52.8.9 problem 9

Internal problem ID [8373]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 7 THE LAPLACE TRANSFORM. EXERCISES 7.5. Page 315
Problem number : 9
Date solved : Sunday, March 30, 2025 at 12:53:14 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+5 y&=\delta \left (t -2 \pi \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.133 (sec). Leaf size: 21

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+5*y(t) = Dirac(t-2*Pi); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \operatorname {Heaviside}\left (t -2 \pi \right ) \sin \left (t \right ) {\mathrm e}^{4 \pi -2 t} \]

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 23

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

\[ y(t)\to e^{4 \pi -2 t} \theta (t-2 \pi ) \sin (t) \]

✓ Sympy. Time used: 2.568 (sec). Leaf size: 82

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

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