60.5.20 problem 1555

Internal problem ID [11554]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1555
Date solved : Tuesday, January 28, 2025 at 06:06:49 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime \prime \prime }+6 x y^{\prime \prime \prime }+6 y^{\prime \prime }-\lambda ^{2} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.177 (sec). Leaf size: 61

dsolve(x^2*diff(diff(diff(diff(y(x),x),x),x),x)+6*x*diff(diff(diff(y(x),x),x),x)+6*diff(diff(y(x),x),x)-lambda^2*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {c_{1} \operatorname {BesselJ}\left (1, 2 \sqrt {\lambda }\, \sqrt {x}\right )+c_3 \operatorname {BesselJ}\left (1, 2 \sqrt {-\lambda }\, \sqrt {x}\right )+c_4 \operatorname {BesselY}\left (1, 2 \sqrt {-\lambda }\, \sqrt {x}\right )+c_{2} \operatorname {BesselY}\left (1, 2 \sqrt {\lambda }\, \sqrt {x}\right )}{\sqrt {x}} \]

✓ Solution by Mathematica

Time used: 0.033 (sec). Leaf size: 156

DSolve[-(\[Lambda]^2*y[x]) + 6*D[y[x],{x,2}] + 6*x*Derivative[3][y][x] + x^2*Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_4 G_{0,4}^{2,0}\left (\frac {x^2 \lambda ^2}{16}| \begin {array}{c} -\frac {1}{2},\frac {1}{2},0,0 \\ \end {array} \right )+c_2 G_{0,4}^{2,0}\left (\frac {x^2 \lambda ^2}{16}| \begin {array}{c} 0,0,-\frac {1}{2},\frac {1}{2} \\ \end {array} \right )+\frac {c_1 \left (\operatorname {BesselJ}\left (1,2 \sqrt {x} \sqrt {\lambda }\right )+\operatorname {BesselI}\left (1,2 \sqrt {x} \sqrt {\lambda }\right )\right )}{2 \sqrt {\lambda } \sqrt {x}}-\frac {i c_3 \left (\operatorname {BesselI}\left (1,2 \sqrt {x} \sqrt {\lambda }\right )-\operatorname {BesselJ}\left (1,2 \sqrt {x} \sqrt {\lambda }\right )\right )}{4 \sqrt {\lambda } \sqrt {x}} \]