86.8.6 problem 6

Internal problem ID [23166]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 5. Linear equations of the second order with constant coefficients. Exercise 5e at page 91
Problem number : 6
Date solved : Thursday, October 02, 2025 at 09:23:41 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} x^{\prime \prime }+3 x^{\prime }&={\mathrm e}^{-3 t} \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 20

ode:=diff(diff(x(t),t),t)+3*diff(x(t),t) = exp(-3*t); 
dsolve(ode,x(t), singsol=all);
 

\[ x = -\frac {\left (3 t +3 c_1 +1\right ) {\mathrm e}^{-3 t}}{9}+c_2 \]

✓ Mathematica. Time used: 0.051 (sec). Leaf size: 26

ode=D[x[t],{t,2}]+3*D[x[t],t]==Exp[-3*t]; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to c_2-\frac {1}{9} e^{-3 t} (3 t+1+3 c_1) \end{align*}

✓ Sympy. Time used: 0.135 (sec). Leaf size: 14

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(3*Derivative(x(t), t) + Derivative(x(t), (t, 2)) - exp(-3*t),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

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