64.20.17 problem 17

Internal problem ID [13660]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 9, The Laplace transform. Section 9.3, Exercises page 452
Problem number : 17
Date solved : Tuesday, January 28, 2025 at 05:54:29 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=\left \{\begin {array}{cc} -4 t +8 \pi & 0<t <2 \pi \\ 0 & 2<t \end {array}\right . \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 12.922 (sec). Leaf size: 50

dsolve([diff(y(t),t$2)+4*y(t)=piecewise(0<t and t<2*Pi,-4*t+8*Pi,t>2,0),y(0) = 2, D(y)(0) = 0],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.040 (sec). Leaf size: 56

DSolve[{D[y[t],{t,2}]+4*y[t]==Piecewise[{{-4*t+8*Pi,0<t<2*Pi},{0,t>2*Pi}}],{y[0]==2,Derivative[1][y][0]==0}},{y[t]},t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 2 \cos (2 t) & t\leq 0 \\ -2 (-1+\pi ) \cos (2 t) & t>2 \pi \\ \frac {1}{2} (-2 t-4 (-1+\pi ) \cos (2 t)+\sin (2 t)+4 \pi ) & \text {True} \\ \end {array} \\ \end {array} \]