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 = 0.0730083 (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 = 0.092 (sec), leaf count = 61
\[ \left \{ y \left ( x \right ) ={\frac {1}{x} \left ( {\it \_C4}\,{{\sl Y}_{2}\left (2\,\sqrt {-a}\sqrt {x}\right )}+{\it \_C2}\,{{\sl Y}_{2}\left (2\,\sqrt {a}\sqrt {x}\right )}+{\it \_C1}\,{{\sl J}_{2}\left (2\,\sqrt {a}\sqrt {x}\right )}+{\it \_C3}\,{{\sl J}_{2}\left (2\,\sqrt {-a}\sqrt {x}\right )} \right ) } \right \} \] 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),'implicit')
Maple raw output
y(x) = (_C4*BesselY(2,2*(-a)^(1/2)*x^(1/2))+_C2*BesselY(2,2*a^(1/2)*x^(1/2))+_C1*BesselJ(2,2*a^(1/2)*x^(1/2))+_C3*BesselJ(2,2*(-a)^(1/2)*x^(1/2)))/x