80.8.1 problem 1

Internal problem ID [21358]
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 8. Qualitative analysis of 2 by 2 systems and nonlinear second order equations. Excercise 8.5 at page 184
Problem number : 1
Date solved : Thursday, October 02, 2025 at 07:28:44 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=1+x\\ y^{\prime }\left (t \right )&=x+3 y \left (t \right )-1 \end{align*}

✓ Maple. Time used: 0.059 (sec). Leaf size: 26

ode:=[diff(x(t),t) = x(t)+1, diff(y(t),t) = x(t)+3*y(t)-1]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= -1+{\mathrm e}^{t} c_2 \\ y \left (t \right ) &= c_1 \,{\mathrm e}^{3 t}-\frac {{\mathrm e}^{t} c_2}{2}+\frac {2}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.072 (sec). Leaf size: 43

ode={D[x[t],t]==x[t]+1,D[y[t],t]==x[t]+3*y[t]-1}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to -1+c_1 e^t\\ y(t)&\to \frac {1}{6} \left (-3 c_1 e^t+3 (c_1+2 c_2) e^{3 t}+4\right ) \end{align*}

✓ Sympy. Time used: 0.073 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t) + Derivative(x(t), t) - 1,0),Eq(-x(t) - 3*y(t) + Derivative(y(t), t) + 1,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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