64.20.16 problem 16

Internal problem ID [13659]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 9, The Laplace transform. Section 9.3, Exercises page 452
Problem number : 16
Date solved : Tuesday, January 28, 2025 at 05:54:27 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+8 y&=\left \{\begin {array}{cc} 3 & 0<t <2 \pi \\ 0 & 2 \pi <t \end {array}\right . \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 13.101 (sec). Leaf size: 109

dsolve([diff(y(t),t$2)+6*diff(y(t),t)+8*y(t)=piecewise(0<t and t<2*Pi,3,t>2*Pi,0),y(0) = 1, D(y)(0) = -1],y(t), singsol=all)
 

\[ y = \frac {\left (\left \{\begin {array}{cc} 40 \,{\mathrm e}^{-3 t} \sinh \left (t \right )+19 \,{\mathrm e}^{-4 t}-14 \,{\mathrm e}^{-2 t}+3 & t <2 \pi \\ 40 \,{\mathrm e}^{-6 \pi } \sinh \left (2 \pi \right )-14 \,{\mathrm e}^{-4 \pi }+19 \,{\mathrm e}^{-8 \pi }+6 & t =2 \pi \\ 40 \,{\mathrm e}^{-3 t} \sinh \left (t \right )+19 \,{\mathrm e}^{-4 t}-14 \,{\mathrm e}^{-2 t}+6 \,{\mathrm e}^{4 \pi -2 t}-3 \,{\mathrm e}^{8 \pi -4 t} & 2 \pi <t \end {array}\right .\right )}{8} \]

✓ Solution by Mathematica

Time used: 0.043 (sec). Leaf size: 94

DSolve[{D[y[t],{t,2}]+6*D[y[t],t]+8*y[t]==Piecewise[{{3,0<t<2*Pi},{0,t>2*Pi}}],{y[0]==1,Derivative[1][y][0]==-1}},{y[t]},t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{2} e^{-4 t} \left (-1+3 e^{2 t}\right ) & t\leq 0 \\ \frac {1}{8} \left (3-e^{-4 t}+6 e^{-2 t}\right ) & 0<t\leq 2 \pi \\ \frac {1}{8} e^{-4 t} \left (-1-3 e^{8 \pi }+6 e^{2 t}+6 e^{2 t+4 \pi }\right ) & \text {True} \\ \end {array} \\ \end {array} \]