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 = 1.53642 (sec), leaf count = 0 , DifferentialRoot result

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

Maple ✓
cpu = 1.897 (sec), leaf count = 113

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

Maple raw output

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