64.1.4 problem 2.1 (iv)

Internal problem ID [15581]
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 (iv)
Date solved : Thursday, October 02, 2025 at 10:20:25 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.112 (sec). Leaf size: 85

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

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{\frac {\left (3+\sqrt {17}\right ) t}{2}}+c_2 \,{\mathrm e}^{-\frac {\left (-3+\sqrt {17}\right ) t}{2}} \\ y \left (t \right ) &= \frac {c_1 \,{\mathrm e}^{\frac {\left (3+\sqrt {17}\right ) t}{2}} \sqrt {17}}{4}-\frac {c_2 \,{\mathrm e}^{-\frac {\left (-3+\sqrt {17}\right ) t}{2}} \sqrt {17}}{4}+\frac {c_1 \,{\mathrm e}^{\frac {\left (3+\sqrt {17}\right ) t}{2}}}{4}+\frac {c_2 \,{\mathrm e}^{-\frac {\left (-3+\sqrt {17}\right ) t}{2}}}{4} \\ \end{align*}

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

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

\begin{align*} x(t)&\to \frac {1}{34} e^{-\frac {1}{2} \left (\sqrt {17}-3\right ) t} \left (c_1 \left (-\left (\sqrt {17}-17\right ) e^{\sqrt {17} t}+17+\sqrt {17}\right )+4 \sqrt {17} c_2 \left (e^{\sqrt {17} t}-1\right )\right )\\ y(t)&\to \frac {1}{34} e^{-\frac {1}{2} \left (\sqrt {17}-3\right ) t} \left (4 \sqrt {17} c_1 \left (e^{\sqrt {17} t}-1\right )+c_2 \left (\left (17+\sqrt {17}\right ) e^{\sqrt {17} t}+17-\sqrt {17}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.118 (sec). Leaf size: 76

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

\[ \left [ x{\left (t \right )} = - \frac {C_{1} \left (1 - \sqrt {17}\right ) e^{\frac {t \left (3 + \sqrt {17}\right )}{2}}}{4} - \frac {C_{2} \left (1 + \sqrt {17}\right ) e^{\frac {t \left (3 - \sqrt {17}\right )}{2}}}{4}, \ y{\left (t \right )} = C_{1} e^{\frac {t \left (3 + \sqrt {17}\right )}{2}} + C_{2} e^{\frac {t \left (3 - \sqrt {17}\right )}{2}}\right ] \]