63.7.6 problem 1(f)

Internal problem ID [13046]
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 : Wednesday, March 05, 2025 at 08:59:23 PM
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*}

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

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

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

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

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

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

✓ Sympy. Time used: 0.207 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(2*x(t)/9 + 5*Derivative(x(t), t)/6 + Derivative(x(t), (t, 2))/2,0) 
ics = {x(0): 1, Subs(Derivative(x(t), t), t, 0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

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