80.7.19 problem B 11

Internal problem ID [21338]
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 7. System of first order equations. Excercise 7.6 at page 162
Problem number : B 11
Date solved : Thursday, October 02, 2025 at 07:28:33 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.047 (sec). Leaf size: 35

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

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 71

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

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

✓ Sympy. Time used: 0.054 (sec). Leaf size: 32

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

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