70.20.7 problem 7

Internal problem ID [19041]
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 5.6 (Differential equations with Discontinuous Forcing Functions). Problems at page 342
Problem number : 7
Date solved : Thursday, October 02, 2025 at 03:37:13 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\operatorname {Heaviside}\left (t -3 \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: 0.203 (sec). Leaf size: 22

ode:=diff(diff(y(t),t),t)+y(t) = Heaviside(t-3*Pi); 
ic:=[y(0) = 1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 21

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

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} \cos (t) & t\leq 3 \pi \\ 2 \cos (t)+1 & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 0.418 (sec). Leaf size: 24

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

\[ y{\left (t \right )} = \left (\theta \left (t - 3 \pi \right ) + 1\right ) \cos {\left (t \right )} + \theta \left (t - 3 \pi \right ) \]