80.10.4 problem 27

Internal problem ID [21399]
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 11. Laplace transform. Excercise 11.7 at page 248
Problem number : 27
Date solved : Thursday, October 02, 2025 at 07:30:48 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }-4 x^{\prime }+3 x&=1 \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.045 (sec). Leaf size: 16

ode:=diff(diff(x(t),t),t)-4*diff(x(t),t)+3*x(t) = 1; 
ic:=[x(0) = 0, D(x)(0) = 0]; 
dsolve([ode,op(ic)],x(t),method='laplace');
 

\[ x = \frac {1}{3}-\frac {{\mathrm e}^{t}}{2}+\frac {{\mathrm e}^{3 t}}{6} \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 21

ode=D[x[t],{t,2}]-4*D[x[t],t]+3*x[t]==1; 
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} \left (e^t-1\right )^2 \left (e^t+2\right ) \end{align*}

✓ Sympy. Time used: 0.105 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(3*x(t) - 4*Derivative(x(t), t) + Derivative(x(t), (t, 2)) - 1,0) 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \frac {e^{3 t}}{6} - \frac {e^{t}}{2} + \frac {1}{3} \]