57.9.9 problem 1(i)

Internal problem ID [14417]
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 : 1(i)
Date solved : Thursday, October 02, 2025 at 09:36:56 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{\prime \prime }+x^{\prime }+x&=4 t +5 \,{\mathrm e}^{-t} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 41

ode:=diff(diff(x(t),t),t)+diff(x(t),t)+x(t) = 4*t+5*exp(-t); 
dsolve(ode,x(t), singsol=all);
 

\[ x = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_2 +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_1 +4 t -4+5 \,{\mathrm e}^{-t} \]

✓ Mathematica. Time used: 1.175 (sec). Leaf size: 59

ode=D[x[t],{t,2}]+D[x[t],t]+x[t]==4*t+5*Exp[-t]; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to 4 t+5 e^{-t}+c_2 e^{-t/2} \cos \left (\frac {\sqrt {3} t}{2}\right )+c_1 e^{-t/2} \sin \left (\frac {\sqrt {3} t}{2}\right )-4 \end{align*}

✓ Sympy. Time used: 0.122 (sec). Leaf size: 41

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-4*t + x(t) + Derivative(x(t), t) + Derivative(x(t), (t, 2)) - 5*exp(-t),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = 4 t + \left (C_{1} \sin {\left (\frac {\sqrt {3} t}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {3} t}{2} \right )}\right ) e^{- \frac {t}{2}} - 4 + 5 e^{- t} \]