70.21.1 problem 1

Internal problem ID [19051]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.7 (Impulse Functions). Problems at page 350
Problem number : 1
Date solved : Thursday, October 02, 2025 at 03:37:29 PM
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 )&=0 \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.276 (sec). Leaf size: 23

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

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

✓ Mathematica. Time used: 0.049 (sec). Leaf size: 56

ode=D[y[t],{t,2}]+2*D[y[t],t]+2*y[t]==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^{-t} \sin (t) \left (-\int _1^t-e^{\pi } \delta (\pi -K[1])dK[1]+\int _1^0-e^{\pi } \delta (\pi -K[1])dK[1]-1\right ) \end{align*}

✓ Sympy. Time used: 1.531 (sec). Leaf size: 68

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): 0, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

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