4.35.15 \(y(x) \left (a+b x^2+c x^4\right )+x^4 y''(x)+x^3 y'(x)=0\)

ODE
\[ y(x) \left (a+b x^2+c x^4\right )+x^4 y''(x)+x^3 y'(x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✗
cpu = 1.29744 (sec), leaf count = 0 , DifferentialRoot result

\[\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{\unicode {f818}''(\unicode {f817}) \unicode {f817}^4+\unicode {f818}'(\unicode {f817}) \unicode {f817}^3+\left (c \unicode {f817}^4+b \unicode {f817}^2+a\right ) \unicode {f818}(\unicode {f817})=0,\unicode {f818}(1)=c_1,\unicode {f818}'(1)=c_2\right \}\right )(x)\right \}\right \}\]

Maple ✓
cpu = 0.319 (sec), leaf count = 81

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

DSolve[(a + b*x^2 + c*x^4)*y[x] + x^3*y'[x] + x^4*y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> DifferentialRoot[Function[{\[FormalY], \[FormalX]}, {(a + \[FormalX]^2
*b + \[FormalX]^4*c)*\[FormalY][\[FormalX]] + \[FormalX]^3*Derivative[1][\[Forma
lY]][\[FormalX]] + \[FormalX]^4*Derivative[2][\[FormalY]][\[FormalX]] == 0, \[Fo
rmalY][1] == C[1], Derivative[1][\[FormalY]][1] == C[2]}]][x]}}

Maple raw input

dsolve(x^4*diff(diff(y(x),x),x)+x^3*diff(y(x),x)+(c*x^4+b*x^2+a)*y(x) = 0, y(x),'implicit')

Maple raw output

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