ODE
\[ x^2 y''(x)-\left (1-x^2\right ) y'(x)-y(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.242227 (sec), leaf count = 35
\[\left \{\left \{y(x)\to e^{-x} \left (c_2 \int _1^xe^{K[1]-\frac {1}{K[1]}}dK[1]+c_1\right )\right \}\right \}\]
Maple ✓
cpu = 1.01 (sec), leaf count = 53
\[\left [y \left (x \right ) = \textit {\_C1} \,{\mathrm e}^{-x} \mathit {HD}\left (4, 3, -8, 5, \frac {x -1}{x +1}\right ) \sqrt {x}+\textit {\_C2} \,{\mathrm e}^{-\frac {1}{x}} \mathit {HD}\left (-4, 3, -8, 5, \frac {x -1}{x +1}\right ) \sqrt {x}\right ]\] Mathematica raw input
DSolve[-y[x] - (1 - x^2)*y'[x] + x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (C[1] + C[2]*Inactive[Integrate][E^(-K[1]^(-1) + K[1]), {K[1], 1, x}])/E^x}}
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x)-(-x^2+1)*diff(y(x),x)-y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*exp(-x)*HeunD(4,3,-8,5,(x-1)/(x+1))*x^(1/2)+_C2*exp(-1/x)*HeunD(-4,3,-8,5,(x-1)/(x+1))*x^(1/2)]