52.8.2 problem 2

Internal problem ID [8366]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 7 THE LAPLACE TRANSFORM. EXERCISES 7.5. Page 315
Problem number : 2
Date solved : Wednesday, March 05, 2025 at 05:36:00 AM
CAS classification : [[_linear, `class A`]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2 \end{align*}

✓ Maple. Time used: 0.667 (sec). Leaf size: 22

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

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

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

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

\[ y(t)\to e^{-t} (e \theta (t-1)+2) \]

✓ Sympy. Time used: 0.629 (sec). Leaf size: 36

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

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