Internal problem ID [5933]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+y-\left (\left \{\begin {array}{cc} 0 & 0\le t <\pi \\ 1 & \pi \le t <2 \pi \\ 0 & 2 \pi \le t \end {array}\right .\right )=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 0, y^{\prime }\relax (0) = 1] \end {align*}
✓ Solution by Maple
Time used: 0.719 (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)
\[ y \relax (t ) = \sin \relax (t )+\left (\left \{\begin {array}{cc} 0 & t <\pi \\ 1+\cos \relax (t ) & t <2 \pi \\ 2 \cos \relax (t ) & 2 \pi \le t \end {array}\right .\right ) \]
✓ Solution by Mathematica
Time used: 0.013 (sec). Leaf size: 35
DSolve[{y''[t]+y[t]==Piecewise[{{0,0<=t<Pi},{1,Pi<=t<2*Pi},{0,t>=2*Pi}}],{y[0]==0,y'[0]==1}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to {cc} \{ & {cc} \sin (t) & t\leq \pi \\ \cos (t)+\sin (t)+1 & \pi <t\leq 2 \pi \\ 2 \cos (t)+\sin (t) & \text {True} \\ \\ \\ \\ \\ \end{align*}