80.5.15 problem B 15

Internal problem ID [21236]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 5. Second order equations. Excercise 5.9 at page 119
Problem number : B 15
Date solved : Thursday, October 02, 2025 at 07:27:10 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }+x^{\prime }-2 x&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 23

ode:=diff(diff(x(t),t),t)+diff(x(t),t)-2*x(t) = 0; 
ic:=[x(0) = a, D(x)(0) = 1]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \frac {\left (a -1\right ) {\mathrm e}^{-2 t}}{3}+\frac {\left (1+2 a \right ) {\mathrm e}^{t}}{3} \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 30

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

\begin{align*} x(t)&\to \frac {1}{3} e^{-2 t} \left (2 a e^{3 t}+a+e^{3 t}-1\right ) \end{align*}

✓ Sympy. Time used: 0.085 (sec). Leaf size: 26

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

\[ x{\left (t \right )} = \left (\frac {a}{3} - \frac {1}{3}\right ) e^{- 2 t} + \left (\frac {2 a}{3} + \frac {1}{3}\right ) e^{t} \]