46.8.10 problem 10

Internal problem ID [9660]
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 : Tuesday, September 30, 2025 at 06:21:59 PM
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: 0.111 (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,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 78

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]
 

\begin{align*} y(t)&\to -e^{-t} \left (-\int _1^t-e \delta (K[1]-1)dK[1]+t \int _1^0e \delta (K[2]-1)dK[2]-t \int _1^te \delta (K[2]-1)dK[2]+\int _1^0-e \delta (K[1]-1)dK[1]\right ) \end{align*}

✓ Sympy. Time used: 0.644 (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)
 

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