52.6.16 problem 66

Internal problem ID [8351]
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 : 66
Date solved : Monday, January 27, 2025 at 03:49:22 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=\left \{\begin {array}{cc} 1 & 0\le t <1 \\ 0 & 1\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: 0.898 (sec). Leaf size: 33

dsolve([diff(y(t),t$2)+4*y(t)=piecewise(0<=t and t<1,1,t>=1,0),y(0) = 0, D(y)(0) = -1],y(t), singsol=all)
 

\[ y = \frac {\left (\left \{\begin {array}{cc} 1 & t <1 \\ \cos \left (2 t -2\right ) & 1\le t \end {array}\right .\right )}{4}-\frac {\sin \left (2 t \right )}{2}-\frac {\cos \left (2 t \right )}{4} \]

✓ Solution by Mathematica

Time used: 0.042 (sec). Leaf size: 65

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

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