52.7.6 problem 14

Internal problem ID [8361]
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 : 14
Date solved : Monday, January 27, 2025 at 03:49:32 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\left \{\begin {array}{cc} 1 & 0\le t <\frac {\pi }{2} \\ \sin \left (t \right ) & \frac {\pi }{2}\le t \end {array}\right . \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.863 (sec). Leaf size: 28

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

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

✓ Solution by Mathematica

Time used: 0.046 (sec). Leaf size: 38

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

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