#### 2.513   ODE No. 513

$y'(x)^2 \sin (y(x))+2 x y'(x) \cos ^3(y(x))-\sin (y(x)) \cos ^4(y(x))=0$ Mathematica : cpu = 0.0750519 (sec), leaf count = 81

$\left \{\left \{y(x)\to \tan ^{-1}\left (2 \left (-\frac {c_1{}^{3/2}}{\sqrt {c_1+x}}-\frac {\sqrt {c_1} x}{\sqrt {c_1+x}}\right )\right )\right \},\left \{y(x)\to \tan ^{-1}\left (2 \left (\frac {c_1{}^{3/2}}{\sqrt {c_1+x}}+\frac {\sqrt {c_1} x}{\sqrt {c_1+x}}\right )\right )\right \}\right \}$ Maple : cpu = 3.5 (sec), leaf count = 1138

$\left \{ [x \left ( {\it \_T} \right ) ={\frac {1}{2\,{\it \_T}} \left ( \left ( \cos \left ( {\frac {1}{2}\arctan \left ( {\frac {1}{3} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}-2\,{\it \_C1}\,{\it \_T}\,\sqrt [3]{{{\it \_C1}}^{3}{{\it \_T}}^{3}+54\,{\it \_C1}\,{\it \_T}+6\,\sqrt {3}\sqrt {{{\it \_C1}}^{2}{{\it \_T}}^{2} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}+27 \right ) }}+ \left ( {{\it \_C1}}^{3}{{\it \_T}}^{3}+54\,{\it \_C1}\,{\it \_T}+6\,\sqrt {3}\sqrt {{{\it \_C1}}^{2}{{\it \_T}}^{2} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}+27 \right ) } \right ) ^{{\frac {2}{3}}} \right ) {\frac {1}{\sqrt [3]{{{\it \_C1}}^{3}{{\it \_T}}^{3}+54\,{\it \_C1}\,{\it \_T}+6\,\sqrt {3}\sqrt {{{\it \_C1}}^{2}{{\it \_T}}^{2} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}+27 \right ) }}}}},-3\,{\frac {1/3\, \left ( {{\it \_C1}}^{3}{{\it \_T}}^{3}+54\,{\it \_C1}\,{\it \_T}+6\,\sqrt {3}\sqrt {{{\it \_C1}}^{2}{{\it \_T}}^{2} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}+27 \right ) } \right ) ^{2/3}+ \left ( {\it \_C1}\,{\it \_T}+1/9\,\sqrt {3}\sqrt {{{\it \_C1}}^{2}{{\it \_T}}^{2} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}+27 \right ) } \right ) \left ( {\it \_C1}\,{\it \_T}-\sqrt [3]{{{\it \_C1}}^{3}{{\it \_T}}^{3}+54\,{\it \_C1}\,{\it \_T}+6\,\sqrt {3}\sqrt {{{\it \_C1}}^{2}{{\it \_T}}^{2} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}+27 \right ) }} \right ) }{ \left ( {{\it \_C1}}^{3}{{\it \_T}}^{3}+54\,{\it \_C1}\,{\it \_T}+6\,\sqrt {3}\sqrt {{{\it \_C1}}^{2}{{\it \_T}}^{2} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}+27 \right ) } \right ) ^{2/3}}} \right ) } \right ) \right ) ^{4}-{{\it \_T}}^{2} \right ) \sin \left ( {\frac {1}{2}\arctan \left ( {\frac {1}{3} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}-2\,{\it \_C1}\,{\it \_T}\,\sqrt [3]{{{\it \_C1}}^{3}{{\it \_T}}^{3}+54\,{\it \_C1}\,{\it \_T}+6\,\sqrt {3}\sqrt {{{\it \_C1}}^{2}{{\it \_T}}^{2} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}+27 \right ) }}+ \left ( {{\it \_C1}}^{3}{{\it \_T}}^{3}+54\,{\it \_C1}\,{\it \_T}+6\,\sqrt {3}\sqrt {{{\it \_C1}}^{2}{{\it \_T}}^{2} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}+27 \right ) } \right ) ^{{\frac {2}{3}}} \right ) {\frac {1}{\sqrt [3]{{{\it \_C1}}^{3}{{\it \_T}}^{3}+54\,{\it \_C1}\,{\it \_T}+6\,\sqrt {3}\sqrt {{{\it \_C1}}^{2}{{\it \_T}}^{2} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}+27 \right ) }}}}},-3\,{\frac {1/3\, \left ( {{\it \_C1}}^{3}{{\it \_T}}^{3}+54\,{\it \_C1}\,{\it \_T}+6\,\sqrt {3}\sqrt {{{\it \_C1}}^{2}{{\it \_T}}^{2} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}+27 \right ) } \right ) ^{2/3}+ \left ( {\it \_C1}\,{\it \_T}+1/9\,\sqrt {3}\sqrt {{{\it \_C1}}^{2}{{\it \_T}}^{2} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}+27 \right ) } \right ) \left ( {\it \_C1}\,{\it \_T}-\sqrt [3]{{{\it \_C1}}^{3}{{\it \_T}}^{3}+54\,{\it \_C1}\,{\it \_T}+6\,\sqrt {3}\sqrt {{{\it \_C1}}^{2}{{\it \_T}}^{2} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}+27 \right ) }} \right ) }{ \left ( {{\it \_C1}}^{3}{{\it \_T}}^{3}+54\,{\it \_C1}\,{\it \_T}+6\,\sqrt {3}\sqrt {{{\it \_C1}}^{2}{{\it \_T}}^{2} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}+27 \right ) } \right ) ^{2/3}}} \right ) } \right ) \left ( \cos \left ( {\frac {1}{2}\arctan \left ( {\frac {1}{3} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}-2\,{\it \_C1}\,{\it \_T}\,\sqrt [3]{{{\it \_C1}}^{3}{{\it \_T}}^{3}+54\,{\it \_C1}\,{\it \_T}+6\,\sqrt {3}\sqrt {{{\it \_C1}}^{2}{{\it \_T}}^{2} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}+27 \right ) }}+ \left ( {{\it \_C1}}^{3}{{\it \_T}}^{3}+54\,{\it \_C1}\,{\it \_T}+6\,\sqrt {3}\sqrt {{{\it \_C1}}^{2}{{\it \_T}}^{2} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}+27 \right ) } \right ) ^{{\frac {2}{3}}} \right ) {\frac {1}{\sqrt [3]{{{\it \_C1}}^{3}{{\it \_T}}^{3}+54\,{\it \_C1}\,{\it \_T}+6\,\sqrt {3}\sqrt {{{\it \_C1}}^{2}{{\it \_T}}^{2} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}+27 \right ) }}}}},-3\,{\frac {1/3\, \left ( {{\it \_C1}}^{3}{{\it \_T}}^{3}+54\,{\it \_C1}\,{\it \_T}+6\,\sqrt {3}\sqrt {{{\it \_C1}}^{2}{{\it \_T}}^{2} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}+27 \right ) } \right ) ^{2/3}+ \left ( {\it \_C1}\,{\it \_T}+1/9\,\sqrt {3}\sqrt {{{\it \_C1}}^{2}{{\it \_T}}^{2} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}+27 \right ) } \right ) \left ( {\it \_C1}\,{\it \_T}-\sqrt [3]{{{\it \_C1}}^{3}{{\it \_T}}^{3}+54\,{\it \_C1}\,{\it \_T}+6\,\sqrt {3}\sqrt {{{\it \_C1}}^{2}{{\it \_T}}^{2} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}+27 \right ) }} \right ) }{ \left ( {{\it \_C1}}^{3}{{\it \_T}}^{3}+54\,{\it \_C1}\,{\it \_T}+6\,\sqrt {3}\sqrt {{{\it \_C1}}^{2}{{\it \_T}}^{2} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}+27 \right ) } \right ) ^{2/3}}} \right ) } \right ) \right ) ^{-3}},y \left ( {\it \_T} \right ) ={\frac {1}{2}\arctan \left ( {\frac {1}{3} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}-2\,{\it \_C1}\,{\it \_T}\,\sqrt [3]{{{\it \_C1}}^{3}{{\it \_T}}^{3}+54\,{\it \_C1}\,{\it \_T}+6\,\sqrt {3}\sqrt {{{\it \_C1}}^{2}{{\it \_T}}^{2} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}+27 \right ) }}+ \left ( {{\it \_C1}}^{3}{{\it \_T}}^{3}+54\,{\it \_C1}\,{\it \_T}+6\,\sqrt {3}\sqrt {{{\it \_C1}}^{2}{{\it \_T}}^{2} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}+27 \right ) } \right ) ^{{\frac {2}{3}}} \right ) {\frac {1}{\sqrt [3]{{{\it \_C1}}^{3}{{\it \_T}}^{3}+54\,{\it \_C1}\,{\it \_T}+6\,\sqrt {3}\sqrt {{{\it \_C1}}^{2}{{\it \_T}}^{2} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}+27 \right ) }}}}},-3\,{\frac {1/3\, \left ( {{\it \_C1}}^{3}{{\it \_T}}^{3}+54\,{\it \_C1}\,{\it \_T}+6\,\sqrt {3}\sqrt {{{\it \_C1}}^{2}{{\it \_T}}^{2} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}+27 \right ) } \right ) ^{2/3}+ \left ( {\it \_C1}\,{\it \_T}+1/9\,\sqrt {3}\sqrt {{{\it \_C1}}^{2}{{\it \_T}}^{2} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}+27 \right ) } \right ) \left ( {\it \_C1}\,{\it \_T}-\sqrt [3]{{{\it \_C1}}^{3}{{\it \_T}}^{3}+54\,{\it \_C1}\,{\it \_T}+6\,\sqrt {3}\sqrt {{{\it \_C1}}^{2}{{\it \_T}}^{2} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}+27 \right ) }} \right ) }{ \left ( {{\it \_C1}}^{3}{{\it \_T}}^{3}+54\,{\it \_C1}\,{\it \_T}+6\,\sqrt {3}\sqrt {{{\it \_C1}}^{2}{{\it \_T}}^{2} \left ( {{\it \_C1}}^{2}{{\it \_T}}^{2}+27 \right ) } \right ) ^{2/3}}} \right ) }] \right \}$