80.10.9 problem 32

Internal problem ID [21404]
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 : 32
Date solved : Thursday, October 02, 2025 at 07:30:52 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+x&=g \left (t \right ) \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: 0.159 (sec). Leaf size: 21

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

\[ x = -\int _{0}^{t}g \left (\textit {\_U1} \right ) \sin \left (-t +\textit {\_U1} \right )d \textit {\_U1} \]

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 77

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

\begin{align*} x(t)&\to -\sin (t) \int _1^0\cos (K[2]) g(K[2])dK[2]+\sin (t) \int _1^t\cos (K[2]) g(K[2])dK[2]+\cos (t) \left (\int _1^t-g(K[1]) \sin (K[1])dK[1]-\int _1^0-g(K[1]) \sin (K[1])dK[1]\right ) \end{align*}

✓ Sympy. Time used: 0.405 (sec). Leaf size: 42

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

\[ x{\left (t \right )} = \left (- \int g{\left (t \right )} \sin {\left (t \right )}\, dt + \int \limits ^{0} g{\left (t \right )} \sin {\left (t \right )}\, dt\right ) \cos {\left (t \right )} + \left (\int g{\left (t \right )} \cos {\left (t \right )}\, dt - \int \limits ^{0} g{\left (t \right )} \cos {\left (t \right )}\, dt\right ) \sin {\left (t \right )} \]