60.5.23 problem 1558

Internal problem ID [11557]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1558
Date solved : Tuesday, January 28, 2025 at 06:06:50 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} 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{align*}

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 67

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)
 

\[ y = \left (\operatorname {BesselK}\left (n -\nu , b \sqrt {x}\right ) c_3 +\operatorname {BesselI}\left (n -\nu , b \sqrt {x}\right ) c_{1} +\operatorname {BesselY}\left (n -\nu , b \sqrt {x}\right ) c_4 +\operatorname {BesselJ}\left (n -\nu , b \sqrt {x}\right ) c_{2} \right ) x^{-\frac {n}{2}+\frac {\nu }{2}} \]

✓ Solution by Mathematica

Time used: 0.090 (sec). Leaf size: 222

DSolve[-1/16*(b^4*y[x]) + (1 + n - nu)*(2 + n - nu)*D[y[x],{x,2}] + (4 + 2*n - 2*nu)*x*Derivative[3][y][x] + x^2*Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to i^{-n} 2^{n-3 \nu -3} b^{\nu -n} x^{\frac {\nu -n}{2}} \left (i^n 4^{\nu } (4 c_1 \operatorname {Gamma}(n-\nu +1)-i c_2 \operatorname {Gamma}(n-\nu +2)) \operatorname {BesselJ}\left (n-\nu ,b \sqrt {x}\right )+i^n 4^{\nu } (4 c_1 \operatorname {Gamma}(n-\nu +1)+i c_2 \operatorname {Gamma}(n-\nu +2)) \operatorname {BesselI}\left (n-\nu ,b \sqrt {x}\right )+4^n i^{\nu } \left ((4 c_3 \operatorname {Gamma}(-n+\nu +1)-i c_4 \operatorname {Gamma}(-n+\nu +2)) \operatorname {BesselJ}\left (\nu -n,b \sqrt {x}\right )+(4 c_3 \operatorname {Gamma}(-n+\nu +1)+i c_4 \operatorname {Gamma}(-n+\nu +2)) \operatorname {BesselI}\left (\nu -n,b \sqrt {x}\right )\right )\right ) \]