80.7.17 problem B 9

Internal problem ID [21336]
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 9
Date solved : Thursday, October 02, 2025 at 07:28:32 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=x\\ y^{\prime }\left (t \right )&=4 x-y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.055 (sec). Leaf size: 23

ode:=[diff(x(t),t) = x(t), diff(y(t),t) = 4*x(t)-y(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 34

ode={D[x[t],t]==x[t],D[y[t],t]==4*x[t]-y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.037 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t) + Derivative(x(t), t),0),Eq(-4*x(t) + y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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