14.17.6 problem 24

Internal problem ID [2684]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.10, Some useful properties of Laplace transform. Excercises page 238
Problem number : 24
Date solved : Monday, January 27, 2025 at 06:06:22 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\left \{\begin {array}{cc} 2 & 0\le t \le 3 \\ 3 t -7 & 3<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: 1.424 (sec). Leaf size: 30

dsolve([diff(y(t),t$2)+y(t)=piecewise(0<=t and t<=3,2,3<t,3*t-7),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = -2 \cos \left (t \right )-\left (\left \{\begin {array}{cc} -2 & t <3 \\ 7+3 \sin \left (t -3\right )-3 t & 3\le t \end {array}\right .\right ) \]

✓ Solution by Mathematica

Time used: 0.045 (sec). Leaf size: 42

DSolve[{D[y[t],{t,2}]+y[t]==Piecewise[{ {2,0<=t<=3},{3*t-7,3<t}}],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 0 & t\leq 0 \\ 2-2 \cos (t) & 0<t\leq 3 \\ 3 t-2 \cos (t)+3 \sin (3-t)-7 & \text {True} \\ \end {array} \\ \end {array} \]