ODE
\[ 32 x (a-b+2) y'''(x)+16 x^2 y''''(x)+16 (a-b+1) (a-b+2) y''(x)+c^4 (-y(x))=0 \] ODE Classification
[[_high_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.16448 (sec), leaf count = 222
\[\left \{\left \{y(x)\to i^{-a} 2^{a-3 b-3} c^{b-a} x^{\frac {b-a}{2}} \left (i^a 4^b \left (4 c_1 \Gamma (a-b+1)-i c_2 \Gamma (a-b+2)\right ) J_{a-b}\left (c \sqrt {x}\right )+i^a 4^b \left (4 c_1 \Gamma (a-b+1)+i c_2 \Gamma (a-b+2)\right ) I_{a-b}\left (c \sqrt {x}\right )+4^a i^b \left (\left (4 c_3 \Gamma (-a+b+1)-i c_4 \Gamma (-a+b+2)\right ) J_{b-a}\left (c \sqrt {x}\right )+\left (4 c_3 \Gamma (-a+b+1)+i c_4 \Gamma (-a+b+2)\right ) I_{b-a}\left (c \sqrt {x}\right )\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.078 (sec), leaf count = 67
\[ \left \{ y \left ( x \right ) ={x}^{-{\frac {a}{2}}+{\frac {b}{2}}} \left ( {{\sl Y}_{a-b}\left (c\sqrt {x}\right )}{\it \_C4}+{{\sl J}_{a-b}\left (c\sqrt {x}\right )}{\it \_C2}+{{\sl K}_{a-b}\left (c\sqrt {x}\right )}{\it \_C3}+{{\sl I}_{a-b}\left (c\sqrt {x}\right )}{\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[-(c^4*y[x]) + 16*(1 + a - b)*(2 + a - b)*y''[x] + 32*(2 + a - b)*x*y'''[x] + 16*x^2*y''''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (2^(-3 + a - 3*b)*c^(-a + b)*x^((-a + b)/2)*(I^a*4^b*BesselJ[a - b, c*Sqrt[x]]*(4*C[1]*Gamma[1 + a - b] - I*C[2]*Gamma[2 + a - b]) + I^a*4^b*BesselI[a - b, c*Sqrt[x]]*(4*C[1]*Gamma[1 + a - b] + I*C[2]*Gamma[2 + a - b]) + I^b*4^a*(BesselJ[-a + b, c*Sqrt[x]]*(4*C[3]*Gamma[1 - a + b] - I*C[4]*Gamma[2 - a + b]) + BesselI[-a + b, c*Sqrt[x]]*(4*C[3]*Gamma[1 - a + b] + I*C[4]*Gamma[2 - a + b]))))/I^a}}
Maple raw input
dsolve(16*x^2*diff(diff(diff(diff(y(x),x),x),x),x)+32*(2+a-b)*x*diff(diff(diff(y(x),x),x),x)+16*(1+a-b)*(2+a-b)*diff(diff(y(x),x),x)-c^4*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = x^(-1/2*a+1/2*b)*(BesselY(a-b,c*x^(1/2))*_C4+BesselJ(a-b,c*x^(1/2))*_C2+BesselK(a-b,c*x^(1/2))*_C3+BesselI(a-b,c*x^(1/2))*_C1)