64.20.14 problem 14

Internal problem ID [13657]
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 : 14
Date solved : Tuesday, January 28, 2025 at 05:54:24 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+5 y^{\prime }+6 y&=\left \{\begin {array}{cc} 6 & 0<t <2 \\ 0 & 2<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: 12.359 (sec). Leaf size: 67

dsolve([diff(y(t),t$2)+5*diff(y(t),t)+6*y(t)=piecewise(0<t and t<2,6,t>2,0),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \left \{\begin {array}{cc} -3 \,{\mathrm e}^{-2 t}+2 \,{\mathrm e}^{-3 t}+1 & t <2 \\ -3 \,{\mathrm e}^{-4}+2 \,{\mathrm e}^{-6}+2 & t =2 \\ -3 \,{\mathrm e}^{-2 t}+2 \,{\mathrm e}^{-3 t}+3 \,{\mathrm e}^{-2 t +4}-2 \,{\mathrm e}^{-3 t +6} & 2<t \end {array}\right . \]

✓ Solution by Mathematica

Time used: 0.042 (sec). Leaf size: 60

DSolve[{D[y[t],{t,2}]+5*D[y[t],t]+6*y[t]==Piecewise[{{6,0<t<2},{0,t>2}}],{y[0]==0,Derivative[1][y][0]==0}},{y[t]},t,IncludeSingularSolutions -> True]
 

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