80.10.3 problem 26

Internal problem ID [21398]
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 : 26
Date solved : Thursday, October 02, 2025 at 07:30:48 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }-2 x^{\prime }+x&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} x \left (0\right )&=0 \\ x^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.041 (sec). Leaf size: 8

ode:=diff(diff(x(t),t),t)-2*diff(x(t),t)+x(t) = 0; 
ic:=[x(0) = 0, D(x)(0) = 1]; 
dsolve([ode,op(ic)],x(t),method='laplace');
 

\[ x = t \,{\mathrm e}^{t} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 10

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

\begin{align*} x(t)&\to e^t t \end{align*}

✓ Sympy. Time used: 0.088 (sec). Leaf size: 7

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

\[ x{\left (t \right )} = t e^{t} \]