74.21.3 problem 17

Internal problem ID [16641]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 5. Applications of Higher Order Equations. Exercises 5.3, page 249
Problem number : 17
Date solved : Tuesday, January 28, 2025 at 09:14:00 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+x&=\left \{\begin {array}{cc} t & 0\le t <1 \\ 2-t & 1\le t <2 \\ 0 & 2\le t \end {array}\right . \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 1.725 (sec). Leaf size: 56

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

\[ x \left (t \right ) = \left \{\begin {array}{cc} 0 & t <0 \\ t -\sin \left (t \right ) & t <1 \\ -t -\sin \left (t \right )+2 \sin \left (t -1\right )+2 & t <2 \\ -\sin \left (t \right )+2 \sin \left (t -1\right )-\sin \left (t -2\right ) & 2\le t \end {array}\right . \]

✓ Solution by Mathematica

Time used: 0.055 (sec). Leaf size: 63

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

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