4.34.13 \(y(x) \left (a+b x+c x^2\right )+x^3 y''(x)+x^2 y'(x)=0\)

ODE
\[ y(x) \left (a+b x+c x^2\right )+x^3 y''(x)+x^2 y'(x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✗
cpu = 1.1325 (sec), leaf count = 0 , DifferentialRoot result

\[\left \{\left \{y(x)\to (x)\right \}\right \}\]

Maple ✓
cpu = 1.269 (sec), leaf count = 131

\[\left [y \left (x \right ) = \textit {\_C1} \mathit {HD}\left (0, 4 c +4 b +4 a , -8 a +8 c , 4 a -4 b +4 c , \frac {x +1}{x -1}\right )+\textit {\_C2} \mathit {HD}\left (0, 4 c +4 b +4 a , -8 a +8 c , 4 a -4 b +4 c , \frac {x +1}{x -1}\right ) \left (\int \frac {1}{x \mathit {HD}\left (0, 4 c +4 b +4 a , -8 a +8 c , 4 a -4 b +4 c , \frac {x +1}{x -1}\right )^{2}}d x \right )\right ]\] Mathematica raw input

DSolve[(a + b*x + c*x^2)*y[x] + x^2*y'[x] + x^3*y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> DifferentialRoot[Function[{\[FormalY], \[FormalX]}, {(a + \[FormalX]*b
 + \[FormalX]^2*c)*\[FormalY][\[FormalX]] + \[FormalX]^2*Derivative[1][\[FormalY
]][\[FormalX]] + \[FormalX]^3*Derivative[2][\[FormalY]][\[FormalX]] == 0, \[Form
alY][1] == C[1], Derivative[1][\[FormalY]][1] == C[2]}]][x]}}

Maple raw input

dsolve(x^3*diff(diff(y(x),x),x)+x^2*diff(y(x),x)+(c*x^2+b*x+a)*y(x) = 0, y(x))

Maple raw output

[y(x) = _C1*HeunD(0,4*c+4*b+4*a,-8*a+8*c,4*a-4*b+4*c,(x+1)/(x-1))+_C2*HeunD(0,4*
c+4*b+4*a,-8*a+8*c,4*a-4*b+4*c,(x+1)/(x-1))*Int(1/x/HeunD(0,4*c+4*b+4*a,-8*a+8*c
,4*a-4*b+4*c,(x+1)/(x-1))^2,x)]