89.16.23 problem 23

Internal problem ID [24673]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 9. Nonhomogeneous Equations: Undetermined coefficients. Exercises at page 140
Problem number : 23
Date solved : Thursday, October 02, 2025 at 10:46:55 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime \prime }-y&={\mathrm e}^{-x} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 30

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

\[ y = \frac {\left (-x +4 c_4 \right ) {\mathrm e}^{-x}}{4}+c_1 \cos \left (x \right )+c_2 \,{\mathrm e}^{x}+c_3 \sin \left (x \right ) \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 50

ode=D[y[x],{x,4}]-y[x]== Exp[-x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.069 (sec). Leaf size: 26

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

\[ y{\left (x \right )} = C_{2} e^{x} + C_{3} \sin {\left (x \right )} + C_{4} \cos {\left (x \right )} + \left (C_{1} - \frac {x}{4}\right ) e^{- x} \]