8.29.2 problem 2

Internal problem ID [2800]
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 : 2
Date solved : Tuesday, September 30, 2025 at 05:52:47 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-3 x \left (t \right )-4 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=2 x \left (t \right )+y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.134 (sec). Leaf size: 57

ode:=[diff(x(t),t) = -3*x(t)-4*y(t), diff(y(t),t) = 2*x(t)+y(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 59

ode={D[x[t],t]==-3*x[t]-4*y[t],D[y[t],t]==2*x[t]+1*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.057 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(3*x(t) + 4*y(t) + Derivative(x(t), t),0),Eq(-2*x(t) - y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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