81.14.5 problem 18-15

Internal problem ID [21690]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 18. Algebra of differential operators. Page 435
Problem number : 18-15
Date solved : Thursday, October 02, 2025 at 08:00:01 PM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} y^{\left (6\right )}+8 y^{\prime \prime \prime }&=a \,{\mathrm e}^{x} \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 68

ode:=diff(diff(diff(diff(diff(diff(y(x),x),x),x),x),x),x)+8*diff(diff(diff(y(x),x),x),x) = a*exp(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {{\mathrm e}^{-x \left (i \sqrt {3}-1\right )} \left (i c_3 +c_2 \right )}{16}+\frac {\left (i c_3 -c_2 \right ) {\mathrm e}^{x \left (1+i \sqrt {3}\right )}}{16}+\frac {c_4 \,x^{2}}{2}+\frac {a \,{\mathrm e}^{x}}{9}-\frac {c_1 \,{\mathrm e}^{-2 x}}{8}+c_5 x +c_6 \]

✓ Mathematica. Time used: 0.32 (sec). Leaf size: 68

ode=D[y[x],{x,6}]+8*D[y[x],{x,3}]==a*Exp[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{72} \left (8 a e^x-9 c_1 e^{-2 x}-9 c_3 e^x \cos \left (\sqrt {3} x\right )-9 c_2 e^x \sin \left (\sqrt {3} x\right )\right )+c_6 x^2+c_5 x+c_4 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a*exp(x) + 18*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 6)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : Cannot find 6 solutions to the homogeneous equation necessary to apply undetermined coefficients to -a*exp(x) + 18*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 6)) (number of terms != order)