63.16.2 problem 8

Internal problem ID [13199]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 3, Laplace transform. Section 3.3 The convolution property. Exercises page 162
Problem number : 8
Date solved : Tuesday, January 28, 2025 at 05:12:18 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{\prime \prime }+3 x^{\prime }+2 x&={\mathrm e}^{-4 t} \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 8.076 (sec). Leaf size: 23

dsolve([diff(x(t),t$2)+3*diff(x(t),t)+2*x(t)=exp(-4*t),x(0) = 0, D(x)(0) = 0],x(t), singsol=all)
 

\[ x \left (t \right ) = -\frac {{\mathrm e}^{-2 t}}{2}+\frac {{\mathrm e}^{-t}}{3}+\frac {{\mathrm e}^{-4 t}}{6} \]

✓ Solution by Mathematica

Time used: 0.029 (sec). Leaf size: 28

DSolve[{D[x[t],{t,2}]+3*D[x[t],t]+2*x[t]==Exp[-4*t],{x[0]==0,Derivative[1][x][0 ]==0}},x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \frac {1}{6} e^{-4 t} \left (e^t-1\right )^2 \left (2 e^t+1\right ) \]