80.11.5 problem 5

Internal problem ID [21413]
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 12. Stability theory. Excercise 12.6 at page 270
Problem number : 5
Date solved : Thursday, October 02, 2025 at 07:30:56 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=-x+4 y \left (t \right )\\ y^{\prime }\left (t \right )&=3 x-5 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.046 (sec). Leaf size: 31

ode:=[diff(x(t),t) = -x(t)+4*y(t), diff(y(t),t) = 3*x(t)-5*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{-7 t}+c_2 \,{\mathrm e}^{t} \\ y \left (t \right ) &= -\frac {3 c_1 \,{\mathrm e}^{-7 t}}{2}+\frac {c_2 \,{\mathrm e}^{t}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 73

ode={D[x[t],t]==-x[t]+4*y[t],D[y[t],t]==3*x[t]-5*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{4} e^{-7 t} \left (c_1 \left (3 e^{8 t}+1\right )+2 c_2 \left (e^{8 t}-1\right )\right )\\ y(t)&\to \frac {1}{8} e^{-7 t} \left (3 c_1 \left (e^{8 t}-1\right )+2 c_2 \left (e^{8 t}+3\right )\right ) \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode=[Eq(x(t) - 4*y(t) + Derivative(x(t), t),0),Eq(-3*x(t) + 5*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

ValueError : 
Input to the funcs should be a list of functions.