7.19.4 problem 30

Internal problem ID [544]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 4. Laplace transform methods. Section 4.3 (Translation and partial fractions). Problems at page 296
Problem number : 30
Date solved : Wednesday, February 05, 2025 at 03:45:41 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{\prime \prime }+4 x^{\prime }+8 x&={\mathrm e}^{-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: 0.232 (sec). Leaf size: 30

dsolve([diff(x(t),t$2)+4*diff(x(t),t)+8*x(t)=exp(-t),x(0) = 0, D(x)(0) = 0],x(t), singsol=all)
 

\[ x \left (t \right ) = \frac {\left (-2 \cos \left (2 t \right )-\sin \left (2 t \right )\right ) {\mathrm e}^{-2 t}}{10}+\frac {{\mathrm e}^{-t}}{5} \]

✓ Solution by Mathematica

Time used: 0.085 (sec). Leaf size: 32

DSolve[{D[x[t],{t,2}]+4*D[x[t],t]+8*x[t]==Exp[-t],{x[0]==0,Derivative[1][x][0] ==0}},x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \frac {1}{10} e^{-2 t} \left (2 e^t-\sin (2 t)-2 \cos (2 t)\right ) \]