67.4.27 problem Problem 4(c)

Internal problem ID [14071]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number : Problem 4(c)
Date solved : Tuesday, January 28, 2025 at 06:13:36 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&=\left \{\begin {array}{cc} 1 & 0\le t <2 \\ -1 & 2\le 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: 13.129 (sec). Leaf size: 45

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

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

✓ Solution by Mathematica

Time used: 0.044 (sec). Leaf size: 68

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

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