Internal problem ID [9136]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1557.
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 }+8 x y^{\prime \prime \prime }+12 y^{\prime \prime }-\lambda ^{2} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 69
dsolve(x^2*diff(diff(diff(diff(y(x),x),x),x),x)+8*x*diff(diff(diff(y(x),x),x),x)+12*diff(diff(y(x),x),x)-lambda^2*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \BesselJ \left (2, 2 \sqrt {\lambda }\, \sqrt {x}\right )}{x}+\frac {c_{2} \BesselY \left (2, 2 \sqrt {\lambda }\, \sqrt {x}\right )}{x}+\frac {c_{3} \BesselJ \left (2, 2 \sqrt {-\lambda }\, \sqrt {x}\right )}{x}+\frac {c_{4} \BesselY \left (2, 2 \sqrt {-\lambda }\, \sqrt {x}\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.035 (sec). Leaf size: 146
DSolve[-(\[Lambda]^2*y[x]) + 12*y''[x] + 8*x*Derivative[3][y][x] + x^2*Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_4 G_{0,4}^{2,0}\left (\frac {x^2 \lambda ^2}{16}| {c} -1,0,-\frac {1}{2},\frac {1}{2} \\ \\ \right )+c_2 G_{0,4}^{2,0}\left (\frac {x^2 \lambda ^2}{16}| {c} -\frac {1}{2},\frac {1}{2},-1,0 \\ \\ \right )-\frac {3 i c_1 \left (I_2\left (2 \sqrt {x} \sqrt {\lambda }\right )-J_2\left (2 \sqrt {x} \sqrt {\lambda }\right )\right )}{4 \lambda x}-\frac {c_3 \left (J_2\left (2 \sqrt {x} \sqrt {\lambda }\right )+I_2\left (2 \sqrt {x} \sqrt {\lambda }\right )\right )}{\lambda x} \\ \end{align*}