64.20.8 problem 8

Internal problem ID [13651]
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 : 8
Date solved : Tuesday, January 28, 2025 at 05:54:20 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 8.262 (sec). Leaf size: 23

dsolve([diff(y(t),t$2)+2*diff(y(t),t)+y(t)=t*exp(-2*t),y(0) = 1, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \left (t +2\right ) {\mathrm e}^{-2 t}+\left (2 t -1\right ) {\mathrm e}^{-t} \]

✓ Solution by Mathematica

Time used: 0.022 (sec). Leaf size: 23

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

\[ y(t)\to e^{-2 t} \left (t+e^t (2 t-1)+2\right ) \]