ODE
\[ y(x) \left (a+b x^2\right )+\left (1-x^2\right ) y''(x)-x y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.162938 (sec), leaf count = 42
\[\left \{\left \{y(x)\to c_1 \text {MathieuC}\left [a+\frac {b}{2},-\frac {b}{4},\cos ^{-1}(x)\right ]+c_2 \text {MathieuS}\left [a+\frac {b}{2},-\frac {b}{4},\cos ^{-1}(x)\right ]\right \}\right \}\]
Maple ✓
cpu = 1.214 (sec), leaf count = 31
\[\left [y \left (x \right ) = \textit {\_C1} \MathieuC \left (\frac {b}{2}+a , -\frac {b}{4}, \arccos \left (x \right )\right )+\textit {\_C2} \MathieuS \left (\frac {b}{2}+a , -\frac {b}{4}, \arccos \left (x \right )\right )\right ]\] Mathematica raw input
DSolve[(a + b*x^2)*y[x] - x*y'[x] + (1 - x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*MathieuC[a + b/2, -1/4*b, ArcCos[x]] + C[2]*MathieuS[a + b/2, -1/4*b, ArcCos[x]]}}
Maple raw input
dsolve((-x^2+1)*diff(diff(y(x),x),x)-x*diff(y(x),x)+(b*x^2+a)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*MathieuC(1/2*b+a,-1/4*b,arccos(x))+_C2*MathieuS(1/2*b+a,-1/4*b,arccos(x))]