76.18.12 problem 24

Internal problem ID [17723]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.2 (Properties of the Laplace transform). Problems at page 309
Problem number : 24
Date solved : Tuesday, January 28, 2025 at 11:01:51 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\left \{\begin {array}{cc} t & 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 )&=0 \end{align*}

✓ Solution by Maple

Time used: 13.676 (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: 0.032 (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]
 

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