ODE
\[ y(x) \left (\sum _{m=0}^n a(m) x^m\right )+4 (1-x) x y''(x)+2 (1-2 x) y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✗
cpu = 1.27644 (sec), leaf count = 0 , could not solve
DSolve[Sum[x^m*a[m], {m, 0, n}]*y[x] + 2*(1 - 2*x)*Derivative[1][y][x] + 4*(1 - x)*x*Derivative[2][y][x] == 0, y[x], x]
Maple ✗
cpu = 9.816 (sec), leaf count = 0 , result contains DESol
\[ \left \{ y \left ( x \right ) ={\it DESol} \left ( \left \{ {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}{\it \_Y} \left ( x \right ) -{\frac { \left ( -2+4\,x \right ) {\frac {\rm d}{{\rm d}x}}{\it \_Y} \left ( x \right ) }{4\,x \left ( 1-x \right ) }}+{\frac {\sum _{m=0}^{n}a \left ( m \right ) {x}^{m}{\it \_Y} \left ( x \right ) }{4\,x \left ( 1-x \right ) }} \right \} , \left \{ {\it \_Y} \left ( x \right ) \right \} \right ) \right \} \]
Mathematica raw input
DSolve[Sum[x^m*a[m], {m, 0, n}]*y[x] + 2*(1 - 2*x)*y'[x] + 4*(1 - x)*x*y''[x] == 0,y[x],x]
Mathematica raw output
DSolve[Sum[x^m*a[m], {m, 0, n}]*y[x] + 2*(1 - 2*x)*Derivative[1][y][x] + 4*(1 - x)*x*Derivative[2][y][x] == 0, y[x], x]
Maple raw input
dsolve(4*x*(1-x)*diff(diff(y(x),x),x)+2*(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)-1/4*(-2+4*x)/x/(1-x)*diff(_Y(x),x)+1/4*sum(a(m)*x^m,m = 0 .. n)/x/(1-x)*_Y(x)},{_Y(x)})