87.18.27 problem 27

Internal problem ID [23668]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 135
Problem number : 27
Date solved : Thursday, October 02, 2025 at 09:44:06 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} y^{\left (5\right )}-y^{\prime }-\frac {4 y}{x}&=0 \end{align*}

✗ Maple

ode:=diff(diff(diff(diff(diff(y(x),x),x),x),x),x)-diff(y(x),x)-4*y(x)/x = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.735 (sec). Leaf size: 113

ode=D[y[x],{x,5}]-D[y[x],x]-4/x*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{24} c_5 e^{-x} \left (e^{2 x} x \operatorname {ExpIntegralEi}(-x)+i e^{(1+i) x} x \operatorname {ExpIntegralEi}(-i x)-i e^{(1-i) x} x \operatorname {ExpIntegralEi}(i x)-x \operatorname {ExpIntegralEi}(x)+4 e^x\right )+c_1 e^{-x} x+c_2 e^{i x} x+c_3 e^{-i x} x+c_4 e^x x \end{align*}

✓ Sympy. Time used: 0.463 (sec). Leaf size: 92

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*y(x) - Derivative(y(x), x) + Derivative(y(x), (x, 5)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{5} e^{x \operatorname {CRootOf} {\left (x^{5} - x - 4, 0\right )}} + \left (C_{1} \sin {\left (x \operatorname {im}{\left (\operatorname {CRootOf} {\left (x^{5} - x - 4, 1\right )}\right )} \right )} + C_{2} \cos {\left (x \operatorname {im}{\left (\operatorname {CRootOf} {\left (x^{5} - x - 4, 1\right )}\right )} \right )}\right ) e^{x \operatorname {re}{\left (\operatorname {CRootOf} {\left (x^{5} - x - 4, 1\right )}\right )}} + \left (C_{3} \sin {\left (x \operatorname {im}{\left (\operatorname {CRootOf} {\left (x^{5} - x - 4, 3\right )}\right )} \right )} + C_{4} \cos {\left (x \operatorname {im}{\left (\operatorname {CRootOf} {\left (x^{5} - x - 4, 3\right )}\right )} \right )}\right ) e^{x \operatorname {re}{\left (\operatorname {CRootOf} {\left (x^{5} - x - 4, 3\right )}\right )}} \]