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