63.8.1 problem 1

Internal problem ID [13126]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 2, Second order linear equations. Section 2.2.4. Applications. Exercises page 99
Problem number : 1
Date solved : Tuesday, January 28, 2025 at 04:53:42 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }+x^{\prime }+x&=0 \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.027 (sec). Leaf size: 28

dsolve([diff(x(t),t$2)+diff(x(t),t)+x(t)=0,x(0) = 1, D(x)(0) = 1],x(t), singsol=all)
 

\[ x \left (t \right ) = {\mathrm e}^{-\frac {t}{2}} \left (\sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{2}\right )+\cos \left (\frac {\sqrt {3}\, t}{2}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.022 (sec). Leaf size: 42

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

\[ x(t)\to e^{-t/2} \left (\sqrt {3} \sin \left (\frac {\sqrt {3} t}{2}\right )+\cos \left (\frac {\sqrt {3} t}{2}\right )\right ) \]