2.168   ODE No. 168

$3 \left (x^2-4\right ) y'(x)+y(x)^2-x y(x)-3=0$ Mathematica : cpu = 0.172918 (sec), leaf count = 234

$\left \{\left \{y(x)\to \frac {3 \left (x^2-4\right ) \left (c_1 \left (\frac {x P_{-\frac {1}{6}}^{\frac {1}{3}}\left (\frac {x}{2}\right )}{6 \left (x^2-4\right )^{11/12}}+\frac {\sqrt [12]{x^2-4} \left (\frac {1}{2} P_{\frac {5}{6}}^{\frac {1}{3}}\left (\frac {x}{2}\right )-\frac {5}{12} x P_{-\frac {1}{6}}^{\frac {1}{3}}\left (\frac {x}{2}\right )\right )}{2 \left (\frac {x^2}{4}-1\right )}\right )+\frac {x Q_{-\frac {1}{6}}^{\frac {1}{3}}\left (\frac {x}{2}\right )}{6 \left (x^2-4\right )^{11/12}}+\frac {\sqrt [12]{x^2-4} \left (\frac {1}{2} Q_{\frac {5}{6}}^{\frac {1}{3}}\left (\frac {x}{2}\right )-\frac {5}{12} x Q_{-\frac {1}{6}}^{\frac {1}{3}}\left (\frac {x}{2}\right )\right )}{2 \left (\frac {x^2}{4}-1\right )}\right )}{c_1 \sqrt [12]{x^2-4} P_{-\frac {1}{6}}^{\frac {1}{3}}\left (\frac {x}{2}\right )+\sqrt [12]{x^2-4} Q_{-\frac {1}{6}}^{\frac {1}{3}}\left (\frac {x}{2}\right )}\right \}\right \}$ Maple : cpu = 0.126 (sec), leaf count = 140

$\left \{ y \left ( x \right ) =-3\,{(x+2) \left ( {\it HeunC} \left ( 0,4/3,-1/3,0,{\frac {25}{36}},4\, \left ( x+2 \right ) ^{-1} \right ) {\it \_C1}-1/3\, \left ( -x/4-1/2 \right ) ^{4/3}{\it HeunC} \left ( 0,-4/3,-1/3,0,{\frac {25}{36}},4\, \left ( x+2 \right ) ^{-1} \right ) \right ) \left ( 4\, \left ( x+2 \right ) \left ( x-5/4 \right ) {\it \_C1}\,{\it HeunC} \left ( 0,4/3,-1/3,0,{\frac {25}{36}},4\, \left ( x+2 \right ) ^{-1} \right ) - \left ( -x/4-1/2 \right ) ^{4/3} \left ( x+2 \right ) {\it HeunC} \left ( 0,-4/3,-1/3,0,{\frac {25}{36}},4\, \left ( x+2 \right ) ^{-1} \right ) +12\, \left ( x-2 \right ) \left ( {\it HeunCPrime} \left ( 0,4/3,-1/3,0,{\frac {25}{36}},4\, \left ( x+2 \right ) ^{-1} \right ) {\it \_C1}-1/3\, \left ( -x/4-1/2 \right ) ^{4/3}{\it HeunCPrime} \left ( 0,-4/3,-1/3,0,{\frac {25}{36}},4\, \left ( x+2 \right ) ^{-1} \right ) \right ) \right ) ^{-1}} \right \}$