11.4.1 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, January 28, 2025 at 02:35:45 PM
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*}

✓ Solution by Maple

Time used: 0.457 (sec). Leaf size: 28

dsolve([diff(y(t),t$2)+y(t)=piecewise(0<=t and t<3*Pi,1,3*Pi<=t and t<infinity,0),y(0) = 0, D(y)(0) = 1],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.031 (sec). Leaf size: 34

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

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