46.7.8 problem 25

Internal problem ID [7369]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 6. Laplace Transforms. Problem set 6.3, page 224
Problem number : 25
Date solved : Monday, January 27, 2025 at 02:51:06 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\left \{\begin {array}{cc} t & 0<t <1 \\ 0 & 1<t \end {array}\right . \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 0.573 (sec). Leaf size: 26

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

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

✓ Solution by Mathematica

Time used: 0.035 (sec). Leaf size: 44

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

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} t-\sin (t) & 0<t\leq 1 \\ \cos (1-t)-\sin (1-t)-\sin (t) & t>1 \\ \end {array} \\ \end {array} \]