\[ -\left (4 n^2-1\right ) x^2 y''(x)+\left (4 n^2-1\right ) x y'(x)+x^4 y^{(4)}(x)-4 x^4 y(x)+4 x^3 y^{(3)}(x)=0 \] ✓ Mathematica : cpu = 0.80731 (sec), leaf count = 140
DSolve[-4*x^4*y[x] + (-1 + 4*n^2)*x*Derivative[1][y][x] - (-1 + 4*n^2)*x^2*Derivative[2][y][x] + 4*x^3*Derivative[3][y][x] + x^4*Derivative[4][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_1 \, _0F_3\left (;\frac {1}{2},1-\frac {n}{2},\frac {n}{2}+1;\frac {x^4}{64}\right )+\frac {1}{8} i c_2 x^2 \, _0F_3\left (;\frac {3}{2},\frac {3}{2}-\frac {n}{2},\frac {n}{2}+\frac {3}{2};\frac {x^4}{64}\right )+c_3 \left (\frac {i}{2}\right )^{-n} \Gamma (1-n)^2 \left (\text {ber}_{-n}(x){}^2+\text {bei}_{-n}(x){}^2\right )+c_4 \left (\frac {i}{2}\right )^n \Gamma (n+1)^2 \left (\text {ber}_n(x){}^2+\text {bei}_n(x){}^2\right )\right \}\right \}\] ✓ Maple : cpu = 0.315 (sec), leaf count = 77
dsolve(x^4*diff(diff(diff(diff(y(x),x),x),x),x)+4*x^3*diff(diff(diff(y(x),x),x),x)-(4*n^2-1)*x^2*diff(diff(y(x),x),x)+(4*n^2-1)*x*diff(y(x),x)-4*y(x)*x^4=0,y(x))
\[y \left (x \right ) = \left (\BesselY \left (n , \left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}\, x \right ) c_{3}+c_{1} \BesselJ \left (n , \left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}\, x \right )\right ) \BesselJ \left (n , \left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {2}\, x \right )+\BesselY \left (n , \left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {2}\, x \right ) \left (\BesselY \left (n , \left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}\, x \right ) c_{4}+c_{2} \BesselJ \left (n , \left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}\, x \right )\right )\]