\[ \left (x \left (-\rho ^2-\sigma ^2+1\right )+16 x^3\right ) y'(x)+y(x) \left (\rho ^2 \sigma ^2+8 x^2\right )+\left (x^2 \left (-\rho ^2-\sigma ^2+7\right )+4 x^4\right ) y''(x)+x^4 y^{(4)}(x)+6 x^3 y^{(3)}(x)=0 \] ✓ Mathematica : cpu = 0.380372 (sec), leaf count = 242
\[\left \{\left \{y(x)\to c_1 x^{-\rho } \, _2F_3\left (\frac {1}{2}-\frac {\rho }{2},1-\frac {\rho }{2};1-\rho ,-\frac {\rho }{2}-\frac {\sigma }{2}+1,-\frac {\rho }{2}+\frac {\sigma }{2}+1;-x^2\right )+c_3 x^{-\sigma } \, _2F_3\left (\frac {1}{2}-\frac {\sigma }{2},1-\frac {\sigma }{2};1-\sigma ,-\frac {\rho }{2}-\frac {\sigma }{2}+1,\frac {\rho }{2}-\frac {\sigma }{2}+1;-x^2\right )+c_4 x^{\sigma } \, _2F_3\left (\frac {\sigma }{2}+\frac {1}{2},\frac {\sigma }{2}+1;-\frac {\rho }{2}+\frac {\sigma }{2}+1,\frac {\rho }{2}+\frac {\sigma }{2}+1,\sigma +1;-x^2\right )+c_2 x^{\rho } \, _2F_3\left (\frac {\rho }{2}+\frac {1}{2},\frac {\rho }{2}+1;\rho +1,\frac {\rho }{2}-\frac {\sigma }{2}+1,\frac {\rho }{2}+\frac {\sigma }{2}+1;-x^2\right )\right \}\right \}\] ✓ Maple : cpu = 0.301 (sec), leaf count = 71
\[y \relax (x ) = \left (\BesselY \left (\frac {\rho }{2}-\frac {\sigma }{2}, x\right ) c_{2}+c_{1} \BesselJ \left (\frac {\rho }{2}-\frac {\sigma }{2}, x\right )\right ) \BesselJ \left (\frac {\rho }{2}+\frac {\sigma }{2}, x\right )+\BesselY \left (\frac {\rho }{2}+\frac {\sigma }{2}, x\right ) \left (\BesselY \left (\frac {\rho }{2}-\frac {\sigma }{2}, x\right ) c_{4}+c_{3} \BesselJ \left (\frac {\rho }{2}-\frac {\sigma }{2}, x\right )\right )\]