80.10.10 problem 33

Internal problem ID [21405]
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 : 33
Date solved : Saturday, October 04, 2025 at 04:13:54 PM
CAS classification : [[_2nd_order, _quadrature]]

\begin{align*} x^{\prime \prime }&=\delta \left (-t +a \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.070 (sec). Leaf size: 26

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

\[ x = \operatorname {Typesetting}\mcoloneq \operatorname {msup}\left (\operatorname {Typesetting}\mcoloneq \operatorname {mi}\left (\text {``$\mathcal \{L\}$''}\right ), \operatorname {Typesetting}\mcoloneq \operatorname {mrow}\left (\operatorname {Typesetting}\mcoloneq \operatorname {mo}\left (\text {``$-$''}\right ), \operatorname {Typesetting}\mcoloneq \operatorname {mn}\left (``1''\right )\right ), \operatorname {Typesetting}\mcoloneq \operatorname {msemantics}=\text {``atomic''}\right )\left (\left \{\begin {array}{cc} 0 & a <0 \\ \frac {{\mathrm e}^{-\textit {\_s1} a}}{\textit {\_s1}^{2}} & 0\le a \end {array}\right ., \textit {\_s1} , t\right )+1 \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 24

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

\begin{align*} x(t)&\to (a-t) (\theta (-a)-\theta (t-a)) \end{align*}

✓ Sympy. Time used: 0.225 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
a = symbols("a") 
x = Function("x") 
ode = Eq(-Dirac(-a + 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)
 

\[ x{\left (t \right )} = t \left (\int \operatorname {Dirac}{\left (- a + t \right )}\, dt - \int \limits ^{0} \operatorname {Dirac}{\left (- a + t \right )}\, dt\right ) - \int t \operatorname {Dirac}{\left (- a + t \right )}\, dt + \int \limits ^{0} t \operatorname {Dirac}{\left (- a + t \right )}\, dt \]