76.18.11 problem 23

Internal problem ID [17722]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.2 (Properties of the Laplace transform). Problems at page 309
Problem number : 23
Date solved : Tuesday, January 28, 2025 at 11:01:50 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+16 y&=\left \{\begin {array}{cc} 1 & 0\le t <\pi \\ 0 & \pi \le t \end {array}\right . \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=9\\ y^{\prime }\left (0\right )&=2 \end{align*}

✓ Solution by Maple

Time used: 16.151 (sec). Leaf size: 32

dsolve([diff(y(t),t$2)+16*y(t)=piecewise(0<=t and t<Pi,1,t>=Pi,0),y(0) = 9, D(y)(0) = 2],y(t), singsol=all)
 

\[ y = \frac {\sin \left (4 t \right )}{2}+\left (\left \{\begin {array}{cc} \frac {1}{16}+\frac {143 \cos \left (4 t \right )}{16} & t <\pi \\ 9 \cos \left (4 t \right ) & \pi \le t \end {array}\right .\right ) \]

✓ Solution by Mathematica

Time used: 0.035 (sec). Leaf size: 41

DSolve[{D[y[t],{t,2}]+16*y[t]==Piecewise[{  {1,0<= t <Pi},{0,t >= Pi}}],{y[0]==0,Derivative[1][y][0] == 2}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{2} \sin (4 t) & t>\pi \lor t\leq 0 \\ \frac {1}{16} (-\cos (4 t)+8 \sin (4 t)+1) & \text {True} \\ \end {array} \\ \end {array} \]