90.28.1 problem 1

Internal problem ID [25423]
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 : 1
Date solved : Friday, October 03, 2025 at 12:01:24 AM
CAS classification : [[_linear, `class A`]]

\begin{align*} y^{\prime }+2 y&=\delta \left (t -1\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 15

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

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

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 17

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

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

✓ Sympy. Time used: 0.432 (sec). Leaf size: 46

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

\[ - \int \operatorname {Dirac}{\left (t - 1 \right )} e^{2 t}\, dt + 2 \int y{\left (t \right )} e^{2 t}\, dt = - \int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{2 t}\, dt + 2 \int \limits ^{0} y{\left (t \right )} e^{2 t}\, dt \]