ODE
\[ x \left (1-x^2\right ) y''(x)-2 \left (1-x^2\right ) y'(x)-2 x y(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.041035 (sec), leaf count = 48
\[\left \{\left \{y(x)\to c_1 \left (-\left (x^2-1\right )\right )-\frac {1}{4} c_2 \left (\left (x^2-1\right ) \log (1-x)-\left (x^2-1\right ) \log (x+1)-2 x\right )\right \}\right \}\]
Maple ✓
cpu = 0.019 (sec), leaf count = 46
\[ \left \{ y \left ( x \right ) ={\frac { \left ( {\it \_C1}\,{x}^{2}-{\it \_C1} \right ) \ln \left ( -1+x \right ) }{4}}+{\frac { \left ( -{\it \_C1}\,{x}^{2}+{\it \_C1} \right ) \ln \left ( 1+x \right ) }{4}}+{\it \_C2}\,{x}^{2}-{\frac {{\it \_C1}\,x}{2}}-{\it \_C2} \right \} \] Mathematica raw input
DSolve[-2*x*y[x] - 2*(1 - x^2)*y'[x] + x*(1 - x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> -((-1 + x^2)*C[1]) - (C[2]*(-2*x + (-1 + x^2)*Log[1 - x] - (-1 + x^2)*Log[1 + x]))/4}}
Maple raw input
dsolve(x*(-x^2+1)*diff(diff(y(x),x),x)-2*(-x^2+1)*diff(y(x),x)-2*x*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = 1/4*(_C1*x^2-_C1)*ln(-1+x)+1/4*(-_C1*x^2+_C1)*ln(1+x)+_C2*x^2-1/2*_C1*x-_C2