57.9.17 problem 2(e)

Internal problem ID [14425]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 2, Second order linear equations. Section 2.3.1 Nonhomogeneous Equations: Undetermined Coefficients. Exercises page 110
Problem number : 2(e)
Date solved : Thursday, October 02, 2025 at 09:37:02 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{\prime \prime }+x&=9 \,{\mathrm e}^{-t} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 19

ode:=diff(diff(x(t),t),t)+x(t) = 9*exp(-t); 
dsolve(ode,x(t), singsol=all);
 

\[ x = \sin \left (t \right ) c_2 +\cos \left (t \right ) c_1 +\frac {9 \,{\mathrm e}^{-t}}{2} \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 25

ode=D[x[t],{t,2}]+x[t]==9*Exp[-t]; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {9 e^{-t}}{2}+c_1 \cos (t)+c_2 \sin (t) \end{align*}

✓ Sympy. Time used: 0.038 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(x(t) + Derivative(x(t), (t, 2)) - 9*exp(-t),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = C_{1} \sin {\left (t \right )} + C_{2} \cos {\left (t \right )} + \frac {9 e^{- t}}{2} \]