74.21.1 problem 15

Internal problem ID [16639]
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 : 15
Date solved : Tuesday, January 28, 2025 at 09:13:51 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+x&=\left \{\begin {array}{cc} 1 & 0\le t <\pi \\ 0 & \pi \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: 1.105 (sec). Leaf size: 25

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

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

✓ Solution by Mathematica

Time used: 0.035 (sec). Leaf size: 29

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

\[ x(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 0 & t\leq 0 \\ 1-\cos (t) & 0<t\leq \pi \\ -2 \cos (t) & \text {True} \\ \end {array} \\ \end {array} \]