11.4.7 problem 7

Internal problem ID [1501]
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 : 7
Date solved : Saturday, March 29, 2025 at 10:56:45 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=\operatorname {Heaviside}\left (t -\pi \right )-\operatorname {Heaviside}\left (t -3 \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: 0.244 (sec). Leaf size: 25

ode:=diff(diff(y(t),t),t)+4*y(t) = Heaviside(t-Pi)-Heaviside(t-3*Pi); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.037 (sec). Leaf size: 25

ode=D[y[t],{t,2}]+4*y[t]==UnitStep[t-Pi]-UnitStep[t-3*Pi]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {\sin ^2(t)}{2} & \pi <t\leq 3 \pi \\ 0 & \text {True} \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 1.787 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) + Heaviside(t - 3*pi) - Heaviside(t - pi) + 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)
 

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