ODE
\[ 16 a^4 x^4 y''''(x)-32 (1-2 a) a^2 x^3 y'''(x)+16 (1-2 a) (1-a) a^2 x^2 y''(x)-b^4 x^{2/a} y(x)=0 \] ODE Classification
[[_high_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.252259 (sec), leaf count = 860
\[\left \{\left \{y(x)\to \left (\frac {i}{16}\right )^a b^{2 a} c_2 \, _0F_3\left (;\frac {a}{2}+1,-\frac {3 a}{4}+2-\frac {\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{4 a}-\frac {1}{2 a},-\frac {3 a}{4}+2+\frac {\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{4 a}-\frac {1}{2 a};\frac {1}{256} b^4 x^{2/a}\right ) \left (x^{2/a}\right )^{a/2}+(-1)^{\frac {5 a}{4}+1-\frac {\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{4 a}} i^{\frac {1}{a}} 4^{-\frac {5 a^2-4 a+\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}+2}{a}} b^{\frac {5 a^2-4 a-\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}+2}{a}} \left (i^{\frac {\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{a}} b^{\frac {2 \sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{a}} c_4 \, _0F_3\left (;\frac {3 a}{4}+\frac {\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{4 a}+\frac {1}{2 a},\frac {5 a}{4}+\frac {\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{4 a}+\frac {1}{2 a},\frac {\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{2 a}+1;\frac {1}{256} b^4 x^{2/a}\right ) \left (x^{2/a}\right )^{\frac {\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{2 a}}+16^{\frac {\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{a}} c_3 \, _0F_3\left (;1-\frac {\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{2 a},\frac {3 a}{4}-\frac {\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{4 a}+\frac {1}{2 a},\frac {5 a}{4}-\frac {\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{4 a}+\frac {1}{2 a};\frac {1}{256} b^4 x^{2/a}\right )\right ) \left (x^{2/a}\right )^{-\frac {-5 a^2+4 a+\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}-2}{4 a}}+c_1 \, _0F_3\left (;1-\frac {a}{2},-\frac {5 a}{4}+2-\frac {\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{4 a}-\frac {1}{2 a},-\frac {5 a}{4}+2+\frac {\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{4 a}-\frac {1}{2 a};\frac {1}{256} b^4 x^{2/a}\right )\right \}\right \}\]
Maple ✓
cpu = 0.5 (sec), leaf count = 508
\[\left [y \left (x \right ) = \textit {\_C1} \hypergeom \left (\left [\right ], \left [-\frac {a}{2}+1, -\frac {5 a^{2}+\sqrt {-7 a^{4}+4 a^{3}+16 a^{2}-16 a +4}-8 a +2}{4 a}, \frac {-5 a^{2}+\sqrt {-7 a^{4}+4 a^{3}+16 a^{2}-16 a +4}+8 a -2}{4 a}\right ], \frac {b^{4} x^{\frac {2}{a}}}{256}\right )+\textit {\_C2} x \hypergeom \left (\left [\right ], \left [\frac {a}{2}+1, -\frac {3 a^{2}+\sqrt {-7 a^{4}+4 a^{3}+16 a^{2}-16 a +4}-8 a +2}{4 a}, \frac {-3 a^{2}+\sqrt {-7 a^{4}+4 a^{3}+16 a^{2}-16 a +4}+8 a -2}{4 a}\right ], \frac {b^{4} x^{\frac {2}{a}}}{256}\right )+\textit {\_C3} \,x^{\frac {5 a^{2}-4 a +2+\sqrt {-7 a^{4}+4 a^{3}+16 a^{2}-16 a +4}}{2 a^{2}}} \hypergeom \left (\left [\right ], \left [\frac {2 a +\sqrt {-7 a^{4}+4 a^{3}+16 a^{2}-16 a +4}}{2 a}, \frac {3 a^{2}+\sqrt {-7 a^{4}+4 a^{3}+16 a^{2}-16 a +4}+2}{4 a}, \frac {5 a^{2}+\sqrt {-7 a^{4}+4 a^{3}+16 a^{2}-16 a +4}+2}{4 a}\right ], \frac {b^{4} x^{\frac {2}{a}}}{256}\right )+\textit {\_C4} \,x^{-\frac {-5 a^{2}+\sqrt {-7 a^{4}+4 a^{3}+16 a^{2}-16 a +4}+4 a -2}{2 a^{2}}} \hypergeom \left (\left [\right ], \left [-\frac {-2 a +\sqrt {-7 a^{4}+4 a^{3}+16 a^{2}-16 a +4}}{2 a}, -\frac {-5 a^{2}+\sqrt {-7 a^{4}+4 a^{3}+16 a^{2}-16 a +4}-2}{4 a}, -\frac {-3 a^{2}+\sqrt {-7 a^{4}+4 a^{3}+16 a^{2}-16 a +4}-2}{4 a}\right ], \frac {b^{4} x^{\frac {2}{a}}}{256}\right )\right ]\] Mathematica raw input
DSolve[-(b^4*x^(2/a)*y[x]) + 16*(1 - 2*a)*(1 - a)*a^2*x^2*y''[x] - 32*(1 - 2*a)*a^2*x^3*y'''[x] + 16*a^4*x^4*y''''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*HypergeometricPFQ[{}, {1 - a/2, 2 - 1/(2*a) - (5*a)/4 - Sqrt[4 - 16*a + 16*a^2 + 4*a^3 - 7*a^4]/(4*a), 2 - 1/(2*a) - (5*a)/4 + Sqrt[4 - 16*a + 16*a^2 + 4*a^3 - 7*a^4]/(4*a)}, (b^4*x^(2/a))/256] + (I/16)^a*b^(2*a)*(x^(2/a))^(a/2)*C[2]*HypergeometricPFQ[{}, {1 + a/2, 2 - 1/(2*a) - (3*a)/4 - Sqrt[4 - 16*a + 16*a^2 + 4*a^3 - 7*a^4]/(4*a), 2 - 1/(2*a) - (3*a)/4 + Sqrt[4 - 16*a + 16*a^2 + 4*a^3 - 7*a^4]/(4*a)}, (b^4*x^(2/a))/256] + ((-1)^(1 + (5*a)/4 - Sqrt[4 - 16*a + 16*a^2 + 4*a^3 - 7*a^4]/(4*a))*I^a^(-1)*b^((2 - 4*a + 5*a^2 - Sqrt[4 - 16*a + 16*a^2 + 4*a^3 - 7*a^4])/a)*(16^(Sqrt[4 - 16*a + 16*a^2 + 4*a^3 - 7*a^4]/a)*C[3]*HypergeometricPFQ[{}, {1 - Sqrt[4 - 16*a + 16*a^2 + 4*a^3 - 7*a^4]/(2*a), 1/(2*a) + (3*a)/4 - Sqrt[4 - 16*a + 16*a^2 + 4*a^3 - 7*a^4]/(4*a), 1/(2*a) + (5*a)/4 - Sqrt[4 - 16*a + 16*a^2 + 4*a^3 - 7*a^4]/(4*a)}, (b^4*x^(2/a))/256] + I^(Sqrt[4 - 16*a + 16*a^2 + 4*a^3 - 7*a^4]/a)*b^((2*Sqrt[4 - 16*a + 16*a^2 + 4*a^3 - 7*a^4])/a)*(x^(2/a))^(Sqrt[4 - 16*a + 16*a^2 + 4*a^3 - 7*a^4]/(2*a))*C[4]*HypergeometricPFQ[{}, {1/(2*a) + (3*a)/4 + Sqrt[4 - 16*a + 16*a^2 + 4*a^3 - 7*a^4]/(4*a), 1/(2*a) + (5*a)/4 + Sqrt[4 - 16*a + 16*a^2 + 4*a^3 - 7*a^4]/(4*a), 1 + Sqrt[4 - 16*a + 16*a^2 + 4*a^3 - 7*a^4]/(2*a)}, (b^4*x^(2/a))/256]))/(4^((2 - 4*a + 5*a^2 + Sqrt[4 - 16*a + 16*a^2 + 4*a^3 - 7*a^4])/a)*(x^(2/a))^((-2 + 4*a - 5*a^2 + Sqrt[4 - 16*a + 16*a^2 + 4*a^3 - 7*a^4])/(4*a)))}}
Maple raw input
dsolve(16*a^4*x^4*diff(diff(diff(diff(y(x),x),x),x),x)-32*a^2*x^3*(1-2*a)*diff(diff(diff(y(x),x),x),x)+16*a^2*x^2*(1-a)*(1-2*a)*diff(diff(y(x),x),x)-b^4*x^(2/a)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*hypergeom([],[-1/2*a+1, -1/4*(5*a^2+(-7*a^4+4*a^3+16*a^2-16*a+4)^(1/2)-8*a+2)/a, 1/4*(-5*a^2+(-7*a^4+4*a^3+16*a^2-16*a+4)^(1/2)+8*a-2)/a],1/256*b^4*x^(2/a))+_C2*x*hypergeom([],[1/2*a+1, -1/4*(3*a^2+(-7*a^4+4*a^3+16*a^2-16*a+4)^(1/2)-8*a+2)/a, 1/4*(-3*a^2+(-7*a^4+4*a^3+16*a^2-16*a+4)^(1/2)+8*a-2)/a],1/256*b^4*x^(2/a))+_C3*x^(1/2/a^2*(5*a^2-4*a+2+(-7*a^4+4*a^3+16*a^2-16*a+4)^(1/2)))*hypergeom([],[1/2*(2*a+(-7*a^4+4*a^3+16*a^2-16*a+4)^(1/2))/a, 1/4*(3*a^2+(-7*a^4+4*a^3+16*a^2-16*a+4)^(1/2)+2)/a, 1/4*(5*a^2+(-7*a^4+4*a^3+16*a^2-16*a+4)^(1/2)+2)/a],1/256*b^4*x^(2/a))+_C4*x^(-1/2*(-5*a^2+(-7*a^4+4*a^3+16*a^2-16*a+4)^(1/2)+4*a-2)/a^2)*hypergeom([],[-1/2*(-2*a+(-7*a^4+4*a^3+16*a^2-16*a+4)^(1/2))/a, -1/4*(-5*a^2+(-7*a^4+4*a^3+16*a^2-16*a+4)^(1/2)-2)/a, -1/4*(-3*a^2+(-7*a^4+4*a^3+16*a^2-16*a+4)^(1/2)-2)/a],1/256*b^4*x^(2/a))]