Internal problem ID [9137]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1558.
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime \prime \prime }+\left (2 n -2 \nu +4\right ) x y^{\prime \prime \prime }+\left (n -\nu +1\right ) \left (n -\nu +2\right ) y^{\prime \prime }-\frac {b^{4} y}{16}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 93
dsolve(x^2*diff(diff(diff(diff(y(x),x),x),x),x)+(2*n-2*nu+4)*x*diff(diff(diff(y(x),x),x),x)+(n-nu+1)*(n-nu+2)*diff(diff(y(x),x),x)-1/16*b^4*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} x^{-\frac {n}{2}+\frac {\nu }{2}} \BesselI \left (n -\nu , b \sqrt {x}\right )+c_{2} x^{-\frac {n}{2}+\frac {\nu }{2}} \BesselJ \left (n -\nu , b \sqrt {x}\right )+c_{3} x^{-\frac {n}{2}+\frac {\nu }{2}} \BesselK \left (n -\nu , b \sqrt {x}\right )+c_{4} x^{-\frac {n}{2}+\frac {\nu }{2}} \BesselY \left (n -\nu , b \sqrt {x}\right ) \]
✓ Solution by Mathematica
Time used: 0.091 (sec). Leaf size: 214
DSolve[-1/16*(b^4*y[x]) + (1 + n - nu)*(2 + n - nu)*y''[x] + (4 + 2*n - 2*nu)*x*Derivative[3][y][x] + x^2*Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to i^{-n} 2^{n-3 \nu -3} b^{\nu -n} x^{\frac {\nu -n}{2}} \left (-i i^n 4^{\nu } (c_2 (n-\nu +1)+4 i c_1) \Gamma (n-\nu +1) J_{n-\nu }\left (b \sqrt {x}\right )+i^{n+1} 4^{\nu } (c_2 (n-\nu +1)-4 i c_1) \Gamma (n-\nu +1) I_{n-\nu }\left (b \sqrt {x}\right )+4^n i^{\nu } \Gamma (-n+\nu +1) \left ((i c_4 (n-\nu -1)+4 c_3) J_{\nu -n}\left (b \sqrt {x}\right )+(i c_4 (-n+\nu +1)+4 c_3) I_{\nu -n}\left (b \sqrt {x}\right )\right )\right ) \\ \end{align*}