46.8.5 problem 5

Internal problem ID [9655]
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 : 5
Date solved : Tuesday, September 30, 2025 at 06:21:55 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\delta \left (t -\frac {\pi }{2}\right )+\delta \left (t -\frac {3 \pi }{2}\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: 0.277 (sec). Leaf size: 22

ode:=diff(diff(y(t),t),t)+y(t) = Dirac(t-1/2*Pi)+Dirac(t-3/2*Pi); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \left (\operatorname {Heaviside}\left (t -\frac {3 \pi }{2}\right )-\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )\right ) \cos \left (t \right ) \]

✓ Mathematica. Time used: 0.036 (sec). Leaf size: 71

ode=D[y[t],{t,2}]+y[t]==DiracDelta[t-1/2*Pi]+DiracDelta[t-3/2*Pi]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \cos (t) \int _1^t-2 (\delta (\pi -2 K[1])+\delta (3 \pi -2 K[1])) \sin (K[1])dK[1]-\cos (t) \int _1^0-2 (\delta (\pi -2 K[1])+\delta (3 \pi -2 K[1])) \sin (K[1])dK[1] \end{align*}

✓ Sympy. Time used: 1.040 (sec). Leaf size: 100

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - 3*pi/2) - Dirac(t - pi/2) + y(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)
 

\[ y{\left (t \right )} = \left (- \int \left (\operatorname {Dirac}{\left (t - \frac {3 \pi }{2} \right )} + \operatorname {Dirac}{\left (t - \frac {\pi }{2} \right )}\right ) \sin {\left (t \right )}\, dt + \int \limits ^{0} \operatorname {Dirac}{\left (t - \frac {3 \pi }{2} \right )} \sin {\left (t \right )}\, dt + \int \limits ^{0} \operatorname {Dirac}{\left (t - \frac {\pi }{2} \right )} \sin {\left (t \right )}\, dt\right ) \cos {\left (t \right )} + \left (\int \left (\operatorname {Dirac}{\left (t - \frac {3 \pi }{2} \right )} + \operatorname {Dirac}{\left (t - \frac {\pi }{2} \right )}\right ) \cos {\left (t \right )}\, dt - \int \limits ^{0} \operatorname {Dirac}{\left (t - \frac {3 \pi }{2} \right )} \cos {\left (t \right )}\, dt - \int \limits ^{0} \operatorname {Dirac}{\left (t - \frac {\pi }{2} \right )} \cos {\left (t \right )}\, dt\right ) \sin {\left (t \right )} \]