ODE
\[ x^3 y''''(x)+2 x^2 y'''(x)+a^4 \left (-x^3\right ) y(x)-x y''(x)=0 \] ODE Classification
[[_high_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.856563 (sec), leaf count = 305
\[\left \{\left \{y(x)\to c_1 \, _0F_3\left (;\frac {3}{4},\frac {5}{8}-\frac {\sqrt {5}}{8},\frac {5}{8}+\frac {\sqrt {5}}{8};\frac {a^4 x^4}{256}\right )+\frac {1}{8} \left ((1+i) \sqrt {2} a c_2 x \, _0F_3\left (;\frac {5}{4},\frac {7}{8}-\frac {\sqrt {5}}{8},\frac {7}{8}+\frac {\sqrt {5}}{8};\frac {a^4 x^4}{256}\right )+(-1)^{\frac {1}{8} \left (3-\sqrt {5}\right )} 2^{-\sqrt {5}} a^{\frac {3}{2}-\frac {\sqrt {5}}{2}} x^{\frac {3}{2}-\frac {\sqrt {5}}{2}} \left (4^{\sqrt {5}} c_3 \, _0F_3\left (;1-\frac {\sqrt {5}}{4},\frac {9}{8}-\frac {\sqrt {5}}{8},\frac {11}{8}-\frac {\sqrt {5}}{8};\frac {a^4 x^4}{256}\right )+(-1)^{\frac {\sqrt {5}}{4}} a^{\sqrt {5}} c_4 x^{\sqrt {5}} \, _0F_3\left (;\frac {9}{8}+\frac {\sqrt {5}}{8},\frac {11}{8}+\frac {\sqrt {5}}{8},1+\frac {\sqrt {5}}{4};\frac {a^4 x^4}{256}\right )\right )\right )\right \}\right \}\]
Maple ✓
cpu = 6.614 (sec), leaf count = 148
\[\left [y \left (x \right ) = \textit {\_C1} \hypergeom \left (\left [\right ], \left [\frac {3}{4}, \frac {5}{8}+\frac {\sqrt {5}}{8}, \frac {5}{8}-\frac {\sqrt {5}}{8}\right ], \frac {a^{4} x^{4}}{256}\right )+\textit {\_C2} x \hypergeom \left (\left [\right ], \left [\frac {5}{4}, \frac {7}{8}+\frac {\sqrt {5}}{8}, \frac {7}{8}-\frac {\sqrt {5}}{8}\right ], \frac {a^{4} x^{4}}{256}\right )+\textit {\_C3} \,x^{\frac {3}{2}-\frac {\sqrt {5}}{2}} \hypergeom \left (\left [\right ], \left [1-\frac {\sqrt {5}}{4}, \frac {9}{8}-\frac {\sqrt {5}}{8}, \frac {11}{8}-\frac {\sqrt {5}}{8}\right ], \frac {a^{4} x^{4}}{256}\right )+\textit {\_C4} \,x^{\frac {3}{2}+\frac {\sqrt {5}}{2}} \hypergeom \left (\left [\right ], \left [1+\frac {\sqrt {5}}{4}, \frac {9}{8}+\frac {\sqrt {5}}{8}, \frac {11}{8}+\frac {\sqrt {5}}{8}\right ], \frac {a^{4} x^{4}}{256}\right )\right ]\] Mathematica raw input
DSolve[-(a^4*x^3*y[x]) - x*y''[x] + 2*x^2*y'''[x] + x^3*y''''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*HypergeometricPFQ[{}, {3/4, 5/8 - Sqrt[5]/8, 5/8 + Sqrt[5]/8}, (a^4*x^4)/256] + ((1 + I)*Sqrt[2]*a*x*C[2]*HypergeometricPFQ[{}, {5/4, 7/8 - Sqrt[5]/8, 7/8 + Sqrt[5]/8}, (a^4*x^4)/256] + ((-1)^((3 - Sqrt[5])/8)*a^(3/2 - Sqrt[5]/2)*x^(3/2 - Sqrt[5]/2)*(4^Sqrt[5]*C[3]*HypergeometricPFQ[{}, {1 - Sqrt[5]/4, 9/8 - Sqrt[5]/8, 11/8 - Sqrt[5]/8}, (a^4*x^4)/256] + (-1)^(Sqrt[5]/4)*a^Sqrt[5]*x^Sqrt[5]*C[4]*HypergeometricPFQ[{}, {9/8 + Sqrt[5]/8, 11/8 + Sqrt[5]/8, 1 + Sqrt[5]/4}, (a^4*x^4)/256]))/2^Sqrt[5])/8}}
Maple raw input
dsolve(x^3*diff(diff(diff(diff(y(x),x),x),x),x)+2*x^2*diff(diff(diff(y(x),x),x),x)-x*diff(diff(y(x),x),x)-a^4*x^3*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*hypergeom([],[3/4, 5/8+1/8*5^(1/2), 5/8-1/8*5^(1/2)],1/256*a^4*x^4)+_C2*x*hypergeom([],[5/4, 7/8+1/8*5^(1/2), 7/8-1/8*5^(1/2)],1/256*a^4*x^4)+_C3*x^(3/2-1/2*5^(1/2))*hypergeom([],[1-1/4*5^(1/2), 9/8-1/8*5^(1/2), 11/8-1/8*5^(1/2)],1/256*a^4*x^4)+_C4*x^(3/2+1/2*5^(1/2))*hypergeom([],[1+1/4*5^(1/2), 9/8+1/8*5^(1/2), 11/8+1/8*5^(1/2)],1/256*a^4*x^4)]