52.6.19 problem 69

Internal problem ID [8354]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 7 THE LAPLACE TRANSFORM. 7.3.1 TRANSLATION ON THE s-AXIS. Page 297
Problem number : 69
Date solved : Monday, January 27, 2025 at 03:49:25 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\left \{\begin {array}{cc} 0 & 0\le t <\pi \\ 1 & \pi \le t <2 \pi \\ 0 & 2 \pi \le t \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: 1.014 (sec). Leaf size: 30

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

\[ y = \sin \left (t \right )+\left (\left \{\begin {array}{cc} 0 & t <\pi \\ \cos \left (t \right )+1 & t <2 \pi \\ 2 \cos \left (t \right ) & 2 \pi \le t \end {array}\right .\right ) \]

✓ Solution by Mathematica

Time used: 0.035 (sec). Leaf size: 35

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

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