90.28.2 problem 2

Internal problem ID [25424]
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 : 2
Date solved : Friday, October 03, 2025 at 12:01:24 AM
CAS classification : [[_linear, `class A`]]

\begin{align*} y^{\prime }-3 y&=3+\delta \left (t -2\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.051 (sec). Leaf size: 17

ode:=diff(y(t),t)-3*y(t) = 3+Dirac(t-2); 
ic:=[y(0) = -1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{3 t -6}-1 \]

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 19

ode=D[y[t],t]-3*y[t]==3+DiracDelta[t-2]; 
ic={y[0]==-1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{3 t-6} \theta (t-2)-1 \end{align*}

✓ Sympy. Time used: 0.406 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - 2) - 3*y(t) + Derivative(y(t), t),0) 
ics = {y(0): -1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ - \int \operatorname {Dirac}{\left (t - 2 \right )} e^{- 3 t}\, dt - 3 \int y{\left (t \right )} e^{- 3 t}\, dt = - \int \limits ^{0} \operatorname {Dirac}{\left (t - 2 \right )} e^{- 3 t}\, dt - 3 \int \limits ^{0} y{\left (t \right )} e^{- 3 t}\, dt \]