90.28.10 problem 10

Internal problem ID [25432]
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 : 10
Date solved : Friday, October 03, 2025 at 12:01:29 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+5 y&=3 \delta \left (t -\pi \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.106 (sec). Leaf size: 25

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+5*y(t) = 3*Dirac(t-Pi); 
ic:=[y(0) = 0, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 29

ode=D[y[t],{t,2}]+4*D[y[t],t]+5*y[t]==3*DiracDelta[t-Pi]; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 1.816 (sec). Leaf size: 83

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-3*Dirac(t - pi) + 5*y(t) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

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