ODE
\[ 2 a^2 y(x)+2 \left (1-x^2\right ) y''(x)-x y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0183003 (sec), leaf count = 66
\[\left \{\left \{y(x)\to \left (x^2-1\right )^{3/8} \left (c_1 P_{\frac {1}{4} \left (\sqrt {16 a^2+1}-2\right )}^{\frac {3}{4}}(x)+c_2 Q_{\frac {1}{4} \left (\sqrt {16 a^2+1}-2\right )}^{\frac {3}{4}}(x)\right )\right \}\right \}\]
Maple ✓
cpu = 0.066 (sec), leaf count = 49
\[ \left \{ y \left ( x \right ) = \left ( {x}^{2}-1 \right ) ^{{\frac {3}{8}}} \left ( {\it LegendreQ} \left ( {\frac {1}{4}\sqrt {16\,{a}^{2}+1}}-{\frac {1}{2}},{\frac {3}{4}},x \right ) {\it \_C2}+{\it LegendreP} \left ( {\frac {1}{4}\sqrt {16\,{a}^{2}+1}}-{\frac {1}{2}},{\frac {3}{4}},x \right ) {\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[2*a^2*y[x] - x*y'[x] + 2*(1 - x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (-1 + x^2)^(3/8)*(C[1]*LegendreP[(-2 + Sqrt[1 + 16*a^2])/4, 3/4, x] + C[2]*LegendreQ[(-2 + Sqrt[1 + 16*a^2])/4, 3/4, x])}}
Maple raw input
dsolve(2*(-x^2+1)*diff(diff(y(x),x),x)-x*diff(y(x),x)+2*a^2*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (x^2-1)^(3/8)*(LegendreQ(1/4*(16*a^2+1)^(1/2)-1/2,3/4,x)*_C2+LegendreP(1/4*(16*a^2+1)^(1/2)-1/2,3/4,x)*_C1)