11.3.10 problem 17

Internal problem ID [1492]
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 : 17
Date solved : Tuesday, January 28, 2025 at 02:35:40 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=\left \{\begin {array}{cc} 1 & 0\le t <\pi \\ 0 & \pi \le t <\infty \end {array}\right . \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.377 (sec). Leaf size: 23

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

\[ y = \left \{\begin {array}{cc} \frac {3 \cos \left (2 t \right )}{4}+\frac {1}{4} & t <\pi \\ \cos \left (2 t \right ) & \pi \le t \end {array}\right . \]

✓ Solution by Mathematica

Time used: 0.039 (sec). Leaf size: 31

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

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \cos (2 t) & t>\pi \lor t\leq 0 \\ \frac {1}{4} (3 \cos (2 t)+1) & \text {True} \\ \end {array} \\ \end {array} \]