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ODE |
Mathematica |
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\[
{}y^{\prime } = f \left (x \right ) y^{2}+\lambda y+a^{2} {\mathrm e}^{2 \lambda x} f \left (x \right )
\] |
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\[
{}y^{\prime } = f \left (x \right ) y^{2}-f \left (x \right ) \left ({\mathrm e}^{\lambda x} a +b \right ) y+a \lambda \,{\mathrm e}^{\lambda x}
\] |
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\[
{}y^{\prime } = {\mathrm e}^{\lambda x} f \left (x \right ) y^{2}+\left (f \left (x \right ) a -\lambda \right ) y+b \,{\mathrm e}^{-\lambda x} f \left (x \right )
\] |
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\[
{}y^{\prime } = f \left (x \right ) y^{2}+g \left (x \right ) y+a \lambda \,{\mathrm e}^{\lambda x}-a \,{\mathrm e}^{\lambda x} g \left (x \right )-a^{2} {\mathrm e}^{2 \lambda x} f \left (x \right )
\] |
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\[
{}y^{\prime } = f \left (x \right ) y^{2}-a \,{\mathrm e}^{\lambda x} g \left (x \right ) y+a \lambda \,{\mathrm e}^{\lambda x}+a^{2} {\mathrm e}^{2 \lambda x} \left (g \left (x \right )-f \left (x \right )\right )
\] |
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\[
{}y^{\prime } = f \left (x \right ) y^{2}+2 a \lambda x \,{\mathrm e}^{\lambda \,x^{2}}-a^{2} f \left (x \right ) {\mathrm e}^{2 \lambda \,x^{2}}
\] |
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\[
{}y^{\prime } = f \left (x \right ) y^{2}+\lambda x y+a f \left (x \right ) {\mathrm e}^{\lambda x}
\] |
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\[
{}y^{\prime } = f \left (x \right ) y^{2}-a \tanh \left (\lambda x \right )^{2} \left (f \left (x \right ) a +\lambda \right )+a \lambda
\] |
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\[
{}y^{\prime } = f \left (x \right ) y^{2}-a \coth \left (\lambda x \right )^{2} \left (f \left (x \right ) a +\lambda \right )+a \lambda
\] |
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\[
{}y^{\prime } = f \left (x \right ) y^{2}-a^{2} f \left (x \right )+a \lambda \sinh \left (\lambda x \right )-a^{2} f \left (x \right ) \sinh \left (\lambda x \right )^{2}
\] |
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\[
{}x y^{\prime } = f \left (x \right ) y^{2}+a -a^{2} f \left (x \right ) \ln \left (x \right )^{2}
\] |
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\[
{}x y^{\prime } = f \left (x \right ) \left (y+a \ln \left (x \right )\right )^{2}-a
\] |
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\[
{}y^{\prime } = f \left (x \right ) y^{2}-a x \ln \left (x \right ) f \left (x \right ) y+a \ln \left (x \right )+a
\] |
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\[
{}y^{\prime } = -a \ln \left (x \right ) y^{2}+a f \left (x \right ) \left (x \ln \left (x \right )-x \right ) y-f \left (x \right )
\] |
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\[
{}y^{\prime } = \lambda \sin \left (\lambda x \right ) y^{2}+f \left (x \right ) \cos \left (\lambda x \right ) y-f \left (x \right )
\] |
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\[
{}y^{\prime } = f \left (x \right ) y^{2}-a^{2} f \left (x \right )+a \lambda \sin \left (\lambda x \right )+a^{2} f \left (x \right ) \sin \left (\lambda x \right )^{2}
\] |
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\[
{}y^{\prime } = f \left (x \right ) y^{2}-a^{2} f \left (x \right )+a \lambda \cos \left (\lambda x \right )+a^{2} f \left (x \right ) \cos \left (\lambda x \right )^{2}
\] |
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\[
{}y^{\prime } = f \left (x \right ) y^{2}-a \tan \left (\lambda x \right )^{2} \left (f \left (x \right ) a -\lambda \right )+a \lambda
\] |
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\[
{}y^{\prime } = f \left (x \right ) y^{2}-a \cot \left (\lambda x \right )^{2} \left (f \left (x \right ) a -\lambda \right )+a \lambda
\] |
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\[
{}y^{\prime } = y^{2}-f \left (x \right )^{2}+f^{\prime }\left (x \right )
\] |
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\[
{}y^{\prime } = f \left (x \right ) y^{2}-f \left (x \right ) g \left (x \right ) y+g^{\prime }\left (x \right )
\] |
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\[
{}y^{\prime } = -f^{\prime }\left (x \right ) y^{2}+f \left (x \right ) g \left (x \right ) y-g \left (x \right )
\] |
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\[
{}y^{\prime } = g \left (x \right ) \left (y-f \left (x \right )\right )^{2}+f^{\prime }\left (x \right )
\] |
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\[
{}y^{\prime } = \frac {f^{\prime }\left (x \right ) y^{2}}{g \left (x \right )}-\frac {g^{\prime }\left (x \right )}{f \left (x \right )}
\] |
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\[
{}f \left (x \right )^{2} y^{\prime }-f^{\prime }\left (x \right ) y^{2}+g \left (x \right ) \left (y-f \left (x \right )\right ) = 0
\] |
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\[
{}y^{\prime } = f^{\prime }\left (x \right ) y^{2}+a \,{\mathrm e}^{\lambda x} f \left (x \right ) y+{\mathrm e}^{\lambda x} a
\] |
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\[
{}y^{\prime } = f \left (x \right ) y^{2}+g^{\prime }\left (x \right ) y+a f \left (x \right ) {\mathrm e}^{2 g \left (x \right )}
\] |
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\[
{}y^{\prime } = y^{2}+a^{2} f \left (a x +b \right )
\] |
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\[
{}y^{\prime } = y^{2}+\frac {f \left (\frac {1}{x}\right )}{x^{4}}
\] |
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\[
{}y^{\prime } = y^{2}+\frac {f \left (\frac {a x +b}{c x +d}\right )}{\left (c x +d \right )^{4}}
\] |
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\[
{}x^{2} y^{\prime } = x^{4} f \left (x \right ) y^{2}+1
\] |
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\[
{}x^{2} y^{\prime } = y^{2} x^{4}+x^{2 n} f \left (a \,x^{n}+b \right )-\frac {n^{2}}{4}+\frac {1}{4}
\] |
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\[
{}y^{\prime } = f \left (x \right ) y^{2}+g \left (x \right ) y+h \left (x \right )
\] |
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\[
{}y^{\prime } = y^{2}+{\mathrm e}^{2 \lambda x} f \left ({\mathrm e}^{\lambda x}\right )-\frac {\lambda ^{2}}{4}
\] |
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\[
{}y^{\prime } = y^{2}-\frac {\lambda ^{2}}{4}+\frac {{\mathrm e}^{2 \lambda x} f \left (\frac {{\mathrm e}^{\lambda x} a +b}{c \,{\mathrm e}^{\lambda x}+d}\right )}{\left (c \,{\mathrm e}^{\lambda x}+d \right )^{4}}
\] |
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\[
{}y^{\prime } = y^{2}-\lambda ^{2}+\frac {f \left (\coth \left (\lambda x \right )\right )}{\sinh \left (\lambda x \right )^{4}}
\] |
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\[
{}y^{\prime } = y^{2}-\lambda ^{2}+\frac {f \left (\tanh \left (\lambda x \right )\right )}{\cosh \left (\lambda x \right )^{4}}
\] |
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\[
{}x^{2} y^{\prime } = x^{2} y^{2}+f \left (a \ln \left (x \right )+b \right )+\frac {1}{4}
\] |
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\[
{}y^{\prime } = y^{2}+\lambda ^{2}+\frac {f \left (\cot \left (\lambda x \right )\right )}{\sin \left (\lambda x \right )^{4}}
\] |
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\[
{}y^{\prime } = y^{2}+\lambda ^{2}+\frac {f \left (\tan \left (\lambda x \right )\right )}{\cos \left (\lambda x \right )^{4}}
\] |
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\[
{}y^{\prime } = y^{2}+\lambda ^{2}+\frac {f \left (\frac {\sin \left (\lambda x +a \right )}{\sin \left (\lambda x +b \right )}\right )}{\sin \left (\lambda x +b \right )^{4}}
\] |
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\[
{}y y^{\prime }-y = A
\] |
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\[
{}y y^{\prime }-y = A x +B
\] |
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\[
{}y y^{\prime }-y = -\frac {2 x}{9}+A +\frac {B}{\sqrt {x}}
\] |
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\[
{}y y^{\prime }-y = 2 A \left (\sqrt {x}+4 A +\frac {3 A^{2}}{\sqrt {x}}\right )
\] |
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\[
{}y y^{\prime }-y = A x +\frac {B}{x}-\frac {B^{2}}{x^{3}}
\] |
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\[
{}y y^{\prime }-y = A \,x^{k -1}-k B \,x^{k}+k \,B^{2} x^{2 k -1}
\] |
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\[
{}y y^{\prime }-y = \frac {A}{x}-\frac {A^{2}}{x^{3}}
\] |
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\[
{}y y^{\prime }-y = A +B \,{\mathrm e}^{-\frac {2 x}{A}}
\] |
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\[
{}y y^{\prime }-y = A \left ({\mathrm e}^{\frac {2 x}{A}}-1\right )
\] |
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\[
{}y y^{\prime }-y = -\frac {2 \left (m +1\right )}{\left (m +3\right )^{2}}+A \,x^{m}
\] |
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\[
{}y y^{\prime }-y = -\frac {2 x}{9}+6 A^{2} \left (1+\frac {2 A}{\sqrt {x}}\right )
\] |
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\[
{}y y^{\prime }-y = \frac {2 m -2}{\left (m -3\right )^{2}}+\frac {2 A \left (m \left (m +3\right ) \sqrt {x}+\left (4 m^{2}+3 m +9\right ) A +\frac {3 m \left (m +3\right ) A^{2}}{\sqrt {x}}\right )}{\left (m -3\right )^{2}}
\] |
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\[
{}y y^{\prime }-y = \frac {\left (2 m +1\right ) x}{4 m^{2}}+\frac {A}{x}-\frac {A^{2}}{x^{3}}
\] |
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\[
{}y y^{\prime }-y = \frac {4}{9} x +2 A \,x^{2}+2 A^{2} x^{3}
\] |
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\[
{}y y^{\prime }-y = -\frac {3 x}{16}+\frac {5 A}{x^{{1}/{3}}}-\frac {12 A^{2}}{x^{{5}/{3}}}
\] |
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\[
{}y y^{\prime }-y = \frac {A}{x}
\] |
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\[
{}y y^{\prime }-y = -\frac {x}{4}+\frac {A \left (\sqrt {x}+5 A +\frac {3 A^{2}}{\sqrt {x}}\right )}{4}
\] |
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\[
{}y y^{\prime }-y = \frac {2 a^{2}}{\sqrt {8 a^{2}+x^{2}}}
\] |
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\[
{}y y^{\prime }-y = 2 x +\frac {A}{x^{2}}
\] |
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\[
{}y y^{\prime }-y = -\frac {6 X}{25}+\frac {2 A \left (2 \sqrt {x}+19 A +\frac {6 A^{2}}{\sqrt {x}}\right )}{25}
\] |
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\[
{}y y^{\prime }-y = \frac {3 x}{8}+\frac {3 \sqrt {a^{2}+x^{2}}}{8}-\frac {a^{2}}{16 \sqrt {a^{2}+x^{2}}}
\] |
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\[
{}y y^{\prime }-y = -\frac {4 x}{25}+\frac {A}{\sqrt {x}}
\] |
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\[
{}y y^{\prime }-y = -\frac {9 x}{100}+\frac {A}{x^{{5}/{3}}}
\] |
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\[
{}y y^{\prime }-y = -\frac {12 x}{49}+\frac {2 A \left (5 \sqrt {x}+34 A +\frac {15 A^{2}}{\sqrt {x}}\right )}{49}
\] |
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\[
{}y y^{\prime }-y = -\frac {12 x}{49}+\frac {A \left (25 \sqrt {x}+41 A +\frac {10 A^{2}}{\sqrt {x}}\right )}{98}
\] |
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\[
{}y y^{\prime }-y = -\frac {2 x}{9}+\frac {A}{\sqrt {x}}
\] |
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\[
{}y y^{\prime }-y = -\frac {5 x}{36}+\frac {A}{x^{{7}/{5}}}
\] |
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\[
{}y y^{\prime }-y = -\frac {12 x}{49}+\frac {6 A \left (-3 \sqrt {x}+23 A +\frac {12 A^{2}}{\sqrt {x}}\right )}{49}
\] |
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\[
{}y y^{\prime }-y = -\frac {30 x}{121}+\frac {3 A \left (21 \sqrt {x}+35 A +\frac {6 A^{2}}{\sqrt {x}}\right )}{242}
\] |
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\[
{}y y^{\prime }-y = -\frac {3 x}{16}+\frac {A}{x^{{5}/{3}}}
\] |
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\[
{}y y^{\prime }-y = -\frac {12 x}{49}+\frac {4 A \left (-10 \sqrt {x}+27 A +\frac {10 A^{2}}{\sqrt {x}}\right )}{49}
\] |
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\[
{}y y^{\prime }-y = \frac {A}{\sqrt {x}}
\] |
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\[
{}y y^{\prime }-y = \frac {A}{x^{2}}
\] |
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\[
{}y y^{\prime }-y = A \left (n +2\right ) \left (\sqrt {x}+2 \left (n +2\right ) A +\frac {\left (n +1\right ) \left (n +3\right ) A^{2}}{\sqrt {x}}\right )
\] |
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\[
{}y y^{\prime }-y = A \left (n +2\right ) \left (\sqrt {x}+2 \left (n +2\right ) A +\frac {\left (2 n +3\right ) A^{2}}{\sqrt {x}}\right )
\] |
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\[
{}y y^{\prime }-y = A \sqrt {x}+2 A^{2}+\frac {B}{\sqrt {x}}
\] |
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\[
{}y y^{\prime }-y = 2 A^{2}-A \sqrt {x}
\] |
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\[
{}y y^{\prime }-y = -\frac {x}{4}+\frac {6 A \left (\sqrt {x}+8 A +\frac {5 A^{2}}{\sqrt {x}}\right )}{49}
\] |
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\[
{}y y^{\prime }-y = -\frac {6 x}{25}+\frac {6 A \left (2 \sqrt {x}+7 A +\frac {4 A^{2}}{\sqrt {x}}\right )}{25}
\] |
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\[
{}y y^{\prime }-y = -\frac {3 x}{16}+\frac {3 A}{x^{{1}/{3}}}-\frac {12 A^{2}}{x^{{5}/{3}}}
\] |
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\[
{}y y^{\prime }-y = \frac {3 x}{8}+\frac {3 \sqrt {b^{2}+x^{2}}}{8}+\frac {3 b^{2}}{16 \sqrt {b^{2}+x^{2}}}
\] |
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\[
{}y y^{\prime }-y = \frac {9 x}{32}+\frac {15 \sqrt {b^{2}+x^{2}}}{32}+\frac {3 b^{2}}{64 \sqrt {b^{2}+x^{2}}}
\] |
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\[
{}y y^{\prime }-y = -\frac {3 x}{32}-\frac {3 \sqrt {a^{2}+x^{2}}}{32}+\frac {15 a^{2}}{64 \sqrt {a^{2}+x^{2}}}
\] |
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\[
{}y y^{\prime }-y = A \,x^{2}-\frac {9}{625 A}
\] |
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\[
{}y y^{\prime }-y = -\frac {6}{25} x -A \,x^{2}
\] |
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\[
{}y y^{\prime }-y = \frac {6}{25} x -A \,x^{2}
\] |
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\[
{}y y^{\prime }-y = 12 x +\frac {A}{x^{{5}/{2}}}
\] |
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\[
{}y y^{\prime }-y = \frac {63 x}{4}+\frac {A}{x^{{5}/{3}}}
\] |
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\[
{}y y^{\prime }-y = 2 x +2 A \left (10 \sqrt {x}+31 A +\frac {30 A^{2}}{\sqrt {x}}\right )
\] |
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\[
{}y y^{\prime }-y = 2 x +2 A \left (-10 \sqrt {x}+19 A +\frac {30 A^{2}}{\sqrt {x}}\right )
\] |
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\[
{}y y^{\prime }-y = -\frac {28 x}{121}+\frac {2 A \left (5 \sqrt {x}+106 A +\frac {65 A^{2}}{\sqrt {x}}\right )}{121}
\] |
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\[
{}y y^{\prime }-y = -\frac {12 x}{49}+\frac {A \left (5 \sqrt {x}+262 A +\frac {65 A^{2}}{\sqrt {x}}\right )}{49}
\] |
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\[
{}y y^{\prime }-y = -\frac {12 x}{49}+A \sqrt {x}
\] |
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\[
{}y y^{\prime }-y = 6 x +\frac {A}{x^{4}}
\] |
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\[
{}y y^{\prime }-y = 20 x +\frac {A}{\sqrt {x}}
\] |
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\[
{}y y^{\prime }-y = \frac {15 x}{4}+\frac {A}{x^{7}}
\] |
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\[
{}y y^{\prime }-y = -\frac {10 x}{49}+\frac {2 A \left (4 \sqrt {x}+61 A +\frac {12 A^{2}}{\sqrt {x}}\right )}{49}
\] |
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\[
{}y y^{\prime }-y = -\frac {12 x}{49}+\frac {2 A \left (\sqrt {x}+166 A +\frac {55 A^{2}}{\sqrt {x}}\right )}{49}
\] |
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\[
{}y y^{\prime }-y = -\frac {4 x}{25}+\frac {A \left (7 \sqrt {x}+49 A +\frac {6 A^{2}}{\sqrt {x}}\right )}{50}
\] |
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