20.23.2 problem Problem 2

Internal problem ID [3974]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 10, The Laplace Transform and Some Elementary Applications. Exercises for 10.8. page 710
Problem number : Problem 2
Date solved : Sunday, March 30, 2025 at 02:13:27 AM
CAS classification : [[_linear, `class A`]]

\begin{align*} y^{\prime }-2 y&=\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: 0.124 (sec). Leaf size: 18

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

\[ y = {\mathrm e}^{2 t} \left (1+{\mathrm e}^{-4} \operatorname {Heaviside}\left (t -2\right )\right ) \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 23

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

\[ y(t)\to e^{2 t-4} \left (\theta (t-2)+3 e^4\right ) \]

✓ Sympy. Time used: 0.673 (sec). Leaf size: 49

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

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