ODE
\[ n^2 y(x)+\left (1-x^2\right ) y''(x)-x y'(x)=0 \] ODE Classification
[_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0288051 (sec), leaf count = 45
\[\left \{\left \{y(x)\to c_1 \cosh \left (n \log \left (\sqrt {x^2-1}+x\right )\right )+i c_2 \sinh \left (n \log \left (\sqrt {x^2-1}+x\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.019 (sec), leaf count = 19
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,\sin \left ( n\arcsin \left ( x \right ) \right ) +{\it \_C2}\,\cos \left ( n\arcsin \left ( x \right ) \right ) \right \} \] Mathematica raw input
DSolve[n^2*y[x] - x*y'[x] + (1 - x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*Cosh[n*Log[x + Sqrt[-1 + x^2]]] + I*C[2]*Sinh[n*Log[x + Sqrt[-1 + x^2]]]}}
Maple raw input
dsolve((-x^2+1)*diff(diff(y(x),x),x)-x*diff(y(x),x)+n^2*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*sin(n*arcsin(x))+_C2*cos(n*arcsin(x))