4.46.5 \(x^2 y''''(x)+8 x y'''(x)+a^2 (-y(x))+12 y''(x)=0\)

ODE
\[ x^2 y''''(x)+8 x y'''(x)+a^2 (-y(x))+12 y''(x)=0 \] ODE Classification

[[_high_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.203316 (sec), leaf count = 146

\[\left \{\left \{y(x)\to c_4 G_{0,4}^{2,0}\left (\frac {a^2 x^2}{16}|\begin {array}{c} -1,0,-\frac {1}{2},\frac {1}{2} \\\end {array}\right )+c_2 G_{0,4}^{2,0}\left (\frac {a^2 x^2}{16}|\begin {array}{c} -\frac {1}{2},\frac {1}{2},-1,0 \\\end {array}\right )-\frac {3 i c_1 \left (I_2\left (2 \sqrt {a} \sqrt {x}\right )-J_2\left (2 \sqrt {a} \sqrt {x}\right )\right )}{4 a x}-\frac {c_3 \left (J_2\left (2 \sqrt {a} \sqrt {x}\right )+I_2\left (2 \sqrt {a} \sqrt {x}\right )\right )}{a x}\right \}\right \}\]

Maple ✓
cpu = 0.088 (sec), leaf count = 69

\[\left [y \left (x \right ) = \frac {\textit {\_C1} \BesselJ \left (2, 2 \sqrt {a}\, \sqrt {x}\right )}{x}+\frac {\textit {\_C2} \BesselY \left (2, 2 \sqrt {a}\, \sqrt {x}\right )}{x}+\frac {\textit {\_C3} \BesselJ \left (2, 2 \sqrt {-a}\, \sqrt {x}\right )}{x}+\frac {\textit {\_C4} \BesselY \left (2, 2 \sqrt {-a}\, \sqrt {x}\right )}{x}\right ]\] Mathematica raw input

DSolve[-(a^2*y[x]) + 12*y''[x] + 8*x*y'''[x] + x^2*y''''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (((-3*I)/4)*(BesselI[2, 2*Sqrt[a]*Sqrt[x]] - BesselJ[2, 2*Sqrt[a]*Sqrt
[x]])*C[1])/(a*x) - ((BesselI[2, 2*Sqrt[a]*Sqrt[x]] + BesselJ[2, 2*Sqrt[a]*Sqrt[
x]])*C[3])/(a*x) + C[4]*MeijerG[{{}, {}}, {{-1, 0}, {-1/2, 1/2}}, (a^2*x^2)/16] 
+ C[2]*MeijerG[{{}, {}}, {{-1/2, 1/2}, {-1, 0}}, (a^2*x^2)/16]}}

Maple raw input

dsolve(x^2*diff(diff(diff(diff(y(x),x),x),x),x)+8*x*diff(diff(diff(y(x),x),x),x)+12*diff(diff(y(x),x),x)-a^2*y(x) = 0, y(x))

Maple raw output

[y(x) = _C1/x*BesselJ(2,2*a^(1/2)*x^(1/2))+_C2/x*BesselY(2,2*a^(1/2)*x^(1/2))+_C
3/x*BesselJ(2,2*(-a)^(1/2)*x^(1/2))+_C4/x*BesselY(2,2*(-a)^(1/2)*x^(1/2))]