4.28.25 \(a y'(x)+y(x) \left (\text {a1}+\text {b1} x+\text {c1} x^2\right )+x y''(x)=0\)

ODE
\[ a y'(x)+y(x) \left (\text {a1}+\text {b1} x+\text {c1} x^2\right )+x y''(x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✗
cpu = 2.5341 (sec), leaf count = 0 , DifferentialRoot result

\[\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{\left (\text {c1} \unicode {f817}^2+\text {b1} \unicode {f817}+\text {a1}\right ) \unicode {f818}(\unicode {f817})+a \unicode {f818}'(\unicode {f817})+\unicode {f817} \unicode {f818}''(\unicode {f817})=0,\unicode {f818}(1)=c_1,\unicode {f818}'(1)=c_2\right \}\right )(x)\right \}\right \}\]

Maple ✗
cpu = 0.295 (sec), leaf count = 0 , result contains DESol

\[ \left \{ y \left ( x \right ) ={\it DESol} \left ( \left \{ {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}{\it \_Y} \left ( x \right ) +{\frac {a{\frac {\rm d}{{\rm d}x}}{\it \_Y} \left ( x \right ) }{x}}+{\frac { \left ( {\it c1}\,{x}^{2}+{\it b1}\,x+{\it a1} \right ) {\it \_Y} \left ( x \right ) }{x}} \right \} , \left \{ {\it \_Y} \left ( x \right ) \right \} \right ) \right \} \]

Mathematica raw input

DSolve[(a1 + b1*x + c1*x^2)*y[x] + a*y'[x] + x*y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> DifferentialRoot[Function[{\[FormalY], \[FormalX]}, {(a1 + \[FormalX]*
b1 + \[FormalX]^2*c1)*\[FormalY][\[FormalX]] + a*Derivative[1][\[FormalY]][\[For
malX]] + \[FormalX]*Derivative[2][\[FormalY]][\[FormalX]] == 0, \[FormalY][1] ==
 C[1], Derivative[1][\[FormalY]][1] == C[2]}]][x]}}

Maple raw input

dsolve(x*diff(diff(y(x),x),x)+a*diff(y(x),x)+(c1*x^2+b1*x+a1)*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = DESol({diff(diff(_Y(x),x),x)+1/x*a*diff(_Y(x),x)+(c1*x^2+b1*x+a1)/x*_Y(x)
},{_Y(x)})