\[ \frac {\left (x^{2}+3 y^{2}\right )^{2}}{4}-2 y^{4} = c_{1} \] Verified OK. |
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\[ {}2 y^{3} y^{\prime } = x^{3}-x y^{2} \] |
\[ \frac {\ln \left (x^{2}+y^{2}\right )}{3}+\frac {\ln \left (2 y^{2}-x^{2}\right )}{6} = c_{1} \] Verified OK. |
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\[ {}y \left (2 y^{2}+1\right ) y^{\prime } = x \left (2 x^{2}+1\right ) \] |
\[ -\frac {\left (2 x^{2}+1\right )^{2}}{8}+\frac {\left (1+2 y^{2}\right )^{2}}{8} = c_{1} \] Verified OK. |
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\[ {}\left (3 x^{2}+2 y^{2}\right ) y y^{\prime }+x^{3} = 0 \] |
\[ \ln \left (x^{2}+2 y^{2}\right )-\frac {\ln \left (x^{2}+y^{2}\right )}{2} = c_{1} \] Verified OK. |
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\[ {}\left (5 x^{2}+2 y^{2}\right ) y y^{\prime }+x \left (x^{2}+5 y^{2}\right ) = 0 \] |
\[ \frac {\left (x^{2}+5 y^{2}\right )^{2}}{4}-\frac {23 y^{4}}{4} = c_{1} \] Verified OK. |
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\[ {}\left (3 x^{3}+6 x^{2} y-3 x y^{2}+20 y^{3}\right ) y^{\prime }+4 x^{3}+9 x^{2} y+6 x y^{2}-y^{3} = 0 \] |
\[ 5 y^{4}-y^{3} x +3 y^{2} x^{2}+3 y x^{3}+x^{4} = c_{1} \] Verified OK. |
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\[ {}x y^{3} y^{\prime } = \left (-x^{2}+1\right ) \left (1+y^{2}\right ) \] |
\[ -\frac {x^{2}}{2}+\ln \left (x \right )-\frac {y^{2}}{2}+\frac {\ln \left (y^{2}+1\right )}{2} = c_{1} \] Verified OK. |
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\[ {}x \left (x -y^{3}\right ) y^{\prime } = \left (3 x +y^{3}\right ) y \] |
\[ -\frac {x^{2} \left (y^{3}+2 x \right )}{2 y} = c_{1} \] Verified OK. |
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\[ {}x \left (2 x^{3}+y^{3}\right ) y^{\prime } = \left (2 x^{3}-x^{2} y+y^{3}\right ) y \] |
\[ \frac {y^{3}-4 x^{3}}{2 x^{2} y} = -\ln \left (x \right )+c_{1} \] Verified OK. |
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\[ {}x \left (2 x^{3}-y^{3}\right ) y^{\prime } = \left (x^{3}-2 y^{3}\right ) y \] |
\[ \frac {-x^{3}-y^{3}}{x^{2} y^{2}} = c_{1} \] Verified OK. |
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\[ {}x \left (x^{3}-2 y^{3}\right ) y^{\prime } = \left (2 x^{3}-y^{3}\right ) y \] |
\[ \frac {-x^{3}-y^{3}}{x y} = c_{1} \] Verified OK. |
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\[ {}x \left (x^{4}-2 y^{3}\right ) y^{\prime }+\left (2 x^{4}+y^{3}\right ) y = 0 \] |
\[ \frac {y \left (2 x^{4}-y^{3}\right )}{2 x^{2}} = c_{1} \] Verified OK. |
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\[ {}x \left (x +y+2 y^{3}\right ) y^{\prime } = \left (x -y\right ) y \] |
\[ \ln \left (x \right )-\frac {x}{y}+y^{2}+\ln \left (y\right ) = c_{1} \] Verified OK. |
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\[ {}\left (5 x -y-7 x y^{3}\right ) y^{\prime }+5 y-y^{4} = 0 \] |
\[ -y \left (y^{3}-5\right )^{2} x -\frac {y^{5}}{5}+\frac {5 y^{2}}{2} = c_{1} \] Verified OK. |
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\[ {}x \left (1-2 x y^{3}\right ) y^{\prime }+\left (1-2 x^{3} y\right ) y = 0 \] |
\[ \frac {-y x^{3}-1}{x y}-y^{2} = c_{1} \] Verified OK. |
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\[ {}\left (2-10 x^{2} y^{3}+3 y^{2}\right ) y^{\prime } = x \left (1+5 y^{4}\right ) \] |
\[ -\frac {x^{2} \left (1+5 y^{4}\right )}{2}+y^{3}+2 y = c_{1} \] Verified OK. |
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\[ {}x \left (a +b x y^{3}\right ) y^{\prime }+\left (a +c \,x^{3} y\right ) y = 0 \] |
\[ \frac {c \,x^{3} y-2 a}{2 x y}+\frac {b y^{2}}{2} = c_{1} \] Verified OK. |
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\[ {}x \left (1-2 x^{2} y^{3}\right ) y^{\prime }+\left (1-2 x^{3} y^{2}\right ) y = 0 \] |
\[ \frac {-2 y^{2} x -\frac {1}{2 x^{2}}}{y^{2}}-2 y = c_{1} \] Verified OK. |
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\[ {}x \left (1-y x \right ) \left (1-y^{2} x^{2}\right ) y^{\prime }+\left (1+y x \right ) \left (1+y^{2} x^{2}\right ) y = 0 \] |
\[ \frac {x^{2} y^{2}-1}{y x}-2 \ln \left (y\right ) = c_{1} \] Verified OK. |
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\[ {}\left (x^{2}-y^{4}\right ) y^{\prime } = y x \] |
\[ -\frac {x^{2}}{2 y^{2}}-\frac {y^{2}}{2} = c_{1} \] Verified OK. |
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\[ {}\left (x^{3}-y^{4}\right ) y^{\prime } = 3 x^{2} y \] |
\[ -\frac {x^{3}}{y}-\frac {y^{3}}{3} = c_{1} \] Verified OK. |
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\[ {}\left (x^{2} a^{2}+\left (x^{2}+y^{2}\right )^{2}\right ) y^{\prime } = a^{2} x y \] |
\[ \frac {\ln \left (\left (-a^{2}+y^{2}\right ) x^{2}+y^{4}\right )-2 \ln \left (y\right )-\ln \left (a^{2}+x^{2}+y^{2}\right )}{2 a^{2}} = c_{1} \] Verified OK. |
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\[ {}2 \left (x -y^{4}\right ) y^{\prime } = y \] |
\[ -\frac {x}{y^{2}}-y^{2} = c_{1} \] Verified OK. |
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\[ {}\left (4 x -x y^{3}-2 y^{4}\right ) y^{\prime } = \left (2+y^{3}\right ) y \] |
\[ -\frac {\left (2+y^{3}\right ) x}{y^{2}}-y^{2} = c_{1} \] Verified OK. |
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\[ {}2 x \left (x^{3}+y^{4}\right ) y^{\prime } = \left (x^{3}+2 y^{4}\right ) y \] |
\[ \frac {-x +\frac {y^{4}}{x^{2}}}{y^{2}} = c_{1} \] Verified OK. |
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\[ {}x \left (1-x^{2} y^{4}\right ) y^{\prime }+y = 0 \] |
\[ -\frac {1}{2 y^{2} x^{2}}-\frac {y^{2}}{2} = c_{1} \] Verified OK. |
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\[ {}\left (x^{2}-y^{5}\right ) y^{\prime } = 2 y x \] |
\[ -\frac {x^{2}}{y}-\frac {y^{4}}{4} = c_{1} \] Verified OK. |
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\[ {}x \left (x^{3}+y^{5}\right ) y^{\prime } = \left (x^{3}-y^{5}\right ) y \] |
\[ -\frac {x \left (-4 y^{5}+x^{3}\right )}{4 y^{4}} = c_{1} \] Verified OK. |
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\[ {}x^{3} \left (1+5 x^{3} y^{7}\right ) y^{\prime }+\left (3 x^{5} y^{5}-1\right ) y^{3} = 0 \] |
\[ x^{3} y^{5}+\frac {1}{2 x^{2}}-\frac {1}{2 y^{2}} = c_{1} \] Verified OK. |
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\[ {}x \left (a +x y^{n}\right ) y^{\prime }+b y = 0 \] |
\[ \frac {a \ln \left (y\right )-b \ln \left (x y^{n}-b n +a \right )+b \ln \left (x \right )-c_{1} \left (-b n +a \right )}{-b n +a} = 0 \] Verified OK. |
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\[ {}y^{\prime } \sqrt {b^{2}+y^{2}} = \sqrt {a^{2}+x^{2}} \] |
\[ -\frac {x \sqrt {a^{2}+x^{2}}}{2}-\frac {a^{2} \ln \left (x +\sqrt {a^{2}+x^{2}}\right )}{2}+\frac {y \sqrt {b^{2}+y^{2}}}{2}+\frac {b^{2} \ln \left (y+\sqrt {b^{2}+y^{2}}\right )}{2} = c_{1} \] Verified OK. |
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\[ {}y^{\prime } \sqrt {b^{2}-y^{2}} = \sqrt {a^{2}-x^{2}} \] |
\[ -\frac {x \sqrt {a^{2}-x^{2}}}{2}-\frac {a^{2} \arctan \left (\frac {x}{\sqrt {a^{2}-x^{2}}}\right )}{2}+\frac {y \sqrt {b^{2}-y^{2}}}{2}+\frac {b^{2} \arctan \left (\frac {y}{\sqrt {b^{2}-y^{2}}}\right )}{2} = c_{1} \] Verified OK. |
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\[ {}y^{\prime } \sqrt {y} = \sqrt {x} \] |
\[ -\frac {2 x^{\frac {3}{2}}}{3}+\frac {2 y^{\frac {3}{2}}}{3} = c_{1} \] Verified OK. |
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\[ {}\left (1+\sqrt {x +y}\right ) y^{\prime }+1 = 0 \] |
\[ 2 \sqrt {x +y}+y = c_{1} \] Verified OK. |
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\[ {}\left (x -2 \sqrt {y x}\right ) y^{\prime } = y \] |
\[ \frac {\ln \left (y\right ) \sqrt {y}+\sqrt {x}}{\sqrt {y}} = c_{1} \] Verified OK. |
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\[ {}\left (y+\sqrt {1+y^{2}}\right ) \left (x^{2}+1\right )^{\frac {3}{2}} y^{\prime } = 1+y^{2} \] |
\[ -\frac {x}{\sqrt {x^{2}+1}}+\frac {\ln \left (1+y^{2}\right )}{2}+\operatorname {arcsinh}\left (y\right ) = c_{1} \] Verified OK. |
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\[ {}\left (y+\sqrt {1+y^{2}}\right ) \left (x^{2}+1\right )^{\frac {3}{2}} y^{\prime } = 1+y^{2} \] |
\[ -\frac {x}{\sqrt {x^{2}+1}}+\frac {\ln \left (1+y^{2}\right )}{2}+\operatorname {arcsinh}\left (y\right ) = c_{1} \] Verified OK. |
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\[ {}\left (x -\sqrt {x^{2}+y^{2}}\right ) y^{\prime } = y \] |
\[ \ln \left (2\right )+\ln \left (x \right )+\ln \left (\sqrt {x^{2}+y^{2}}+x \right ) = \ln \left (x \right )+c_{1} \] Verified OK. |
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\[ {}x \left (x +\sqrt {x^{2}+y^{2}}\right ) y^{\prime }+y \sqrt {x^{2}+y^{2}} = 0 \] |
\[ \ln \left (x \right ) = -\frac {\ln \left (\frac {4 y^{2}}{x^{2}}+3\right )}{12}-\frac {\ln \left (\frac {y}{x}\right )}{3}+\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {\frac {y^{2}}{x^{2}}+1}}\right )}{3}-\frac {\operatorname {arctanh}\left (2 \sqrt {\frac {y^{2}}{x^{2}}+1}\right )}{6}+c_{1} \] Verified OK. |
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\[ {}x y \left (x +\sqrt {x^{2}-y^{2}}\right ) y^{\prime } = x y^{2}-\left (x^{2}-y^{2}\right )^{\frac {3}{2}} \] |
\[ \frac {-2 x \sqrt {x^{2}-y^{2}}+y^{2}}{2 x^{2}} = -\ln \left (x \right )+c_{1} \] Verified OK. |
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\[ {}y^{\prime } \cos \left (y\right ) \left (\cos \left (y\right )-\sin \left (A \right ) \sin \left (x \right )\right )+\cos \left (x \right ) \left (\cos \left (x \right )-\sin \left (A \right ) \sin \left (y\right )\right ) = 0 \] |
\[ \frac {\sin \left (x \right ) \left (-2 \sin \left (A \right ) \sin \left (y\right )+\cos \left (x \right )\right )}{2}+\frac {x}{2}+\frac {\sin \left (y\right ) \cos \left (y\right )}{2}+\frac {y}{2} = c_{1} \] Verified OK. |
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\[ {}\left (a \cos \left (b x +a y\right )-b \sin \left (x a +b y\right )\right ) y^{\prime }+b \cos \left (b x +a y\right )-a \sin \left (x a +b y\right ) = 0 \] |
\[ \cos \left (a x +b y\right )+\sin \left (b x +a y\right ) = c_{1} \] Verified OK. |
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\[ {}\left (x +\cos \left (x \right ) \sec \left (y\right )\right ) y^{\prime }+\tan \left (y\right )-y \sin \left (x \right ) \sec \left (y\right ) = 0 \] |
\[ y \cos \left (x \right )+x \sin \left (y\right ) = c_{1} \] Verified OK. |
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\[ {}\left (1+\left (x +y\right ) \tan \left (y\right )\right ) y^{\prime }+1 = 0 \] |
\[ \sec \left (y\right ) x +\frac {y}{\cos \left (y\right )} = c_{1} \] Verified OK. |
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\[ {}x \left (x -y \tan \left (\frac {y}{x}\right )\right ) y^{\prime }+\left (x +y \tan \left (\frac {y}{x}\right )\right ) y = 0 \] |
\[ -\ln \left (\frac {1}{x}\right )-\frac {\ln \left (\sec \left (\frac {y}{x}\right )^{2}\right )}{2}+\ln \left (y\right ) = c_{1} \] Verified OK. |
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\[ {}\left (\sinh \left (x \right )+x \cosh \left (y\right )\right ) y^{\prime }+y \cosh \left (x \right )+\sinh \left (y\right ) = 0 \] |
\[ y \sinh \left (x \right )+x \sinh \left (y\right ) = c_{1} \] Verified OK. |
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\[ {}{y^{\prime }}^{2}+f \left (x \right ) \left (y-a \right )^{2} \left (y-b \right ) = 0 \] |
\[
\int _{}^{y}\frac {1}{\sqrt {\left (a -y \right )^{2} \left (b -y \right )}}d \textit {\_a} = \int \sqrt {f \left (x \right )}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}-\frac {1}{\sqrt {\left (a -y \right )^{2} \left (b -y \right )}}d \textit {\_a} = \int \sqrt {f \left (x \right )}d x +c_{1}
\] Verified OK. |
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\[ {}{y^{\prime }}^{2}+f \left (x \right ) \left (y-a \right ) \left (y-b \right ) \left (y-c \right ) = 0 \] |
\[
\int _{}^{y}\frac {1}{\sqrt {\left (c -y \right ) \left (b -y \right ) \left (a -y \right )}}d \textit {\_a} = \int \sqrt {f \left (x \right )}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}-\frac {1}{\sqrt {\left (c -y \right ) \left (b -y \right ) \left (a -y \right )}}d \textit {\_a} = \int \sqrt {f \left (x \right )}d x +c_{1}
\] Verified OK. |
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\[ {}{y^{\prime }}^{2}+f \left (x \right ) \left (y-a \right )^{2} \left (y-b \right ) \left (y-c \right ) = 0 \] |
\[
\int _{}^{y}\frac {1}{\sqrt {\left (a -y \right )^{2} \left (c -y \right ) \left (b -y \right )}}d \textit {\_a} = \int \sqrt {-f \left (x \right )}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}-\frac {1}{\sqrt {\left (a -y \right )^{2} \left (c -y \right ) \left (b -y \right )}}d \textit {\_a} = \int \sqrt {-f \left (x \right )}d x +c_{1}
\] Verified OK. |
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\[ {}{y^{\prime }}^{2} = f \left (x \right )^{2} \left (y-a \right ) \left (y-b \right ) \left (y-c \right )^{2} \] |
\[
\int _{}^{y}\frac {1}{\sqrt {\left (c -y \right )^{2} \left (b -y \right ) \left (a -y \right )}}d \textit {\_a} = \int \sqrt {f \left (x \right )^{2}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}-\frac {1}{\sqrt {\left (c -y \right )^{2} \left (b -y \right ) \left (a -y \right )}}d \textit {\_a} = \int \sqrt {f \left (x \right )^{2}}d x +c_{1}
\] Verified OK. |
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\[ {}{y^{\prime }}^{2}-x y^{\prime }-y = 0 \] |
\[ y = 0 \] Verified OK. \[ x = \frac {x}{3}+\frac {\sqrt {x^{2}+4 y}}{3}+\frac {2 c_{1}}{\sqrt {2 x +2 \sqrt {x^{2}+4 y}}} \] Verified OK. \[ x = \frac {x}{3}-\frac {\sqrt {x^{2}+4 y}}{3}+\frac {2 c_{1}}{\sqrt {2 x -2 \sqrt {x^{2}+4 y}}} \] Verified OK. |
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\[ {}{y^{\prime }}^{2}+x y^{\prime }+x -y = 0 \] |
\[ x = x -\sqrt {x^{2}+4 y-4 x}+2+c_{1} {\mathrm e}^{\frac {x}{2}-\frac {\sqrt {x^{2}+4 y-4 x}}{2}} \] Verified OK. \[ x = x +\sqrt {x^{2}+4 y-4 x}+2+c_{1} {\mathrm e}^{\frac {x}{2}+\frac {\sqrt {x^{2}+4 y-4 x}}{2}} \] Verified OK. |
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\[ {}{y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
\[ y = 0 \] Verified OK. \[ x = \frac {\left (-8 x^{2}-2 y\right ) \sqrt {x^{2}+y}+8 x^{3}+6 y x +3 c_{1}}{3 \left (x -\sqrt {x^{2}+y}\right )^{2}} \] Verified OK. \[ x = \frac {\left (8 x^{2}+2 y\right ) \sqrt {x^{2}+y}+8 x^{3}+6 y x +3 c_{1}}{3 \left (x +\sqrt {x^{2}+y}\right )^{2}} \] Verified OK. |
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\[ {}{y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
\[ y = 0 \] Verified OK. \[ x = \frac {\left (-8 x^{2}-2 y\right ) \sqrt {x^{2}+y}+8 x^{3}+6 y x +3 c_{1}}{3 \left (x -\sqrt {x^{2}+y}\right )^{2}} \] Verified OK. \[ x = \frac {\left (8 x^{2}+2 y\right ) \sqrt {x^{2}+y}+8 x^{3}+6 y x +3 c_{1}}{3 \left (x +\sqrt {x^{2}+y}\right )^{2}} \] Verified OK. |
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\[ {}{y^{\prime }}^{2}+2 \left (1-x \right ) y^{\prime }-2 x +2 y = 0 \] |
\[ x = -1+x +\sqrt {x^{2}-2 y+1}+c_{1} {\mathrm e}^{1-x -\sqrt {x^{2}-2 y+1}} \] Verified OK. \[ x = -1+x -\sqrt {x^{2}-2 y+1}+c_{1} {\mathrm e}^{1-x +\sqrt {x^{2}-2 y+1}} \] Verified OK. |
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\[ {}{y^{\prime }}^{2}+3 x y^{\prime }-y = 0 \] |
\[ y = 0 \] Verified OK. \[ x = \frac {3 x}{5}-\frac {\sqrt {9 x^{2}+4 y}}{5}+\frac {8 c_{1}}{\left (-6 x +2 \sqrt {9 x^{2}+4 y}\right )^{\frac {3}{2}}} \] Verified OK. \[ x = \frac {3 x}{5}+\frac {\sqrt {9 x^{2}+4 y}}{5}+\frac {8 c_{1}}{\left (-6 x -2 \sqrt {9 x^{2}+4 y}\right )^{\frac {3}{2}}} \] Verified OK. |
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\[ {}{y^{\prime }}^{2}-2 y y^{\prime }-2 x = 0 \] |
\[ y = -\frac {1}{2} i-i x \] Verified OK. \[ y = \frac {1}{2} i+i x \] Verified OK. \[ x = \frac {\left (y+\sqrt {y^{2}+2 x}\right ) \left (\operatorname {arcsinh}\left (y+\sqrt {y^{2}+2 x}\right )+2 c_{1} \right )}{2 \sqrt {2 y^{2}+2 x +2 y \sqrt {y^{2}+2 x}+1}} \] Verified OK. \[ x = -\frac {\left (-y+\sqrt {y^{2}+2 x}\right ) \left (-\operatorname {arcsinh}\left (-y+\sqrt {y^{2}+2 x}\right )+2 c_{1} \right )}{2 \sqrt {2 y^{2}+2 x -2 y \sqrt {y^{2}+2 x}+1}} \] Verified OK. |
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\[ {}{y^{\prime }}^{2}+a y y^{\prime }-x a = 0 \] |
\[ y = -\frac {x a -1}{a} \] Verified OK. \[ y = \frac {x a -1}{a} \] Verified OK. \[ x = \left (-y a +\sqrt {a \left (a y^{2}+4 x \right )}\right ) \left (\frac {\ln \left (2\right )-\ln \left (-y a +\sqrt {a \left (a y^{2}+4 x \right )}+\sqrt {2 y^{2} a^{2}-2 y a \sqrt {a \left (a y^{2}+4 x \right )}+4 x a -4}\right )}{\sqrt {2 y^{2} a^{2}-2 y a \sqrt {a \left (a y^{2}+4 x \right )}+4 x a -4}\, a}+\frac {2 c_{1}}{\sqrt {-2 y a +2 \sqrt {a \left (a y^{2}+4 x \right )}+4}\, \sqrt {-2 y a +2 \sqrt {a \left (a y^{2}+4 x \right )}-4}}\right ) \] Verified OK. \[ x = \left (y a +\sqrt {a \left (a y^{2}+4 x \right )}\right ) \left (-\frac {\ln \left (2\right )-\ln \left (-y a -\sqrt {a \left (a y^{2}+4 x \right )}+\sqrt {2 y^{2} a^{2}+2 y a \sqrt {a \left (a y^{2}+4 x \right )}+4 x a -4}\right )}{\sqrt {2 y^{2} a^{2}+2 y a \sqrt {a \left (a y^{2}+4 x \right )}+4 x a -4}\, a}-\frac {2 c_{1}}{\sqrt {-2 y a -2 \sqrt {a \left (a y^{2}+4 x \right )}+4}\, \sqrt {-2 y a -2 \sqrt {a \left (a y^{2}+4 x \right )}-4}}\right ) \] Verified OK. |
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\[ {}{y^{\prime }}^{2}-a y y^{\prime }-x a = 0 \] |
\[ y = \frac {-i x a -i}{a} \] Verified OK. \[ y = \frac {i x a +i}{a} \] Verified OK. \[ x = \frac {\left (y a +\sqrt {a \left (a y^{2}+4 x \right )}\right ) \left (c_{1} a +\operatorname {arcsinh}\left (\frac {y a}{2}+\frac {\sqrt {a \left (a y^{2}+4 x \right )}}{2}\right )\right )}{\sqrt {2 y^{2} a^{2}+2 y a \sqrt {a \left (a y^{2}+4 x \right )}+4 x a +4}\, a} \] Verified OK. \[ x = \frac {\left (y a -\sqrt {a \left (a y^{2}+4 x \right )}\right ) \left (c_{1} a +\operatorname {arcsinh}\left (\frac {y a}{2}-\frac {\sqrt {a \left (a y^{2}+4 x \right )}}{2}\right )\right )}{\sqrt {2 y^{2} a^{2}-2 y a \sqrt {a \left (a y^{2}+4 x \right )}+4 x a +4}\, a} \] Verified OK. |
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\[ {}2 {y^{\prime }}^{2}+x y^{\prime }-2 y = 0 \] |
\[ y = 0 \] Verified OK. \[ x = -\frac {\left (x -\sqrt {x^{2}+16 y}\right ) \left (-8 \ln \left (2\right )+4 \ln \left (-x +\sqrt {x^{2}+16 y}\right )+c_{1} \right )}{4} \] Verified OK. \[ x = -\frac {\left (x +\sqrt {x^{2}+16 y}\right ) \left (-8 \ln \left (2\right )+4 \ln \left (-x -\sqrt {x^{2}+16 y}\right )+c_{1} \right )}{4} \] Verified OK. |
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\[ {}3 {y^{\prime }}^{2}-2 x y^{\prime }+y = 0 \] |
\[ y = 0 \] Verified OK. \[ x = \frac {\left (8 x^{2}-6 y\right ) \sqrt {x^{2}-3 y}+8 x^{3}-18 y x +27 c_{1}}{3 \left (x +\sqrt {x^{2}-3 y}\right )^{2}} \] Verified OK. \[ x = \frac {\left (-8 x^{2}+6 y\right ) \sqrt {x^{2}-3 y}+8 x^{3}-18 y x +27 c_{1}}{3 \left (x -\sqrt {x^{2}-3 y}\right )^{2}} \] Verified OK. |
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\[ {}5 {y^{\prime }}^{2}+3 x y^{\prime }-y = 0 \] |
\[ y = 0 \] Verified OK. \[ x = \frac {3 x}{5}-\frac {\sqrt {9 x^{2}+20 y}}{5}+\frac {1000 c_{1}}{\left (-30 x +10 \sqrt {9 x^{2}+20 y}\right )^{\frac {3}{2}}} \] Verified OK. \[ x = \frac {3 x}{5}+\frac {\sqrt {9 x^{2}+20 y}}{5}+\frac {1000 c_{1}}{\left (-30 x -10 \sqrt {9 x^{2}+20 y}\right )^{\frac {3}{2}}} \] Verified OK. |
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\[ {}5 {y^{\prime }}^{2}+6 x y^{\prime }-2 y = 0 \] |
\[ y = 0 \] Verified OK. \[ x = \frac {3 x}{5}-\frac {\sqrt {9 x^{2}+10 y}}{5}+\frac {125 c_{1}}{\left (-15 x +5 \sqrt {9 x^{2}+10 y}\right )^{\frac {3}{2}}} \] Verified OK. \[ x = \frac {3 x}{5}+\frac {\sqrt {9 x^{2}+10 y}}{5}+\frac {125 c_{1}}{\left (-15 x -5 \sqrt {9 x^{2}+10 y}\right )^{\frac {3}{2}}} \] Verified OK. |
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\[ {}x {y^{\prime }}^{2}+x -2 y = 0 \] |
\[ y = x \] Verified OK. \[ x = \frac {c_{2} {\mathrm e}^{\frac {2 x}{\sqrt {-x \left (-2 y+x \right )}-x}} x^{2}}{\left (\sqrt {-x \left (-2 y+x \right )}-x \right )^{2}} \] Verified OK. \[ x = \frac {c_{2} {\mathrm e}^{-\frac {2 x}{\sqrt {-x \left (-2 y+x \right )}+x}} x^{2}}{\left (\sqrt {-x \left (-2 y+x \right )}+x \right )^{2}} \] Verified OK. |
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\[ {}x {y^{\prime }}^{2}+y^{\prime } = y \] |
\[ y = 0 \] Verified OK. \[ y = x +1 \] Verified OK. \[ x = -\frac {2 \left (-2 \ln \left (\frac {-1+\sqrt {1+4 y x}}{x}\right ) x +2 \ln \left (2\right ) x -2 c_{1} x +\sqrt {1+4 y x}-1\right ) x}{\left (-1+\sqrt {1+4 y x}-2 x \right )^{2}} \] Verified OK. \[ x = \frac {2 \left (2 \ln \left (\frac {-1-\sqrt {1+4 y x}}{x}\right ) x -2 \ln \left (2\right ) x +2 c_{1} x +\sqrt {1+4 y x}+1\right ) x}{\left (1+\sqrt {1+4 y x}+2 x \right )^{2}} \] Verified OK. |
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\[ {}x {y^{\prime }}^{2}+2 y^{\prime }-y = 0 \] |
\[ y = 0 \] Verified OK. \[ y = x +2 \] Verified OK. \[ x = \frac {\left (2 \ln \left (\frac {-1+\sqrt {y x +1}}{x}\right ) x +c_{1} x -2 \sqrt {y x +1}+2\right ) x}{\left (1-\sqrt {y x +1}+x \right )^{2}} \] Verified OK. \[ x = \frac {\left (2 \ln \left (\frac {-1-\sqrt {y x +1}}{x}\right ) x +c_{1} x +2 \sqrt {y x +1}+2\right ) x}{\left (1+\sqrt {y x +1}+x \right )^{2}} \] Verified OK. |
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\[ {}x {y^{\prime }}^{2}-2 y^{\prime }-y = 0 \] |
\[ y = 0 \] Verified OK. \[ y = x -2 \] Verified OK. \[ x = \frac {\left (-2 \ln \left (\frac {1+\sqrt {y x +1}}{x}\right ) x +c_{1} x +2 \sqrt {y x +1}+2\right ) x}{\left (-1-\sqrt {y x +1}+x \right )^{2}} \] Verified OK. \[ x = \frac {\left (-2 \ln \left (\frac {1-\sqrt {y x +1}}{x}\right ) x +c_{1} x -2 \sqrt {y x +1}+2\right ) x}{\left (-1+\sqrt {y x +1}+x \right )^{2}} \] Verified OK. |
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\[ {}x {y^{\prime }}^{2}+4 y^{\prime }-2 y = 0 \] |
\[ y = 0 \] Verified OK. \[ y = 2 x +4 \] Verified OK. \[ x = \frac {\left (8 \ln \left (\frac {-2+\sqrt {4+2 y x}}{x}\right ) x +c_{1} x -4 \sqrt {4+2 y x}+8\right ) x}{\left (2 x +2-\sqrt {4+2 y x}\right )^{2}} \] Verified OK. \[ x = \frac {\left (8 \ln \left (\frac {-2-\sqrt {4+2 y x}}{x}\right ) x +c_{1} x +4 \sqrt {4+2 y x}+8\right ) x}{\left (2 x +2+\sqrt {4+2 y x}\right )^{2}} \] Verified OK. |
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\[ {}x {y^{\prime }}^{2}+x y^{\prime }-y = 0 \] |
\[ y = 0 \] 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. |
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\[ {}x {y^{\prime }}^{2}+y y^{\prime }+a = 0 \] |
\[ x = -\frac {4 a \,x^{2}}{3 \left (-y+\sqrt {y^{2}-4 x a}\right )^{2}}+\frac {c_{1} \sqrt {2}}{\sqrt {\frac {-y+\sqrt {y^{2}-4 x a}}{x}}} \] Verified OK. \[ x = -\frac {4 a \,x^{2}}{3 \left (y+\sqrt {y^{2}-4 x a}\right )^{2}}+\frac {c_{1} \sqrt {2}}{\sqrt {\frac {-y-\sqrt {y^{2}-4 x a}}{x}}} \] Verified OK. |
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\[ {}x {y^{\prime }}^{2}-y y^{\prime }+x a = 0 \] |
\[ x = \frac {c_{2} \left (y+\sqrt {y^{2}-4 x^{2} a}\right ) {\mathrm e}^{-\frac {\left (y+\sqrt {y^{2}-4 x^{2} a}\right )^{2}}{8 a \,x^{2}}}}{2 x} \] Verified OK. \[ x = -\frac {c_{2} \left (-y+\sqrt {y^{2}-4 x^{2} a}\right ) {\mathrm e}^{-\frac {\left (-y+\sqrt {y^{2}-4 x^{2} a}\right )^{2}}{8 a \,x^{2}}}}{2 x} \] Verified OK. |
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\[ {}x {y^{\prime }}^{2}-y y^{\prime }+a y = 0 \] |
\[ y = 0 \] Verified OK. \[ x = \frac {c_{2} \left (-2 x a +y+\sqrt {y^{2}-4 x a y}\right ) {\mathrm e}^{-\frac {y+\sqrt {y^{2}-4 x a y}}{2 x a}}}{2 x} \] Verified OK. \[ x = -\frac {c_{2} \left (2 x a -y+\sqrt {y^{2}-4 x a y}\right ) {\mathrm e}^{\frac {-y+\sqrt {y^{2}-4 x a y}}{2 x a}}}{2 x} \] Verified OK. |
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\[ {}x {y^{\prime }}^{2}-\left (3 x -y\right ) y^{\prime }+y = 0 \] |
\[ y = 0 \] Verified OK. \[ y = x \] Verified OK. \[ x = \frac {c_{3} \left (5 x -y+\sqrt {y^{2}-10 y x +9 x^{2}}\right ) \sqrt {2}}{\left (3 x -y+\sqrt {y^{2}-10 y x +9 x^{2}}\right ) \sqrt {\frac {3 x -y+\sqrt {y^{2}-10 y x +9 x^{2}}}{x}}} \] Verified OK. \[ x = \frac {c_{3} \left (-5 x +y+\sqrt {y^{2}-10 y x +9 x^{2}}\right ) \sqrt {2}}{\left (-3 x +y+\sqrt {y^{2}-10 y x +9 x^{2}}\right ) \sqrt {\frac {3 x -y-\sqrt {y^{2}-10 y x +9 x^{2}}}{x}}} \] Verified OK. |
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\[ {}x {y^{\prime }}^{2}-2 y y^{\prime }+a = 0 \] |
\[ x = \frac {-3 c_{1} \left (x a -4 y^{2}\right ) \sqrt {y^{2}-x a}+\left (-9 x a y+12 y^{3}\right ) c_{1} +x^{3} a}{3 \left (y+\sqrt {y^{2}-x a}\right )^{2} x} \] Verified OK. \[ x = \frac {3 c_{1} \left (x a -4 y^{2}\right ) \sqrt {y^{2}-x a}+\left (-9 x a y+12 y^{3}\right ) c_{1} +x^{3} a}{3 \left (y-\sqrt {y^{2}-x a}\right )^{2} x} \] Verified OK. |
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\[ {}x {y^{\prime }}^{2}-a y y^{\prime }+b = 0 \] |
\[ x = \frac {2 \left (a y \left (a 2^{\frac {a -2}{a -1}}-2^{-\frac {1}{a -1}}\right ) \sqrt {y^{2} a^{2}-4 x b}+\left (a^{3} y^{2}+x b \right ) 2^{\frac {a -2}{a -1}}-a 2^{-\frac {1}{a -1}} \left (a y^{2}+4 x b \right )\right ) c_{1} {\left (\frac {y a +\sqrt {y^{2} a^{2}-4 x b}}{x}\right )}^{\frac {1}{a -1}}+4 b \,x^{2}}{\left (2 a -1\right ) \left (y a +\sqrt {y^{2} a^{2}-4 x b}\right )^{2}} \] Warning, solution could not be verified \[ x = \frac {4 c_{1} \left (a -\frac {1}{2}\right ) \left (y^{2} a^{2}-y \sqrt {y^{2} a^{2}-4 x b}\, a -2 x b \right ) {\left (-\frac {-y a +\sqrt {y^{2} a^{2}-4 x b}}{2 x}\right )}^{\frac {1}{a -1}}+4 b \,x^{2}}{\left (2 a -1\right ) \left (y a -\sqrt {y^{2} a^{2}-4 x b}\right )^{2}} \] Verified OK. |
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\[ {}2 x {y^{\prime }}^{2}+\left (2 x -y\right ) y^{\prime }+1-y = 0 \] |
\[ y = 1 \] Verified OK. \[ x = \frac {32 \left (\left (x +\frac {y}{2}+\frac {\sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}}{2}\right ) \ln \left (\frac {2 x +y+\sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}}{x}\right )+\left (\frac {c_{1}}{2}-\ln \left (2\right )\right ) \sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}+\left (-2 x -y\right ) \ln \left (2\right )+\left (c_{1} +2\right ) x +\frac {c_{1} y}{2}\right ) x^{2}}{\left (2 x +y+\sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}\right ) \left (2 x -y-\sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}\right )^{2}} \] Verified OK. \[ x = \frac {32 x^{2} \left (\left (x +\frac {y}{2}-\frac {\sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}}{2}\right ) \ln \left (\frac {2 x +y-\sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}}{x}\right )+\left (-\frac {c_{1}}{2}+\ln \left (2\right )\right ) \sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}+\left (-2 x -y\right ) \ln \left (2\right )+\left (c_{1} +2\right ) x +\frac {c_{1} y}{2}\right )}{\left (2 x +y-\sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}\right ) \left (2 x -y+\sqrt {4 x^{2}+\left (4 y-8\right ) x +y^{2}}\right )^{2}} \] Verified OK. |
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\[ {}\left (5+3 x \right ) {y^{\prime }}^{2}-\left (3+3 y\right ) y^{\prime }+y = 0 \] |
\[ y = 0 \] Verified OK. \[ x = \frac {-18 \left (c_{1} +3 \,\operatorname {expIntegral}_{1}\left (\frac {-9-9 y-3 \sqrt {9+9 y^{2}+\left (-12 x -2\right ) y}}{10+6 x}\right )\right ) \left (x -\frac {3 y}{2}-\frac {\sqrt {9+9 y^{2}+\left (-12 x -2\right ) y}}{2}+\frac {1}{6}\right ) {\mathrm e}^{\frac {-9-9 y-3 \sqrt {9+9 y^{2}+\left (-12 x -2\right ) y}}{10+6 x}}+24 x +40}{30+18 x} \] Verified OK. \[ x = \frac {-18 \left (c_{1} +3 \,\operatorname {expIntegral}_{1}\left (\frac {-9 y-9+3 \sqrt {9+9 y^{2}+\left (-12 x -2\right ) y}}{10+6 x}\right )\right ) \left (x -\frac {3 y}{2}+\frac {\sqrt {9+9 y^{2}+\left (-12 x -2\right ) y}}{2}+\frac {1}{6}\right ) {\mathrm e}^{\frac {-9 y-9+3 \sqrt {9+9 y^{2}+\left (-12 x -2\right ) y}}{10+6 x}}+24 x +40}{30+18 x} \] Verified OK. |
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\[ {}4 x {y^{\prime }}^{2}-3 y y^{\prime }+3 = 0 \] |
\[ x = \frac {64 x^{2} \left (64 c_{1} x^{2}+9 y \sqrt {9 y^{2}-48 x}+27 y^{2}-72 x \right )}{\left (3 y+\sqrt {9 y^{2}-48 x}\right )^{4}} \] Verified OK. \[ x = -\frac {64 x^{2} \left (-64 c_{1} x^{2}+9 y \sqrt {9 y^{2}-48 x}-27 y^{2}+72 x \right )}{\left (-3 y+\sqrt {9 y^{2}-48 x}\right )^{4}} \] Verified OK. |
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\[ {}4 x {y^{\prime }}^{2}+4 y y^{\prime } = 1 \] |
\[ x = \frac {x^{2}}{3 \left (-y+\sqrt {y^{2}+x}\right )^{2}}+\frac {c_{1} \sqrt {2}}{\sqrt {\frac {-y+\sqrt {y^{2}+x}}{x}}} \] Verified OK. \[ x = \frac {x^{2}}{3 \left (y+\sqrt {y^{2}+x}\right )^{2}}+\frac {c_{1} \sqrt {2}}{\sqrt {\frac {-y-\sqrt {y^{2}+x}}{x}}} \] Verified OK. |
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\[ {}x^{2} {y^{\prime }}^{2}+x^{2}-y^{2} = 0 \] |
\[ x = c_{2} {\mathrm e}^{\frac {\operatorname {arcsinh}\left (\frac {\sqrt {y^{2}-x^{2}}}{x}\right ) x^{2}-\sqrt {\frac {y^{2}}{x^{2}}}\, \sqrt {y^{2}-x^{2}}\, x +x^{2}-y^{2}}{2 x^{2}}} \] Verified OK. \[ x = c_{2} {\mathrm e}^{\frac {\sqrt {\frac {y^{2}}{x^{2}}}\, \sqrt {y^{2}-x^{2}}\, x -\operatorname {arcsinh}\left (\frac {\sqrt {y^{2}-x^{2}}}{x}\right ) x^{2}+x^{2}-y^{2}}{2 x^{2}}} \] Verified OK. \[ x = c_{4} {\mathrm e}^{\frac {\sqrt {\frac {y^{2}}{x^{2}}}\, \sqrt {y^{2}-x^{2}}\, x -\operatorname {arcsinh}\left (\frac {\sqrt {y^{2}-x^{2}}}{x}\right ) x^{2}+x^{2}-y^{2}}{2 x^{2}}} \] Verified OK. \[ x = c_{4} {\mathrm e}^{\frac {\operatorname {arcsinh}\left (\frac {\sqrt {y^{2}-x^{2}}}{x}\right ) x^{2}-\sqrt {\frac {y^{2}}{x^{2}}}\, \sqrt {y^{2}-x^{2}}\, x +x^{2}-y^{2}}{2 x^{2}}} \] Verified OK. |
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\[ {}a \,x^{2} {y^{\prime }}^{2}-2 a x y y^{\prime }+a \left (-a +1\right ) x^{2}+y^{2} = 0 \] |
\[ y = -\left (-a \right )^{\frac {3}{2}} x +\sqrt {-a^{3}+2 a^{2}-a}\, x \] Verified OK. \[ y = \left (-a \right )^{\frac {3}{2}} x +\sqrt {-a^{3}+2 a^{2}-a}\, x \] Verified OK. \[ x = c_{2} {\left (\frac {\sqrt {\frac {\left (a^{2} x^{2}+2 y^{2} a +2 \sqrt {a \left (a -1\right ) \left (a \,x^{2}+y^{2}\right )}\, y-y^{2}\right ) \left (a -1\right )}{x^{2}}}\, a x +\sqrt {\left (a -1\right ) a}\, a y+\sqrt {a \left (a -1\right ) \left (a \,x^{2}+y^{2}\right )}\, \sqrt {\left (a -1\right ) a}}{a x}\right )}^{-\frac {a}{\sqrt {\left (a -1\right ) a}}} \] Verified OK. \[ x = c_{2} {\left (\frac {\sqrt {\frac {\left (a^{2} x^{2}+2 y^{2} a -2 \sqrt {a \left (a -1\right ) \left (a \,x^{2}+y^{2}\right )}\, y-y^{2}\right ) \left (a -1\right )}{x^{2}}}\, a x +\sqrt {\left (a -1\right ) a}\, \left (a y-\sqrt {a \left (a -1\right ) \left (a \,x^{2}+y^{2}\right )}\right )}{a x}\right )}^{-\frac {a}{\sqrt {\left (a -1\right ) a}}} \] Verified OK. \[ y = -\left (-a \right )^{\frac {3}{2}} x -\sqrt {-a^{3}+2 a^{2}-a}\, x \] Verified OK. \[ y = \left (-a \right )^{\frac {3}{2}} x -\sqrt {-a^{3}+2 a^{2}-a}\, x \] Verified OK. \[ x = c_{4} {\left (\frac {\sqrt {\frac {\left (a^{2} x^{2}+2 y^{2} a +2 \sqrt {a \left (a -1\right ) \left (a \,x^{2}+y^{2}\right )}\, y-y^{2}\right ) \left (a -1\right )}{x^{2}}}\, a x +\sqrt {\left (a -1\right ) a}\, a y+\sqrt {a \left (a -1\right ) \left (a \,x^{2}+y^{2}\right )}\, \sqrt {\left (a -1\right ) a}}{a x}\right )}^{\frac {a}{\sqrt {\left (a -1\right ) a}}} \] Verified OK. \[ x = c_{4} {\left (\frac {\sqrt {\frac {\left (a^{2} x^{2}+2 y^{2} a -2 \sqrt {a \left (a -1\right ) \left (a \,x^{2}+y^{2}\right )}\, y-y^{2}\right ) \left (a -1\right )}{x^{2}}}\, a x +\sqrt {\left (a -1\right ) a}\, \left (a y-\sqrt {a \left (a -1\right ) \left (a \,x^{2}+y^{2}\right )}\right )}{a x}\right )}^{\frac {a}{\sqrt {\left (a -1\right ) a}}} \] Verified OK. |
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\[ {}\left (-a^{2}+1\right ) x^{2} {y^{\prime }}^{2}-2 x y y^{\prime }-x^{2} a^{2}+y^{2} = 0 \] |
\[ y = -i x \] Verified OK. \[ y = i x \] Verified OK. \[ x = \frac {c_{2} a^{-\frac {1}{a}} {\left (\frac {\left (a^{2}-1\right ) x \sqrt {\frac {-2 y \sqrt {-a^{2} \left (x^{2} \left (a^{2}-1\right )-y^{2}\right )}+\left (-x^{2}+y^{2}\right ) a^{2}+x^{2}+y^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}-y+\sqrt {-a^{2} \left (x^{2} \left (a^{2}-1\right )-y^{2}\right )}}{\left (a^{2}-1\right ) x}\right )}^{-\frac {1}{a}}}{\sqrt {\frac {-2 y \sqrt {-a^{2} \left (x^{2} \left (a^{2}-1\right )-y^{2}\right )}+\left (-x^{2}+y^{2}\right ) a^{2}+x^{2}+y^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}} \] Warning, solution could not be verified \[ x = \frac {c_{2} a^{-\frac {1}{a}} {\left (\frac {\left (a^{2}-1\right ) x \sqrt {\frac {2 y \sqrt {-a^{2} \left (x^{2} \left (a^{2}-1\right )-y^{2}\right )}+\left (-x^{2}+y^{2}\right ) a^{2}+x^{2}+y^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}-y-\sqrt {-a^{2} \left (x^{2} \left (a^{2}-1\right )-y^{2}\right )}}{\left (a^{2}-1\right ) x}\right )}^{-\frac {1}{a}}}{\sqrt {\frac {2 y \sqrt {-a^{2} \left (x^{2} \left (a^{2}-1\right )-y^{2}\right )}+\left (-x^{2}+y^{2}\right ) a^{2}+x^{2}+y^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}} \] Warning, solution could not be verified \[ y = -i x \] Verified OK. \[ y = i x \] Verified OK. \[ x = \frac {c_{4} {\left (\frac {\left (a^{2}-1\right ) x \sqrt {-\frac {a^{2} \left (\left (x^{2}-y^{2}\right ) a^{2}+2 \sqrt {-a^{2} x^{2}+x^{2}+y^{2}}\, a y-x^{2}-y^{2}\right )}{\left (a^{2}-1\right )^{2} x^{2}}}+a \left (a \sqrt {-a^{2} x^{2}+x^{2}+y^{2}}-y\right )}{\left (a^{2}-1\right ) x}\right )}^{\frac {1}{a}}}{\sqrt {\frac {\left (-x^{2}+y^{2}\right ) a^{2}-2 \sqrt {-a^{2} x^{2}+x^{2}+y^{2}}\, a y+x^{2}+y^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}} \] Warning, solution could not be verified \[ x = \frac {c_{4} {\left (\frac {\left (a^{2}-1\right ) x \sqrt {\frac {a^{2} \left (-a^{2} x^{2}+a^{2} y^{2}+2 \sqrt {-a^{2} x^{2}+x^{2}+y^{2}}\, a y+x^{2}+y^{2}\right )}{\left (a^{2}-1\right )^{2} x^{2}}}-\sqrt {-a^{2} x^{2}+x^{2}+y^{2}}\, a^{2}-a y}{\left (a^{2}-1\right ) x}\right )}^{\frac {1}{a}}}{\sqrt {\frac {\left (-x^{2}+y^{2}\right ) a^{2}+2 \sqrt {-a^{2} x^{2}+x^{2}+y^{2}}\, a y+x^{2}+y^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}} \] Warning, solution could not be verified |
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\[ {}y {y^{\prime }}^{2} = x \,a^{2} \] |
\[ y = a^{\frac {2}{3}} x \] Verified OK. \[ y = \frac {4 a^{\frac {2}{3}} x}{-2+2 i \sqrt {3}} \] Verified OK. \[ y = \frac {4 a^{\frac {2}{3}} x}{-2-2 i \sqrt {3}} \] Verified OK. \[ x = \frac {c_{3} x \,a^{2}}{\left (\frac {a^{2} \left (a x \sqrt {y x}-y^{2}\right )}{y^{2}}\right )^{\frac {2}{3}} y} \] Verified OK. \[ x = \frac {c_{3} x \,a^{2}}{\left (-\frac {a^{2} \left (a x \sqrt {y x}+y^{2}\right )}{y^{2}}\right )^{\frac {2}{3}} y} \] Verified OK. |
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\[ {}y {y^{\prime }}^{2}+2 a x y^{\prime }-a y = 0 \] |
\[ y = 0 \] Verified OK. \[ y = -\frac {a x}{\sqrt {-a}} \] Verified OK. \[ y = \frac {a x}{\sqrt {-a}} \] Verified OK. \[ x = -\frac {2 c_{3} a x}{-a x +\sqrt {a \left (a \,x^{2}+y^{2}\right )}} \] Verified OK. \[ x = \frac {2 c_{3} a x}{a x +\sqrt {a \left (a \,x^{2}+y^{2}\right )}} \] Verified OK. |
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\[ {}y {y^{\prime }}^{2}-4 a^{2} x y^{\prime }+a^{2} y = 0 \] |
\[ y = 0 \] Verified OK. \[ y = \sqrt {3}\, a x \] Verified OK. \[ y = -\sqrt {3}\, a x \] Verified OK. \[ x = \frac {2 c_{3} 2^{\frac {1}{3}} a^{2} x}{\left (\frac {2 x^{2} a^{4}+a^{3} x \sqrt {4 a^{2} x^{2}-y^{2}}-a^{2} y^{2}}{y^{2}}\right )^{\frac {1}{3}} y {\left (\frac {\left (2 a x +\sqrt {4 a^{2} x^{2}-y^{2}}\right ) a}{y}\right )}^{\frac {1}{3}}} \] Verified OK. \[ x = \frac {2 c_{3} 2^{\frac {1}{3}} a^{2} x}{\left (\frac {2 x^{2} a^{4}-a^{3} x \sqrt {4 a^{2} x^{2}-y^{2}}-a^{2} y^{2}}{y^{2}}\right )^{\frac {1}{3}} y {\left (\frac {\left (2 a x -\sqrt {4 a^{2} x^{2}-y^{2}}\right ) a}{y}\right )}^{\frac {1}{3}}} \] Verified OK. |
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\[ {}y {y^{\prime }}^{2}-\left (-2 b x +a \right ) y^{\prime }-b y = 0 \] |
\[ y = 0 \] Verified OK. \[ y = \frac {-2 b x +a}{2 \sqrt {-b}} \] Verified OK. \[ y = \frac {2 b x -a}{2 \sqrt {-b}} \] Verified OK. \[ x = \frac {2 c_{3} \left (-2 b x +a \right ) \sqrt {4 b^{2} x^{2}+4 y^{2} b -4 a b x +a^{2}}+2 \left (-2 b x +a \right )^{2} c_{3} +2 a y^{2}}{\left (-2 b x +a +\sqrt {4 b^{2} x^{2}+4 y^{2} b -4 a b x +a^{2}}\right )^{2}} \] Verified OK. \[ x = \frac {-2 c_{3} \left (-2 b x +a \right ) \sqrt {4 b^{2} x^{2}+4 y^{2} b -4 a b x +a^{2}}+2 \left (-2 b x +a \right )^{2} c_{3} +2 a y^{2}}{\left (-2 b x +a -\sqrt {4 b^{2} x^{2}+4 y^{2} b -4 a b x +a^{2}}\right )^{2}} \] Verified OK. |
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\[ {}y {y^{\prime }}^{2}-\left (x +y\right ) y^{\prime }+y = 0 \] |
\[ y = 0 \] Verified OK. \[ y = x \] Verified OK. \[ x = \frac {2 c_{2} x \,{\mathrm e}^{\frac {2 y}{x +y+\sqrt {\left (x +3 y\right ) \left (x -y\right )}}}}{x +y+\sqrt {\left (x +3 y\right ) \left (x -y\right )}} \] Verified OK. \[ x = \frac {2 c_{2} x \,{\mathrm e}^{\frac {2 y}{x +y-\sqrt {\left (x +3 y\right ) \left (x -y\right )}}}}{x +y-\sqrt {\left (x +3 y\right ) \left (x -y\right )}} \] Verified OK. |
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\[ {}\left (x +y\right ) {y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
\[ y = 0 \] Verified OK. \[ y = -\frac {i \sqrt {3}\, x}{2}-\frac {x}{2} \] Verified OK. \[ y = \frac {i \sqrt {3}\, x}{2}-\frac {x}{2} \] Verified OK. \[ x = \frac {x \left (-2 \sqrt {y^{2}+y x +x^{2}}+x -y\right ) c_{3}}{\left (x -\sqrt {y^{2}+y x +x^{2}}\right )^{2}} \] Verified OK. \[ x = \frac {x \left (2 \sqrt {y^{2}+y x +x^{2}}+x -y\right ) c_{3}}{\left (x +\sqrt {y^{2}+y x +x^{2}}\right )^{2}} \] Verified OK. |
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\[ {}\left (2 x -y\right ) {y^{\prime }}^{2}-2 \left (1-x \right ) y^{\prime }+2-y = 0 \] |
\[ y = 2 \] Verified OK. \[ y = \frac {3 \sqrt {2}\, x -\sqrt {2}-4 x}{\sqrt {2}-2} \] Verified OK. \[ y = \frac {4 x +3 \sqrt {2}\, x -\sqrt {2}}{2+\sqrt {2}} \] Verified OK. \[ x = \frac {-2 c_{3} \left (-1+x \right ) \sqrt {x^{2}+\left (2 y-6\right ) x -y^{2}+2 y+1}+6 \left (x -\frac {y}{3}-\frac {1}{3}\right ) \left (-1+x \right ) c_{3} -4 \left (x -\frac {y}{2}\right )^{2}}{\left (-1+x -\sqrt {x^{2}+\left (2 y-6\right ) x -y^{2}+2 y+1}\right )^{2}} \] Verified OK. \[ x = \frac {2 c_{3} \left (-1+x \right ) \sqrt {x^{2}+\left (2 y-6\right ) x -y^{2}+2 y+1}+6 \left (x -\frac {y}{3}-\frac {1}{3}\right ) \left (-1+x \right ) c_{3} -4 \left (x -\frac {y}{2}\right )^{2}}{\left (-1+x +\sqrt {x^{2}+\left (2 y-6\right ) x -y^{2}+2 y+1}\right )^{2}} \] Verified OK. |
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\[ {}2 y {y^{\prime }}^{2}+\left (5-4 x \right ) y^{\prime }+2 y = 0 \] |
\[ y = -x +\frac {5}{4} \] Verified OK. \[ y = 0 \] Verified OK. \[ y = x -\frac {5}{4} \] Verified OK. \[ x = \frac {\left (8 x -10\right ) c_{3} \sqrt {-16 y^{2}+16 x^{2}-40 x +25}+32 \left (x -\frac {5}{4}\right )^{2} c_{3} -20 y^{2}}{\left (-5+4 x +\sqrt {-16 y^{2}+16 x^{2}-40 x +25}\right )^{2}} \] Verified OK. \[ x = \frac {\left (-8 x +10\right ) c_{3} \sqrt {-16 y^{2}+16 x^{2}-40 x +25}+32 \left (x -\frac {5}{4}\right )^{2} c_{3} -20 y^{2}}{\left (-5+4 x -\sqrt {-16 y^{2}+16 x^{2}-40 x +25}\right )^{2}} \] Verified OK. |
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\[ {}x \left (x -2 y\right ) {y^{\prime }}^{2}+6 x y y^{\prime }-2 y x +y^{2} = 0 \] |
\[ y = -x \sqrt {3}+2 x \] Verified OK. \[ y = x \sqrt {3}+2 x \] Verified OK. \[ x = \frac {c_{2} {\left (\frac {x \left (x -2 y\right ) \sqrt {\frac {-4 \sqrt {2}\, \left (x -\frac {y}{2}\right ) \sqrt {y x \left (x +y\right )^{2}}+x^{3}+10 y x^{2}-7 y^{2} x +2 y^{3}}{x \left (x -2 y\right )^{2}}}-2 x^{2}+y x +\sqrt {2}\, \sqrt {y x \left (x +y\right )^{2}}}{x \left (x -2 y\right )}\right )}^{\frac {\sqrt {\frac {\left (-4 \sqrt {2}\, \left (x -\frac {y}{2}\right ) \sqrt {y x \left (x +y\right )^{2}}+x^{3}+10 y x^{2}-7 y^{2} x +2 y^{3}\right ) \left (-\sqrt {2}\, \sqrt {y x \left (x +y\right )^{2}}+x \left (x +y\right )\right )^{2}}{x^{3} \left (x -2 y\right )^{4}}}\, x \left (x -2 y\right )}{\sqrt {\frac {-4 \sqrt {2}\, \left (x -\frac {y}{2}\right ) \sqrt {y x \left (x +y\right )^{2}}+x^{3}+10 y x^{2}-7 y^{2} x +2 y^{3}}{x \left (x -2 y\right )^{2}}}\, \left (-\sqrt {2}\, \sqrt {y x \left (x +y\right )^{2}}+x \left (x +y\right )\right )}}}{\sqrt {\frac {-4 \sqrt {2}\, \left (x -\frac {y}{2}\right ) \sqrt {y x \left (x +y\right )^{2}}+x^{3}+10 y x^{2}-7 y^{2} x +2 y^{3}}{x \left (x -2 y\right )^{2}}}} \] Warning, solution could not be verified \[ x = \frac {c_{2} {\left (\frac {x \left (x -2 y\right ) \sqrt {\frac {4 \sqrt {2}\, \left (x -\frac {y}{2}\right ) \sqrt {y x \left (x +y\right )^{2}}+x^{3}+10 y x^{2}-7 y^{2} x +2 y^{3}}{x \left (x -2 y\right )^{2}}}-2 x^{2}+y x -\sqrt {2}\, \sqrt {y x \left (x +y\right )^{2}}}{x \left (x -2 y\right )}\right )}^{\frac {\sqrt {\frac {\left (4 \sqrt {2}\, \left (x -\frac {y}{2}\right ) \sqrt {y x \left (x +y\right )^{2}}+x^{3}+10 y x^{2}-7 y^{2} x +2 y^{3}\right ) \left (\sqrt {2}\, \sqrt {y x \left (x +y\right )^{2}}+x \left (x +y\right )\right )^{2}}{x^{3} \left (x -2 y\right )^{4}}}\, x \left (x -2 y\right )}{\left (\sqrt {2}\, \sqrt {y x \left (x +y\right )^{2}}+x^{2}+y x \right ) \sqrt {\frac {4 \sqrt {2}\, \left (x -\frac {y}{2}\right ) \sqrt {y x \left (x +y\right )^{2}}+x^{3}+10 y x^{2}-7 y^{2} x +2 y^{3}}{x \left (x -2 y\right )^{2}}}}}}{\sqrt {\frac {4 \sqrt {2}\, \left (x -\frac {y}{2}\right ) \sqrt {y x \left (x +y\right )^{2}}+x^{3}+10 y x^{2}-7 y^{2} x +2 y^{3}}{x \left (x -2 y\right )^{2}}}} \] Warning, solution could not be verified \[ y = 0 \] Verified OK. \[ y = -x \sqrt {3}+2 x \] Verified OK. \[ y = x \sqrt {3}+2 x \] Verified OK. \[ x = \frac {c_{4} {\left (\frac {x \left (x -2 y\right ) \sqrt {\frac {-4 \sqrt {2}\, \left (x -\frac {y}{2}\right ) \sqrt {y x \left (x +y\right )^{2}}+x^{3}+10 y x^{2}-7 y^{2} x +2 y^{3}}{x \left (x -2 y\right )^{2}}}-2 x^{2}+y x +\sqrt {2}\, \sqrt {y x \left (x +y\right )^{2}}}{x \left (x -2 y\right )}\right )}^{-\frac {\sqrt {\frac {\left (-4 \sqrt {2}\, \left (x -\frac {y}{2}\right ) \sqrt {y x \left (x +y\right )^{2}}+x^{3}+10 y x^{2}-7 y^{2} x +2 y^{3}\right ) \left (-\sqrt {2}\, \sqrt {y x \left (x +y\right )^{2}}+x \left (x +y\right )\right )^{2}}{x^{3} \left (x -2 y\right )^{4}}}\, x \left (x -2 y\right )}{\sqrt {\frac {-4 \sqrt {2}\, \left (x -\frac {y}{2}\right ) \sqrt {y x \left (x +y\right )^{2}}+x^{3}+10 y x^{2}-7 y^{2} x +2 y^{3}}{x \left (x -2 y\right )^{2}}}\, \left (-\sqrt {2}\, \sqrt {y x \left (x +y\right )^{2}}+x \left (x +y\right )\right )}}}{\sqrt {\frac {-4 \sqrt {2}\, \left (x -\frac {y}{2}\right ) \sqrt {y x \left (x +y\right )^{2}}+x^{3}+10 y x^{2}-7 y^{2} x +2 y^{3}}{x \left (x -2 y\right )^{2}}}} \] Warning, solution could not be verified \[ x = \frac {c_{4} {\left (\frac {x \left (x -2 y\right ) \sqrt {\frac {4 \sqrt {2}\, \left (x -\frac {y}{2}\right ) \sqrt {y x \left (x +y\right )^{2}}+x^{3}+10 y x^{2}-7 y^{2} x +2 y^{3}}{x \left (x -2 y\right )^{2}}}-2 x^{2}+y x -\sqrt {2}\, \sqrt {y x \left (x +y\right )^{2}}}{x \left (x -2 y\right )}\right )}^{-\frac {\sqrt {\frac {\left (4 \sqrt {2}\, \left (x -\frac {y}{2}\right ) \sqrt {y x \left (x +y\right )^{2}}+x^{3}+10 y x^{2}-7 y^{2} x +2 y^{3}\right ) \left (\sqrt {2}\, \sqrt {y x \left (x +y\right )^{2}}+x \left (x +y\right )\right )^{2}}{x^{3} \left (x -2 y\right )^{4}}}\, x \left (x -2 y\right )}{\left (\sqrt {2}\, \sqrt {y x \left (x +y\right )^{2}}+x^{2}+y x \right ) \sqrt {\frac {4 \sqrt {2}\, \left (x -\frac {y}{2}\right ) \sqrt {y x \left (x +y\right )^{2}}+x^{3}+10 y x^{2}-7 y^{2} x +2 y^{3}}{x \left (x -2 y\right )^{2}}}}}}{\sqrt {\frac {4 \sqrt {2}\, \left (x -\frac {y}{2}\right ) \sqrt {y x \left (x +y\right )^{2}}+x^{3}+10 y x^{2}-7 y^{2} x +2 y^{3}}{x \left (x -2 y\right )^{2}}}} \] Warning, solution could not be verified |
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\[ {}\left (x +y\right )^{2} {y^{\prime }}^{2} = y^{2} \] |
\[ \ln \left (\frac {y}{x}\right )-\frac {x}{y}+\ln \left (x \right )-c_{5} = 0 \] Verified OK. |
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