46.8.6 problem 6

Internal problem ID [9656]
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 : 6
Date solved : Tuesday, September 30, 2025 at 06:21:56 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\delta \left (t -2 \pi \right )+\delta \left (t -4 \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: 0.236 (sec). Leaf size: 25

ode:=diff(diff(y(t),t),t)+y(t) = Dirac(t-2*Pi)+Dirac(t-4*Pi); 
ic:=[y(0) = 1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \sin \left (t \right ) \operatorname {Heaviside}\left (t -4 \pi \right )+\sin \left (t \right ) \operatorname {Heaviside}\left (t -2 \pi \right )+\cos \left (t \right ) \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 67

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

\begin{align*} y(t)&\to -\sin (t) \int _1^0\cos (K[1]) (\delta (K[1]-4 \pi )+\delta (K[1]-2 \pi ))dK[1]+\sin (t) \int _1^t\cos (K[1]) (\delta (K[1]-4 \pi )+\delta (K[1]-2 \pi ))dK[1]+\cos (t) \end{align*}

✓ Sympy. Time used: 0.995 (sec). Leaf size: 95

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - 4*pi) - Dirac(t - 2*pi) + y(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)
 

\[ y{\left (t \right )} = \left (\int \left (\operatorname {Dirac}{\left (t - 4 \pi \right )} + \operatorname {Dirac}{\left (t - 2 \pi \right )}\right ) \cos {\left (t \right )}\, dt - \int \limits ^{0} \operatorname {Dirac}{\left (t - 4 \pi \right )} \cos {\left (t \right )}\, dt - \int \limits ^{0} \operatorname {Dirac}{\left (t - 2 \pi \right )} \cos {\left (t \right )}\, dt\right ) \sin {\left (t \right )} + \left (- \int \left (\operatorname {Dirac}{\left (t - 4 \pi \right )} + \operatorname {Dirac}{\left (t - 2 \pi \right )}\right ) \sin {\left (t \right )}\, dt + \int \limits ^{0} \operatorname {Dirac}{\left (t - 4 \pi \right )} \sin {\left (t \right )}\, dt + \int \limits ^{0} \operatorname {Dirac}{\left (t - 2 \pi \right )} \sin {\left (t \right )}\, dt + 1\right ) \cos {\left (t \right )} \]