4.46.7 \(32 x (a-b+2) y'''(x)+16 x^2 y''''(x)+16 (a-b+1) (a-b+2) y''(x)+c^4 (-y(x))=0\)

ODE
\[ 32 x (a-b+2) y'''(x)+16 x^2 y''''(x)+16 (a-b+1) (a-b+2) y''(x)+c^4 (-y(x))=0 \] ODE Classification

[[_high_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.275276 (sec), leaf count = 222

\[\left \{\left \{y(x)\to i^{-a} 2^{a-3 b-3} c^{b-a} x^{\frac {b-a}{2}} \left (i^a 4^b (4 c_1 \Gamma (a-b+1)-i c_2 \Gamma (a-b+2)) J_{a-b}\left (c \sqrt {x}\right )+i^a 4^b (4 c_1 \Gamma (a-b+1)+i c_2 \Gamma (a-b+2)) I_{a-b}\left (c \sqrt {x}\right )+4^a i^b \left ((4 c_3 \Gamma (-a+b+1)-i c_4 \Gamma (-a+b+2)) J_{b-a}\left (c \sqrt {x}\right )+(4 c_3 \Gamma (-a+b+1)+i c_4 \Gamma (-a+b+2)) I_{b-a}\left (c \sqrt {x}\right )\right )\right )\right \}\right \}\]

Maple ✓
cpu = 0.123 (sec), leaf count = 93

\[\left [y \left (x \right ) = \textit {\_C1} \,x^{\frac {b}{2}-\frac {a}{2}} \BesselI \left (a -b , c \sqrt {x}\right )+\textit {\_C2} \,x^{\frac {b}{2}-\frac {a}{2}} \BesselJ \left (a -b , c \sqrt {x}\right )+\textit {\_C3} \,x^{\frac {b}{2}-\frac {a}{2}} \BesselK \left (a -b , c \sqrt {x}\right )+\textit {\_C4} \,x^{\frac {b}{2}-\frac {a}{2}} \BesselY \left (a -b , c \sqrt {x}\right )\right ]\] 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))

Maple raw output

[y(x) = _C1*x^(1/2*b-1/2*a)*BesselI(a-b,c*x^(1/2))+_C2*x^(1/2*b-1/2*a)*BesselJ(a
-b,c*x^(1/2))+_C3*x^(1/2*b-1/2*a)*BesselK(a-b,c*x^(1/2))+_C4*x^(1/2*b-1/2*a)*Bes
selY(a-b,c*x^(1/2))]