8.29.4 problem 4

Internal problem ID [2802]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 4. Qualitative theory of differential equations. Section 4.2 (Stability of linear systems). Page 383
Problem number : 4
Date solved : Tuesday, September 30, 2025 at 05:52:48 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )-4 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=4 x \left (t \right )-7 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.121 (sec). Leaf size: 34

ode:=[diff(x(t),t) = x(t)-4*y(t), diff(y(t),t) = 4*x(t)-7*y(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 46

ode={D[x[t],t]==1*x[t]-4*y[t],D[y[t],t]==4*x[t]-7*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.052 (sec). Leaf size: 42

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t) + 4*y(t) + Derivative(x(t), t),0),Eq(-4*x(t) + 7*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = 4 C_{1} t e^{- 3 t} + \left (C_{1} + 4 C_{2}\right ) e^{- 3 t}, \ y{\left (t \right )} = 4 C_{1} t e^{- 3 t} + 4 C_{2} e^{- 3 t}\right ] \]