46.7.7 problem 24

Internal problem ID [7368]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 6. Laplace Transforms. Problem set 6.3, page 224
Problem number : 24
Date solved : Monday, January 27, 2025 at 02:51:05 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&=\left \{\begin {array}{cc} 1 & 0<t <1 \\ 0 & 1<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: 0.324 (sec). Leaf size: 63

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

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

✓ Solution by Mathematica

Time used: 0.044 (sec). Leaf size: 57

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

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