60.5.24 problem 1559

Internal problem ID [11558]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1559
Date solved : Tuesday, January 28, 2025 at 06:06:51 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} x^{3} y^{\prime \prime \prime \prime }+2 x^{2} y^{\prime \prime \prime }-x y^{\prime \prime }+y^{\prime }-a^{4} x^{3} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.013 (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)
 

\[ y = c_{1} \operatorname {BesselI}\left (0, a x \right )+c_{2} \operatorname {BesselJ}\left (0, a x \right )+c_3 \operatorname {BesselK}\left (0, a x \right )+c_4 \operatorname {BesselY}\left (0, a x \right ) \]

✓ Solution by Mathematica

Time used: 0.142 (sec). Leaf size: 100

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

\[ y(x)\to c_4 G_{0,4}^{2,0}\left (\frac {a^4 x^4}{256}| \begin {array}{c} 0,0,\frac {1}{2},\frac {1}{2} \\ \end {array} \right )+c_2 G_{0,4}^{2,0}\left (\frac {a^4 x^4}{256}| \begin {array}{c} \frac {1}{2},\frac {1}{2},0,0 \\ \end {array} \right )+\frac {1}{8} i c_1 (\operatorname {BesselI}(0,a x)-\operatorname {BesselJ}(0,a x))+\frac {1}{2} c_3 (\operatorname {BesselJ}(0,a x)+\operatorname {BesselI}(0,a x)) \]