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 = 8.95038 (sec), leaf count = 34
\[\left \{\left \{y(x)\to e^{-x} \left (c_2 \int _1^x e^{K[1]-\frac {1}{K[1]}} \, dK[1]+c_1\right )\right \}\right \}\]
Maple ✓
cpu = 0.176 (sec), leaf count = 51
\[ \left \{ y \left ( x \right ) =\sqrt {x} \left ( {{\rm e}^{-x}}{\it HeunD} \left ( 4,3,-8,5,{\frac {-1+x}{1+x}} \right ) {\it \_C1}+{{\rm e}^{-{x}^{-1}}}{\it HeunD} \left ( -4,3,-8,5,{\frac {-1+x}{1+x}} \right ) {\it \_C2} \right ) \right \} \] 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]*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),'implicit')
Maple raw output
y(x) = x^(1/2)*(exp(-x)*HeunD(4,3,-8,5,(-1+x)/(1+x))*_C1+exp(-1/x)*HeunD(-4,3,-8,5,(-1+x)/(1+x))*_C2)