46.7.5 problem 22

Internal problem ID [7366]
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 : 22
Date solved : Tuesday, January 28, 2025 at 03:10:44 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&=\left \{\begin {array}{cc} 4 t & 0<t <1 \\ 8 & 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: 0.397 (sec). Leaf size: 71

dsolve([diff(y(t),t$2)+3*diff(y(t),t)+2*y(t)=piecewise(0<t and t<1,4*t,t>1,8),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \left \{\begin {array}{cc} 2 t -3-{\mathrm e}^{-2 t}+4 \,{\mathrm e}^{-t} & t <1 \\ 1-{\mathrm e}^{-2}+4 \,{\mathrm e}^{-1} & t =1 \\ 3 \,{\mathrm e}^{2-2 t}-8 \,{\mathrm e}^{1-t}+4-{\mathrm e}^{-2 t}+4 \,{\mathrm e}^{-t} & 1<t \end {array}\right . \]

✓ Solution by Mathematica

Time used: 0.047 (sec). Leaf size: 70

DSolve[{D[y[t],{t,2}]+3*D[y[t],t]+2*y[t]==Piecewise[{{4*t,0<t<1},{8,t>1}}],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

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