Internal problem ID [9138]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1559.
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{3} y^{\prime \prime \prime \prime }+2 x^{2} y^{\prime \prime \prime }-y^{\prime \prime } x +y^{\prime }-a^{4} x^{3} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 33
dsolve(x^3*diff(diff(diff(diff(y(x),x),x),x),x)+2*x^2*diff(diff(diff(y(x),x),x),x)-x*diff(diff(y(x),x),x)+diff(y(x),x)-a^4*x^3*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \BesselI \left (0, a x \right )+c_{2} \BesselJ \left (0, a x \right )+c_{3} \BesselK \left (0, a x \right )+c_{4} \BesselY \left (0, a x \right ) \]
✓ Solution by Mathematica
Time used: 0.14 (sec). Leaf size: 100
DSolve[-(a^4*x^3*y[x]) + y'[x] - x*y''[x] + 2*x^2*Derivative[3][y][x] + x^3*Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_4 G_{0,4}^{2,0}\left (\frac {a^4 x^4}{256}| {c} 0,0,\frac {1}{2},\frac {1}{2} \\ \\ \right )+c_2 G_{0,4}^{2,0}\left (\frac {a^4 x^4}{256}| {c} \frac {1}{2},\frac {1}{2},0,0 \\ \\ \right )+\frac {1}{8} i c_1 (I_0(a x)-J_0(a x))+\frac {1}{2} c_3 (J_0(a x)+I_0(a x)) \\ \end{align*}