5.5.1 problem 1

Internal problem ID [1506]
Book : Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section : Chapter 6.5, The Laplace Transform. Impulse functions. page 273
Problem number : 1
Date solved : Tuesday, September 30, 2025 at 04:34:59 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.305 (sec). Leaf size: 26

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

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

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 29

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

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

✓ Sympy. Time used: 1.958 (sec). Leaf size: 70

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

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