#### 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 = 0.173873 (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 = 0.168 (sec), leaf count = 231

$\left \{ y \left ( x \right ) ={\frac {1}{4\,a \left ( 1+x \right ) } \left ( 8\, \left ( \left ( a-1/2 \right ) x-a/2+1/2 \right ) \left ( 1+x \right ) {\it \_C1}\,{\it HeunC} \left ( 0,-2\,a+1,0,0,{a}^{2}-a+1/2,2\, \left ( 1+x \right ) ^{-1} \right ) -a \left ( -{\frac {x}{2}}-{\frac {1}{2}} \right ) ^{-2\,a+1} \left ( 1+x \right ) {\it HeunC} \left ( 0,2\,a-1,0,0,{a}^{2}-a+{\frac {1}{2}},2\, \left ( 1+x \right ) ^{-1} \right ) -8\, \left ( x-1 \right ) \left ( {\it HeunCPrime} \left ( 0,-2\,a+1,0,0,{a}^{2}-a+1/2,2\, \left ( 1+x \right ) ^{-1} \right ) {\it \_C1}-1/4\, \left ( -x/2-1/2 \right ) ^{-2\,a+1}{\it HeunCPrime} \left ( 0,2\,a-1,0,0,{a}^{2}-a+1/2,2\, \left ( 1+x \right ) ^{-1} \right ) \right ) \right ) \left ( {\it HeunC} \left ( 0,-2\,a+1,0,0,{a}^{2}-a+{\frac {1}{2}},2\, \left ( 1+x \right ) ^{-1} \right ) {\it \_C1}-{\frac {1}{4} \left ( -{\frac {x}{2}}-{\frac {1}{2}} \right ) ^{-2\,a+1}{\it HeunC} \left ( 0,2\,a-1,0,0,{a}^{2}-a+{\frac {1}{2}},2\, \left ( 1+x \right ) ^{-1} \right ) } \right ) ^{-1}} \right \}$