2.157   ODE No. 157

\[ a \left (y(x)^2-2 x y(x)+1\right )+\left (x^2-1\right ) y'(x)=0 \] Mathematica : cpu = 0.21159 (sec), leaf count = 158


\[\left \{\left \{y(x)\to \frac {\left (x^2-1\right ) \left (c_1 \left (a x \left (x^2-1\right )^{\frac {a}{2}-1} P_{a-1}(x)+\left (x^2-1\right )^{\frac {a}{2}-1} (a P_a(x)-a x P_{a-1}(x))\right )+a x \left (x^2-1\right )^{\frac {a}{2}-1} Q_{a-1}(x)+\left (x^2-1\right )^{\frac {a}{2}-1} (a Q_a(x)-a x Q_{a-1}(x))\right )}{a \left (\left (x^2-1\right )^{a/2} Q_{a-1}(x)+c_1 \left (x^2-1\right )^{a/2} P_{a-1}(x)\right )}\right \}\right \}\] Maple : cpu = 0.265 (sec), leaf count = 231


\[y \relax (x ) = \frac {8 c_{1} \left (\left (a -\frac {1}{2}\right ) x -\frac {a}{2}+\frac {1}{2}\right ) \left (1+x \right ) \HeunC \left (0, -2 a +1, 0, 0, a^{2}-a +\frac {1}{2}, \frac {2}{1+x}\right )-a \left (-\frac {x}{2}-\frac {1}{2}\right )^{-2 a +1} \left (1+x \right ) \HeunC \left (0, 2 a -1, 0, 0, a^{2}-a +\frac {1}{2}, \frac {2}{1+x}\right )-8 \left (\HeunCPrime \left (0, -2 a +1, 0, 0, a^{2}-a +\frac {1}{2}, \frac {2}{1+x}\right ) c_{1}-\frac {\left (-\frac {x}{2}-\frac {1}{2}\right )^{-2 a +1} \HeunCPrime \left (0, 2 a -1, 0, 0, a^{2}-a +\frac {1}{2}, \frac {2}{1+x}\right )}{4}\right ) \left (x -1\right )}{4 a \left (\HeunC \left (0, -2 a +1, 0, 0, a^{2}-a +\frac {1}{2}, \frac {2}{1+x}\right ) c_{1}-\frac {\left (-\frac {x}{2}-\frac {1}{2}\right )^{-2 a +1} \HeunC \left (0, 2 a -1, 0, 0, a^{2}-a +\frac {1}{2}, \frac {2}{1+x}\right )}{4}\right ) \left (1+x \right )}\]