70.21.2 problem 2

Internal problem ID [19052]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.7 (Impulse Functions). Problems at page 350
Problem number : 2
Date solved : Thursday, October 02, 2025 at 03:37:30 PM
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 )&=0 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.321 (sec). Leaf size: 25

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

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

✓ Mathematica. Time used: 0.045 (sec). Leaf size: 83

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

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

✓ Sympy. Time used: 1.949 (sec). Leaf size: 128

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): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

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