\[ \lambda ^2 (-y(x))+x^2 y^{(4)}(x)+8 x y^{(3)}(x)+12 y''(x)=0 \] ✓ Mathematica : cpu = 0.048755 (sec), leaf count = 146
DSolve[-(lambda^2*y[x]) + 12*Derivative[2][y][x] + 8*x*Derivative[3][y][x] + x^2*Derivative[4][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_4 G_{0,4}^{2,0}\left (\frac {\lambda ^2 x^2}{16}|\begin {array}{c} -1,0,-\frac {1}{2},\frac {1}{2} \\\end {array}\right )+c_2 G_{0,4}^{2,0}\left (\frac {\lambda ^2 x^2}{16}|\begin {array}{c} -\frac {1}{2},\frac {1}{2},-1,0 \\\end {array}\right )-\frac {3 i c_1 \left (I_2\left (2 \sqrt {\lambda } \sqrt {x}\right )-J_2\left (2 \sqrt {\lambda } \sqrt {x}\right )\right )}{4 \lambda x}-\frac {c_3 \left (J_2\left (2 \sqrt {\lambda } \sqrt {x}\right )+I_2\left (2 \sqrt {\lambda } \sqrt {x}\right )\right )}{\lambda x}\right \}\right \}\] ✓ Maple : cpu = 0.091 (sec), leaf count = 61
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)-lambda^2*y(x)=0,y(x))
\[y \left (x \right ) = \frac {c_{3} \BesselJ \left (2, 2 \sqrt {-\lambda }\, \sqrt {x}\right )+c_{1} \BesselJ \left (2, 2 \sqrt {\lambda }\, \sqrt {x}\right )+c_{4} \BesselY \left (2, 2 \sqrt {-\lambda }\, \sqrt {x}\right )+c_{2} \BesselY \left (2, 2 \sqrt {\lambda }\, \sqrt {x}\right )}{x}\]