5.4.1 problem 1

Internal problem ID [1495]
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 : 1
Date solved : Tuesday, September 30, 2025 at 04:34:39 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\left \{\begin {array}{cc} 1 & 0\le t <3 \pi \\ 0 & 3 \pi \le t <\infty \end {array}\right . \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.198 (sec). Leaf size: 28

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

\[ y = \sin \left (t \right )-\left (\left \{\begin {array}{cc} \cos \left (t \right )-1 & t <3 \pi \\ 2 \cos \left (t \right ) & 3 \pi \le t \end {array}\right .\right ) \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 34

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

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} \sin (t) & t\leq 0 \\ \sin (t)-2 \cos (t) & t>3 \pi \\ -\cos (t)+\sin (t)+1 & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 0.282 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((1, (t >= 0) & (t < 3*pi)), (0, (t < oo) & (t >= 3*pi))) + y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \begin {cases} 1 & \text {for}\: t \geq 0 \wedge t < 3 \pi \\0 & \text {for}\: t \geq 3 \pi \wedge t < \infty \\\text {NaN} & \text {otherwise} \end {cases} + \sin {\left (t \right )} - \cos {\left (t \right )} \]