90.28.4 problem 4

Internal problem ID [25426]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 6. Discontinuous Functions and the Laplace Transform. Exercises at page 437
Problem number : 4
Date solved : Friday, October 03, 2025 at 12:01:25 AM
CAS classification : [[_linear, `class A`]]

\begin{align*} y+y^{\prime }&=\delta \left (t -1\right )-\delta \left (-3+t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.059 (sec). Leaf size: 24

ode:=diff(y(t),t)+y(t) = Dirac(t-1)-Dirac(t-3); 
ic:=[y(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \left (\operatorname {Heaviside}\left (t -1\right )-\operatorname {Heaviside}\left (t -3\right ) {\mathrm e}^{2}\right ) {\mathrm e}^{-t +1} \]

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 27

ode=D[y[t],t]+y[t]==DiracDelta[t-1]-DiracDelta[t-3]; 
ic={y[0]==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{1-t} \left (\theta (t-1)-e^2 \theta (t-3)\right ) \end{align*}

✓ Sympy. Time used: 0.472 (sec). Leaf size: 56

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

\[ \int \operatorname {Dirac}{\left (t - 3 \right )} e^{t}\, dt - \int \operatorname {Dirac}{\left (t - 1 \right )} e^{t}\, dt + \int y{\left (t \right )} e^{t}\, dt = \int \limits ^{0} \operatorname {Dirac}{\left (t - 3 \right )} e^{t}\, dt - \int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{t}\, dt + \int \limits ^{0} y{\left (t \right )} e^{t}\, dt \]