57.17.6 problem 9

Internal problem ID [14486]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 3, Laplace transform. Section 3.4 Impulsive sources. Exercises page 173
Problem number : 9
Date solved : Thursday, October 02, 2025 at 09:37:46 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+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: 0.185 (sec). Leaf size: 28

ode:=diff(diff(y(t),t),t)+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');
 

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

✓ Mathematica. Time used: 0.052 (sec). Leaf size: 40

ode=D[y[t],{t,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 \frac {2 e^{\frac {1}{2}-\frac {t}{2}} \theta (t-1) \sin \left (\frac {1}{2} \sqrt {3} (t-1)\right )}{\sqrt {3}} \end{align*}

✓ Sympy. Time used: 1.407 (sec). Leaf size: 150

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - 1) + y(t) + 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)
 

\[ y{\left (t \right )} = \left (\left (- \frac {2 \sqrt {3} \int \operatorname {Dirac}{\left (t - 1 \right )} e^{\frac {t}{2}} \sin {\left (\frac {\sqrt {3} t}{2} \right )}\, dt}{3} + \frac {2 \sqrt {3} \int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{\frac {t}{2}} \sin {\left (\frac {\sqrt {3} t}{2} \right )}\, dt}{3}\right ) \cos {\left (\frac {\sqrt {3} t}{2} \right )} + \left (\frac {2 \sqrt {3} \int \operatorname {Dirac}{\left (t - 1 \right )} e^{\frac {t}{2}} \cos {\left (\frac {\sqrt {3} t}{2} \right )}\, dt}{3} - \frac {2 \sqrt {3} \int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{\frac {t}{2}} \cos {\left (\frac {\sqrt {3} t}{2} \right )}\, dt}{3}\right ) \sin {\left (\frac {\sqrt {3} t}{2} \right )}\right ) e^{- \frac {t}{2}} \]