80.5.49 problem C 25

Internal problem ID [21270]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 5. Second order equations. Excercise 5.9 at page 119
Problem number : C 25
Date solved : Thursday, October 02, 2025 at 07:27:32 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }+4 x^{\prime }+x&=k \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=0 \\ x^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.052 (sec). Leaf size: 42

ode:=diff(diff(x(t),t),t)+4*diff(x(t),t)+x(t) = k; 
ic:=[x(0) = 0, D(x)(0) = 0]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = -\frac {k \left ({\mathrm e}^{-\left (2+\sqrt {3}\right ) t} \left (3-2 \sqrt {3}\right )+{\mathrm e}^{\left (-2+\sqrt {3}\right ) t} \left (3+2 \sqrt {3}\right )-6\right )}{6} \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 65

ode=D[x[t],{t,2}]+4*D[x[t],t]+x[t]==k; 
ic={x[0]==0,Derivative[1][x][0] == 0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{6} k e^{-\left (\left (2+\sqrt {3}\right ) t\right )} \left (-\left (3+2 \sqrt {3}\right ) e^{2 \sqrt {3} t}+6 e^{\left (2+\sqrt {3}\right ) t}-3+2 \sqrt {3}\right ) \end{align*}

✓ Sympy. Time used: 0.128 (sec). Leaf size: 49

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

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