52.7.5 problem 13

Internal problem ID [8360]
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. 7.4.1 DERIVATIVES OF A TRANSFORM. Page 309
Problem number : 13
Date solved : Monday, January 27, 2025 at 03:49:31 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+16 y&=\left \{\begin {array}{cc} \cos \left (4 t \right ) & 0\le t <\pi \\ 0 & \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: 0.872 (sec). Leaf size: 21

dsolve([diff(y(t),t$2)+16*y(t)=piecewise(0<=t and t<Pi,cos(4*t),t>= Pi,0),y(0) = 0, D(y)(0) = 1],y(t), singsol=all)
 
\[ y = \frac {\sin \left (4 t \right ) \left (2+\left (\left \{\begin {array}{cc} t & t <\pi \\ \pi & \pi \le t \end {array}\right .\right )\right )}{8} \]

Solution by Mathematica

Time used: 0.092 (sec). Leaf size: 60

DSolve[{D[y[t],{t,2}]+16*y[t]==Piecewise[{{Cos[4*t],0<=t<Pi},{0,t>=Pi}}],{y[0]==1,Derivative[1][y][0] ==1}},y[t],t,IncludeSingularSolutions -> True]
 
\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \cos (4 t)+\frac {1}{4} \sin (4 t) & t\leq 0 \\ \cos (4 t)+\frac {1}{8} (2+\pi ) \sin (4 t) & t>\pi \\ \cos (4 t)+\frac {1}{8} (t+2) \sin (4 t) & \text {True} \\ \end {array} \\ \end {array} \]