64.20.13 problem 13

Internal problem ID [13656]
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 : 13
Date solved : Tuesday, January 28, 2025 at 05:54:23 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-3 y^{\prime }+2 y&=\left \{\begin {array}{cc} 2 & 0<t <4 \\ 0 & 4<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: 14.187 (sec). Leaf size: 36

dsolve([diff(y(t),t$2)-3*diff(y(t),t)+2*y(t)=piecewise(0<t and t<4,2,t>4,0),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = {\mathrm e}^{2 t}-2 \,{\mathrm e}^{t}+\left (\left \{\begin {array}{cc} 1 & t \le 4 \\ 2 \,{\mathrm e}^{t -4}-{\mathrm e}^{-8+2 t} & 4<t \end {array}\right .\right ) \]

✓ Solution by Mathematica

Time used: 0.035 (sec). Leaf size: 51

DSolve[{D[y[t],{t,2}]-3*D[y[t],t]+2*y[t]==Piecewise[{{2,0<t<4},{0,t>4}}],{y[0]==0,Derivative[1][y][0]==0}},{y[t]},t,IncludeSingularSolutions -> True]
 

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