11.4.3 problem 3

Internal problem ID [1497]
Book : Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section : Chapter 6.4, The Laplace Transform. Differential equations with discontinuous forcing functions. page 268
Problem number : 3
Date solved : Saturday, March 29, 2025 at 10:56:35 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=\sin \left (t \right )-\operatorname {Heaviside}\left (t -2 \pi \right ) \sin \left (t \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: 0.337 (sec). Leaf size: 20

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

\[ y = \frac {\sin \left (t \right ) \left (\cos \left (t \right )-1\right ) \left (-1+\operatorname {Heaviside}\left (t -2 \pi \right )\right )}{3} \]

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 27

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

\[ y(t)\to \frac {2}{3} \theta (2 \pi -t) \sin ^2\left (\frac {t}{2}\right ) \sin (t) \]

✓ Sympy. Time used: 3.308 (sec). Leaf size: 100

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

\[ y{\left (t \right )} = - \frac {2 \sin ^{5}{\left (t \right )} \theta \left (t - 2 \pi \right )}{3} + \frac {5 \sin ^{3}{\left (t \right )} \theta \left (t - 2 \pi \right )}{6} + \frac {\sin {\left (t \right )} \cos {\left (t \right )} \cos {\left (3 t \right )} \theta \left (t - 2 \pi \right )}{6} + \frac {\sin {\left (t \right )} \cos {\left (t \right )} \theta \left (t - 2 \pi \right )}{3} - \frac {\sin {\left (t \right )} \theta \left (t - 2 \pi \right )}{2} + \frac {\sin {\left (t \right )}}{3} - \frac {\sin {\left (2 t \right )}}{6} \]