60.5.22 problem 1559

Internal problem ID [11519]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1559
Date solved : Sunday, March 30, 2025 at 08:24:05 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*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 33

ode:=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; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \operatorname {BesselI}\left (0, a x \right )+c_2 \operatorname {BesselJ}\left (0, a x \right )+c_3 \operatorname {BesselK}\left (0, a x \right )+c_4 \operatorname {BesselY}\left (0, a x \right ) \]

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

ode=-(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; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a**4*x**3*y(x) + x**3*Derivative(y(x), (x, 4)) + 2*x**2*Derivative(y(x), (x, 3)) - x*Derivative(y(x), (x, 2)) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -x*(a**4*x**2*y(x) - x**2*Derivative(y(x), (x, 4)) - 2*x*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 2))) + Derivative(y(x), x) cannot be solved by the factorable group method