11.3.11 problem 18

Internal problem ID [1493]
Book : Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section : Chapter 6.2, The Laplace Transform. Solution of Initial Value Problems. page 255
Problem number : 18
Date solved : Tuesday, January 28, 2025 at 02:35:41 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=\left \{\begin {array}{cc} 1 & 0\le t <1 \\ 0 & 1\le t <\infty \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: 0.548 (sec). Leaf size: 27

dsolve([diff(y(t),t$2)+4*y(t)=piecewise(0<=t and t<1,1,1<=t and t<infinity,0),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \frac {\left (\left \{\begin {array}{cc} 1 & t <1 \\ \cos \left (-2+2 t \right ) & 1\le t \end {array}\right .\right )}{4}-\frac {\cos \left (2 t \right )}{4} \]

✓ Solution by Mathematica

Time used: 0.038 (sec). Leaf size: 39

DSolve[{D[y[t],{t,2}]+4*y[t]==Piecewise[{{1,0<t<1},{0,1<=t<Infinity}}],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

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