90.28.8 problem 8

Internal problem ID [25430]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 6. Discontinuous Functions and the Laplace Transform. Exercises at page 437
Problem number : 8
Date solved : Friday, October 03, 2025 at 12:01:28 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 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 )&=1 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.162 (sec). Leaf size: 33

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

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

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 33

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

\begin{align*} y(t)&\to -\theta (t-2 \pi ) \sin (t) \cos (t)+\theta (t-\pi ) \sin (t) \cos (t)+\cos (2 t) \end{align*}

✓ Sympy. Time used: 1.961 (sec). Leaf size: 129

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

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