14.19.1 problem 2

Internal problem ID [2694]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.12, Dirac delta function. Excercises page 250
Problem number : 2
Date solved : Sunday, March 30, 2025 at 12:14:56 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+5 y&=\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 )&=0 \end{align*}

✓ Maple. Time used: 0.301 (sec). Leaf size: 27

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

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

✓ Mathematica. Time used: 0.043 (sec). Leaf size: 33

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

\[ y(t)\to e^{-2 t} \left (-e^2 \theta (t-1) \sin (1-t)+2 \sin (t)+\cos (t)\right ) \]

✓ Sympy. Time used: 2.555 (sec). Leaf size: 78

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - 1) + 5*y(t) + 4*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)
 

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