28.3.8 problem 6.43

Internal problem ID [4521]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 6. The Laplace Transform and Its Applications. Problems at page 291
Problem number : 6.43
Date solved : Monday, January 27, 2025 at 09:22:40 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }-2 y&=54 t \,{\mathrm e}^{-2 t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=6\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 3.053 (sec). Leaf size: 24

dsolve([diff(y(t),t$2)+diff(y(t),t)-2*y(t)=54*t*exp(-2*t),y(0) = 6, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \left (-9 t^{2}-6 t +6 \,{\mathrm e}^{3 t}\right ) {\mathrm e}^{-2 t} \]

✓ Solution by Mathematica

Time used: 0.031 (sec). Leaf size: 24

DSolve[{D[y[t],{t,2}]+D[y[t],t]-2*y[t]==54*t*Exp[-2*t],{y[0]==6,Derivative[1][y][0] == 0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to 6 e^t-3 e^{-2 t} t (3 t+2) \]