ODE
\[ y(x) \left (\sum _{m=0}^n a(m) x^m\right )+\left (1-x^2\right )^2 y''(x)-2 x \left (1-x^2\right ) y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✗
cpu = 4.78176 (sec), leaf count = 0 , could not solve
DSolve[Sum[x^m*a[m], {m, 0, n}]*y[x] - 2*x*(1 - x^2)*Derivative[1][y][x] + (1 - x^2)^2*Derivative[2][y][x] == 0, y[x], x]
Maple ✗
cpu = 4.512 (sec), leaf count = 0 , result contains DESol
\[ \left \{ y \left ( x \right ) ={\it DESol} \left ( \left \{ {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}{\it \_Y} \left ( x \right ) -2\,{\frac {x{\frac {\rm d}{{\rm d}x}}{\it \_Y} \left ( x \right ) }{-{x}^{2}+1}}+{\frac {\sum _{m=0}^{n}a \left ( m \right ) {x}^{m}{\it \_Y} \left ( x \right ) }{ \left ( -{x}^{2}+1 \right ) ^{2}}} \right \} , \left \{ {\it \_Y} \left ( x \right ) \right \} \right ) \right \} \]
Mathematica raw input
DSolve[Sum[x^m*a[m], {m, 0, n}]*y[x] - 2*x*(1 - x^2)*y'[x] + (1 - x^2)^2*y''[x] == 0,y[x],x]
Mathematica raw output
DSolve[Sum[x^m*a[m], {m, 0, n}]*y[x] - 2*x*(1 - x^2)*Derivative[1][y][x] + (1 - x^2)^2*Derivative[2][y][x] == 0, y[x], x]
Maple raw input
dsolve((-x^2+1)^2*diff(diff(y(x),x),x)-2*x*(-x^2+1)*diff(y(x),x)+sum(a(m)*x^m,m = 0 .. n)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = DESol({diff(diff(_Y(x),x),x)-2*x/(-x^2+1)*diff(_Y(x),x)+sum(a(m)*x^m,m = 0 .. n)/(-x^2+1)^2*_Y(x)},{_Y(x)})