63.7.6 problem 1(f)

Internal problem ID [13125]
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 2.2.3 Complex eigenvalues. Exercises page 94
Problem number : 1(f)
Date solved : Tuesday, January 28, 2025 at 04:53:40 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} \frac {x^{\prime \prime }}{2}+\frac {5 x^{\prime }}{6}+\frac {2 x}{9}&=0 \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 17

dsolve([1/2*diff(x(t),t$2)+5/6*diff(x(t),t)+2/9*x(t)=0,x(0) = 1, D(x)(0) = 0],x(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 23

DSolve[{1/2*D[x[t],{t,2}]+5/6*D[x[t],t]+2/9*x[t]==0,{x[0]==1,Derivative[1][x][0 ]==0}},x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \frac {1}{3} e^{-4 t/3} \left (4 e^t-1\right ) \]