40.8.3 problem 5

Internal problem ID [8660]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 6. Laplace Transforms. Problem set 6.4, page 230
Problem number : 5
Date solved : Tuesday, September 30, 2025 at 05:40:22 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\delta \left (t -\pi \right )-\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 )&=1 \\ \end{align*}

✓ Maple. Time used: 0.304 (sec). Leaf size: 25

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

\[ y = \sin \left (t \right ) \left (1-\operatorname {Heaviside}\left (t -2 \pi \right )-\operatorname {Heaviside}\left (t -\pi \right )\right ) \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 73

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

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

✓ Sympy. Time used: 1.178 (sec). Leaf size: 99

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

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