4.25.16 \(y(x) \left (\text {a0}+\text {a1} x+\text {a2} x^2+\text {a3} x^3+\text {a4} x^4+x^8\right )+y''(x)=0\)

ODE
\[ y(x) \left (\text {a0}+\text {a1} x+\text {a2} x^2+\text {a3} x^3+\text {a4} x^4+x^8\right )+y''(x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✗
cpu = 2.13896 (sec), leaf count = 0 , DifferentialRoot result

\[\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{\left (\unicode {f817}^8+\text {a4} \unicode {f817}^4+\text {a3} \unicode {f817}^3+\text {a2} \unicode {f817}^2+\text {a1} \unicode {f817}+\text {a0}\right ) \unicode {f818}(\unicode {f817})+\unicode {f818}''(\unicode {f817})=0,\unicode {f818}(0)=c_1,\unicode {f818}'(0)=c_2\right \}\right )(x)\right \}\right \}\]

Maple ✗
cpu = 1.177 (sec), leaf count = 0 , result contains DESol

\[ \left \{ y \left ( x \right ) ={\it DESol} \left ( \left \{ {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}{\it \_Y} \left ( x \right ) +\sum _{n=0}^{m}a \left ( n \right ) {x}^{n}{\it \_Y} \left ( x \right ) \right \} , \left \{ {\it \_Y} \left ( x \right ) \right \} \right ) \right \} \]

Mathematica raw input

DSolve[(a0 + a1*x + a2*x^2 + a3*x^3 + a4*x^4 + x^8)*y[x] + y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> DifferentialRoot[Function[{\[FormalY], \[FormalX]}, {(\[FormalX]^8 + a
0 + \[FormalX]*a1 + \[FormalX]^2*a2 + \[FormalX]^3*a3 + \[FormalX]^4*a4)*\[Forma
lY][\[FormalX]] + Derivative[2][\[FormalY]][\[FormalX]] == 0, \[FormalY][0] == C
[1], Derivative[1][\[FormalY]][0] == C[2]}]][x]}}

Maple raw input

dsolve(diff(diff(y(x),x),x)+sum(a(n)*x^n,n = 0 .. m)*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = DESol({diff(diff(_Y(x),x),x)+sum(a(n)*x^n,n = 0 .. m)*_Y(x)},{_Y(x)})