90.28.9 problem 9

Internal problem ID [25431]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 6. Discontinuous Functions and the Laplace Transform. Exercises at page 437
Problem number : 9
Date solved : Friday, October 03, 2025 at 12:01:28 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+4 y&=3 \delta \left (t -1\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=-1 \\ y^{\prime }\left (0\right )&=3 \\ \end{align*}

✓ Maple. Time used: 0.054 (sec). Leaf size: 22

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+4*y(t) = 3*Dirac(t-1); 
ic:=[y(0) = -1, D(y)(0) = 3]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 25

ode=D[y[t],{t,2}]+4*D[y[t],t]+4*y[t]==3*DiracDelta[t-1]; 
ic={y[0]==-1,Derivative[1][y][0] ==3}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-2 t} (t-1) \left (3 e^2 \theta (t-1)+1\right ) \end{align*}

✓ Sympy. Time used: 0.795 (sec). Leaf size: 70

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-3*Dirac(t - 1) + 4*y(t) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): -1, Subs(Derivative(y(t), t), t, 0): 3} 
dsolve(ode,func=y(t),ics=ics)
 

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