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 = 0.980815 (sec), leaf count = 0 , DifferentialRoot result

\[\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{\unicode {f818}''(\unicode {f817}) \unicode {f817}^3+\unicode {f818}'(\unicode {f817}) \unicode {f817}^2+\left (c \unicode {f817}^2+b \unicode {f817}+a\right ) \unicode {f818}(\unicode {f817})=0,\unicode {f818}(1)=c_1,\unicode {f818}'(1)=c_2\right \}\right )(x)\right \}\right \}\]

Maple ✓
cpu = 0.322 (sec), leaf count = 93

\[ \left \{ y \left ( x \right ) ={\it HeunD} \left ( 0,4\,c+4\,b+4\,a,-8\,a+8\,c,4\,a-4\,b+4\,c,{\frac {1+x}{-1+x}} \right ) \left ( \int \!{\frac {1}{x} \left ( {\it HeunD} \left ( 0,4\,c+4\,b+4\,a,-8\,a+8\,c,4\,a-4\,b+4\,c,{\frac {1+x}{-1+x}} \right ) \right ) ^{-2}}\,{\rm d}x{\it \_C2}+{\it \_C1} \right ) \right \} \] 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),'implicit')

Maple raw output

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