ODE
\[ y(x) \left (\sum _{m=0}^n a(m) x^m\right )+a (1-2 x) y'(x)+x (x+1) y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✗
cpu = 4.67855 (sec), leaf count = 0 , could not solve
DSolve[Sum[x^m*a[m], {m, 0, n}]*y[x] + a*(1 - 2*x)*Derivative[1][y][x] + x*(1 + x)*Derivative[2][y][x] == 0, y[x], x]
Maple ✗
cpu = 23.138 (sec), leaf count = 0 , result contains DESol
\[ \left \{ y \left ( x \right ) ={\it DESol} \left ( \left \{ {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}{\it \_Y} \left ( x \right ) +{\frac {a \left ( 1-2\,x \right ) {\frac {\rm d}{{\rm d}x}}{\it \_Y} \left ( x \right ) }{ \left ( 1+x \right ) x}}+{\frac {\sum _{m=0}^{n}a \left ( m \right ) {x}^{m}{\it \_Y} \left ( x \right ) }{ \left ( 1+x \right ) x}} \right \} , \left \{ {\it \_Y} \left ( x \right ) \right \} \right ) \right \} \]
Mathematica raw input
DSolve[Sum[x^m*a[m], {m, 0, n}]*y[x] + a*(1 - 2*x)*y'[x] + x*(1 + x)*y''[x] == 0,y[x],x]
Mathematica raw output
DSolve[Sum[x^m*a[m], {m, 0, n}]*y[x] + a*(1 - 2*x)*Derivative[1][y][x] + x*(1 + x)*Derivative[2][y][x] == 0, y[x], x]
Maple raw input
dsolve(x*(1+x)*diff(diff(y(x),x),x)+a*(1-2*x)*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)+a*(1-2*x)/x/(1+x)*diff(_Y(x),x)+sum(a(m)*x^m,m = 0 .. n)/x/(1+x)*_Y(x)},{_Y(x)})