88.20.6 problem 6

Internal problem ID [24144]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 5. Special Techniques for Linear Equations. Exercises at page 146
Problem number : 6
Date solved : Thursday, October 02, 2025 at 10:00:11 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\left (8\right )}+y&=x^{15} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 177

ode:=diff(diff(diff(diff(diff(diff(diff(diff(y(x),x),x),x),x),x),x),x),x)+y(x) = x^15; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (c_5 \,{\mathrm e}^{-\frac {\sqrt {2+\sqrt {2}}\, x}{2}}+c_7 \,{\mathrm e}^{\frac {\sqrt {2+\sqrt {2}}\, x}{2}}\right ) \cos \left (\frac {\sqrt {2-\sqrt {2}}\, x}{2}\right )+\left (c_1 \cos \left (\frac {\sqrt {2+\sqrt {2}}\, x}{2}\right )+c_2 \sin \left (\frac {\sqrt {2+\sqrt {2}}\, x}{2}\right )\right ) {\mathrm e}^{-\frac {\sqrt {2-\sqrt {2}}\, x}{2}}+\left (c_3 \cos \left (\frac {\sqrt {2+\sqrt {2}}\, x}{2}\right )+c_4 \sin \left (\frac {\sqrt {2+\sqrt {2}}\, x}{2}\right )\right ) {\mathrm e}^{\frac {\sqrt {2-\sqrt {2}}\, x}{2}}+\left ({\mathrm e}^{-\frac {\sqrt {2+\sqrt {2}}\, x}{2}} c_6 +{\mathrm e}^{\frac {\sqrt {2+\sqrt {2}}\, x}{2}} c_8 \right ) \sin \left (\frac {\sqrt {2-\sqrt {2}}\, x}{2}\right )+x^{15}-259459200 x^{7} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 180

ode=D[y[x],{x,8}]+y[x]==x^15; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to x^{15}-259459200 x^7+e^{-x \sin \left (\frac {\pi }{8}\right )} \cos \left (x \cos \left (\frac {\pi }{8}\right )\right ) \left (c_2 e^{2 x \sin \left (\frac {\pi }{8}\right )}+c_3\right )+e^{-x \cos \left (\frac {\pi }{8}\right )} \left (c_1 e^{2 x \cos \left (\frac {\pi }{8}\right )}+c_4\right ) \cos \left (x \sin \left (\frac {\pi }{8}\right )\right )+c_6 e^{-x \sin \left (\frac {\pi }{8}\right )} \sin \left (x \cos \left (\frac {\pi }{8}\right )\right )+c_7 e^{x \sin \left (\frac {\pi }{8}\right )} \sin \left (x \cos \left (\frac {\pi }{8}\right )\right )+c_5 \sin \left (x \sin \left (\frac {\pi }{8}\right )\right ) e^{-x \cos \left (\frac {\pi }{8}\right )}+c_8 \sin \left (x \sin \left (\frac {\pi }{8}\right )\right ) e^{x \cos \left (\frac {\pi }{8}\right )} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**15 + y(x) + Derivative(y(x), (x, 8)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : Cannot find 8 solutions to the homogeneous equation necessary to apply undeter