52.8.10 problem 10

Internal problem ID [8374]
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 : 10
Date solved : Wednesday, March 05, 2025 at 05:36:10 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.664 (sec). Leaf size: 18

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

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

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 20

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

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

✓ Sympy. Time used: 1.063 (sec). Leaf size: 51

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

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