ODE
\[ \left (1-x^2\right ) x^2 y''(x)-\left (2-x^2\right ) x y'(x)+\left (2-x^2\right ) y(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.18697 (sec), leaf count = 71
\[\left \{\left \{y(x)\to \frac {x \sqrt [4]{x^2-1} \left (-c_2 \log \left (1-\frac {x}{\sqrt {x^2-1}}\right )+c_2 \log \left (\frac {x}{\sqrt {x^2-1}}+1\right )+2 c_1\right )}{2 \sqrt [4]{1-x^2}}\right \}\right \}\]
Maple ✓
cpu = 0.071 (sec), leaf count = 21
\[\left [y \left (x \right ) = \textit {\_C1} x +\textit {\_C2} x \ln \left (x +\sqrt {x^{2}-1}\right )\right ]\] Mathematica raw input
DSolve[(2 - x^2)*y[x] - x*(2 - x^2)*y'[x] + x^2*(1 - x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (x*(-1 + x^2)^(1/4)*(2*C[1] - C[2]*Log[1 - x/Sqrt[-1 + x^2]] + C[2]*Log[1 + x/Sqrt[-1 + x^2]]))/(2*(1 - x^2)^(1/4))}}
Maple raw input
dsolve(x^2*(-x^2+1)*diff(diff(y(x),x),x)-x*(-x^2+2)*diff(y(x),x)+(-x^2+2)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*x+_C2*x*ln(x+(x^2-1)^(1/2))]