64.1.3 problem 2.1 (iii)

Internal problem ID [15580]
Book : Nonlinear Ordinary Differential Equations by D.W.Jordna and P.Smith. 4th edition 1999. Oxford Univ. Press. NY
Section : Chapter 2. Plane autonomous systems and linearization. Problems page 79
Problem number : 2.1 (iii)
Date solved : Thursday, October 02, 2025 at 10:20:24 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.101 (sec). Leaf size: 82

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

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 143

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

\begin{align*} x(t)&\to \frac {1}{14} e^{-\left (\left (3+\sqrt {7}\right ) t\right )} \left (c_1 \left (-\left (\sqrt {7}-7\right ) e^{2 \sqrt {7} t}+7+\sqrt {7}\right )+2 \sqrt {7} c_2 \left (e^{2 \sqrt {7} t}-1\right )\right )\\ y(t)&\to \frac {1}{14} e^{-\left (\left (3+\sqrt {7}\right ) t\right )} \left (3 \sqrt {7} c_1 \left (e^{2 \sqrt {7} t}-1\right )+c_2 \left (\left (7+\sqrt {7}\right ) e^{2 \sqrt {7} t}+7-\sqrt {7}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.121 (sec). Leaf size: 70

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

\[ \left [ x{\left (t \right )} = - \frac {C_{1} \left (1 - \sqrt {7}\right ) e^{- t \left (3 - \sqrt {7}\right )}}{3} - \frac {C_{2} \left (1 + \sqrt {7}\right ) e^{- t \left (\sqrt {7} + 3\right )}}{3}, \ y{\left (t \right )} = C_{1} e^{- t \left (3 - \sqrt {7}\right )} + C_{2} e^{- t \left (\sqrt {7} + 3\right )}\right ] \]